[ { "id": "https://authors.library.caltech.edu/records/w3v6a-pqr34", "eprint_status": "archive", "datestamp": "2026-01-15 00:34:41", "lastmod": "2026-03-09 22:09:27", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "name": { "family": "Conlon", "given": "David" } }, { "name": { "family": "Fox", "given": "Jacob" } }, { "id": "Pham-Huy-Tuan", "name": { "family": "Pham", "given": "Huy Tuan" }, "orcid": "0000-0003-4659-4345" }, { "name": { "family": "Yepremyan", "given": "Liana" } } ] }, "title": "On the clique number of random Cayley graphs and related topics", "ispublished": "unpub", "full_text_status": "public", "keywords": "Combinatorics (math.CO); Discrete Mathematics (cs.DM); Number Theory (math.NT); Probability (math.PR); FOS: Mathematics; FOS: Computer and information sciences", "abstract": "We prove that a random Cayley graph on a group of order $N$ has clique number $O(\\log N \\log \\log N)$ with high probability. This bound is best possible up to the constant factor for certain groups, including~$\\mathbb{F}_2^n$, and improves the longstanding upper bound of $O(\\log^2 N)$ due to Alon. Our proof does not make use of the underlying group structure and is purely combinatorial, with the key result being an essentially best possible upper bound for the number of subsets of given order that contain at most a given number of colors in a properly edge-colored complete graph. As a further application of this result, we study a conjecture of Alon stating that every group of order $N$ has a Cayley graph whose clique number and independence number are both $O(\\log N)$, proving the conjecture for all abelian groups of order $N$ for almost all $N$. For finite vector spaces of order $N$ with characteristic congruent to $1 \\pmod 4$, we prove the existence of a self-complementary Cayley graph on the vector space whose clique number and independence number are both at most $(2+o(1))\\log N$. This matches the lower bound for Ramsey numbers coming from random graphs and solves, in a strong form, a problem of Alon and Orlitsky motivated by information theory.", "date": "2024", "date_type": "published", "publisher": "arXiv", "official_url": "https://authors.library.caltech.edu/records/w3v6a-pqr34", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arxiv.2412.21194", "pub_year": "2024", "author_list": "Conlon, David; Fox, Jacob; et al." }, { "id": "https://authors.library.caltech.edu/records/aydgt-f5883", "eprint_status": "archive", "datestamp": "2026-01-15 00:35:19", "lastmod": "2026-03-09 22:09:40", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "name": { "family": "Nenadov", "given": "Rajko" }, "orcid": "0009-0003-3777-021X" }, { "id": "Pham-Huy-Tuan", "name": { "family": "Pham", "given": "Huy Tuan" }, "orcid": "0000-0003-4659-4345" } ] }, "title": "Spread blow-up lemma with an application to perturbed random graphs", "ispublished": "unpub", "full_text_status": "public", "keywords": "Combinatorics (math.CO); Discrete Mathematics (cs.DM); Probability (math.PR); FOS: Mathematics; FOS: Computer and information sciences", "abstract": "Combining ideas of Pham, Sah, Sawhney, and Simkin on spread perfect matchings in super-regular bipartite graphs with an algorithmic blow-up lemma, we prove a spread version of the blow-up lemma. Intuitively, this means that there exists a probability measure over copies of a desired spanning graph $H$ in a given system of super-regular pairs which does not heavily pin down any subset of vertices. This allows one to complement the use of the blow-up lemma with the recently resolved Kahn-Kalai conjecture. As an application, we prove an approximate version of a conjecture of B\u00f6ttcher, Parczyk, Sgueglia, and Skokan on the threshold for appearance of powers of Hamilton cycles in perturbed random graphs.", "date": "2024", "date_type": "published", "publisher": "arXiv", "official_url": "https://authors.library.caltech.edu/records/aydgt-f5883", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arxiv.2410.06132", "pub_year": "2024", "author_list": "Nenadov, Rajko and Pham, Huy Tuan" }, { "id": "https://authors.library.caltech.edu/records/nh023-sdw60", "eprint_status": "archive", "datestamp": "2026-01-15 00:35:25", "lastmod": "2026-03-09 22:09:30", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "name": { "family": "Fox", "given": "Jacob" } }, { "name": { "family": "Nenadov", "given": "Rajko" }, "orcid": "0009-0003-3777-021X" }, { "id": "Pham-Huy-Tuan", "name": { "family": "Pham", "given": "Huy Tuan" }, "orcid": "0000-0003-4659-4345" } ] }, "title": "The largest subgraph without a forbidden induced subgraph", "ispublished": "unpub", "full_text_status": "public", "keywords": "Combinatorics (math.CO); Probability (math.PR); FOS: Mathematics", "abstract": "We initiate the systematic study of the following Tur\u00e1n-type question. Suppose $\u0393$ is a graph with $n$ vertices such that the edge density between any pair of subsets of vertices of size at least $t$ is at most $1 - c$, for some $t$ and $c > 0$. What is the largest number of edges in a subgraph $G \\subseteq \u0393$ which does not contain a fixed graph $H$ as an induced subgraph or, more generally, which belongs to a hereditary property $\\mathcal{P}$? This provides a common generalization of two recently studied cases, namely $\u0393$ being a (pseudo-)random graph and a graph without a large complete bipartite subgraph. We focus on the interesting case where $H$ is a bipartite graph.\n We determine the answer up to a constant factor with respect to $n$ and $t$, for certain bipartite $H$ and for $\u0393$ either a dense random graph or a Paley graph with a square number of vertices. In particular, our bounds match if $H$ is a tree, or if one part of $H$ has $d$ vertices complete to the other part, all other vertices in that part have degree at most $d$, and the other part has sufficiently many vertices. As applications of the latter result, we answer a question of Alon, Krivelevich, and Samotij on the largest subgraph with a hereditary property which misses a bipartite graph, and determine up to a constant factor the largest number of edges in a string subgraph of $\u0393$. The proofs are based on a variant of the dependent random choice and a novel approach for finding induced copies by inductively defining probability distributions supported on induced copies of smaller subgraphs.", "date": "2024", "date_type": "published", "publisher": "arXiv", "official_url": "https://authors.library.caltech.edu/records/nh023-sdw60", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arxiv.2405.05902", "pub_year": "2024", "author_list": "Fox, Jacob; Nenadov, Rajko; et al." }, { "id": "https://authors.library.caltech.edu/records/2vghy-wsp47", "eprint_status": "archive", "datestamp": "2026-01-15 00:35:03", "lastmod": "2026-03-09 22:09:34", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "name": { "family": "Balogh", "given": "J\u00f3zsef" } }, { "name": { "family": "Bernshteyn", "given": "Anton" } }, { "name": { "family": "Delcourt", "given": "Michelle" } }, { "name": { "family": "Ferber", "given": "Asaf" }, "orcid": "0000-0002-0568-4523" }, { "id": "Pham-Huy-Tuan", "name": { "family": "Pham", "given": "Huy Tuan" }, "orcid": "0000-0003-4659-4345" } ] }, "title": "Sunflowers in set systems with small VC-dimension", "ispublished": "unpub", "full_text_status": "public", "keywords": "Combinatorics (math.CO); Discrete Mathematics (cs.DM); Probability (math.PR); FOS: Mathematics; FOS: Computer and information sciences", "abstract": "A family of $r$ distinct sets $\\{A_1,\\ldots, A_r\\}$ is an $r$-sunflower if for all $1 \\leqslant i < j \\leqslant r$ and $1 \\leqslant i' < j' \\leqslant r$, we have $A_i \\cap A_j = A_{i'} \\cap A_{j'}$. Erd\u0151s and Rado conjectured in 1960 that every family $\\mathcal{H}$ of $\\ell$-element sets of size at least $K(r)^\\ell$ contains an $r$-sunflower, where $K(r)$ is some function that depends only on $r$. We prove that if $\\mathcal{H}$ is a family of $\\ell$-element sets of VC-dimension at most $d$ and $|\\mathcal{H}| > (C r (\\log d+\\log^\\ast \\ell))^\\ell$ for some absolute constant $C > 0$, then $\\mathcal{H}$ contains an $r$-sunflower. This improves a recent result of Fox, Pach, and Suk. When $d=1$, we obtain a sharp bound, namely that $|\\mathcal{H}| > (r-1)^\\ell$ is sufficient. Along the way, we establish a strengthening of the Kahn-Kalai conjecture for set families of bounded VC-dimension, which is of independent interest.", "date": "2024", "date_type": "published", "publisher": "arXiv", "official_url": "https://authors.library.caltech.edu/records/2vghy-wsp47", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arxiv.2408.04165", "pub_year": "2024", "author_list": "Balogh, J\u00f3zsef; Bernshteyn, Anton; et al." }, { "id": "https://authors.library.caltech.edu/records/3c516-6z150", "eprint_status": "archive", "datestamp": "2026-01-15 00:34:57", "lastmod": "2026-03-09 22:09:25", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Pham-Huy-Tuan", "name": { "family": "Pham", "given": "Huy Tuan" }, "orcid": "0000-0003-4659-4345" }, { "name": { "family": "Zakharov", "given": "Dmitrii" } } ] }, "title": "Sharp bound for the Erd\u0151s-Straus non-averaging set problem", "ispublished": "unpub", "full_text_status": "public", "keywords": "Combinatorics (math.CO); Number Theory (math.NT); FOS: Mathematics", "abstract": "A set of integers $A$ is non-averaging if there is no element $a$ in $A$ which can be written as an average of a subset of $A$ not containing $a$. We show that the largest non-averaging subset of $\\{1, \\ldots, n\\}$ has size $n^{1/4+o(1)}$, thus solving the Erd\u0151s-Straus problem. We also determine the largest size of a non-averaging set in a $d$-dimensional box for any fixed $d$. Our main tool includes the structure theorem for the set of subset sums due to Conlon, Fox and the first author, together with a result about the structure of a point set in nearly convex position.", "date": "2024", "date_type": "published", "publisher": "arXiv", "official_url": "https://authors.library.caltech.edu/records/3c516-6z150", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arxiv.2410.14624", "pub_year": "2024", "author_list": "Pham, Huy Tuan and Zakharov, Dmitrii" }, { "id": "https://authors.library.caltech.edu/records/jzzp0-cpa20", "eprint_status": "archive", "datestamp": "2026-01-15 00:35:08", "lastmod": "2026-03-09 22:09:32", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "name": { "family": "Huang", "given": "Brice" } }, { "name": { "family": "Montanari", "given": "Andrea" } }, { "id": "Pham-Huy-Tuan", "name": { "family": "Pham", "given": "Huy Tuan" }, "orcid": "0000-0003-4659-4345" } ] }, "title": "Sampling from Spherical Spin Glasses in Total Variation via Algorithmic Stochastic Localization", "ispublished": "unpub", "full_text_status": "public", "keywords": "Probability (math.PR); Disordered Systems and Neural Networks (cond-mat.dis-nn); Mathematical Physics (math-ph); FOS: Mathematics; FOS: Physical sciences", "abstract": "We consider the problem of algorithmically sampling from the Gibbs measure of a mixed $p$-spin spherical spin glass. We give a polynomial-time algorithm that samples from the Gibbs measure up to vanishing total variation error, for any model whose mixture satisfies $$\u03be''(s) < \\frac{1}{(1-s)^2}, \\qquad \\forall s\\in [0,1).$$ This includes the pure $p$-spin glasses above a critical temperature that is within an absolute ($p$-independent) constant of the so-called shattering phase transition. Our algorithm follows the algorithmic stochastic localization approach introduced in (Alaoui, Montanari, Sellke, 20022). A key step of this approach is to estimate the mean of a sequence of tilted measures. We produce an improved estimator for this task by identifying a suitable correction to the TAP fixed point selected by approximate message passing (AMP). As a consequence, we improve the algorithm's guarantee over previous work, from normalized Wasserstein to total variation error. In particular, the new algorithm and analysis opens the way to perform inference about one-dimensional projections of the measure.", "date": "2024", "date_type": "published", "publisher": "arXiv", "official_url": "https://authors.library.caltech.edu/records/jzzp0-cpa20", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arxiv.2404.15651", "pub_year": "2024", "author_list": "Huang, Brice; Montanari, Andrea; et al." }, { "id": "https://authors.library.caltech.edu/records/ybh4f-j2f14", "eprint_status": "archive", "datestamp": "2026-01-15 00:35:45", "lastmod": "2026-03-09 22:09:38", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "name": { "family": "Conlon", "given": "David" } }, { "name": { "family": "Fox", "given": "Jacob" } }, { "id": "Pham-Huy-Tuan", "name": { "family": "Pham", "given": "Huy Tuan" }, "orcid": "0000-0003-4659-4345" } ] }, "title": "Homogeneous structures in subset sums and non-averaging sets", "ispublished": "unpub", "full_text_status": "public", "keywords": "Combinatorics (math.CO); Number Theory (math.NT); FOS: Mathematics", "abstract": "We show that for every positive integer $k$ there are positive constants $C$ and $c$ such that if $A$ is a subset of $\\{1, 2, \\dots, n\\}$ of size at least $C n^{1/k}$, then, for some $d \\leq k-1$, the set of subset sums of $A$ contains a homogeneous $d$-dimensional generalized arithmetic progression of size at least $c|A|^{d+1}$. This strengthens a result of Szemer\u00e9di and Vu, who proved a similar statement without the homogeneity condition. As an application, we make progress on the Erd\u0151s--Straus non-averaging sets problem, showing that every subset $A$ of $\\{1, 2, \\dots, n\\}$ of size at least $n^{\\sqrt{2} - 1 + o(1)}$ contains an element which is the average of two or more other elements of $A$. This gives the first polynomial improvement on a result of Erd\u0151s and S\u00e1rk\u00f6zy from 1990.", "date": "2023", "date_type": "published", "publisher": "arXiv", "official_url": "https://authors.library.caltech.edu/records/ybh4f-j2f14", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arxiv.2311.01416", "pub_year": "2023", "author_list": "Conlon, David; Fox, Jacob; et al." }, { "id": "https://authors.library.caltech.edu/records/w9vm2-wpd47", "eprint_id": 116319, "eprint_status": "archive", "datestamp": "2023-08-20 07:38:54", "lastmod": "2026-03-18 18:06:33", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Gukov-S", "name": { "family": "Gukov", "given": "Sergei" }, "orcid": "0000-0002-9486-1762" }, { "id": "Lee-Vincent-S-H", "name": { "family": "Lee", "given": "Vincent S. H." } }, { "id": "Zurek-K-M", "name": { "family": "Zurek", "given": "Kathryn M." }, "orcid": "0000-0002-2629-337X" } ] }, "title": "Near-Horizon Quantum Dynamics of 4-d Einstein Gravity from 2-d JT Gravity", "ispublished": "unpub", "full_text_status": "public", "note": "We thank Tom Banks, Temple He, Cynthia Keeler, Juan Maldacena, Allic Sivaramakrishnan and Erik Verlinde for discussion on these directions. KZ and VL are supported by the Heising-Simons Foundation \"Observational Signatures of Quantum Gravity\" collaboration grant 2021-2817, and by a Simons Investigator award. The work of SG and KZ is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632.\n\n

Submitted - 2205.02233.pdf

", "abstract": "We study quantum fluctuations in the lightcone metric of the 4-d Einstein-Hilbert action via dimensional reduction to Jackiw-Teitelboim (JT) gravity. In particular, we show that, in Einstein gravity, the causal development of a region in flat Minkowski spacetime, near a horizon defined by light sheets, can be described by an effective two-dimensional dilaton theory. This enables us to make use of known solutions of the JT action, where the spacetime position of a horizon has quantum uncertainty due to metric fluctuations. This quantum uncertainty can be then directly related to the original 4-d light cone coordinates, allowing us to compute the uncertainty in the time of a photon to travel from tip-to-tip of a causal diamond in flat 4-d Minkowski space. We find that both Planck and infrared scales (with the latter set by the size of the causal diamond) enter the uncertainty in photon travel time, such that the quantum fluctuation in the arrival time may be observably large.", "date": "2022-05-04", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20220816-192438269", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220816-192438269", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Heising-Simons Foundation", "grant_number": "2021-2817" }, { "agency": "Simons Foundation" }, { "agency": "Department of Energy (DOE)", "grant_number": "DE-SC0011632" } ] }, "local_group": { "items": [ { "id": "Walter-Burke-Institute-for-Theoretical-Physics" }, { "id": "Mathematics-Department" }, { "id": "Physics-Department" } ] }, "doi": "10.48550/arXiv.2205.02233", "primary_object": { "basename": "2205.02233.pdf", "url": "https://authors.library.caltech.edu/records/w9vm2-wpd47/files/2205.02233.pdf" }, "pub_year": "2022", "author_list": "Gukov, Sergei; Lee, Vincent S. H.; et al." }, { "id": "https://authors.library.caltech.edu/records/bvc21-att86", "eprint_id": 117590, "eprint_status": "archive", "datestamp": "2023-08-20 07:03:25", "lastmod": "2026-03-09 23:57:04", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Song-Antoine", "name": { "family": "Song", "given": "Antoine" } } ] }, "title": "The spherical Plateau problem for group homology", "ispublished": "unpub", "full_text_status": "public", "note": "I am grateful to G\u00e9rard Besson, Gilles Courtois, John Lott, Ian Agol, Richard Bamler, Song Sun, Jason Manning, St\u00e9phane Sabourau, Camillo De Lellis, Xin Zhou, Alexander Nabutovsky, Shi Wang, Ben Lowe for many insightful and stimulating discussions during the writing of this article. \n\nThis research was conducted during the period A.S. served as a Clay Research Fellow.", "abstract": "Given a group homology class h of a countable group \u0393, we study a corresponding homological Plateau problem inside a canonical classifying space of \u0393, which is defined using the regular representation of \u0393 and which is locally isometric to a Hilbert unit sphere. We investigate the relation between group theoretic properties of the pair (\u0393,h) and the geometry of its Plateau solutions. For instance, we prove that for a closed oriented 3-manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M endowed with one third of its hyperbolic metric.", "date": "2022-02-22", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20221026-538906000.1", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221026-538906000.1", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Clay Mathematics Institute" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2202.10636", "pub_year": "2022", "author_list": "Song, Antoine" }, { "id": "https://authors.library.caltech.edu/records/ppmbc-mkv85", "eprint_id": 113595, "eprint_status": "archive", "datestamp": "2023-09-15 07:33:24", "lastmod": "2026-03-09 02:37:43", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" }, { "id": "Radzihovsky-Leo", "name": { "family": "Radzihovsky", "given": "Leo" }, "orcid": "0000-0002-2281-0835" } ] }, "title": "Piezo-superconductivity: new effects in non-centrosymmetric superconductors", "ispublished": "unpub", "full_text_status": "public", "note": "Attribution 4.0 International (CC BY 4.0).\n\nWe acknowledge financial support through the Simons Investigator Awards by the James Simons Foundation. A. K. was also supported in part by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632. L.R. thanks D. Agterberg and L. Fu for discussions.\n\n

Submitted - 2201.06583.pdf

", "abstract": "We study novel effects in non-centrosymmetric superconductors arising from their unique coupling of Cooper-pair condensate and elasticity. We show that although the much discussed Lifshitz coupling is not observable in a uniform bulk state, it strikingly endows dislocations with a fractional magnetic flux. We also predict a generation of voltage-free strain by a DC current in a P and T-breaking Josephson junction. Viewing superconductors through the lens of higher-form symmetries we identify the Lifshitz coupling as a chemical potential for the approximately conserved winding number, drawing an analogy with pyroelectric insulators.", "date": "2022-01-17", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20220224-200904432", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220224-200904432", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Simons Foundation" }, { "agency": "Department of Energy (DOE)", "grant_number": "DE-SC0011632" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2201.06583", "primary_object": { "basename": "2201.06583.pdf", "url": "https://authors.library.caltech.edu/records/ppmbc-mkv85/files/2201.06583.pdf" }, "pub_year": "2022", "author_list": "Kapustin, Anton and Radzihovsky, Leo" }, { "id": "https://authors.library.caltech.edu/records/5qyvs-abm77", "eprint_id": 115370, "eprint_status": "archive", "datestamp": "2023-08-20 06:19:11", "lastmod": "2026-03-29 17:57:58", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Galeotti-Andrea", "name": { "family": "Galeotti", "given": "Andrea" }, "orcid": "0000-0002-3071-9767" }, { "id": "Golub-Benjamin", "name": { "family": "Golub", "given": "Benjamin" } }, { "id": "Goyal-Sanjeev", "name": { "family": "Goyal", "given": "Sanjeev" } }, { "id": "Talam\u00e0s-Eduard", "name": { "family": "Talam\u00e0s", "given": "Eduard" }, "orcid": "0000-0002-9128-0532" }, { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "Omer" }, "orcid": "0000-0002-0111-0418" } ] }, "title": "Taxes and Market Power: A Principal Components Approach", "ispublished": "unpub", "full_text_status": "public", "note": "Attribution 4.0 International (CC BY 4.0).\n\nThis work was supported by the ERC grant 724356 (Galeotti), the Pershing Square Fund for Research on the Foundations of Human Behavior (Golub), the National Science Foundation (SES-1658940, SES-1629446, Golub & DMS-1944153, Tamuz), the Sloan Foundation (Tamuz) and the US-Israel Binational Science Foundation (2018397, Tamuz).\n\n

Submitted - 2112.08153.pdf

", "abstract": "Suppliers of differentiated goods make simultaneous pricing decisions, which are strategically linked. Because of market power, the equilibrium is inefficient. We study how a policymaker should target a budget-balanced tax-and-subsidy policy to increase welfare. A key tool is a certain basis for the goods space, determined by the network of interactions among suppliers. It consists of eigenbundles -- orthogonal in the sense that a tax on any eigenbundle passes through only to its own price -- with pass-through coefficients determined by associated eigenvalues. Our basis permits a simple characterization of optimal interventions. A planner maximizing consumer surplus should tax eigenbundles with low pass-through and subsidize ones with high pass-through. The Pigouvian leverage of the system -- the gain in consumer surplus achievable by an optimal tax scheme -- depends only on the dispersion of the eigenvalues of the matrix of strategic interactions. We interpret these results in terms of the network structure of the market.", "date": "2021-12-15", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20220707-170547552", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220707-170547552", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "European Research Council (ERC)", "grant_number": "724356" }, { "agency": "Pershing Square Fund for Research on the Foundations of Human Behavior" }, { "agency": "NSF", "grant_number": "SES-1658940" }, { "agency": "NSF", "grant_number": "SES-1629446" }, { "agency": "NSF", "grant_number": "DMS-1944153" }, { "agency": "Alfred P. Sloan Foundation" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2018397" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2112.08153", "primary_object": { "basename": "2112.08153.pdf", "url": "https://authors.library.caltech.edu/records/5qyvs-abm77/files/2112.08153.pdf" }, "pub_year": "2021", "author_list": "Galeotti, Andrea; Golub, Benjamin; et al." }, { "id": "https://authors.library.caltech.edu/records/ygk81-2jx08", "eprint_id": 112233, "eprint_status": "archive", "datestamp": "2023-08-20 05:58:16", "lastmod": "2026-03-09 02:20:18", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Gukov-S", "name": { "family": "Gukov", "given": "Sergei" }, "orcid": "0000-0002-9486-1762" }, { "id": "Pei-Du", "name": { "family": "Pei", "given": "Du" }, "orcid": "0000-0001-5587-6905" }, { "id": "Reid-Charles", "name": { "family": "Reid", "given": "Charles" }, "orcid": "0000-0003-4598-5635" }, { "id": "Shehper-Ali", "name": { "family": "Shehper", "given": "Ali" }, "orcid": "0000-0002-3440-1721" } ] }, "title": "Symmetries of 2d TQFTs and Equivariant Verlinde Formulae for General Groups", "ispublished": "unpub", "full_text_status": "public", "note": "We would like to thank Dan Freed, Po-Shen Hsin, Andrew Neitzke, and Sebastian Schulz for useful discussions. We would also like to thank Simons Center for Geometry and Physics for generous hospitality during Graduate Summer School on the Mathematics and Physics of Hitchin Systems and the Simons Summer Workshop 2021. The work of SG is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632, and by the National Science Foundation under Grant No. NSF DMS 1664227. The work of AS was supported by NSF grants DMS-2005312 and DMS1711692. The work of DP was supported by the Center for Mathematical Sciences and Application. This paper is partly a result of the ERC-SyG project, Recursive and Exact New Quantum Theory (ReNewQuantum) which received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 810573.\n\n

Submitted - 2111.08032.pdf

", "abstract": "We study (generalized) discrete symmetries of 2d semisimple TQFTs. These are 2d TQFTs whose fusion rules can be diagonalized. We show that, in this special basis, the 0-form symmetries always act as permutations while 1-form symmetries act by phases. This leads to an explicit description of the gauging of these symmetries. One application of our results is a generalization of the equivariant Verlinde formula to the case of general Lie groups. The generalized formula leads to many predictions for the geometry of Hitchin moduli spaces, which we explicitly check in several cases with low genus and SO(3) gauge group.", "date": "2021-11-15", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20211206-191012374", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20211206-191012374", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-SC0011632" }, { "agency": "NSF", "grant_number": "DMS-1664227" }, { "agency": "NSF", "grant_number": "DMS-2005312" }, { "agency": "NSF", "grant_number": "DMS-1711692" }, { "agency": "European Research Council (ERC)", "grant_number": "810573" } ] }, "other_numbering_system": { "items": [ { "id": "2021-041", "name": "CALT-TH" } ] }, "local_group": { "items": [ { "id": "Walter-Burke-Institute-for-Theoretical-Physics" }, { "id": "Mathematics-Department" } ] }, "primary_object": { "basename": "2111.08032.pdf", "url": "https://authors.library.caltech.edu/records/ygk81-2jx08/files/2111.08032.pdf" }, "pub_year": "2021", "author_list": "Gukov, Sergei; Pei, Du; et al." }, { "id": "https://authors.library.caltech.edu/records/hp8yn-7bc98", "eprint_id": 111533, "eprint_status": "archive", "datestamp": "2023-08-20 05:06:16", "lastmod": "2026-03-29 15:31:01", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Gontier-David", "name": { "family": "Gontier", "given": "David" }, "orcid": "0000-0001-8648-7910" }, { "id": "Lewin-Mathieu", "name": { "family": "Lewin", "given": "Mathieu" }, "orcid": "0000-0002-1755-0207" } ] }, "title": "Optimizers for the finite-rank Lieb-Thirring inequality", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2021 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\nThis project has received funding from the U.S. National Science Foundation (DMS-1363432 and DMS-1954995 of R.L.F.), from the German Research Foundation (EXC-2111-390814868 of R.L.F.), and from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (MDFT 725528 of M.L.).\n\n

Submitted - 2109.05984.pdf

", "abstract": "The finite-rank Lieb-Thirring inequality provides an estimate on a Riesz sum of the N lowest eigenvalues of a Schr\u00f6dinger operator \u2212\u0394\u2212V(x) in terms of an L\u1d56(R\u1d48) norm of the potential V. We prove here the existence of an optimizing potential for each N, discuss its qualitative properties and the Euler--Lagrange equation (which is a system of coupled nonlinear Schr\u00f6dinger equations) and study in detail the behavior of optimizing sequences. In particular, under the condition \u03b3>max{0,2 \u2212 d/2} on the Riesz exponent in the inequality, we prove the compactness of all the optimizing sequences up to translations. We also show that the optimal Lieb-Thirring constant cannot be stationary in N, which sheds a new light on a conjecture of Lieb-Thirring. In dimension d = 1 at \u03b3 = 3/2, we show that the optimizers with N negative eigenvalues are exactly the Korteweg-de Vries N--solitons and that optimizing sequences must approach the corresponding manifold. Our work covers the critical case \u03b3 = 0 in dimension d \u2265 3 (Cwikel-Lieb-Rozenblum inequality) for which we exhibit and use a link with invariants of the Yamabe problem.", "date": "2021-09-13", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20211018-185313891", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20211018-185313891", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1363432" }, { "agency": "NSF", "grant_number": "DMS-1954995" }, { "agency": "Deutsche Forschungsgemeinschaft (DFG)", "grant_number": "EXC-2111-390814868" }, { "agency": "European Research Council (ERC)", "grant_number": "725528" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2109.05984", "primary_object": { "basename": "2109.05984.pdf", "url": "https://authors.library.caltech.edu/records/hp8yn-7bc98/files/2109.05984.pdf" }, "pub_year": "2021", "author_list": "Frank, Rupert L.; Gontier, David; et al." }, { "id": "https://authors.library.caltech.edu/records/bkex1-ej117", "eprint_status": "archive", "datestamp": "2023-08-20 04:41:22", "lastmod": "2026-03-29 19:21:41", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Larson-Simon", "name": { "family": "Larson", "given": "Simon" }, "orcid": "0000-0002-0057-8211" } ] }, "title": "Discrete Schr\u00f6dinger operators with decaying and oscillating potentials", "ispublished": "pub", "full_text_status": "public", "keywords": "Spectrum; almost Mathieu operator; Laplacian", "note": "

© 2024 American Mathematical Society.

\n\n

The first author is grateful to J. Breuer and H. Krüger for discussions on the topic of
this paper.

\n\n

Partial support through U.S. National Science Foundation grants DMS-1363432 and DMS-1954995 (R.L.F.), through the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), through Germany’s Excellence Strategy EXC–2111–390814868 (R.L.F.), and through Knut and Alice Wallenberg Foundation grant KAW 2018.0281 and KAW 2021.0193 (S.L.) is acknowledged.

", "abstract": "

The paper is devoted to a family of discrete one-dimensional Schrödinger operators with power-like decaying potentials with rapid oscillations. In particular, for the potential V\u2061(n)=λ\u2062n^(−α)\u2062 cos\u2061(π\u2062ω\u2062n^β) with 1<β<2\u2062α, it is proved that the spectrum is purely absolutely continuous on the spectrum of the Laplacian.

", "date": "2021-08-11", "date_type": "published", "publication": "St. Petersburg Mathematical Journal", "volume": "35", "number": "1", "publisher": "American Mathematical Society (AMS)", "pagerange": "233-244", "issn": "1061-0022", "official_url": "https://authors.library.caltech.edu/records/bkex1-ej117", "funders": { "items": [ { "grant_number": "DMS-1363432" }, { "grant_number": "DMS-1954995" }, { "agency": "Deutsche Forschungsgemeinschaft (DFG)", "grant_number": "EXC-2111-390814868" }, { "agency": "Knut and Alice Wallenberg Foundation", "grant_number": "KAW 2018.0281" }, { "grant_number": "KAW 2021.0193" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.1090/spmj/1803", "primary_object": { "basename": "2108.05083.pdf", "url": "https://authors.library.caltech.edu/records/bkex1-ej117/files/2108.05083.pdf" }, "pub_year": "2021", "author_list": "Frank, Rupert L. and Larson, Simon" }, { "id": "https://authors.library.caltech.edu/records/h9k7j-fzm15", "eprint_id": 111041, "eprint_status": "archive", "datestamp": "2023-08-22 10:35:59", "lastmod": "2026-03-30 05:14:50", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" }, { "id": "Michta-Emmanuel", "name": { "family": "Michta", "given": "Emmanuel" }, "orcid": "0000-0001-7222-0422" }, { "id": "Slade-Gordon", "name": { "family": "Slade", "given": "Gordon" }, "orcid": "0000-0001-9389-9497" } ] }, "title": "High-dimensional near-critical percolation and the torus plateau", "ispublished": "unpub", "full_text_status": "public", "keywords": "percolation; lace expansion; two-point function; one-arm exponent; triangle condition; torus plateau", "note": "This work was carried out primarily while TH was a Senior Research Associate at the University of Cambridge, during which time he was supported by ERC starting grant 804166 (SPRS). The work of EM and GS was supported in part by NSERC of Canada.\n\n

Submitted - 2107.12971.pdf

", "abstract": "We consider percolation on Z^d and on the d-dimensional discrete torus, in dimensions d \u2265 11 for the nearest-neighbour model and in dimensions d > 6 for spread-out models. For \u2124^d, we employ a wide range of techniques and previous results to prove that there exist positive constants c and C such that the slightly subcritical two-point function and one-arm probabilities satisfy\n\u2119_(p_c \u2212 \u03b5) (0\u2194x) \u2264 C/(\u2016x\u2016^(d\u22122)) e^(\u2212c\u03b5^(1/2)\u2016x\u2016) and c/r^2 e^(\u2212C\u03b5^((1/2)r( \u2264 \u2119_(p_c \u2212 \u03b5)(0\u2194\u2202[\u2212r,r]^d) \u2264 C/r^2 e^(\u2212c\u03b5^((1/2)r)). \n\nUsing this, we prove that throughout the critical window the torus two-point function has a \"plateau,\" meaning that it decays for small x as \u2016x\u2016^(\u2212(d\u22122)) but for large x is essentially constant and of order V^(\u22122/3) where V is the volume of the torus. The plateau for the two-point function leads immediately to a proof of the torus triangle condition, which is known to have many implications for the critical behaviour on the torus, and also leads to a proof that the critical values on the torus and on \u2124^d are separated by a multiple of V^(\u22121/3). The torus triangle condition and the size of the separation of critical points have been proved previously, but our proofs are different and are direct consequences of the bound on the \u2124^d two-point function. In particular, we use results derived from the lace expansion on \u2124^d, but in contrast to previous work on high-dimensional torus percolation we do not need or use a separate torus lace expansion.", "date": "2021-07-27", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202253573", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202253573", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "European Research Council (ERC)", "grant_number": "804166" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2107.12971", "primary_object": { "basename": "2107.12971.pdf", "url": "https://authors.library.caltech.edu/records/h9k7j-fzm15/files/2107.12971.pdf" }, "pub_year": "2021", "author_list": "Hutchcroft, Tom; Michta, Emmanuel; et al." }, { "id": "https://authors.library.caltech.edu/records/zvb1x-m6j72", "eprint_id": 111521, "eprint_status": "archive", "datestamp": "2023-08-20 04:19:40", "lastmod": "2026-03-09 00:47:34", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" } ] }, "title": "Degenerate stability of some Sobolev inequalities", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2021 by the author. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\nThe author wishes to thank R. Neumayer for several discussions on the topic of this paper and her seminar talk in January 2021 at Caltech which motivated this work. J. Dolbeault's help with references is much appreciated. Partial support through US National Science Foundation grants DMS-1363432 and DMS-1954995 and through German Research Foundation grant EXC-2111-390814868 is acknowledged.\n\n

Submitted - 2107.11608.pdf

", "abstract": "We show that on S\u00b9(1/\u221a(d\u22122)) \u00d7 S^(d\u22121)(1) the conformally invariant Sobolev inequality holds with a remainder term that is the fourth power of the distance to the optimizers. The fourth power is best possible. This is in contrast to the more usual vanishing to second order and is motivated by work of Engelstein, Neumayer and Spolaor. A similar phenomenon arises for subcritical Sobolev inequalities on S^d. Our proof proceeds by an iterated Bianchi-Egnell strategy.", "date": "2021-07-24", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20211018-185230397", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20211018-185230397", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1363432" }, { "agency": "NSF", "grant_number": "DMS-1954995" }, { "agency": "Deutsche Forschungsgemeinschaft (DFG)", "grant_number": "EXC-2111-390814868" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2107.11608", "primary_object": { "basename": "2107.11608.pdf", "url": "https://authors.library.caltech.edu/records/zvb1x-m6j72/files/2107.11608.pdf" }, "pub_year": "2021", "author_list": "Frank, Rupert L." }, { "id": "https://authors.library.caltech.edu/records/3z2pr-3dv08", "eprint_id": 110556, "eprint_status": "archive", "datestamp": "2023-08-20 04:16:01", "lastmod": "2026-03-30 13:49:08", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Combe-No\u00e9mie", "name": { "family": "Combe", "given": "No\u00e9mie" } }, { "id": "Manin-Yuri-I", "name": { "family": "Manin", "given": "Yuri I." } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Moufang Patterns and Geometry of Information", "ispublished": "unpub", "full_text_status": "public", "keywords": "Probability distributions, convex cones, Moufang loops, quasigroups, Malcev algebras, error\u2013correcting codes, asymptotic bound, code loops, perfect tensors, tensor networks, CRSS quantum codes", "note": "Dedicated to Don Zagier. \n\nN. C. Combe acknowledges support from the Minerva Fast track grant from the Max Planck Institute for Mathematics in the Sciences, in Leipzig. \n\nYu. Manin acknowledges the continuing strong support from the Max Planck Institute for Mathematics in Bonn. \n\nM. Marcolli acknowledges support from NSF grants DMS\u20131707882 and DMS-2104330.\n\n

Accepted Version - 2107.07486.pdf

", "abstract": "Technology of data collection and information transmission is based on various mathematical models of encoding. The words \"Geometry of information\" refer to such models, whereas the words \"Moufang patterns\" refer to various sophisticated symmetries appearing naturally in such models. In this paper we show that the symmetries of spaces of probability distributions, endowed with their canonical Riemannian metric of information geometry, have the structure of a commutative Moufang loop. We also show that the F-manifold structure on the space of probability distribution can be described in terms of differential 3-webs and Malcev algebras. We then present a new construction of (noncommutative) Moufang loops associated to almost-symplectic structures over finite fields, and use then to construct a new class of code loops with associated quantum error-correcting codes and networks of perfect tensors.", "date": "2021-07-15", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210825-184621604", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184621604", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Max Planck Institute for Mathematics in the Sciences" }, { "agency": "Max Planck Institute for Mathematics" }, { "agency": "NSF", "grant_number": "DMS-1707882" }, { "agency": "NSF", "grant_number": "DMS-2104330" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2107.07486", "primary_object": { "basename": "2107.07486.pdf", "url": "https://authors.library.caltech.edu/records/3z2pr-3dv08/files/2107.07486.pdf" }, "pub_year": "2021", "author_list": "Combe, No\u00e9mie; Manin, Yuri I.; et al." }, { "id": "https://authors.library.caltech.edu/records/t2p6s-ma745", "eprint_id": 111517, "eprint_status": "archive", "datestamp": "2023-08-20 04:14:17", "lastmod": "2026-03-09 02:14:03", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Loss-Michael", "name": { "family": "Loss", "given": "Michael" } } ] }, "title": "Existence of optimizers in a Sobolev inequality for vector fields", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2021 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\nPartial support through US National Science Foundation grants DMS-1363432 and DMS-1954995 (R.L.F.) and DMS-1856645 (M.L.) is acknowledged.\n\n

Submitted - 2107.06450.pdf

", "abstract": "We consider the minimization problem corresponding to a Sobolev inequality for vector fields and show that minimizing sequences are relatively compact up to the symmetries of the problem. In particular, there is a minimizer. An ingredient in our proof is a version of the Rellich--Kondrachov compactness theorem for sequences satisfying a nonlinear constraint.", "date": "2021-07-14", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20211018-185216561", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20211018-185216561", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1363432" }, { "agency": "NSF", "grant_number": "DMS-1954995" }, { "agency": "NSF", "grant_number": "DMS-1856645" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2107.06450", "primary_object": { "basename": "2107.06450.pdf", "url": "https://authors.library.caltech.edu/records/t2p6s-ma745/files/2107.06450.pdf" }, "pub_year": "2021", "author_list": "Frank, Rupert L. and Loss, Michael" }, { "id": "https://authors.library.caltech.edu/records/9ak9w-nca68", "eprint_id": 111040, "eprint_status": "archive", "datestamp": "2023-08-20 03:40:07", "lastmod": "2026-03-09 02:35:11", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" } ] }, "title": "On the derivation of mean-field percolation critical exponents from the triangle condition", "ispublished": "unpub", "full_text_status": "public", "note": "The author was supported in part by ERC starting grant 804166 (SPRS). We thank Vivek Dewan, Emmanuel Michta, Stephen Muirhead, and Gordon Slade for helpful comments on a previous version of the manuscript.\n\n

Submitted - 2106.06400.pdf

", "abstract": "We give a new derivation of mean-field percolation critical behaviour from the triangle condition that is quantitatively much better than previous proofs when the triangle diagram \u2207_(p_c) is large. In contrast to earlier methods, our approach continues to yield bounds of reasonable order when the triangle diagram \u2207^p is unbounded but diverges slowly as p \u2191 p_c, as is expected to occur in percolation on \u2124^d at the upper-critical dimension d=6. Indeed, we show in particular that if the triangle diagram diverges polylogarithmically as p\u2191pc then mean-field critical behaviour holds to within a polylogarithmic factor. We apply the methods we develop to deduce that for long-range percolation on the hierarchical lattice, mean-field critical behaviour holds to within polylogarithmic factors at the upper-critical dimension. \n\nAs part of the proof, we introduce a new method for comparing diagrammatic sums on general transitive graphs that may be of independent interest.", "date": "2021-06-11", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202147400", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202147400", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "European Research Council (ERC)", "grant_number": "804166" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2106.06400v2", "primary_object": { "basename": "2106.06400.pdf", "url": "https://authors.library.caltech.edu/records/9ak9w-nca68/files/2106.06400.pdf" }, "pub_year": "2021", "author_list": "Hutchcroft, Tom" }, { "id": "https://authors.library.caltech.edu/records/bepy3-mey72", "eprint_id": 121743, "eprint_status": "archive", "datestamp": "2023-08-20 03:37:40", "lastmod": "2026-03-08 17:36:17", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" }, { "id": "Fox-Jacob", "name": { "family": "Fox", "given": "Jacob" }, "orcid": "0000-0002-0664-497X" }, { "id": "Sudakov-Benny", "name": { "family": "Sudakov", "given": "Benny" }, "orcid": "0000-0003-3307-9475" }, { "id": "Zhao-Yufei", "name": { "family": "Zhao", "given": "Yufei" }, "orcid": "0000-0002-1995-3755" } ] }, "title": "Which graphs can be counted in C\u2084-free graphs?", "ispublished": "unpub", "full_text_status": "public", "note": "Part of this work was completed in the summer of 2019 while Yufei Zhao was generously hosted by FIM (the Institute for Mathematical Research) during a visit to Benny Sudakov at ETH Z\u00fcrich. \n\nDavid Conlon was supported by NSF Award DMS-2054452. \n\nJacob Fox was supported by a Packard Fellowship and by NSF Award DMS-1855635. \n\nBenny Sudakov is supported in part by SNSF grant 200021_196965. \n\nYufei Zhao is supported by NSF Award DMS-1764176, the MIT Solomon Buchsbaum Fund, and a Sloan Research Fellowship.\n\n

Submitted - 2106.03261.pdf

", "abstract": "For which graphs F is there a sparse F-counting lemma in C\u2084-free graphs? We are interested in identifying graphs F with the property that, roughly speaking, if G is an n-vertex C\u2084-free graph with on the order of n^(3/2) edges, then the density of F in G, after a suitable normalization, is approximately at least the density of F in an \u03b5-regular approximation of G. In recent work, motivated by applications in extremal and additive combinatorics, we showed that C\u2085 has this property. Here we construct a family of graphs with the property.", "date": "2021-06-06", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20230606-676918000.2", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230606-676918000.2", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-2054452" }, { "agency": "NSF", "grant_number": "DMS-1855635" }, { "agency": "David and Lucile Packard Foundation" }, { "agency": "Swiss National Science Foundation (SNSF)", "grant_number": "200021_196965" }, { "agency": "NSF", "grant_number": "DMS-1764176" }, { "agency": "Massachusetts Institute of Technology (MIT)" }, { "agency": "Alfred P. Sloan Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "primary_object": { "basename": "2106.03261.pdf", "url": "https://authors.library.caltech.edu/records/bepy3-mey72/files/2106.03261.pdf" }, "pub_year": "2021", "author_list": "Conlon, David; Fox, Jacob; et al." }, { "id": "https://authors.library.caltech.edu/records/qye4a-re870", "eprint_id": 115367, "eprint_status": "archive", "datestamp": "2023-08-20 03:19:32", "lastmod": "2026-03-29 21:49:46", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Bikhchandani-Sushil", "name": { "family": "Bikhchandani", "given": "Sushil" }, "orcid": "0000-0003-0554-072X" }, { "id": "Hirshleifer-David", "name": { "family": "Hirshleifer", "given": "David" }, "orcid": "0000-0003-0280-8882" }, { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "Omer" }, "orcid": "0000-0002-0111-0418" }, { "id": "Welch-Ivo", "name": { "family": "Welch", "given": "Ivo" }, "orcid": "0000-0002-4347-7250" } ] }, "title": "Information Cascades and Social Learning", "ispublished": "unpub", "full_text_status": "public", "keywords": "Information Cascades, Social Learning, Herding, Fads, Fashion, Conformity, Culture", "note": "We thank the editor, Steven Durlauf; and three anonymous referees for very helpful comments.\n\n

Submitted - 2105.11044.pdf

", "abstract": "We review the theory of information cascades and social learning. Our goal is to describe in a relatively integrated and accessible way the more important themes, insights and applications of the literature as it has developed over the last thirty years. We also highlight open questions and promising directions for further theoretical and empirical exploration.", "date": "2021-05-23", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20220707-170537442", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220707-170537442", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2105.11044", "primary_object": { "basename": "2105.11044.pdf", "url": "https://authors.library.caltech.edu/records/qye4a-re870/files/2105.11044.pdf" }, "pub_year": "2021", "author_list": "Bikhchandani, Sushil; Hirshleifer, David; et al." }, { "id": "https://authors.library.caltech.edu/records/6y1c3-3gg58", "eprint_id": 117591, "eprint_status": "archive", "datestamp": "2023-08-20 03:10:12", "lastmod": "2026-03-28 21:09:59", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Sackel-Kevin", "name": { "family": "Sackel", "given": "Kevin" } }, { "id": "Song-Antoine", "name": { "family": "Song", "given": "Antoine" } }, { "id": "Varolgunes-Umut", "name": { "family": "Varolgunes", "given": "Umut" } }, { "id": "Zhu-Jonathan-J", "name": { "family": "Zhu", "given": "Jonathan J." } } ] }, "title": "On certain quantifications of Gromov's non-squeezing theorem", "ispublished": "unpub", "full_text_status": "public", "note": "Revision includes an appendix by J. Brendel. Version accepted in Geometry & Topology. \n\nWe thank Michael Usher for a useful e-mail correspondence. We also thank Felix Schlenk and Jo\u00b4e Brendel for their interest and helpful comments our paper. K.S. thanks Larry Guth for originally suggesting the version of this problem involving Lipschitz constants which provided the initial impetus for this project. U.V. thanks Grigory Mikhalkin for very useful discussions on Theorem 1.3 and also sketching a proof of Corollary 6.5 that we ended up not using; and Kyler Siegel for a discussion regarding Section 6.2. \n\nK.S. was partially supported by the National Science Foundation under grant DMS-1547145. This research was conducted during the period A.S. served as a Clay Research Fellow. J.Z. was supported in part by the National Science Foundation under grant DMS-1802984 and the Australian Research Council under grant FL150100126.", "abstract": "Let R > 1 and let B be the Euclidean 4-ball of radius R with a closed subset E removed. Suppose that B embeds symplectically into the unit cylinder D\u00b2 \u00d7 R\u00b2. By Gromov's non-squeezing theorem, E must be non-empty. We prove that the Minkowski dimension of E is at least 2, and we exhibit an explicit example showing that this result is optimal at least for R \u2264 \u221a2\u0305. In an appendix by Jo\u00e9 Brendel, it is shown that the lower bound is optimal for R < \u221a3\u0305. We also discuss the minimum volume of E in the case that the symplectic embedding extends, with bounded Lipschitz constant, to the entire ball.", "date": "2021-05-03", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20221026-539095000.2", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221026-539095000.2", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1547145" }, { "agency": "Clay Mathematics Institute" }, { "agency": "NSF", "grant_number": "DMS-1802984" }, { "agency": "Australian Research Council", "grant_number": "FL150100126" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2105.00586", "pub_year": "2021", "author_list": "Sackel, Kevin; Song, Antoine; et al." }, { "id": "https://authors.library.caltech.edu/records/7xe47-39q18", "eprint_id": 111207, "eprint_status": "archive", "datestamp": "2023-08-20 02:47:26", "lastmod": "2026-03-29 16:46:08", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Larson-Simon", "name": { "family": "Larson", "given": "Simon" }, "orcid": "0000-0002-0057-8211" } ] }, "title": "On the spectrum of the Kronig-Penney model in a constant electric field", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2021 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\nU.S. National Science Foundation grants DMS-1363432 and DMS-1954995 (R.L.F.) and Knut and Alice Wallenberg Foundation grant KAW 2018.0281 (S.L.) are acknowledged.\n\n

Submitted - 2104.10256.pdf

", "abstract": "We are interested in the nature of the spectrum of the one-dimensional Schr\u00f6dinger operator \n\n\u2212d\u00b2/dx\u00b2 \u2212 Fx + \u2211_(n\u2208\u2124) g_n\u03b4(x\u2212n)in L\u00b2(\u211d) \n\nwith F > 0 and two different choices of the coupling constants {g_n}n \u2208 \u2124. In the first model g\u00b2 \u2261 \u03bb and we prove that if F \u2208 \u03c0\u00b2\u211a then the spectrum is \u211d and is furthermore absolutely continuous away from an explicit discrete set of points. In the second model g_n are independent random variables with mean zero and variance \u03bb\u00b2. Under certain assumptions on the distribution of these random variables we prove that almost surely the spectrum is \u211d and it is dense pure point if F < \u03bb\u00b2/2 and purely singular continuous if F > \u03bb\u00b2/2.", "date": "2021-04-20", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20211004-232828060", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20211004-232828060", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1363432" }, { "agency": "NSF", "grant_number": "DMS-1954995" }, { "agency": "Knut and Alice Wallenberg Foundation", "grant_number": "2018.0281" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2104.10256", "primary_object": { "basename": "2104.10256.pdf", "url": "https://authors.library.caltech.edu/records/7xe47-39q18/files/2104.10256.pdf" }, "pub_year": "2021", "author_list": "Frank, Rupert L. and Larson, Simon" }, { "id": "https://authors.library.caltech.edu/records/4713m-5zz85", "eprint_id": 110554, "eprint_status": "archive", "datestamp": "2023-08-20 02:46:26", "lastmod": "2026-03-28 21:09:47", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Gakkhar-Sitanshu", "name": { "family": "Gakkhar", "given": "Sitanshu" } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Syntactic structures and the general Markov models", "ispublished": "unpub", "full_text_status": "public", "note": "We would like to thank Andrea Ceolin for thoughtful feedback on the previous version of this paper that motivated this revision.\n\n

Submitted - 2104.08462.pdf

", "abstract": "We further the theme of studying syntactic structures data from Longobardi (2017b), Collins (2010), Ceolin et al. (2020) and Koopman (2011) using general Markov models initiated in Shu et al. (2017), exploring the question of how consistent the data is with the idea that general Markov models. The ideas explored in the present paper are more generally applicable than to the setting of syntactic structures, and can be used when analyzing consistency of data with general Markov models. Additionally, we give an interpretation of the methods of Ceolin et al. (2020) as an infinite sites evolutionary model and compare it to the Markov model and explore each in the context of evolutionary processes acting on human language syntax.", "date": "2021-04-17", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210825-184618189", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184618189", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2104.08462", "primary_object": { "basename": "2104.08462.pdf", "url": "https://authors.library.caltech.edu/records/4713m-5zz85/files/2104.08462.pdf" }, "pub_year": "2021", "author_list": "Gakkhar, Sitanshu and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/r0bnm-b8563", "eprint_id": 111039, "eprint_status": "archive", "datestamp": "2023-08-20 02:42:50", "lastmod": "2026-03-28 22:23:56", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" }, { "id": "Tointon-Matthew", "name": { "family": "Tointon", "given": "Matthew" }, "orcid": "0000-0001-8086-9280" } ] }, "title": "Non-triviality of the phase transition for percolation on finite transitive graphs", "ispublished": "unpub", "full_text_status": "public", "note": "The first author was supported in part by ERC starting grant 804166 (SPRS) and thanks Gabor Pete for helpful discussions. The second author was partially supported by the Stokes Research Fellowship at Pembroke College, Cambridge. He is also grateful to Itai Benjamini and Ariel Yadin for introducing him to the problems considered in this paper, and to Romain Tessera for helpful conversations.\n\n

Submitted - 2104.05607.pdf

", "abstract": "We prove that if (G_n)_(n \u2265 1) = ((V_n,E_n))_(n \u2265 1) is a sequence of\nfinite, vertex-transitive graphs with bounded degrees and |V_n|\u2192\u221e that\nis at least (1+\u03f5)-dimensional for some \u03f5 > 0 in the sense that\ndiam(G_n)=O(|V_n|^(1/(1+\u03f5) as n \u2192 \u221e then this sequence of graphs has a non-trivial phase transition\nfor Bernoulli bond percolation. More precisely, we prove under these conditions\nthat for each 0<\u03b1<1 there exists p_c(\u03b1) < 1 such that for each\np \u2265 p_c(\u03b1), Bernoulli-p bond percolation on G_n has a cluster of\nsize at least \u03b1|V_n| with probability tending to 1 as n \u2192 \u221e.\nIn fact, we prove more generally that there exists a universal constant a\nsuch that the same conclusion holds whenever diam(G_n) = 0(|V_n|/(log|V_n|\u03b1) as n \u2192 \u221e.\nThis verifies a conjecture of Benjamini up to the value of the constant a,\nwhich he suggested should be 1.\nWe also prove a generalization of this result to quasitransitive graph\nsequences with a bounded number of vertex orbits and prove that one may indeed\ntake a = 1 when the graphs G_n are all Cayley graphs of Abelian groups. A key\nstep in our proof is to adapt the methods of Duminil-Copin, Goswami, Raoufi,\nSevero, and Yadin from infinite graphs to finite graphs. This adaptation also\nleads to an isoperimetric criterion for infinite graphs to have a nontrivial\nuniqueness phase (i.e., to have p_u < 1) which is of independent interest. We\nalso prove that the set of possible values of the critical probability of an\ninfinite quasitransitive graph has a gap at 1 in the sense that for every\nk,n < \u221e there exists \u03f5 > 0 such that every infinite graph G of\ndegree at most k whose vertex set has at most n orbits under Aut(G)\neither has p_c = 1 or p_c \u2264 1 - \u03f5.", "date": "2021-04-12", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202143857", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202143857", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "European Research Council (ERC)", "grant_number": "804166" }, { "agency": "Pembroke College, Cambridge" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2104.05607", "primary_object": { "basename": "2104.05607.pdf", "url": "https://authors.library.caltech.edu/records/r0bnm-b8563/files/2104.05607.pdf" }, "pub_year": "2021", "author_list": "Hutchcroft, Tom and Tointon, Matthew" }, { "id": "https://authors.library.caltech.edu/records/qstde-m2w06", "eprint_id": 111038, "eprint_status": "archive", "datestamp": "2023-08-20 02:27:06", "lastmod": "2026-03-09 02:36:00", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" } ] }, "title": "The critical two-point function for long-range percolation on the hierarchical lattice", "ispublished": "unpub", "full_text_status": "public", "note": "This research was supported by ERC starting grant 804166 (SPRS). We thank Gordon Slade for helpful comments on a previous version of the manuscript.\n\n

Submitted - 2103.17013.pdf

", "abstract": "We prove up-to-constants bounds on the two-point function (i.e.,\npoint-to-point connection probabilities) for critical long-range percolation on\nthe d-dimensional hierarchical lattice. More precisely, we prove that if we\nconnect each pair of points x and y by an edge with probability\n1-exp(-\u03b2||x-y||^(-d-\u03b1)), where 0 < \u03b1 < d is fixed and \u03b2 \u2265 0 is a parameter, then the critical two-point function satisfies P_(\u03b2_c)(x \u2194 y)||x-y||^(-d+\u03b1) for\nevery pair of distinct points x and y. We deduce in particular that the\nmodel has mean-field critical behaviour when \u03b1 < d/3 and does not have\nmean-field critical behaviour when \u03b1 > d/3.", "date": "2021-03-31", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202140311", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202140311", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "European Research Council (ERC)", "grant_number": "804166" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2103.17013", "primary_object": { "basename": "2103.17013.pdf", "url": "https://authors.library.caltech.edu/records/qstde-m2w06/files/2103.17013.pdf" }, "pub_year": "2021", "author_list": "Hutchcroft, Tom" }, { "id": "https://authors.library.caltech.edu/records/7b0pr-tw127", "eprint_id": 117592, "eprint_status": "archive", "datestamp": "2023-08-20 02:13:52", "lastmod": "2026-03-09 23:56:53", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Song-Antoine", "name": { "family": "Song", "given": "Antoine" } } ] }, "title": "Bounded sectional curvature and essential minimal volume", "ispublished": "unpub", "full_text_status": "public", "note": "I am grateful to John Lott for numerous discussions that improved the results. I would also like to thank Song Sun, Aaron Naber, Xiaochun Rong, Ruobing Zhang, Ben Lowe for helpful conversations, and Claude LeBrun, Zolt\u00e1n Szab\u00f3 for comments. \n\nThis research was conducted during the period the author served as a Clay Research Fellow.", "abstract": "For a closed smooth manifold M, we consider a closure of the set of metrics on M with sectional curvature bounded between \u22121 and 1. We introduce a variant of Gromov's minimal volume, called essential minimal volume, defined as the infimum of the volume over this closure. We study metrics achieving the essential minimal volume, and prove estimates for negatively curved manifolds, Einstein 4-manifolds and complex surfaces with nonnegative Kodaira dimension.", "date": "2021-03-09", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20221026-539117000.3", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221026-539117000.3", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Clay Mathematics Institute" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2103.05659", "pub_year": "2021", "author_list": "Song, Antoine" }, { "id": "https://authors.library.caltech.edu/records/kh4s6-1kn92", "eprint_id": 111204, "eprint_status": "archive", "datestamp": "2023-08-20 01:55:00", "lastmod": "2026-03-29 17:58:27", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "K\u00f6nig-Tobias", "name": { "family": "K\u00f6nig", "given": "Tobias" }, "orcid": "0000-0002-8808-897X" }, { "id": "Kova\u0159\u00edk-Hynek", "name": { "family": "Kova\u0159\u00edk", "given": "Hynek" }, "orcid": "0000-0003-3647-8447" } ] }, "title": "Blow-up of solutions of critical elliptic equation in three dimensions", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2021 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\nPartial support through US National Science Foundation grants DMS-1363432 and DMS-1954995 (R.L.F.) and through ANR BLADE-JC ANR-18-CE40-002 (T.K.) is acknowledged. T.K. thanks Paul Laurain for several useful discussions.\n\n

Submitted - 2102.10525.pdf

", "abstract": "We describe the asymptotic behavior of positive solutions u\u03f5 of the equation \u2212\u0394u + au = 3u^(5\u2212\u03f5) in \u03a9 \u2282 \u211d\u00b3 with a homogeneous Dirichlet boundary condition. The function a is assumed to be critical in the sense of Hebey and Vaugon and the functions u_\u03f5 are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Br\u00e9zis and Peletier (1989). Similar results are also obtained for solutions of the equation \u2212\u0394u +(a+\u03f5V)u = 3u\u2075 in \u03a9.", "date": "2021-02-21", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20211004-232817904", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20211004-232817904", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1363432" }, { "agency": "NSF", "grant_number": "DMS-1954995" }, { "agency": "Agence Nationale pour la Recherche (ANR)", "grant_number": "ANR-18-CE40-002" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2102.10525", "primary_object": { "basename": "2102.10525.pdf", "url": "https://authors.library.caltech.edu/records/kh4s6-1kn92/files/2102.10525.pdf" }, "pub_year": "2021", "author_list": "Frank, Rupert L.; K\u00f6nig, Tobias; et al." }, { "id": "https://authors.library.caltech.edu/records/yzhga-z9829", "eprint_id": 110552, "eprint_status": "archive", "datestamp": "2023-08-22 08:24:12", "lastmod": "2026-03-30 08:21:53", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Manin-Yuri-I", "name": { "family": "Manin", "given": "Yuri I." } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Homotopy Spectra and Diophantine Equations", "ispublished": "unpub", "full_text_status": "public", "keywords": "Algebraic Geometry (math.AG); Number Theory (math.NT); Topology (math.AT)", "note": "To Xenia and Paolo, from Yuri and Matilde, with all our love and gratitude. \n\nM. Marcolli acknowledges support from NSF grants DMS\u20131707882 and DMS\u20132104330 and from NSERC grants RGPIN\u20132018\u201304937 and RGPAS\u20132018\u2013522593. \n\nYu. Manin acknowledges the excellent scientific environment of the Max Planck Institute for Mathematics in Bonn and permanent support of its administration and of the Max Planck Society. \n\nWe thank the three anonymous referees for a very careful reading of the paper and for providing many detailed comments and suggestions that greatly improved the paper.\n\n

Submitted - 2101.00197.pdf

", "abstract": "Arguably, the first bridge between vast, ancient, but disjoint domains of mathematical knowledge, - topology and number theory, - was built only during the last fifty years. This bridge is the theory of spectra in stable homotopy theory. This connection poses the challenge: discover new information in number theory using the independently-developed machinery of homotopy theory. In this combined research/survey paper we suggest to apply homotopy spectra to the problem of distribution of rational points upon algebraic manifolds.", "date": "2021-01-01", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210825-184614782", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184614782", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1707882" }, { "agency": "NSF", "grant_number": "DMS-2104330" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "RGPIN-2018-04937" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "RGPAS-2018-522593" }, { "agency": "Max Planck Institute for Mathematics" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2101.00197", "primary_object": { "basename": "2101.00197.pdf", "url": "https://authors.library.caltech.edu/records/yzhga-z9829/files/2101.00197.pdf" }, "pub_year": "2021", "author_list": "Manin, Yuri I. and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/rznf7-sqp73", "eprint_status": "archive", "datestamp": "2026-01-15 00:36:55", "lastmod": "2026-03-29 18:43:37", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "name": { "family": "Anari", "given": "Nima" }, "orcid": "0000-0002-4394-3530" }, { "name": { "family": "Jain", "given": "Vishesh" }, "orcid": "0000-0002-7275-3218" }, { "name": { "family": "Koehler", "given": "Frederic" }, "orcid": "0000-0001-5220-9680" }, { "id": "Pham-Huy-Tuan", "name": { "family": "Pham", "given": "Huy Tuan" }, "orcid": "0000-0003-4659-4345" }, { "name": { "family": "Vuong", "given": "Thuy-Duong" }, "orcid": "0000-0003-0271-9687" } ] }, "title": "Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models", "ispublished": "unpub", "full_text_status": "public", "keywords": "Data Structures and Algorithms (cs.DS); Discrete Mathematics (cs.DM); Mathematical Physics (math-ph); Probability (math.PR); FOS: Computer and information sciences; FOS: Physical sciences; FOS: Mathematics", "abstract": "We introduce a notion called entropic independence that is an entropic analog of spectral notions of high-dimensional expansion. Informally, entropic independence of a background distribution $\u03bc$ on $k$-sized subsets of a ground set of elements says that for any (possibly randomly chosen) set $S$, the relative entropy of a single element of $S$ drawn uniformly at random carries at most $O(1/k)$ fraction of the relative entropy of $S$. Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mixing time bounds, overcoming the lossy nature of spectral analysis of Markov chains on exponential-sized state spaces. In our main technical result, we show a general way of deriving entropy contraction, a.k.a. modified log-Sobolev inequalities, for down-up random walks from spectral notions. We show that spectral independence of a distribution under arbitrary external fields automatically implies entropic independence. To derive our results, we relate entropic independence to properties of polynomials: $\u03bc$ is entropically independent exactly when a transformed version of the generating polynomial of $\u03bc$ is upper bounded by its linear tangent; this property is implied by concavity of the said transformation, which was shown by prior work to be locally equivalent to spectral independence. We apply our results to obtain tight modified log-Sobolev inequalities and mixing times for multi-step down-up walks on fractionally log-concave distributions. As our flagship application, we establish the tight mixing time of $O(n\\log n)$ for Glauber dynamics on Ising models whose interaction matrix has eigenspectrum lying within an interval of length smaller than $1$, improving upon the prior quadratic dependence on $n$.", "date": "2021", "date_type": "published", "publisher": "arXiv", "official_url": "https://authors.library.caltech.edu/records/rznf7-sqp73", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arxiv.2106.04105", "pub_year": "2021", "author_list": "Anari, Nima; Jain, Vishesh; et al." }, { "id": "https://authors.library.caltech.edu/records/aensr-td180", "eprint_status": "archive", "datestamp": "2026-01-15 00:36:49", "lastmod": "2026-03-29 15:39:33", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "name": { "family": "Anari", "given": "Nima" }, "orcid": "0000-0002-4394-3530" }, { "name": { "family": "Jain", "given": "Vishesh" }, "orcid": "0000-0002-7275-3218" }, { "name": { "family": "Koehler", "given": "Frederic" }, "orcid": "0000-0001-5220-9680" }, { "id": "Pham-Huy-Tuan", "name": { "family": "Pham", "given": "Huy Tuan" }, "orcid": "0000-0003-4659-4345" }, { "name": { "family": "Vuong", "given": "Thuy-Duong" }, "orcid": "0000-0003-0271-9687" } ] }, "title": "Entropic Independence II: Optimal Sampling and Concentration via Restricted Modified Log-Sobolev Inequalities", "ispublished": "unpub", "full_text_status": "public", "keywords": "Data Structures and Algorithms (cs.DS); Mathematical Physics (math-ph); Probability (math.PR); FOS: Computer and information sciences; FOS: Physical sciences; FOS: Mathematics", "abstract": "We introduce a framework for obtaining tight mixing times for Markov chains based on what we call restricted modified log-Sobolev inequalities. Modified log-Sobolev inequalities (MLSI) quantify the rate of relative entropy contraction for the Markov operator, and are notoriously difficult to establish. However, infinitesimally close to stationarity, entropy contraction becomes equivalent to variance contraction, a.k.a. a Poincare inequality, which is significantly easier to establish through, e.g., spectral analysis. Motivated by this observation, we study restricted modified log-Sobolev inequalities that guarantee entropy contraction not for all starting distributions, but for those in a large neighborhood of the stationary distribution. We show how to sample from the hardcore and Ising models on $n$-node graphs that have a constant $\u03b4$ relative gap to the tree-uniqueness threshold, in nearly-linear time $\\widetilde O_\u03b4(n)$. Notably, our bound does not depend on the maximum degree $\u0394$, and is therefore optimal even for high-degree graphs. This improves on prior mixing time bounds of $\\widetilde O_{\u03b4, \u0394}(n)$ and $\\widetilde O_\u03b4(n^2)$, established via (non-restricted) modified log-Sobolev and Poincare inequalities respectively. We further show that optimal concentration inequalities can still be achieved from the restricted form of modified log-Sobolev inequalities. To establish restricted entropy contraction, we extend the entropic independence framework of Anari, Jain, Koehler, Pham, and Vuong to the setting of distributions that are spectrally independent under a restricted set of external fields. We also develop an orthogonal trick that might be of independent interest: utilizing Bernoulli factories we show how to implement Glauber dynamics updates on high-degree graphs in $O(1)$ time, assuming standard adjacency array representation of the graph.", "date": "2021", "date_type": "published", "publisher": "arXiv", "official_url": "https://authors.library.caltech.edu/records/aensr-td180", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arxiv.2111.03247", "pub_year": "2021", "author_list": "Anari, Nima; Jain, Vishesh; et al." }, { "id": "https://authors.library.caltech.edu/records/13k16-93r85", "eprint_id": 110550, "eprint_status": "archive", "datestamp": "2023-08-20 01:08:49", "lastmod": "2026-03-28 21:08:29", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Combe-No\u00e9mie", "name": { "family": "Combe", "given": "No\u00e9mie C." } }, { "id": "Manin-Y-I", "name": { "family": "Manin", "given": "Yuri I." } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Birational maps and Nori motives", "ispublished": "unpub", "full_text_status": "public", "keywords": "Birational geometry, equivariance, Nori motives", "note": "

Submitted - 2012.13814.pdf

", "abstract": "In this note, we sketch an approach to the problems of equivariant birational geometry developed by M. Kontsevich and Yu. Tschinkel, where Burnside invariants were introduced. We are making explicit the role of Nori constructions in this environment.", "date": "2020-12-26", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210825-184611333", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184611333", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2012.13814", "primary_object": { "basename": "2012.13814.pdf", "url": "https://authors.library.caltech.edu/records/13k16-93r85/files/2012.13814.pdf" }, "pub_year": "2020", "author_list": "Combe, No\u00e9mie C.; Manin, Yuri I.; et al." }, { "id": "https://authors.library.caltech.edu/records/fenqz-ggn88", "eprint_id": 110548, "eprint_status": "archive", "datestamp": "2023-08-20 00:47:44", "lastmod": "2026-03-09 21:45:53", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" }, { "id": "Penrose-Roger", "name": { "family": "Penrose", "given": "Roger" } } ] }, "title": "Gluing Noncommutative Twistor Spaces", "ispublished": "unpub", "full_text_status": "public", "note": "In memory of Sir Michael Atiyah. \n\nThe first author is partially supported by NSF grant DMS-1707882, and by NSERC Discovery Grant RGPIN-2018-04937 and Accelerator Supplement grant RGPAS-2018-522593.\n\n

Submitted - 2012.02823.pdf

", "abstract": "We describe a general procedure, based on Gerstenhaber-Schack complexes, for extending to quantized twistor spaces the Donaldson-Friedman gluing of twistor spaces via deformation theory of singular spaces. We consider in particular various possible quantizations of twistor spaces that leave the underlying spacetime manifold classical, including the geometric quantization of twistor spaces originally constructed by the second author, as well as some variants based on noncommutative geometry. We discuss specific aspects of the gluing construction for these different quantization procedures.", "date": "2020-12-04", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210825-184607934", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184607934", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1707882" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "RGPIN-2018-04937" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "RGPAS-2018-522593" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2012.02823v1", "primary_object": { "basename": "2012.02823.pdf", "url": "https://authors.library.caltech.edu/records/fenqz-ggn88/files/2012.02823.pdf" }, "pub_year": "2020", "author_list": "Marcolli, Matilde and Penrose, Roger" }, { "id": "https://authors.library.caltech.edu/records/ksvt4-kdg91", "eprint_id": 111037, "eprint_status": "archive", "datestamp": "2023-08-20 00:21:33", "lastmod": "2026-03-09 02:35:38", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" } ] }, "title": "Transience and recurrence of sets for branching random walk via non-standard stochastic orders", "ispublished": "unpub", "full_text_status": "public", "note": "We thank Toby Johnson and Matt Junge for helpful discussions.\n\n

Submitted - 2011.06402.pdf

", "abstract": "We study how the recurrence and transience of space-time sets for a branching random walk on a graph depends on the offspring distribution. Here, we say that a space-time set A is recurrent if it is visited infinitely often almost surely on the event that the branching random walk survives forever, and say that A is transient if it is visited at most finitely often almost surely. We prove that if \u03bc and \u03bd are supercritical offspring distributions with means \u03bc\u00af<\u03bd\u00af then every space-time set that is recurrent with respect to the offspring distribution \u03bc is also recurrent with respect to the offspring distribution \u03bd and similarly that every space-time set that is transient with respect to the offspring distribution \u03bd is also transient with respect to the offspring distribution \u03bc. To prove this, we introduce a new order on probability measures that we call the germ order and prove more generally that the same result holds whenever \u03bc is smaller than \u03bd in the germ order. Our work is inspired by the work of Johnson and Junge (AIHP 2018), who used related stochastic orders to study the frog model.", "date": "2020-11-12", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202136797", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202136797", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2011.06402", "primary_object": { "basename": "2011.06402.pdf", "url": "https://authors.library.caltech.edu/records/ksvt4-kdg91/files/2011.06402.pdf" }, "pub_year": "2020", "author_list": "Hutchcroft, Tom" }, { "id": "https://authors.library.caltech.edu/records/vw7fw-2em04", "eprint_id": 111036, "eprint_status": "archive", "datestamp": "2023-08-19 23:59:46", "lastmod": "2026-03-30 07:23:55", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" }, { "id": "Sousi-Perla", "name": { "family": "Sousi", "given": "Perla" } } ] }, "title": "Logarithmic corrections to scaling in the four-dimensional uniform spanning tree", "ispublished": "unpub", "full_text_status": "public", "note": "Perla Sousi's research was supported by the Engineering and Physical Sciences Research Council: EP/R022615/1.\n\n

Submitted - 2010.15830.pdf

", "abstract": "We compute the precise logarithmic corrections to mean-field scaling for various quantities describing the uniform spanning tree of the four-dimensional hypercubic lattice \u2124\u2074. We are particularly interested in the distribution of the past of the origin, that is, the finite piece of the tree that is separated from infinity by the origin. We prove that the probability that the past contains a path of length n is of order (log n)^(1/3)n\u207b\u00b9, that the probability that the past contains at least n vertices is of order (log n)^(1/6)n^(\u22121/2), and that the probability that the past reaches the boundary of the box [\u2212n,n]\u2074 is of order (log n)^(2/3+o)(1))n\u207b\u00b2. An important part of our proof is to prove concentration estimates for the capacity of the four-dimensional loop-erased random walk which may be of independent interest. \n\nOur results imply that the Abelian sandpile model also exhibits non-trivial polylogarithmic corrections to mean-field scaling in four dimensions, although it remains open to compute the precise order of these corrections.", "date": "2020-10-29", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202133236", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202133236", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Engineering and Physical Sciences Research Council (EPSRC)", "grant_number": "EP/R022615/1" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2010.15830", "primary_object": { "basename": "2010.15830.pdf", "url": "https://authors.library.caltech.edu/records/vw7fw-2em04/files/2010.15830.pdf" }, "pub_year": "2020", "author_list": "Hutchcroft, Tom and Sousi, Perla" }, { "id": "https://authors.library.caltech.edu/records/2wy8c-yg798", "eprint_id": 108298, "eprint_status": "archive", "datestamp": "2023-08-19 23:53:32", "lastmod": "2026-03-30 08:20:41", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Mu-Xiaosheng", "name": { "family": "Mu", "given": "Xiaosheng" }, "orcid": "0000-0002-2868-5182" }, { "id": "Pomatto-L", "name": { "family": "Pomatto", "given": "Luciano" }, "orcid": "0000-0002-4331-8436" }, { "id": "Strack-Philipp", "name": { "family": "Strack", "given": "Philipp" }, "orcid": "0000-0002-7960-9243" }, { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "Omer" }, "orcid": "0000-0002-0111-0418" } ] }, "title": "Background Risk and Small-Stakes Risk Aversion", "ispublished": "unpub", "full_text_status": "public", "note": "We want to thank Ned Augenblick, Nick Barberis, Sebastian Ebert, Armin Falk, Stefano DellaVigna, Paul Heidhues, Matthew Rabin, Todd Sarver, Charlie Sprenger and Juuso V\u00e4lim\u00e4ki for helpful comments and discussions. \n\nXiaosheng Mu acknowledges the hospitality of Columbia University and the Cowles Foundation at Yale University, which hosted him during parts of this research. \n\nOmer Tamuz was supported by a grant from the Simons Foundation (#419427), by a BSF grant (#2018397) and by a Sloan fellowship.\n\n

Submitted - 2010.08033.pdf

", "abstract": "We show that under plausible levels of background risk, no theory of choice under risk---such as expected utility theory, prospect theory, or rank dependent utility---can simultaneously satisfy the following three economic postulates: (i) Decision makers are risk-averse over small gambles, (ii) they respect stochastic dominance, and (iii) they account for background risk.", "date": "2020-10-15", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210303-154333226", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210303-154333226", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Simons Foundation", "grant_number": "419427" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2018397" }, { "agency": "Alfred P. Sloan Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2010.08033", "primary_object": { "basename": "2010.08033.pdf", "url": "https://authors.library.caltech.edu/records/2wy8c-yg798/files/2010.08033.pdf" }, "pub_year": "2020", "author_list": "Mu, Xiaosheng; Pomatto, Luciano; et al." }, { "id": "https://authors.library.caltech.edu/records/855yq-8df47", "eprint_id": 111198, "eprint_status": "archive", "datestamp": "2023-08-19 23:51:14", "lastmod": "2026-03-29 19:51:02", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Larson-Simon", "name": { "family": "Larson", "given": "Simon" }, "orcid": "0000-0002-0057-8211" } ] }, "title": "Semiclassical asymptotics for a class of singular Schr\u00f6dinger operators", "ispublished": "unpub", "full_text_status": "public", "keywords": "Schr\u00f6dinger operator, Semiclassical asymptotics", "note": "We are deeply grateful to Ari Laptev for sharing his fascination for spectral estimates and Hardy's inequality with us and we would like to dedicate this paper to him on the occasion of his 70th birthday. \n\nU.S. National Science Foundation grants DMS-1363432 and DMS-1954995 (R.L.F.) and Knut and Alice Wallenberg Foundation grant KAW 2018.0281 (S.L.) are acknowledged.\n\n

Submitted - 2010.05417.pdf

", "abstract": "Let \u03a9 \u2282 \u211d^d be bounded with C\u00b9 boundary. In this paper we consider Schr\u00f6dinger operators \u2212\u0394 + W on \u03a9 with W(x) \u2248 dist(x,\u2202\u03a9)\u207b\u00b2 as dist(x,\u2202\u03a9) \u2192 0. Under weak assumptions on W we derive a two-term asymptotic formula for the sum of the eigenvalues of such operators.", "date": "2020-10-12", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20211004-222658897", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20211004-222658897", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1363432" }, { "agency": "NSF", "grant_number": "DMS-1954995" }, { "agency": "Knut and Alice Wallenberg Foundation", "grant_number": "2018.0281" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2010.05417", "primary_object": { "basename": "2010.05417.pdf", "url": "https://authors.library.caltech.edu/records/855yq-8df47/files/2010.05417.pdf" }, "pub_year": "2020", "author_list": "Frank, Rupert L. and Larson, Simon" }, { "id": "https://authors.library.caltech.edu/records/hfhhh-rj092", "eprint_id": 111197, "eprint_status": "archive", "datestamp": "2023-08-19 23:47:59", "lastmod": "2026-03-29 19:54:44", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Gontier-David", "name": { "family": "Gontier", "given": "David" }, "orcid": "0000-0001-8648-7910" }, { "id": "Lewin-Mathieu", "name": { "family": "Lewin", "given": "Mathieu" }, "orcid": "0000-0002-1755-0207" } ] }, "title": "The periodic Lieb-Thirring inequality", "ispublished": "unpub", "full_text_status": "public", "keywords": "Lieb-Thirring inequality, periodic Schr\u00a8odinger operators", "note": "This project has received funding from the U.S. National Science Foundation (grant agreements DMS-1363432 and DMS-1954995 of R.L.F.) and from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement MDFT 725528 of M.L.).\n\n

Submitted - 2010.02981.pdf

", "abstract": "We discuss the Lieb-Thirring inequality for periodic systems, which has the same optimal constant as the original inequality for finite systems. This allows us to formulate a new conjecture about the value of its best constant. To demonstrate the importance of periodic states, we prove that the 1D Lieb-Thirring inequality at the special exponent \u03b3 = 3/2 admits a one-parameter family of periodic optimizers, interpolating between the one-bound state and the uniform potential. Finally, we provide numerical simulations in 2D which support our conjecture that optimizers could be periodic.", "date": "2020-10-06", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20211004-222655493", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20211004-222655493", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1363432" }, { "agency": "NSF", "grant_number": "DMS-1954995" }, { "agency": "European Research Council (ERC)", "grant_number": "725528" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2010.02981", "primary_object": { "basename": "2010.02981.pdf", "url": "https://authors.library.caltech.edu/records/hfhhh-rj092/files/2010.02981.pdf" }, "pub_year": "2020", "author_list": "Frank, Rupert L.; Gontier, David; et al." }, { "id": "https://authors.library.caltech.edu/records/gpwcf-nz689", "eprint_id": 111195, "eprint_status": "archive", "datestamp": "2023-08-19 23:17:16", "lastmod": "2026-03-29 19:18:29", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Merz-Konstantin", "name": { "family": "Merz", "given": "Konstantin" } }, { "id": "Siedentop-Heinz", "name": { "family": "Siedentop", "given": "Heinz" } } ] }, "title": "Relativistic Strong Scott Conjecture: A Short Proof", "ispublished": "unpub", "full_text_status": "public", "note": "The authors warmly thank Barry Simon for his initial contributions and continuing support and interest in the relativistic strong Scott conjecture. They also acknowledge partial support by the U.S. National Science Foundation through grants DMS-1363432 and DMS-1954995 (R.L.F.), by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through grant SI 348/15-1 (H.S.) and through Germany's Excellence Strategy EXC-2111 390814868 (R.L.F., H.S.). One of us (K.M.) would like to thank the organizers of the program Density Functionals for Many-Particle Systems: Mathematical Theory and Physical Applications of Effective Equations, which took place at the Institute for Mathematical Sciences (IMS) at the National University of Singapore (NUS), for their invitation to speak, their kind hospitality, as well as for generous financial support by the Julian Schwinger foundation that made his stay possible.\n\n

Submitted - 2009.02474.pdf

", "abstract": "We consider heavy neutral atoms of atomic number Z modeled with kinetic energy (c\u00b2p\u00b2 + c\u2074)^(1/2) \u2212 c\u00b2 used already by Chandrasekhar. We study the behavior of the one-particle ground state density on the length scale Z\u207b\u00b9 in the limit Z,c \u2192 \u221e keeping Z/c fixed. We give a short proof of a recent result by the authors and Barry Simon showing the convergence of the density to the relativistic hydrogenic density on this scale.", "date": "2020-09-05", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20211004-222648670", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20211004-222648670", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1363432" }, { "agency": "NSF", "grant_number": "DMS-1954995" }, { "agency": "Deutsche Forschungsgemeinschaft (DFG)", "grant_number": "SI 348/15-1" }, { "agency": "Deutsche Forschungsgemeinschaft (DFG)", "grant_number": "EXC-2111 390814868" }, { "agency": "Julian Schwinger Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2009.02474", "primary_object": { "basename": "2009.02474.pdf", "url": "https://authors.library.caltech.edu/records/gpwcf-nz689/files/2009.02474.pdf" }, "pub_year": "2020", "author_list": "Frank, Rupert L.; Merz, Konstantin; et al." }, { "id": "https://authors.library.caltech.edu/records/rsb7g-z9w39", "eprint_id": 111034, "eprint_status": "archive", "datestamp": "2023-08-19 23:02:37", "lastmod": "2026-03-09 02:34:43", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" } ] }, "title": "Power-law bounds for critical long-range percolation below the upper-critical dimension", "ispublished": "unpub", "full_text_status": "public", "note": "Dedicated to Harry Kesten, November 19, 1931 - March 29, 2019. \n\nWe thank Jonathan Hermon for his careful reading of an earlier version of this manuscript, and thank Gordon Slade for helpful discussions on the physics literature. We also thank the anonymous referee for their helpful comments and corrections.\n\n

Accepted Version - 2008.11197.pdf

", "abstract": "We study long-range Bernoulli percolation on \u2124d in which each two vertices x and y are connected by an edge with probability 1\u2212exp(\u2212\u03b2\u2016x\u2212y\u2016\u2212d\u2212\u03b1). It is a theorem of Noam Berger (CMP, 2002) that if 0<\u03b1", "date": "2020-08-25", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202126385", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202126385", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2008.11197", "primary_object": { "basename": "2008.11197.pdf", "url": "https://authors.library.caltech.edu/records/rsb7g-z9w39/files/2008.11197.pdf" }, "pub_year": "2020", "author_list": "Hutchcroft, Tom" }, { "id": "https://authors.library.caltech.edu/records/7vjfb-ybc90", "eprint_id": 110840, "eprint_status": "archive", "datestamp": "2023-08-19 22:52:08", "lastmod": "2026-03-28 21:07:28", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hacking-Paul", "name": { "family": "Hacking", "given": "Paul" } }, { "id": "Keel-Sean", "name": { "family": "Keel", "given": "Sean" } }, { "id": "Yu-Tony-Yue", "name": { "family": "Yu", "given": "Tony Yue" }, "orcid": "0000-0002-6019-8552" } ] }, "title": "Secondary fan, theta functions and moduli of Calabi-Yau pairs", "ispublished": "unpub", "full_text_status": "public", "note": "We enjoyed fruitful conversations with P. Achinger, V. Alexeev, M. Baker, V. Berkovich, F. Charles, A. Corti, A. Durcos, W. Gubler, J. Koll\u00e1r, M. Porta, J. Rabinoff, D. Ranganathan, M. Robalo and Y. Soibelman. We were heavily inspired and influenced by our long-term collaborations with M. Gross, M. Kontsevich and B. Siebert. Hacking was supported by NSF grants DMS-1601065 and DMS-1901970. Keel was supported by NSF grant DMS-1561632. T.Y. Yu was supported by the Clay Mathematics Institute as Clay Research Fellow. Some of the research was conducted while Keel and Yu visited the Institute for Advanced Study in Princeton, and some while the three authors visited the Institut des Hautes \u00c9tudes Scientifiques in Bures-sur-Yvette.\n\n

Submitted - 2008.02299.pdf

", "abstract": "We conjecture that any connected component Q of the moduli space of triples (X,E=E\u2081+\u22ef+E_n,\u0398) where X is a smooth projective variety, E is a normal crossing anti-canonical divisor with a 0-stratum, every E_i is smooth, and \u0398 is an ample divisor not containing any 0-stratum of E, is unirational. More precisely: note that Q has a natural embedding into the Koll\u00e1r-Shepherd-Barron-Alexeev moduli space of stable pairs, we conjecture that its closure admits a finite cover by a complete toric variety. We construct the associated complete toric fan, generalizing the Gelfand-Kapranov-Zelevinski secondary fan for reflexive polytopes. Inspired by mirror symmetry, we speculate a synthetic construction of the universal family over this toric variety, as the Proj of a sheaf of graded algebras with a canonical basis, whose structure constants are given by counts of non-archimedean analytic disks. In the Fano case and under the assumption that the mirror contains a Zariski open torus, we construct the conjectural universal family, generalizing the families of Kapranov-Sturmfels-Zelevinski and Alexeev in the toric case. In the case of del Pezzo surfaces with an anti-canonical cycle of (\u22121)-curves, we prove the full conjecture.", "date": "2020-08-05", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210914-164517148", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210914-164517148", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1601065" }, { "agency": "NSF", "grant_number": "DMS-1901970" }, { "agency": "NSF", "grant_number": "DMS-1561632" }, { "agency": "Clay Mathematics Institute" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2008.02299", "primary_object": { "basename": "2008.02299.pdf", "url": "https://authors.library.caltech.edu/records/7vjfb-ybc90/files/2008.02299.pdf" }, "pub_year": "2020", "author_list": "Hacking, Paul; Keel, Sean; et al." }, { "id": "https://authors.library.caltech.edu/records/2xfrr-8xy91", "eprint_id": 111033, "eprint_status": "archive", "datestamp": "2023-08-19 22:34:30", "lastmod": "2026-03-09 02:34:30", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" } ] }, "title": "Continuity of the Ising phase transition on nonamenable groups", "ispublished": "unpub", "full_text_status": "public", "note": "We thank Jonathan Hermon for making us aware of Freedman's work on maximal inequalities for martingales [33], which inspired Lemmas 3.4 and 3.5. We also thank Hugo Duminil-Copin, Geoffrey Grimmett, and Russ Lyons for helpful comments on an earlier version of the manuscript.\n\n

Submitted - 2007.15625.pdf

", "abstract": "We prove rigorously that the ferromagnetic Ising model on any nonamenable Cayley graph undergoes a continuous (second-order) phase transition in the sense that there is a unique Gibbs measure at the critical temperature. The proof of this theorem is quantitative and also yields power-law bounds on the magnetization at and near criticality. Indeed, we prove more generally that the magnetization \u27e8\u03c3o\u27e9+\u03b2,h is a locally H\u00f6lder-continuous function of the inverse temperature \u03b2 and external field h throughout the non-negative quadrant (\u03b2,h)\u2208[0,\u221e)2. As a second application of the methods we develop, we also prove that the free energy of Bernoulli percolation is twice differentiable at pc on any transitive nonamenable graph.", "date": "2020-07-30", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202122977", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202122977", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2007.15625", "primary_object": { "basename": "2007.15625.pdf", "url": "https://authors.library.caltech.edu/records/2xfrr-8xy91/files/2007.15625.pdf" }, "pub_year": "2020", "author_list": "Hutchcroft, Tom" }, { "id": "https://authors.library.caltech.edu/records/24qn3-vsa64", "eprint_id": 105032, "eprint_status": "archive", "datestamp": "2023-08-19 22:28:48", "lastmod": "2026-03-09 02:11:26", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" } ] }, "title": "The Lieb-Thirring inequalities: Recent results and open problems", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2020 by the author. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\nPartial support through US National Science Foundation grant DMS-1363432 is acknowledged. \n\nThe author would like to thank all his collaborators on the topic of Lieb\u2013Thirring inequalities and, in particular, A. Laptev, S. Larson, M. Lewin, E. H. Lieb, P. T. Nam and T. Weidl for helpful remarks on a preliminary version of this review.\n\n

Submitted - 2007.09326.pdf

", "abstract": "This review celebrates the generous gift by Ronald and Maxine Linde for the remodeling of the Caltech mathematics department and the author is very grateful to the editors of this volume for the invitation to contribute. We attempt to survey recent results and open problems connected to Lieb\u2013Thirring inequalities. In view of several excellent existing reviews [132, 17, 113, 91, 108] as well as highly recommended textbooks [134, 136], we sometimes put our focus on developments during the past decade.", "date": "2020-07-18", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20200819-122507869", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200819-122507869", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1363432" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2007.09326", "primary_object": { "basename": "2007.09326.pdf", "url": "https://authors.library.caltech.edu/records/24qn3-vsa64/files/2007.09326.pdf" }, "pub_year": "2020", "author_list": "Frank, Rupert L." }, { "id": "https://authors.library.caltech.edu/records/jqdzp-8w086", "eprint_id": 110546, "eprint_status": "archive", "datestamp": "2023-08-19 22:27:22", "lastmod": "2026-03-30 06:48:40", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aluffi-Paolo", "name": { "family": "Aluffi", "given": "Paolo" }, "orcid": "0000-0002-8249-685X" }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" }, { "id": "Qaisar-Waleed", "name": { "family": "Qaisar", "given": "Waleed" } } ] }, "title": "Motives of melonic graphs", "ispublished": "unpub", "full_text_status": "public", "note": "The first author acknowledges support from a Simons Foundation Collaboration Grant, award number 625561, and thanks the University of\nToronto for hospitality. The second author is partially supported by NSF grant DMS-1707882, and by NSERC Discovery Grant RGPIN-2018-04937 and Accelerator Supplement grant RGPAS-2018-522593, and by the Perimeter Institute for Theoretical\nPhysics. The third author worked on parts of this project as summer undergraduate\nresearch at the University of Toronto.", "abstract": "We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence-4 internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure, in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space M_(0,4). We also conjecture that the corresponding polynomials are log-concave, on the basis of hundreds of explicit computations.", "date": "2020-07-16", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210825-184604511", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184604511", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Simons Foundation", "grant_number": "625561" }, { "agency": "NSF", "grant_number": "DMS-1707882" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "RGPIN-2018-04937" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "RGPAS-2018-522593" }, { "agency": "Perimeter Institute for Theoretical Physics" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2007.08565", "primary_object": { "basename": "2007.08565.pdf", "url": "https://authors.library.caltech.edu/records/jqdzp-8w086/files/2007.08565.pdf" }, "pub_year": "2020", "author_list": "Aluffi, Paolo; Marcolli, Matilde; et al." }, { "id": "https://authors.library.caltech.edu/records/kwdr4-s2945", "eprint_id": 105037, "eprint_status": "archive", "datestamp": "2023-08-19 22:20:57", "lastmod": "2026-03-30 08:58:15", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Sabin-J", "name": { "family": "Sabin", "given": "Julien" } } ] }, "title": "Sharp Weyl laws with singular potentials", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2020 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.\n\n

Submitted - 2007.04284.pdf

", "abstract": "We consider the Laplace--Beltrami operator on a three-dimensional Riemannian manifold perturbed by a potential from the Kato class and study whether various forms of Weyl's law remain valid under this perturbation. We show that a pointwise Weyl law holds, modified by an additional term, for any Kato class potential with the standard sharp remainder term. The additional term is always of lower order than the leading term, but it may or may not be of lower order than the sharp remainder term. In particular, we provide examples of singular potentials for which this additional term violates the sharp pointwise Weyl law of the standard Laplace-Beltrami operator. For the proof we extend the method of Avakumovi\u0107 to the case of Schr\u00f6dinger operators with singular potentials.", "date": "2020-07-08", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20200819-151324737", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200819-151324737", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2007.04284", "primary_object": { "basename": "2007.04284.pdf", "url": "https://authors.library.caltech.edu/records/kwdr4-s2945/files/2007.04284.pdf" }, "pub_year": "2020", "author_list": "Frank, Rupert L. and Sabin, Julien" }, { "id": "https://authors.library.caltech.edu/records/ns6ke-29z07", "eprint_id": 110544, "eprint_status": "archive", "datestamp": "2023-08-19 22:00:19", "lastmod": "2026-03-30 08:55:32", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" }, { "id": "Panangaden-Jane", "name": { "family": "Panangaden", "given": "Jane" } } ] }, "title": "Quantum Statistical Mechanics and the Boundary of Modular Curves", "ispublished": "unpub", "full_text_status": "public", "note": "The first author is partially supported by NSF grant DMS-1707882, by NSERC Discovery Grant RGPIN-2018-04937 and Accelerator Supplement grant RGPAS-2018-522593, and the Perimeter Institute for Theoretical Physics.\n\n

Submitted - 2006.16897.pdf

", "abstract": "The theory of limiting modular symbols provides a noncommutative geometric model of the boundary of modular curves that includes irrational points in addition to cusps. A noncommutative space associated to this boundary is constructed, as part of a family of noncommutative spaces associated to different continued fractions algorithms, endowed with the structure of a quantum statistical mechanical system. Two special cases of this family of quantum systems can be interpreted as a boundary of the system associated to the Shimura variety of GL\u2082 and an analog for SL\u2082. The structure of KMS states for this family of systems is discussed. In the geometric cases, the ground states evaluated on boundary arithmetic elements are given by pairings of cusp forms and limiting modular symbols.", "date": "2020-06-30", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210825-184601111", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184601111", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1707882" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "RGPIN-2018-04937" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "RGPAS-2018-522593" }, { "agency": "Perimeter Institute for Theoretical Physics" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2006.16897", "primary_object": { "basename": "2006.16897.pdf", "url": "https://authors.library.caltech.edu/records/ns6ke-29z07/files/2006.16897.pdf" }, "pub_year": "2020", "author_list": "Marcolli, Matilde and Panangaden, Jane" }, { "id": "https://authors.library.caltech.edu/records/88t62-g9926", "eprint_id": 107323, "eprint_status": "archive", "datestamp": "2023-08-19 21:57:57", "lastmod": "2026-03-30 07:01:45", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Ooguri-H", "name": { "family": "Ooguri", "given": "Hirosi" }, "orcid": "0000-0001-6021-3778" }, { "id": "Takanayagi-Tadashi", "name": { "family": "Takayanagi", "given": "Tadashi" } } ] }, "title": "Cobordism Conjecture in AdS", "ispublished": "unpub", "full_text_status": "public", "note": "We thank Jacob McNamara, Cumrun Vafa, and Mark Van Raamsdonk for their comments on this paper. We also thank Mark Van Raamsdonk for sharing a draft of their paper [14] prior to publication. H.O. thanks participants of the Boot-strapping String Theory Workshop of the Simons Collaboration on the Nonpertubative Bootstrap for discussion. The work of H.O. is supported in part by U.S. Department of Energy grant de-sc0011632, by the World Premier International Research Center Initiative, MEXT, Japan, by JSPS Grant-in-Aid for Scientific Research C-26400240, and by JSPS Grant-in-Aid for Scientific Research on Innovative Areas 15H05895. T.T. is supported by Inamori Research Institute for Science and World Premier International Research Center Initiative (WPI Initiative) from the Japan Ministry of Education, Culture, Sports, Science and Technology (MEXT). T.T. is also supported by the Simons Foundation through the \"It from Qubit\" collaboration, JSPS Grant-in-Aid for Scientific Research (A) No.16H02182 and by JSPS Grant-in-Aid for Challenging Research (Exploratory) 18K18766.\n\n

Submitted - 2006.13953.pdf

", "abstract": "McNamara and Vafa conjectured that any pair of consistent quantum gravity theories can be connected by a domain wall. We test the conjecture in the context of the AdS/CFT correspondence. There are topological constraints on existence of an interface between the corresponding conformal field theories. We discuss how to construct domain walls in AdS predicted by the conjecture when the corresponding conformal interfaces are prohibited by topological obstructions.", "date": "2020-06-24", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210105-133413836", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210105-133413836", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-SC0011632" }, { "agency": "Ministry of Education, Culture, Sports, Science and Technology (MEXT)" }, { "agency": "Japan Society for the Promotion of Science (JSPS)", "grant_number": "C-26400240" }, { "agency": "Japan Society for the Promotion of Science (JSPS)", "grant_number": "15H05895" }, { "agency": "Simons Foundation" }, { "agency": "Japan Society for the Promotion of Science (JSPS)", "grant_number": "16H02182" }, { "agency": "Japan Society for the Promotion of Science (JSPS)", "grant_number": "18K18766" } ] }, "other_numbering_system": { "items": [ { "id": "2020-028", "name": "CALT-TH" } ] }, "local_group": { "items": [ { "id": "Walter-Burke-Institute-for-Theoretical-Physics" }, { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2006.13953", "primary_object": { "basename": "2006.13953.pdf", "url": "https://authors.library.caltech.edu/records/88t62-g9926/files/2006.13953.pdf" }, "pub_year": "2020", "author_list": "Ooguri, Hirosi and Takayanagi, Tadashi" }, { "id": "https://authors.library.caltech.edu/records/099r5-twh05", "eprint_id": 110540, "eprint_status": "archive", "datestamp": "2023-08-19 21:47:17", "lastmod": "2026-03-28 21:08:25", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Combe-No\u00e9mie", "name": { "family": "Combe", "given": "No\u00e9mie C." } }, { "id": "Manin-Yuri-I", "name": { "family": "Manin", "given": "Yuri I." } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Dessins for Modular Operad and Grothendieck-Teichm\u00fcller Group", "ispublished": "unpub", "full_text_status": "public", "note": "N. C. Combe acknowledges support from the Minerva Fast track grant from the Max Planck Institute for Mathematics in the Sciences, in Leipzig. \n\nM. Marcolli acknowledges support from NSF grant DMS-1707882 and NSERC grants RGPIN\u20132018\u201304937 and RGPAS\u20132018\u2013522593.\n\n

Submitted - 2006.13663.pdf

", "abstract": "A part of Grothendieck's program for studying the Galois group G_\u211a of the field of all algebraic numbers \u211a emerged from his insight that one should lift its action upon \u211a to the action of G_\u211a upon the (appropriately defined) profinite completion of \u03c0\u2081(\u2119\u00b9\u2216{0,1,\u221e}). The latter admits a good combinatorial encoding via finite graphs \"dessins d'enfant\". \n\nThis part was actively developing during the last decades, starting with foundational works of A. Belyi, V. Drinfeld and Y. Ihara. \n\nOur brief note concerns another part of Grothendieck program, in which its geometric environment is extended to moduli spaces of algebraic curves, more specifically, stable curves of genus zero with marked/labelled points. Our main goal is to show that dual graphs of such curves may play the role of \"modular dessins\" in an appropriate operadic context.", "date": "2020-06-03", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210825-184554285", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184554285", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Max Planck Institute for Mathematics in the Sciences" }, { "agency": "NSF", "grant_number": "DMS-1707882" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "RGPIN-2018-04937" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "RGPAS-2018-522593" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2006.13663", "primary_object": { "basename": "2006.13663.pdf", "url": "https://authors.library.caltech.edu/records/099r5-twh05/files/2006.13663.pdf" }, "pub_year": "2020", "author_list": "Combe, No\u00e9mie C.; Manin, Yuri I.; et al." }, { "id": "https://authors.library.caltech.edu/records/b78n3-3jm42", "eprint_id": 103290, "eprint_status": "archive", "datestamp": "2023-08-19 20:36:58", "lastmod": "2026-03-28 20:49:03", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" }, { "id": "Spodyneiko-L", "name": { "family": "Spodyneiko", "given": "Lev" }, "orcid": "0000-0002-6099-7717" } ] }, "title": "Higher-dimensional generalizations of the Thouless charge pump", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 2003.09519.pdf

", "abstract": "We define and study analogs of the Thouless charge pump for many-body gapped systems in dimension D. We show how to attach a topological invariant to a D-dimensional family of such systems, provided all of them have an on-site U(1) symmetry. For a large class of families we argue that this topological invariant is an integer. In the case of gapped systems of free fermions in two dimensions, the invariant can be expressed in terms of the curvature of the Bloch-Berry connection. We also obtain a new formula for the Thouless charge pump in 1d which involves only static linear response and is analogous to the Streda formula for Hall conductivity.", "date": "2020-03-20", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20200518-151948447", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200518-151948447", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2003.09519", "primary_object": { "basename": "2003.09519.pdf", "url": "https://authors.library.caltech.edu/records/b78n3-3jm42/files/2003.09519.pdf" }, "pub_year": "2020", "author_list": "Kapustin, Anton and Spodyneiko, Lev" }, { "id": "https://authors.library.caltech.edu/records/kppwz-x0f53", "eprint_id": 118799, "eprint_status": "archive", "datestamp": "2023-08-19 20:28:26", "lastmod": "2026-03-18 04:01:15", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Vigneaux-Juan-Pablo", "name": { "family": "Vigneaux", "given": "Juan Pablo" }, "orcid": "0000-0003-4696-4537" } ] }, "title": "A homological characterization of generalized multinomial coefficients related to the entropic chain rule", "ispublished": "unpub", "full_text_status": "public", "note": "The work presented here was part of my doctoral dissertation. I want to thank my Ph.D. advisor, Prof. Daniel Bennequin, for his support.\n\n

Submitted - 2003.02021.pdf

", "abstract": "There is an asymptotic relationship between the multiplicative relations among multinomial coefficients and the (additive) recurrence property of Shannon entropy known as the chain rule. We show that both types of identities are manifestations of a unique algebraic construction: a 1-cocycle condition in information cohomology, an algebraic invariant of phesheaves of modules on information structures (categories of observables). Baudot and Bennequin introduced this cohomology and proved that Shannon entropy represents the only nontrivial cohomology class in degree 1 when the coefficients are a natural presheaf of probabilistic functionals. The author obtained later a 1-parameter family of deformations of that presheaf, in such a way that each Tsallis \u03b1-entropy appears as the unique 1-cocycle associated to the parameter \u03b1. In this article, we introduce a new presheaf of combinatorial functionals, which are measurable functions of finite arrays of integers; these arrays represent histograms associated to random experiments. In this case, the only cohomology class in degree 0 is generated by the exponential function and 1-cocycles are Fonten\u00e9-Ward generalized multinomial coefficients. As a byproduct, we get a simple combinatorial analogue of the fundamental equation of information theory that characterizes the generalized binomial coefficients. The asymptotic relationship mentioned above is extended to a correspondence between certain generalized multinomial coefficients and any \u03b1-entropy, that sheds new light on the meaning of the chain rule and its deformations.", "date": "2020-03-04", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20230117-165349882", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230117-165349882", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" }, { "id": "Physics-Department" } ] }, "doi": "10.48550/arXiv.2003.02021", "primary_object": { "basename": "2003.02021.pdf", "url": "https://authors.library.caltech.edu/records/kppwz-x0f53/files/2003.02021.pdf" }, "pub_year": "2020", "author_list": "Vigneaux, Juan Pablo" }, { "id": "https://authors.library.caltech.edu/records/b7n9w-z9n48", "eprint_id": 111032, "eprint_status": "archive", "datestamp": "2023-08-19 20:03:59", "lastmod": "2026-03-28 20:44:07", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Angel-Omer", "name": { "family": "Angel", "given": "Omer" }, "orcid": "0000-0002-6451-8242" }, { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" }, { "id": "J\u00e1rai-Antal-A", "name": { "family": "J\u00e1rai", "given": "Antal A." }, "orcid": "0000-0003-3522-498X" } ] }, "title": "On the tail of the branching random walk local time", "ispublished": "unpub", "full_text_status": "public", "note": "The authors are grateful to the organizers of the Oberwolfach Workshop Strongly Correlated Interacting Processes, where this work was initiated. We thank Ed Perkins and Jean-Fran\u00e7ois Le Gall for helpful discussions on the literature. OA is supported in part by an NSERC discovery grant.\n\n

Submitted - 2002.12188.pdf

", "abstract": "Consider a critical branching random walk on \u2124^d, d\u22651, started with a single particle at the origin, and let L(x) be the total number of particles that ever visit a vertex x. We study the tail of L(x) under suitable conditions on the offspring distribution. In particular, our results hold if the offspring distribution has an exponential moment.", "date": "2020-02-27", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202119558", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202119558", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2002.12188", "primary_object": { "basename": "2002.12188.pdf", "url": "https://authors.library.caltech.edu/records/b7n9w-z9n48/files/2002.12188.pdf" }, "pub_year": "2020", "author_list": "Angel, Omer; Hutchcroft, Tom; et al." }, { "id": "https://authors.library.caltech.edu/records/8bzn7-k1747", "eprint_id": 111031, "eprint_status": "archive", "datestamp": "2023-08-19 19:55:55", "lastmod": "2026-03-09 02:35:40", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" } ] }, "title": "Slightly supercritical percolation on nonamenable graphs I: The distribution of finite clusters", "ispublished": "unpub", "full_text_status": "public", "note": "We thank Jonathan Hermon and Asaf Nachmias for many helpful discussions, and thank Remco van der Hofstad for helpful comments on an earlier version of this manuscript. We also thank Antoine Godin for sharing his simplified proof of Proposition 3.1 with us.\n\n

Submitted - 2002.02916.pdf

", "abstract": "We study the distribution of finite clusters in slightly supercritical (p\u2193pc) Bernoulli bond percolation on transitive nonamenable graphs, proving in particular that if G is a transitive nonamenable graph satisfying the L2 boundedness condition (pc0 such that\nPp(n\u2264|K|<\u221e)\u224dn\u22121/2exp[\u2212\u0398(|p\u2212pc|2n)]\nand\nPp(r\u2264Rad(K)<\u221e)\u224dr\u22121exp[\u2212\u0398(|p\u2212pc|r)]\nfor every p\u2208(pc\u2212\u03b4,pc+\u03b4) and n,r\u22651, where all implicit constants depend only on G. We deduce in particular that the critical exponents \u03b3\u2032 and \u0394\u2032 describing the rate of growth of the moments of a finite cluster as p\u2193pc take their mean-field values of 1 and 2 respectively.\nThese results apply in particular to Cayley graphs of nonelementary hyperbolic groups, to products with trees, and to transitive graphs of spectral radius \u03c1<1/2. In particular, every finitely generated nonamenable group has a Cayley graph to which these results apply. They are new for graphs that are not trees. The corresponding facts are yet to be understood on \u2124d even for d very large. In a second paper in this series, we will apply these results to study the geometric and spectral properties of infinite slightly supercritical clusters in the same setting.", "date": "2020-02-07", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202116146", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202116146", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2002.02916", "primary_object": { "basename": "2002.02916.pdf", "url": "https://authors.library.caltech.edu/records/8bzn7-k1747/files/2002.02916.pdf" }, "pub_year": "2020", "author_list": "Hutchcroft, Tom" }, { "id": "https://authors.library.caltech.edu/records/14tsm-ebs32", "eprint_id": 110839, "eprint_status": "archive", "datestamp": "2023-08-19 19:36:39", "lastmod": "2026-03-10 00:02:30", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Porta-Mauro", "name": { "family": "Porta", "given": "Mauro" }, "orcid": "0000-0002-1239-3409" }, { "id": "Yu-Tony-Yue", "name": { "family": "Yu", "given": "Tony Yue" }, "orcid": "0000-0002-6019-8552" } ] }, "title": "Non-archimedean quantum K-invariants", "ispublished": "unpub", "full_text_status": "public", "keywords": "Quantum K-invariant, Gromov-Witten, non-archimedean geometry, \nrigid analytic geometry", "note": "We are very grateful to Federico Binda, Antoine Chambert-Loir, Benjamin Hennion, Maxim Kontsevich, G\u00e9rard Laumon, Y.P. Lee, Jacob Lurie, Etienne Mann, Tony Pantev, Francesco Sala, Carlos Simpson, Georg Tamme, Bertrand To\u00ebn and Gabriele Vezzosi for valuable discussions. We would like to thank Marco Robalo in particular for many detailed discussions and for his enthusiasm for our work. The authors would also like to thank each other for the joint effort. This research was partially conducted during the period when T.Y. Yu served as a Clay Research Fellow. We have also received supports from the National Science Foundation under Grant No. 1440140, while we were in residence at the Mathematical Sciences Research Institute in Berkeley, California, and from the Agence Nationale de la Recherce under Grant ANR-17-CE40-0014, while we were at the Universit\u00e9 Paris-Sud in Orsay, France.\n\n

Submitted - 2001.05515.pdf

", "abstract": "We construct quantum K-invariants in non-archimedean analytic geometry. Our approach differs from the classical one in algebraic geometry via perfect obstruction theory. Instead, we build on our previous works on the foundation of derived non-archimedean geometry and Gromov compactness. We obtain a list of natural geometric relations of the stacks of stable maps, directly at the derived level, with respect to elementary operations on graphs, namely, products, cutting edges, forgetting tails and contracting edges. They imply immediately the corresponding properties of K-theoretic invariants. The derived approach produces highly intuitive statements and functorial proofs, without the need to manipulate perfect obstruction theories. The flexibility of our derived approach to quantum K-invariants allows us to impose not only simple incidence conditions for marked points, but also incidence conditions with multiplicities, which we discuss in the final section of the paper. This leads to a new set of enumerative invariants, which is not yet considered in the literature, to the best of our knowledge. For the proofs, we had to further develop the foundations of derived non-archimedean geometry in this paper: we study derived lci morphisms, relative analytification, and deformation to the normal bundle. Our motivations come from non-archimedean enumerative geometry and mirror symmetry.", "date": "2020-01-15", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210914-164513718", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210914-164513718", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Clay Mathematics Institute" }, { "agency": "NSF", "grant_number": "DMS-1440140" }, { "agency": "Agence Nationale de la Recherche (ANR)", "grant_number": "ANR-17-CE40-0014" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.2001.05515", "primary_object": { "basename": "2001.05515.pdf", "url": "https://authors.library.caltech.edu/records/14tsm-ebs32/files/2001.05515.pdf" }, "pub_year": "2020", "author_list": "Porta, Mauro and Yu, Tony Yue" }, { "id": "https://authors.library.caltech.edu/records/eyaza-xse95", "eprint_id": 105389, "eprint_status": "archive", "datestamp": "2023-08-19 19:11:55", "lastmod": "2026-03-31 15:48:55", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" }, { "id": "Gishboliner-Lior", "name": { "family": "Gishboliner", "given": "Lior" }, "orcid": "0000-0003-0688-8111" }, { "id": "Levanzov-Yevgeny", "name": { "family": "Levanzov", "given": "Yevgeny" } }, { "id": "Shapira-Asaf", "name": { "family": "Shapira", "given": "Asaf" } } ] }, "title": "A New Bound for the Brown-Erd\u0151s-S\u00f3s Problem", "ispublished": "unpub", "full_text_status": "public", "note": "Supported in part by ISF Grant 1028/16 and ERC Starting Grant 633509.\n\n

Submitted - 1912.08834.pdf

", "abstract": "Let f(n,v,e) denote the maximum number of edges in a 3-uniform hypergraph not containing e edges spanned by at most v vertices. One of the most influential open problems in extremal combinatorics then asks, for a given number of edges e\u22653, what is the smallest integer d=d(e) so that f(n,e+d,e)=o(n\u00b2)? This question has its origins in work of Brown, Erd\u0151s and S\u00f3s from the early 70's and the standard conjecture is that d(e)=3 for every e\u22653. The state of the art result regarding this problem was obtained in 2004 by S\u00e1rk\u00f6zy and Selkow, who showed that f(n,e+2+\u230alog\u2082e\u230b,e)=o(n\u00b2). The only improvement over this result was a recent breakthrough of Solymosi and Solymosi, who improved the bound for d(10) from 5 to 4. We obtain the first asymptotic improvement over the S\u00e1rk\u00f6zy--Selkow bound, showing that\nf(n,e+O(loge/logloge),e)=o(n\u00b2).", "date": "2019-12-18", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20200915-144809437", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200915-144809437", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Israel Science Foundation", "grant_number": "1028/16" }, { "agency": "European Research Council (ERC)", "grant_number": "633509" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1912.08834", "primary_object": { "basename": "1912.08834.pdf", "url": "https://authors.library.caltech.edu/records/eyaza-xse95/files/1912.08834.pdf" }, "pub_year": "2019", "author_list": "Conlon, David; Gishboliner, Lior; et al." }, { "id": "https://authors.library.caltech.edu/records/z89d1-ffk59", "eprint_id": 100877, "eprint_status": "archive", "datestamp": "2023-08-19 19:04:02", "lastmod": "2026-03-09 23:59:28", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Fritz-Tobias", "name": { "family": "Fritz", "given": "Tobias" } }, { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "Omer" }, "orcid": "0000-0002-0111-0418" } ] }, "title": "There are no monotone homomorphisms out of the convolution semigroup", "ispublished": "unpub", "full_text_status": "public", "note": "T. Fritz would like to thank Lu Chen for discussion. O. Tamuz was supported by the Simons Foundation (#419427) and by the BSF (#2018397).\n\n

Submitted - 1912.01733.pdf

", "abstract": "We prove that there is no nonzero way of assigning real numbers to probability measures on R in a way which is monotone under first-order stochastic dominance and additive under convolution.", "date": "2019-12-03", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20200123-105749369", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200123-105749369", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Simons Foundation", "grant_number": "419427" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2018397" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1912.01733", "primary_object": { "basename": "1912.01733.pdf", "url": "https://authors.library.caltech.edu/records/z89d1-ffk59/files/1912.01733.pdf" }, "pub_year": "2019", "author_list": "Fritz, Tobias and Tamuz, Omer" }, { "id": "https://authors.library.caltech.edu/records/eaxj1-wty15", "eprint_id": 117594, "eprint_status": "archive", "datestamp": "2023-08-19 18:47:34", "lastmod": "2026-03-09 23:57:28", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Song-Antoine", "name": { "family": "Song", "given": "Antoine" } } ] }, "title": "Morse index, Betti numbers and singular set of bounded area minimal hypersurfaces", "ispublished": "unpub", "full_text_status": "public", "note": "I am grateful to Fernando Cod\u00e1 Marques and Andr\u00e9 Neves for their continued support. This work benefited from extended discussions with Jonathan J. Zhu. I thank Otis Chodosh, Chao Li, Davi Maximo, Brian White, Xin Zhou for interesting conversations, and Hans-Joachim Hein for explaining to me his unpublished work with Aaron Naber on certain constructions of K\u00e4hler-Einstein metrics. I also thank Giada Franz and Santiago Cordero Misteli for correcting a mistake in Lemma 25. I am indebted to the reviewers, whose constructive comments and numerous corrections substantially improved the writing of this article. \n\nThe author was partially supported by NSF-DMS-1509027. This research was partially conducted during the period the author served as a Clay Research Fellow.", "abstract": "We introduce a combinatorial argument to study closed minimal hypersurfaces of bounded area and high Morse index. Let (M\u207f\u207a\u00b9,g) be a closed Riemannian manifold and \u03a3 subset M$ be a closed embedded minimal hypersurface with area at most A > 0 and with a singular set of Hausdorff dimension at most n - 7. We show the following bounds: there is C_A > 0 depending only on n, g, and A so that \u03a3\u1d62\u208c\u2080\u207f b\u1da6(\u03a3) \u2264 C_A (1 + index(\u03a3)) if 3 \u2264 n + 1 \u2264 7, H\u207f\u207b\u2077(Sing(\u03a3)) \u2264 C_A (1 + index(\u03a3))^(7/n) if n + 1 \u2265 8, where b\u1da6 denote the Betti numbers over any field, H\u207f\u207b\u2077 is the (n - 7)-dimensional Hausdorff measure and Sing(\u03a3) is the singular set of \u03a3. In fact in dimension n + 1 = 3, C_A depends linearly on A. We list some open problems at the end of the paper.", "date": "2019-11-20", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20221026-539125000.6", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221026-539125000.6", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1509027" }, { "agency": "Clay Mathematics Institute" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1911.09166", "pub_year": "2019", "author_list": "Song, Antoine" }, { "id": "https://authors.library.caltech.edu/records/ezp7p-t3102", "eprint_id": 117595, "eprint_status": "archive", "datestamp": "2023-08-19 18:46:07", "lastmod": "2026-03-09 23:57:38", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Ketover-Daniel", "name": { "family": "Ketover", "given": "Daniel" } }, { "id": "Liokumovich-Yevgeny", "name": { "family": "Liokumovich", "given": "Yevgeny" } }, { "id": "Song-Antoine", "name": { "family": "Song", "given": "Antoine" } } ] }, "title": "On the existence of minimal Heegaard surfaces", "ispublished": "unpub", "full_text_status": "public", "note": "D.K. was partially supported by an NSF Postdoctoral Research fellowship as well as ERC-2011-StG-278940. Y.L. was partially supported by NSF DMS-1711053 and NSERC Discovery grants. A.S. was partially supported by NSF-DMS-1509027. This research was partially conducted during the period A.S. served as a Clay Research Fellow.", "abstract": "Let H be a strongly irreducible Heegaard surface in a closed oriented Riemannian 3-manifold. We prove that H is either isotopic to a minimal surface of index at most one or isotopic to the boundary of a tubular neighborhood about a non-orientable minimal surface with a vertical handle attached. This confirms a long-standing conjecture of J. Pitts and J.H. Rubinstein. In the case of positive scalar curvature, we show for spherical space forms not diffeomorphic to S\u00b3 or RP\u00b3 that any strongly irreducible Heegaard splitting is isotopic to a minimal surface, and that there is a minimal Heegaard splitting of area less than $4\u03c0$ if R \u2265 6.", "date": "2019-11-17", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20221026-539131000.7", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221026-539131000.7", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF Postdoctoral Fellowship" }, { "agency": "European Research Council (ERC)", "grant_number": "278940" }, { "agency": "NSF", "grant_number": "DMS-1711053" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)" }, { "agency": "NSF", "grant_number": "DMS-1509027" }, { "agency": "Clay Mathematics Institute" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1911.07161", "pub_year": "2019", "author_list": "Ketover, Daniel; Liokumovich, Yevgeny; et al." }, { "id": "https://authors.library.caltech.edu/records/k4eb1-c9n17", "eprint_id": 111030, "eprint_status": "archive", "datestamp": "2023-08-19 18:12:44", "lastmod": "2026-03-09 02:36:11", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" } ] }, "title": "Non-intersection of transient branching random walks", "ispublished": "unpub", "full_text_status": "public", "note": "We thank Itai Benjamini, Jonathan Hermon, Asaf Nachmias, and Elisabetta Candellero for useful discussions. In particular, we thank Asaf for discussions that led to a substantially simpler proof of Theorem 3.3. We also thank the anonymous referee for their careful reading and helpful suggestions.\n\n

Accepted Version - 1910.01018.pdf

", "abstract": "Let G be a Cayley graph of a nonamenable group with spectral radius \u03c1<1. It is known that branching random walk on G with offspring distribution \u03bc is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring \u03bc\u00af satisfies \u03bc\u00af\u2264\u03c1\u22121. Benjamini and M\u00fcller (2010) conjectured that throughout the transient supercritical phase 1<\u03bc\u00af\u2264\u03c1\u22121, and in particular at the recurrence threshold \u03bc\u00af=\u03c1\u22121, the trace of the branching random walk is tree-like in the sense that it is infinitely-ended almost surely on the event that the walk survives forever. This is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely. We prove this conjecture, along with several other related conjectures made by the same authors.\nA central contribution of this work is the introduction of the notion of local unimodularity, which we expect to have several further applications in the future.", "date": "2019-10-02", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202112742", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202112742", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1910.01018", "primary_object": { "basename": "1910.01018.pdf", "url": "https://authors.library.caltech.edu/records/k4eb1-c9n17/files/1910.01018.pdf" }, "pub_year": "2019", "author_list": "Hutchcroft, Tom" }, { "id": "https://authors.library.caltech.edu/records/ea1m3-7tq44", "eprint_id": 110842, "eprint_status": "archive", "datestamp": "2023-08-19 17:28:21", "lastmod": "2026-03-31 15:55:16", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Keel-Sean", "name": { "family": "Keel", "given": "Sean" } }, { "id": "Yu-Tony-Yue", "name": { "family": "Yu", "given": "Tony Yue" }, "orcid": "0000-0002-6019-8552" } ] }, "title": "The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus", "ispublished": "unpub", "full_text_status": "public", "keywords": "Frobenius structure, mirror symmetry, log Calabi-Yau, skeletal curve, non-archimedean geometry, rigid analytic geometry, cluster algebra, scattering diagram, wall-crossing, broken lines", "note": "We benefited tremendously from profound, detailed technical discussions with Mark Gross, Paul Hacking, Johannes Nicaise and Maxim Kontsevich. The beautiful Frobenius structure conjecture is due to Hacking, as is the idea of using degeneration to the toric case to prove non-degeneracy of the trace pairing. We enjoyed fruitful conversations with M. Baker, V. Berkovich, M. Brown, A. Chambert-Loir, F. Charles, A. Corti, A. Durcos, W. Gubler, E. Mazzon, M. Porta, J. Rabinoff, M. Robalo, B. Siebert, Y. Soibelman, M. Temkin and J. Xie. Keel would like to especially thank B. Conrad and S. Payne for detailed email tutorials on rigid geometry. Keel was supported by NSF grant DMS-1561632. T.Y. Yu was supported by the Clay Mathematics Institute as Clay Research Fellow. Much of the research was carried out during the authors' trips to IHES and IAS.\n\n

Submitted - 1908.09861.pdf

", "abstract": "We show that the naive counts of rational curves in any affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a compatible multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich.", "date": "2019-08-26", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210914-164521010", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210914-164521010", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1561632" }, { "agency": "Clay Mathematics Institute" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1908.09861", "primary_object": { "basename": "1908.09861.pdf", "url": "https://authors.library.caltech.edu/records/ea1m3-7tq44/files/1908.09861.pdf" }, "pub_year": "2019", "author_list": "Keel, Sean and Yu, Tony Yue" }, { "id": "https://authors.library.caltech.edu/records/3j2gs-bf147", "eprint_id": 98021, "eprint_status": "archive", "datestamp": "2023-08-19 17:22:42", "lastmod": "2026-03-08 18:13:57", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" } ] }, "title": "Graphs with few paths of prescribed length between any two vertices", "ispublished": "unpub", "full_text_status": "public", "note": "Research supported by a Royal Society University Research Fellowship. \n\nI would like to thank Boris Bukh, Gal Kronenberg, Rudi Mrazovi\u0107 and Lisa Sauermann for a number of valuable comments on an earlier draft of this paper. I would also like to thank the anonymous referee for their considered review.\n\n

Submitted - 1411.0856.pdf

", "abstract": "We use a variant of Bukh's random algebraic method to show that for every natural number k \u2265 2 there exists a natural number \u2113 such that, for every n, there is a graph with n vertices and \u03a9_(k)(n^(1 + 1/k)) edges with at most \u2113 paths of length k between any two vertices. A result of Faudree and Simonovits shows that the bound on the number of edges is tight up to the implied constant.", "date": "2019-08-16", "date_type": "published", "id_number": "CaltechAUTHORS:20190819-170853333", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170853333", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Royal Society" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1411.0856", "primary_object": { "basename": "1411.0856.pdf", "url": "https://authors.library.caltech.edu/records/3j2gs-bf147/files/1411.0856.pdf" }, "pub_year": "2019", "author_list": "Conlon, David" }, { "id": "https://authors.library.caltech.edu/records/2rh1h-48t12", "eprint_id": 103931, "eprint_status": "archive", "datestamp": "2023-08-19 17:16:52", "lastmod": "2026-03-31 14:44:25", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frisch-J", "name": { "family": "Frisch", "given": "Joshua" } }, { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "Omer" }, "orcid": "0000-0002-0111-0418" } ] }, "title": "Characteristic measures of symbolic dynamical systems", "ispublished": "unpub", "full_text_status": "public", "note": "J. Frisch was funded by NSF Grant DMS-1464475. O. Tamuz was supported by the Simons Foundation (#419427) and by a BSF award (#2018397).\n\n

Submitted - 1908.02930.pdf

", "abstract": "A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques to show that automorphism groups of minimal zero entropy shifts are sofic.", "date": "2019-08-08", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20200615-160518971", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200615-160518971", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1464475" }, { "agency": "Simons Foundation", "grant_number": "419427" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2018397" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1908.02930", "primary_object": { "basename": "1908.02930.pdf", "url": "https://authors.library.caltech.edu/records/2rh1h-48t12/files/1908.02930.pdf" }, "pub_year": "2019", "author_list": "Frisch, Joshua and Tamuz, Omer" }, { "id": "https://authors.library.caltech.edu/records/4jhes-wet92", "eprint_id": 98030, "eprint_status": "archive", "datestamp": "2023-08-19 17:14:24", "lastmod": "2026-03-08 04:08:27", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" } ] }, "title": "The Ramsey number of books", "ispublished": "unpub", "full_text_status": "public", "note": "Research supported by a Royal Society University Research Fellowship and ERC Starting Grant 676632. \n\nThis paper was partially written while I was visiting the California Institute of Technology as a Moore Distinguished Scholar and I am extremely grateful for their kind support. I am also much indebted to Jacob Fox, Lisa Sauermann and Yuval Wigderson for pointing out a subtle error in the first version of this paper. Finally, I would like to thank the anonymous reviewers for several helpful remarks which improved the presentation.\n\n

Submitted - 1808.03157.pdf

", "abstract": "We show that in every two-colouring of the edges of the complete graph K_N there is a monochromatic K_k which can be extended in at least (1 + o_(k)(1))2^(-k)N ways to a monochromatic K_(k+1). This result is asymptotically best possible, as may be seen by considering a random colouring. Equivalently, defining the book B_n^(k) to be the graph consisting of n copies of K_(k+1) all sharing a common K_k, we show that the Ramsey number r(B_n^(k)) = 2^(k)n + o_(k)(n). In this form, our result answers a question of Erd\u0151s, Faudree, Rousseau and Schelp and establishes an asymptotic version of a conjecture of Thomason.", "date": "2019-08-06", "date_type": "published", "id_number": "CaltechAUTHORS:20190819-170924800", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170924800", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Royal Society" }, { "agency": "European Research Council (ERC)", "grant_number": "676632" }, { "agency": "Gordon and Betty Moore Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1808.03157", "primary_object": { "basename": "1808.03157.pdf", "url": "https://authors.library.caltech.edu/records/4jhes-wet92/files/1808.03157.pdf" }, "pub_year": "2019", "author_list": "Conlon, David" }, { "id": "https://authors.library.caltech.edu/records/427jk-z8f82", "eprint_id": 98027, "eprint_status": "archive", "datestamp": "2023-08-19 16:24:42", "lastmod": "2026-03-08 18:13:49", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" }, { "id": "Fox-J", "name": { "family": "Fox", "given": "Jacob" } }, { "id": "Kwan-Matthew", "name": { "family": "Kwan", "given": "Matthew" } }, { "id": "Sudakov-B", "name": { "family": "Sudakov", "given": "Benny" } } ] }, "title": "Hypergraph cuts above the average", "ispublished": "unpub", "full_text_status": "public", "note": "Conlon research supported by a Royal Society University Research Fellowship and by ERC Starting Grant 676632. Fox research supported by a Packard Fellowship and by NSF Career Award DMS-1352121. Sudakov research supported in part by SNSF grant 200021-175573.\n\n

Submitted - 1803.08462.pdf

", "abstract": "An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut of size m/2 + \u03a9(\u221am), and this is best possible. That is, there exist cuts which exceed the expected size of a random cut by some multiple of the standard deviation. We study analogues of this and related results in hypergraphs. First, we observe that similarly to graphs, every m-edge k-uniform hypergraph has an r-cut whose size is \u03a9(\u221am) larger than the expected size of a random r-cut. Moreover, in the case where k = 3 and r = 2 this bound is best possible and is attained by Steiner triple systems. Surprisingly, for all other cases (that is, if k \u2265 4 or r \u2265 3), we show that every m-edge k-uniform hypergraph has an r-cut whose size is \u03a9(m^(5/9)) larger than the expected size of a random r-cut. This is a significant difference in behaviour, since the amount by which the size of the largest cut exceeds the expected size of a random cut is now considerably larger than the standard deviation.", "date": "2019-06-27", "date_type": "published", "id_number": "CaltechAUTHORS:20190819-170914451", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170914451", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Royal Society" }, { "agency": "European Research Council (ERC)", "grant_number": "676632" }, { "agency": "David and Lucile Packard Foundation" }, { "agency": "NSF", "grant_number": "DMS-1352121" }, { "agency": "Swiss National Science Foundation (SNSF)", "grant_number": "200021-175573" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1803.08462", "primary_object": { "basename": "1803.08462.pdf", "url": "https://authors.library.caltech.edu/records/427jk-z8f82/files/1803.08462.pdf" }, "pub_year": "2019", "author_list": "Conlon, David; Fox, Jacob; et al." }, { "id": "https://authors.library.caltech.edu/records/3fpfb-rxa32", "eprint_id": 98025, "eprint_status": "archive", "datestamp": "2023-08-19 16:22:47", "lastmod": "2026-03-08 03:20:26", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" } ] }, "title": "Hypergraph expanders from Cayley graphs", "ispublished": "unpub", "full_text_status": "public", "note": "Research supported by a Royal Society University Research Fellowship and by ERC Starting Grant 676632. \n\nThe author gratefully acknowledges the support of the Simons Institute for the Theory of Computing during part of the period when this paper was written. The author is also indebted to Noga Alon, who brought the problem of constructing high-dimensional expanders to his attention, and to Rajko Nenadov, Jonathan Tidor and Yufei Zhao for several valuable discussions.\n\n

Submitted - 1709.10006.pdf

", "abstract": "We present a simple mechanism, which can be randomised, for constructing sparse 3-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over \u2124^(t)_(2) and have vertex degree which is polylogarithmic in the number of vertices. Their expansion properties, which are derived from the underlying Cayley graphs, include analogues of vertex and edge expansion in graphs, rapid mixing of the random walk on the edges of the skeleton graph, uniform distribution of edges on large vertex subsets and the geometric overlap property.", "date": "2019-06-24", "date_type": "published", "id_number": "CaltechAUTHORS:20190819-170907486", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170907486", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Royal Society" }, { "agency": "European Research Council (ERC)", "grant_number": "676632" }, { "agency": "Simons Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1709.10006", "primary_object": { "basename": "1709.10006.pdf", "url": "https://authors.library.caltech.edu/records/3fpfb-rxa32/files/1709.10006.pdf" }, "pub_year": "2019", "author_list": "Conlon, David" }, { "id": "https://authors.library.caltech.edu/records/gw2tv-g4730", "eprint_id": 110538, "eprint_status": "archive", "datestamp": "2023-08-19 14:51:51", "lastmod": "2026-03-31 15:28:42", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Port-Alexander", "name": { "family": "Port", "given": "Alexander" }, "orcid": "0000-0002-2862-6895" }, { "id": "Karidi-Taelin", "name": { "family": "Karidi", "given": "Taelin" } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Topological Analysis of Syntactic Structures", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 1903.05181.pdf

", "abstract": "We use the persistent homology method of topological data analysis and dimensional analysis techniques to study data of syntactic structures of world languages. We analyze relations between syntactic parameters in terms of dimensionality, of hierarchical clustering structures, and of non-trivial loops. We show there are relations that hold across language families and additional relations that are family-specific. We then analyze the trees describing the merging structure of persistent connected components for languages in different language families and we show that they partly correlate to historical phylogenetic trees but with significant differences. We also show the existence of interesting non-trivial persistent first homology groups in various language families. We give examples where explicit generators for the persistent first homology can be identified, some of which appear to correspond to homoplasy phenomena, while others may have an explanation in terms of historical linguistics, corresponding to known cases of syntactic borrowing across different language subfamilies.", "date": "2019-03-12", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210825-184550813", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210825-184550813", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1903.05181", "primary_object": { "basename": "1903.05181.pdf", "url": "https://authors.library.caltech.edu/records/gw2tv-g4730/files/1903.05181.pdf" }, "pub_year": "2019", "author_list": "Port, Alexander; Karidi, Taelin; et al." }, { "id": "https://authors.library.caltech.edu/records/hr7ja-y2861", "eprint_id": 100888, "eprint_status": "archive", "datestamp": "2023-08-19 14:44:59", "lastmod": "2026-03-09 23:58:46", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "Omer" }, "orcid": "0000-0002-0111-0418" }, { "id": "Zheng-Tianyi", "name": { "family": "Zheng", "given": "Tianyi" } } ] }, "title": "On the spectrum of asymptotic entropies of random walks", "ispublished": "unpub", "full_text_status": "public", "note": "Omer Tamuz was supported by a grant from the Simons Foundation (#419427).\n\n

Submitted - 1903.01312.pdf

", "abstract": "Given a random walk on a free group, we study the random walks it induces on the group's quotients. We show that the spectrum of asymptotic entropies of the induced random walks has no isolated points, except perhaps its maximum.", "date": "2019-03-04", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20200124-085648355", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200124-085648355", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Simons Foundation", "grant_number": "419427" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1903.01312", "primary_object": { "basename": "1903.01312.pdf", "url": "https://authors.library.caltech.edu/records/hr7ja-y2861/files/1903.01312.pdf" }, "pub_year": "2019", "author_list": "Tamuz, Omer and Zheng, Tianyi" }, { "id": "https://authors.library.caltech.edu/records/hgkxn-1da10", "eprint_id": 117596, "eprint_status": "archive", "datestamp": "2023-08-19 14:21:51", "lastmod": "2026-03-09 23:57:33", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Song-Antoine", "name": { "family": "Song", "given": "Antoine" } } ] }, "title": "A dichotomy for minimal hypersurfaces in manifolds thick at infinity", "ispublished": "unpub", "full_text_status": "public", "note": "The author was partially supported by NSF-DMS-1509027. \n\nI am grateful to my advisor Fernando Cod\u00e1 Marques for his crucial guidance. I thank Yevgeny Liokumovich for explaining [2] to me and mentioning [17], [37]. I am thankful to Misha Gromov for exchanges about [17]. I also want to thank Franco Vargas Pallete for discussing with me Yau's conjecture for finite volume hyperbolic 3-manifolds, a result of which he was also aware. Moreover, a very careful reading by the referees improved the writing of this article.", "abstract": "Let (M,g) be a complete (n + 1)-dimensional Riemannian manifold with 2 \u2264 n \u2264 6. Our main theorem generalizes the solution of S.-T. Yau's conjecture on the abundance of minimal surfaces and builds on a result of M. Gromov. Suppose that (M,g) has bounded geometry, or more generally is thick at infinity. Then the following dichotomy holds for the space of closed hypersurfaces in M: either there are infinitely many saddle points of the n-volume functional, or there is none. Additionally, we give a new short proof of the existence of a finite volume minimal hypersurface in finite volume manifolds, we check Yau's conjecture for finite volume hyperbolic 3-manifolds and we extend the density result due to Irie-Marques-Neves when (M,g) is shrinking to zero at infinity.", "date": "2019-02-18", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20221026-539140000.8", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221026-539140000.8", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1509027" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1902.06767", "pub_year": "2019", "author_list": "Song, Antoine" }, { "id": "https://authors.library.caltech.edu/records/kzw07-h3e65", "eprint_id": 98029, "eprint_status": "archive", "datestamp": "2023-08-19 14:15:55", "lastmod": "2026-03-08 17:36:27", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" }, { "id": "Lee-Joonkyung", "name": { "family": "Lee", "given": "Joonkyung" } } ] }, "title": "On the extremal number of subdivisions", "ispublished": "unpub", "full_text_status": "public", "note": "Conlon research supported by a Royal Society University Research Fellowship and by ERC Starting Grant 676632. Lee research supported by ERC Starting Grant 676632. \n\nThis paper was partially written while the first author was visiting the California Institute of Technology as a Moore Distinguished Scholar and he is extremely grateful for their kind support.\n\n

Submitted - 1807.05008.pdf

", "abstract": "One of the cornerstones of extremal graph theory is a result of F\u00fcredi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if H is a bipartite graph with maximum degree r on one side, then there is a constant C such that every graph with n vertices and Cn^(2 - (1/r)) edges contains a copy of H. This result is tight up to the constant when H contains a copy of K_(r,s) with s sufficiently large in terms of r. We conjecture that this is essentially the only situation in which F\u00fcredi's result can be tight and prove this conjecture for r = 2. More precisely, we show that if H is a C_(4)-free bipartite graph with maximum degree 2 on one side, then there are positive constants C and \u03b4 such that every graph with n vertices and Cn^(3/2) - \u03b4) edges contains a copy of H. This answers a question of Erd\u0151s from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest.", "date": "2019-02-08", "date_type": "published", "id_number": "CaltechAUTHORS:20190819-170921338", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170921338", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Royal Society" }, { "agency": "European Research Council (ERC)", "grant_number": "676632" }, { "agency": "Gordon and Betty Moore Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1807.05008", "primary_object": { "basename": "1807.05008.pdf", "url": "https://authors.library.caltech.edu/records/kzw07-h3e65/files/1807.05008.pdf" }, "pub_year": "2019", "author_list": "Conlon, David and Lee, Joonkyung" }, { "id": "https://authors.library.caltech.edu/records/xam4m-fzc19", "eprint_id": 98034, "eprint_status": "archive", "datestamp": "2023-08-19 13:55:26", "lastmod": "2026-03-31 05:33:42", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" }, { "id": "Fox-J", "name": { "family": "Fox", "given": "Jacob" } }, { "id": "Sudakov-B", "name": { "family": "Sudakov", "given": "Benny" } } ] }, "title": "Independent arithmetic progressions", "ispublished": "unpub", "full_text_status": "public", "note": "Conlon research supported by ERC Starting Grant 676632. Fox research supported by a Packard Fellowship and by NSF Career Award DMS-1352121. Sudakov research supported in part by SNSF grant 200021-175573. \n\nThis note was first written in May 2015, predating a recent paper of Geneson [A note on long rainbow arithmetic progressions, arXiv:1811.07989] showing that T_k \u2264 k^((5/2) + o(1)), and will form part of the forthcoming paper Short proofs of some extremal results III. We would like to thank Kevin Ford for some helpful discussions. We would also like to mention that recently J\u00f3zsef Balogh, Will Linz and Mina Nahvi independently investigated the question of estimating T_k and showed that T_k = k^(2 + o(1)).\n\n

Submitted - 1901.05084.pdf

", "abstract": "We show that there is a positive constant c such that any graph on vertex set [n] with at most cn^(2)/k^(2) log k edges contains an independent set of order k whose vertices form an arithmetic progression. We also present applications of this result to several questions in Ramsey theory.", "date": "2019-01-15", "date_type": "published", "id_number": "CaltechAUTHORS:20190819-170939635", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170939635", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "European Research Council (ERC)", "grant_number": "676632" }, { "agency": "David and Lucile Packard Foundation" }, { "agency": "NSF", "grant_number": "DMS-1352121" }, { "agency": "Swiss National Science Foundation (SNSF)", "grant_number": "200021-175573" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1901.05084", "primary_object": { "basename": "1901.05084.pdf", "url": "https://authors.library.caltech.edu/records/xam4m-fzc19/files/1901.05084.pdf" }, "pub_year": "2019", "author_list": "Conlon, David; Fox, Jacob; et al." }, { "id": "https://authors.library.caltech.edu/records/vj9xq-qqb28", "eprint_id": 103268, "eprint_status": "archive", "datestamp": "2023-08-19 13:28:05", "lastmod": "2026-03-31 19:49:56", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Lieber-J-F", "name": { "family": "Lieber", "given": "Joshua F." } }, { "id": "Manin-Y-I", "name": { "family": "Manin", "given": "Yuri I." } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Bost-Connes systems and F\u2081-structures in Grothendieck rings, spectra, and Nori motives", "ispublished": "unpub", "full_text_status": "public", "note": "The first and third authors were supported in part by the Perimeter Institute for Theoretical Physics. The third author is also partially supported by NSF grant DMS-1707882, and by NSERC Discovery Grant RGPIN-2018-04937 and Accelerator Supplement grant RGPAS-2018-522593.\n\n

Submitted - 1901.00020.pdf

", "abstract": "We construct geometric lifts of the Bost-Connes algebra to Grothendieck rings and to the associated assembler categories and spectra, as well as to certain categories of Nori motives. These categorifications are related to the integral Bost-Connes algebra via suitable Euler characteristic type maps and zeta functions, and in the motivic case via fiber functors. We also discuss aspects of F\u2081-geometry, in the framework of torifications, that fit into this general setting.", "date": "2018-12-31", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20200518-093742753", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200518-093742753", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Perimeter Institute for Theoretical Physics" }, { "agency": "NSF", "grant_number": "DMS-1707882" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "RGPIN-2018-04937" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "RGPAS-2018-522593" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1901.00020", "primary_object": { "basename": "1901.00020.pdf", "url": "https://authors.library.caltech.edu/records/vj9xq-qqb28/files/1901.00020.pdf" }, "pub_year": "2018", "author_list": "Lieber, Joshua F.; Manin, Yuri I.; et al." }, { "id": "https://authors.library.caltech.edu/records/ew8zj-5j465", "eprint_id": 98031, "eprint_status": "archive", "datestamp": "2023-08-19 13:26:46", "lastmod": "2026-03-08 03:39:39", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" }, { "id": "Lee-Joonkyung", "name": { "family": "Lee", "given": "Joonkyung" } } ] }, "title": "Sidorenko's conjecture for blow-ups", "ispublished": "unpub", "full_text_status": "public", "note": "Conlon research supported by a Royal Society University Research Fellowship and by ERC Starting Grant 676632. Lee research supported by ERC Consolidator Grant PEPCo 724903. \n\nWe are greatly indebted to Yufei Zhao and also to Leonardo Nagami Coregliano and Sasha Razborov for spotting a substantial error in an earlier version of this paper. The upshot of the resulting changes is the divisibility condition in Theorem 1.1, which was not present in the previous version. \n\nThis paper was partially written while the second author was working as a postdoctoral research associate at the University of Oxford and he would like to acknowledge the support of ERC Starting Grant 676632 during that period.\n\n

Submitted - 1809.01259.pdf

", "abstract": "A celebrated conjecture of Sidorenko and Erd\u0151s-Simonovits states that, for all bipartite graphs H, quasirandom graphs contain asymptotically the minimum number of copies of H taken over all graphs with the same order and edge density. This conjecture has attracted considerable interest over the last decade and is now known to hold for a broad range of bipartite graphs, with the overall trend saying that a graph satisfies the conjecture if it can be built from simple building blocks such as trees in a certain recursive fashion. \n\nOur contribution here, which goes beyond this paradigm, is to show that the conjecture holds for any bipartite graph H with bipartition A \u222a B where the number of vertices in B of degree k satisfies a certain divisibility condition for each k. As a corollary, we have that for every bipartite graph H with bipartition A \u222a B, there is a positive integer p such that the blow-up H_(A)^(p) formed by taking p vertex-disjoint copies of H and gluing all copies of A along corresponding vertices satisfies the conjecture. \n\nAnother way of viewing this latter result is that for every bipartite H there is a positive integer p such that an L^(p)-version of Sidorenko's conjecture holds for H.", "date": "2018-12-21", "date_type": "published", "id_number": "CaltechAUTHORS:20190819-170928301", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170928301", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Royal Society" }, { "agency": "European Research Council (ERC)", "grant_number": "676632" }, { "agency": "European Research Council (ERC)", "grant_number": "724903" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1809.01259", "primary_object": { "basename": "1809.01259.pdf", "url": "https://authors.library.caltech.edu/records/ew8zj-5j465/files/1809.01259.pdf" }, "pub_year": "2018", "author_list": "Conlon, David and Lee, Joonkyung" }, { "id": "https://authors.library.caltech.edu/records/p8dqv-h2k91", "eprint_id": 98028, "eprint_status": "archive", "datestamp": "2023-08-19 12:38:03", "lastmod": "2026-04-01 02:56:03", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" }, { "id": "Fox-J", "name": { "family": "Fox", "given": "Jacob" } }, { "id": "Grinshpun-A", "name": { "family": "Grinshpun", "given": "Andrey" } }, { "id": "He-Xiaoyu", "name": { "family": "He", "given": "Xiaoyu" } } ] }, "title": "Online Ramsey Numbers and the Subgraph Query Problem", "ispublished": "unpub", "full_text_status": "public", "note": "Conlon research supported by a Royal Society University Research Fellowship and by ERC Starting Grant 676632. Fox research supported by a Packard Fellowship and by NSF Career Award DMS-1352121. \n\nWe are extremely grateful to Joel Spencer for pointing out a serious flaw in our previous proof of Theorem 4 which had been based on a generalization of the Lov\u00e1sz Local Lemma [P. Erd\u0151s and J. Spencer, Lopsided Lov\u00e1sz local lemma and Latin transversals, Discrete Appl. Math. 30 (1991), 151\u2013154]. In the current version, we have a correct proof using a different approach. We would also like to thank the anonymous referee for some helpful remarks and Benny Sudakov for bringing the paper of Krivelevich [Bounding Ramsey numbers through large deviation inequalities, Random Structures Algorithms 7 (1995), 145\u2013155] to our attention.\n\n

Submitted - 1806.09726.pdf

", "abstract": "The (m,n)-online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge on each turn and Painter immediately paints it red or blue. Builder's goal is to force Painter to create either a red K_m or a blue K_n using as few turns as possible. The online Ramsey number [equation; see abstract in PDF for details] is the minimum number of edges Builder needs to guarantee a win in the (m,n)-online Ramsey game. By analyzing the special case where Painter plays randomly, we obtain an exponential improvement \n\n[equation; see abstract in PDF for details]\n\nfor the lower bound on the diagonal online Ramsey number, as well as a corresponding improvement \n\n[equation; see abstract in PDF for details]\n\nfor the off-diagonal case, where m \u2265 3 is fixed and n \u2192 \u221e. Using a different randomized Painter strategy, we prove that [equation; see abstract in PDF for details], determining this function up to a polylogarithmic factor. We also improve the upper bound in the off-diagonal case for m \u2265 4.\n\nIn connection with the online Ramsey game with a random Painter, we study the problem of finding a copy of a target graph H in a sufficiently large unknown Erd\u0151s-R\u00e9nyi random graph G(N,p) using as few queries as possible, where each query reveals whether or not a particular pair of vertices are adjacent. We call this problem the Subgraph Query Problem. We determine the order of the number of queries needed for complete graphs up to five vertices and prove general bounds for this problem.", "date": "2018-11-04", "date_type": "published", "id_number": "CaltechAUTHORS:20190819-170917920", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170917920", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Royal Society" }, { "agency": "European Research Council (ERC)", "grant_number": "676632" }, { "agency": "David and Lucile Packard Foundation" }, { "agency": "NSF", "grant_number": "DMS-1352121" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1806.09726", "primary_object": { "basename": "1806.09726.pdf", "url": "https://authors.library.caltech.edu/records/p8dqv-h2k91/files/1806.09726.pdf" }, "pub_year": "2018", "author_list": "Conlon, David; Fox, Jacob; et al." }, { "id": "https://authors.library.caltech.edu/records/z6mhx-cmf35", "eprint_id": 108280, "eprint_status": "archive", "datestamp": "2023-08-19 12:36:48", "lastmod": "2026-03-31 20:16:19", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Li-Wanlin", "name": { "family": "Li", "given": "Wanlin" } }, { "id": "Mantovan-E", "name": { "family": "Mantovan", "given": "Elena" } }, { "id": "Pries-Rachel", "name": { "family": "Pries", "given": "Rachel" }, "orcid": "0000-0001-5987-0324" }, { "id": "Tang-Yunqing", "name": { "family": "Tang", "given": "Yunqing" } } ] }, "title": "Newton polygon stratification of the Torelli locus in PEL-type Shimura varieties", "ispublished": "unpub", "full_text_status": "public", "keywords": "curve, cyclic cover, Jacobian, abelian variety, moduli space, Shimura variety,\nPEL-type, reduction, Frobenius, supersingular, Newton polygon, p-rank, Dieudonn\u00e9\nmodule, p-divisible group", "note": "

Submitted - 1811.00604.pdf

", "abstract": "We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo p reduction of certain PEL-type Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is non-empty. This allows us to give a positive answer, under some compatibility conditions, to a question of Oort about smooth curves in characteristic p whose Newton polygons are an amalgamate sum. As an application, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic covers of the projective line. Most of these arise in inductive systems which demonstrate unlikely intersections of the open Torelli locus with the Newton polygon stratification in Siegel modular varieties. In addition, for the twenty special PEL-type Shimura varieties found in Moonen's work, we prove that all Newton polygon strata intersect the open Torelli locus (if p>>0 in the supersingular cases).", "date": "2018-11-01", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210302-154930394", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210302-154930394", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1811.00604", "primary_object": { "basename": "1811.00604.pdf", "url": "https://authors.library.caltech.edu/records/z6mhx-cmf35/files/1811.00604.pdf" }, "pub_year": "2018", "author_list": "Li, Wanlin; Mantovan, Elena; et al." }, { "id": "https://authors.library.caltech.edu/records/86jzm-jh215", "eprint_id": 98026, "eprint_status": "archive", "datestamp": "2023-08-19 10:35:43", "lastmod": "2026-03-31 21:49:21", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" }, { "id": "Kam\u010dev-N", "name": { "family": "Kam\u010dev", "given": "Nina" } } ] }, "title": "Intervals in the Hales-Jewett theorem", "ispublished": "unpub", "full_text_status": "public", "note": "Conlon research supported by a Royal Society University Research Fellowship and by ERC Starting Grant 676632. \n\nThe second author would like to thank Sarah Gales, Hannah and Sven Eggimann for hosting her in Oxford while this research was conducted. We would also like to thank the anonymous referee for a number of helpful remarks.\n\n

Submitted - 1801.08919.pdf

", "abstract": "The Hales-Jewett theorem states that for any m and r there exists an n such that any r-colouring of the elements of [m]^n contains a monochromatic combinatorial line. We study the structure of the wildcard set S \u2286 [n] which determines this monochromatic line, showing that when r is odd there are r-colourings of [3]^n where the wildcard set of a monochromatic line cannot be the union of fewer than r intervals. This is tight, as for n sufficiently large there are always monochromatic lines whose wildcard set is the union of at most r intervals.", "date": "2018-07-26", "date_type": "published", "id_number": "CaltechAUTHORS:20190819-170911002", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170911002", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Royal Society" }, { "agency": "European Research Council (ERC)", "grant_number": "676632" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1801.08919", "primary_object": { "basename": "1801.08919.pdf", "url": "https://authors.library.caltech.edu/records/86jzm-jh215/files/1801.08919.pdf" }, "pub_year": "2018", "author_list": "Conlon, David and Kam\u010dev, Nina" }, { "id": "https://authors.library.caltech.edu/records/b9843-67586", "eprint_id": 95611, "eprint_status": "archive", "datestamp": "2023-08-19 10:27:42", "lastmod": "2026-04-01 03:03:51", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Carlen-E-A", "name": { "family": "Carlen", "given": "Eric A." }, "orcid": "0000-0003-2613-187X" }, { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Ivanisvili-P", "name": { "family": "Ivanisvili", "given": "Paata" } }, { "id": "Lieb-E-H", "name": { "family": "Lieb", "given": "Elliott H." }, "orcid": "0000-0001-5843-3587" } ] }, "title": "Inequalities for L^p-norms that sharpen the triangle inequality and complement Hanner's Inequality", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2018 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\nWork partially supported by NSF grants DMS-1501007 (E.A.C.), DMS-1363432 (R.L.F.), PHY-1265118 (E.H.L.).\n\n

Submitted - 1807.05599.pdf

", "abstract": "In 2006 Carbery raised a question about an improvement on the na\u00efve norm inequality \u2225f+g\u2225^p_p \u2264 2^(p\u22121)(\u2225f\u2225^p_p+\u2225g\u2225^p_p) for two functions in L^p of any measure space. When f=g this is an equality, but when the supports of f and g are disjoint the factor 2^(p\u22121) is not needed. Carbery's question concerns a proposed interpolation between the two situations for p>2. The interpolation parameter measuring the overlap is \u2225fg\u2225_(p/2). We prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all p.", "date": "2018-07-15", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20190520-141054764", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190520-141054764", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1501007" }, { "agency": "NSF", "grant_number": "DMS-1363432" }, { "agency": "NSF", "grant_number": "PHY-1265118" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1807.05599", "primary_object": { "basename": "1807.05599.pdf", "url": "https://authors.library.caltech.edu/records/b9843-67586/files/1807.05599.pdf" }, "pub_year": "2018", "author_list": "Carlen, Eric A.; Frank, Rupert L.; et al." }, { "id": "https://authors.library.caltech.edu/records/rn537-f8138", "eprint_id": 117597, "eprint_status": "archive", "datestamp": "2023-08-19 09:53:05", "lastmod": "2026-03-09 23:57:15", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Song-Antoine", "name": { "family": "Song", "given": "Antoine" } } ] }, "title": "Existence of infinitely many minimal hypersurfaces in closed manifolds", "ispublished": "unpub", "full_text_status": "public", "note": "Attribution 4.0 International (CC BY 4.0) \n\nThe author was partially supported by NSF-DMS-1509027. \n\nI am very grateful to my advisor Fernando Cod\u00e1 Marques for his constant support, his generosity and inspiring discussions during the course of this work. I also thank him for pointing out references [39] and [5]. I would like to thank Andr\u00e9 Neves for many valuable conversations.\n\n

Accepted Version - 1806.08816.pdf

", "abstract": "Using min-max theory, we show that in any closed Riemannian manifold of dimension at least 3 and at most 7, there exist infinitely many smoothly embedded closed minimal hypersurfaces. It proves a conjecture of S.-T. Yau. This paper builds on the methods developed by F. C. Marques and A. Neves.", "date": "2018-06-22", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20221026-539144000.10", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221026-539144000.10", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1509027" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1806.08816", "primary_object": { "basename": "1806.08816.pdf", "url": "https://authors.library.caltech.edu/records/rn537-f8138/files/1806.08816.pdf" }, "pub_year": "2018", "author_list": "Song, Antoine" }, { "id": "https://authors.library.caltech.edu/records/7e9nc-jny65", "eprint_id": 98054, "eprint_status": "archive", "datestamp": "2023-08-19 09:09:42", "lastmod": "2026-03-07 23:29:42", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" }, { "id": "Kim-Jeong-Han", "name": { "family": "Kim", "given": "Jeong Han" } }, { "id": "Lee-Choongbum", "name": { "family": "Lee", "given": "Choongbum" } }, { "id": "Lee-Joonkyung", "name": { "family": "Lee", "given": "Joonkyung" } } ] }, "title": "Sidorenko's conjecture for higher tree decompositions", "ispublished": "unpub", "full_text_status": "public", "note": "Conlon research supported by a Royal Society University Research Fellowship. Kim was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP) (NRF-2012R1A2A2A01018585) and KIAS internal Research Fund CG046001. C. Lee research supported by NSF Grant DMS-1362326. J. Lee supported by ILJU Foundation of Education and Culture.\n\n

Submitted - 1805.02238.pdf

", "abstract": "This is a companion note to our paper 'Some advances on Sidorenko's conjecture', elaborating on a remark in that paper that the approach which proves Sidorenko's conjecture for strongly tree-decomposable graphs may be extended to a broader class, comparable to that given in work of Szegedy, through further iteration.", "date": "2018-05-06", "date_type": "published", "publisher": "Caltech Library", "id_number": "CaltechAUTHORS:20190820-161459631", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190820-161459631", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Royal Society" }, { "agency": "National Research Foundation of Korea", "grant_number": "NRF-2012R1A2A2A01018585" }, { "agency": "Korea Institute for Advanced Study", "grant_number": "CG046001" }, { "agency": "NSF", "grant_number": "DMS-1362326" }, { "agency": "ILJU Foundation of Education and Culture" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "primary_object": { "basename": "1805.02238.pdf", "url": "https://authors.library.caltech.edu/records/7e9nc-jny65/files/1805.02238.pdf" }, "pub_year": "2018", "author_list": "Conlon, David; Kim, Jeong Han; et al." }, { "id": "https://authors.library.caltech.edu/records/9mw93-kpz90", "eprint_id": 110837, "eprint_status": "archive", "datestamp": "2023-08-19 07:28:03", "lastmod": "2026-03-10 00:02:20", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Porta-Mauro", "name": { "family": "Porta", "given": "Mauro" }, "orcid": "0000-0002-1239-3409" }, { "id": "Yu-Tony-Yue", "name": { "family": "Yu", "given": "Tony Yue" }, "orcid": "0000-0002-6019-8552" } ] }, "title": "Derived Hom spaces in rigid analytic geometry", "ispublished": "unpub", "full_text_status": "public", "keywords": "mapping stack, Hom scheme, representability, derived geometry, rigid analytic geometry, non-archimedean geometry, Tate acyclicity, projection formula, proper base change", "note": "We are very grateful to Antoine Chambert-Loir, Maxim Kontsevich, Jacob Lurie, Tony Pantev, Marco Robalo, Carlos Simpson, Bertrand To\u00ebn and Gabriele Vezzosi for valuable discussions. The authors would also like to thank each other for the joint effort. Various stages of this research received supports from the Clay Mathematics Institute, Simons Foundation grant number 347070, and from the Ky Fan and Yu-Fen Fan Membership Fund and the S.-S. Chern Endowment Fund of the Institute for Advanced Study.\n\n

Submitted - 1801.07730.pdf

", "abstract": "We construct a derived enhancement of Hom spaces between rigid analytic spaces. It encodes the hidden deformation-theoretic informations of the underlying classical moduli space. The main tool in our construction is the representability theorem in derived analytic geometry, which has been established in our previous work. The representability theorem provides us sufficient and necessary conditions for an analytic moduli functor to possess the structure of a derived analytic stack. In order to verify the conditions of the representability theorem, we prove several general results in the context of derived non-archimedean analytic geometry: derived Tate acyclicity, projection formula, and proper base change. These results also deserve independent interest themselves. Our main motivation comes from non-archimedean enumerative geometry. In our subsequent works, we will apply the derived mapping stacks to obtain non-archimedean analytic Gromov-Witten invariants.", "date": "2018-01-23", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210914-164506886", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210914-164506886", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Clay Mathematics Institute" }, { "agency": "Simons Foundation", "grant_number": "347070" }, { "agency": "Ky Fan and Yu-Fen Fan Membership Fund" }, { "agency": "Institute for Advanced Study" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1801.07730", "primary_object": { "basename": "1801.07730.pdf", "url": "https://authors.library.caltech.edu/records/9mw93-kpz90/files/1801.07730.pdf" }, "pub_year": "2018", "author_list": "Porta, Mauro and Yu, Tony Yue" }, { "id": "https://authors.library.caltech.edu/records/6d8gp-d2472", "eprint_id": 88616, "eprint_status": "archive", "datestamp": "2023-08-19 07:00:26", "lastmod": "2026-03-18 00:08:03", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Puhalskii-A", "name": { "family": "Puhalskii", "given": "A." } }, { "id": "Simon-B", "name": { "family": "Simon", "given": "Barry" }, "orcid": "0000-0003-2561-8539" } ] }, "title": "A large-population limit for a Markovian model of group-structured populations", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 1712.09119.pdf

", "abstract": "A Markovian model of group-structured (two-level) population dynamics features births, deaths, and migrations of individuals, and fission and extinction of groups. These models are useful for studying group selection and other evolutionary processes that occur when individuals live in distinct groups. We show that the sample paths of a properly scaled sequence of these models converge in an appropriate Skorohod space to a deterministic trajectory that is a unique solution to a quasilinear evolution equation. The PDE model can therefore be justified as an approximation to the Markovian one.", "date": "2017-12-25", "date_type": "published", "id_number": "CaltechAUTHORS:20180806-145540097", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180806-145540097", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" }, { "id": "Physics-Department" } ] }, "doi": "10.48550/arXiv.1712.09119", "primary_object": { "basename": "1712.09119.pdf", "url": "https://authors.library.caltech.edu/records/6d8gp-d2472/files/1712.09119.pdf" }, "pub_year": "2017", "author_list": "Puhalskii, A. and Simon, Barry" }, { "id": "https://authors.library.caltech.edu/records/db3yw-1n481", "eprint_id": 103270, "eprint_status": "archive", "datestamp": "2023-08-19 07:00:14", "lastmod": "2026-04-01 07:05:29", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Motivic Information", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 1712.08703.pdf

", "abstract": "We introduce notions of information/entropy and information loss associated to exponentiable motivic measures. We show that they satisfy appropriate analogs to the Khinchin-type properties that characterize information loss in the context of measures on finite sets.", "date": "2017-12-23", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20200518-094532333", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200518-094532333", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1712.08703", "primary_object": { "basename": "1712.08703.pdf", "url": "https://authors.library.caltech.edu/records/db3yw-1n481/files/1712.08703.pdf" }, "pub_year": "2017", "author_list": "Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/bpdne-0ep45", "eprint_id": 87364, "eprint_status": "archive", "datestamp": "2023-08-19 05:45:36", "lastmod": "2026-03-09 02:36:28", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Isett-Philip", "name": { "family": "Isett", "given": "Philip" }, "orcid": "0000-0001-9038-5546" } ] }, "title": "Nonuniqueness and existence of continuous, globally dissipative Euler flows", "ispublished": "unpub", "full_text_status": "public", "note": "The work of P. Isett is supported by the National Science Foundation under Award No. DMS-1402370.\n\n

Submitted - 1710.11186.pdf

", "abstract": "We show that H\u00f6lder continuous, globally dissipative incompressible Euler flows (solutions obeying the local energy inequality) are nonunique and contain examples that strictly dissipate energy. The collection of such solutions emanating from a single initial data may have positive Hausdorff dimension in the energy space even if the local energy equality is imposed, and the set of initial data giving rise to such an infinite family of solutions is C^0 dense in the space of continuous, divergence free vector fields on the torus T^3.", "date": "2017-10-30", "date_type": "published", "id_number": "CaltechAUTHORS:20180626-160542363", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180626-160542363", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1402370" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1710.11186", "primary_object": { "basename": "1710.11186.pdf", "url": "https://authors.library.caltech.edu/records/bpdne-0ep45/files/1710.11186.pdf" }, "pub_year": "2017", "author_list": "Isett, Philip" }, { "id": "https://authors.library.caltech.edu/records/1vhrq-rej68", "eprint_id": 96245, "eprint_status": "archive", "datestamp": "2023-08-19 05:42:45", "lastmod": "2026-03-18 00:06:53", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Simon-B", "name": { "family": "Simon", "given": "Barry" }, "orcid": "0000-0003-2561-8539" } ] }, "title": "Tosio Kato's Work on Non-Relativistic Quantum Mechanics: An Outline", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 1710.06999.pdf

", "abstract": "Based at a talk given at the Kato Centennial Symposium in Sept. 2017, this article discusses the scientific life and some of the scientific work of T. Kato.", "date": "2017-10-20", "date_type": "published", "id_number": "CaltechAUTHORS:20190610-130001875", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190610-130001875", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" }, { "id": "Physics-Department" } ] }, "doi": "10.48550/arXiv.1710.06999", "primary_object": { "basename": "1710.06999.pdf", "url": "https://authors.library.caltech.edu/records/1vhrq-rej68/files/1710.06999.pdf" }, "pub_year": "2017", "author_list": "Simon, Barry" }, { "id": "https://authors.library.caltech.edu/records/0hr4y-96b77", "eprint_id": 87365, "eprint_status": "archive", "datestamp": "2023-08-19 03:35:35", "lastmod": "2026-03-09 02:36:30", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Isett-Philip", "name": { "family": "Isett", "given": "Philip" }, "orcid": "0000-0001-9038-5546" } ] }, "title": "On the Endpoint Regularity in Onsager's Conjecture", "ispublished": "unpub", "full_text_status": "public", "note": "The work of P. Isett is supported by the National Science Foundation under Award No. DMS-1402370.\n\n

Submitted - 1706.01549.pdf

", "abstract": "Onsager's conjecture states that the conservation of energy may fail for 3D incompressible Euler flows with H\u00f6lder regularity below 1/3. This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question with further connections to turbulence theory. In this work, we construct energy non-conserving solutions to the 3D incompressible Euler equations with space-time H\u00f6lder regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents [0,1/3). \nOur construction improves the author's previous result towards the endpoint case. To obtain this improvement, we introduce a new method for optimizing the regularity that can be achieved by a general convex integration scheme. A crucial point is to avoid power-losses in frequency in the estimates of the iteration. This goal is achieved using localization techniques of [IO16b] to modify the convex integration scheme. \nWe also prove results on general solutions at the critical regularity that may not conserve energy. These include the fact that singularites of positive space-time Lebesgue measure are necessary for any energy non-conserving solution to exist while having critical regularity of an integrability exponent greater than three.", "date": "2017-06-05", "date_type": "published", "id_number": "CaltechAUTHORS:20180626-161143819", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180626-161143819", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1402370" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1706.01549", "primary_object": { "basename": "1706.01549.pdf", "url": "https://authors.library.caltech.edu/records/0hr4y-96b77/files/1706.01549.pdf" }, "pub_year": "2017", "author_list": "Isett, Philip" }, { "id": "https://authors.library.caltech.edu/records/7p4y5-z2m03", "eprint_id": 77558, "eprint_status": "archive", "datestamp": "2023-08-19 03:04:18", "lastmod": "2026-04-01 19:43:23", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Dedushenko-M", "name": { "family": "Dedushenko", "given": "Mykola" }, "orcid": "0000-0002-9273-7602" }, { "id": "Gukov-S", "name": { "family": "Gukov", "given": "Sergei" }, "orcid": "0000-0002-9486-1762" }, { "id": "Putrov-P", "name": { "family": "Putrov", "given": "Pavel" } } ] }, "title": "Vertex algebras and 4-manifold invariants", "ispublished": "unpub", "full_text_status": "public", "note": "We would like to thank J. Bryan, A. Dabholkar, A. Gadde, A. Haydys, M. Mari\u02dcno,\nV. Mikhaylov, J.W. Morgan, H. Nakajima, H. Ooguri, J. Rasmussen, S. Schafer-Nameki,\nA. Soldatenkov, M. Stosic, E. Verlinde, H. Verlinde, J. Wong, and K. Ye for useful discussions\nand comments. The work of M.D. and S.G. is supported in part by the U.S.\nDepartment of Energy, Office of Science, Office of High Energy Physics, under Award\nNumber DE-SC0011632. In addition, the work of S.G. is supported in part by the ERC\nStarting Grant no. 335739 \"Quantum fields and knot homologies\" funded by the European\nResearch Council under the European Union Seventh Framework Programme. P.P.\ngratefully acknowledges the support from Marvin L. Goldberger Fellowship and the DOE\nGrant DE-SC0009988. Opinions and conclusions expressed here are those of the authors\nand do not necessarily reflect the views of funding agencies.\n\n

Submitted - 1705.01645.pdf

", "abstract": "We propose a way of computing 4-manifold invariants, old and new, as chiral\ncorrelation functions in half-twisted 2d N = (0, 2) theories that arise from compactification of fivebranes. Such formulation gives a new interpretation of some known statements\nabout Seiberg-Witten invariants, such as the basic class condition, and gives a prediction\nfor structural properties of the multi-monopole invariants and their non-abelian generalizations.", "date": "2017-05-03", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20170518-093106050", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170518-093106050", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-SC0011632" }, { "agency": "European Research Council (ERC)", "grant_number": "335739" }, { "agency": "Department of Energy (DOE)", "grant_number": "DE-SC0009988" }, { "agency": "Marvin L. Goldberger Fellowship" } ] }, "other_numbering_system": { "items": [ { "id": "2017-008", "name": "CALT-TH" } ] }, "local_group": { "items": [ { "id": "Walter-Burke-Institute-for-Theoretical-Physics" }, { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1705.01645", "primary_object": { "basename": "1705.01645.pdf", "url": "https://authors.library.caltech.edu/records/7p4y5-z2m03/files/1705.01645.pdf" }, "pub_year": "2017", "author_list": "Dedushenko, Mykola; Gukov, Sergei; et al." }, { "id": "https://authors.library.caltech.edu/records/njjv8-zag17", "eprint_id": 100892, "eprint_status": "archive", "datestamp": "2023-08-19 01:57:38", "lastmod": "2026-04-01 18:11:02", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Biringer-I", "name": { "family": "Biringer", "given": "Ian" } }, { "id": "Bowen-L", "name": { "family": "Bowen", "given": "Lewis" } }, { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "Omer" }, "orcid": "0000-0002-0111-0418" } ] }, "title": "Invariant random subgroups of semidirect products", "ispublished": "unpub", "full_text_status": "public", "note": "Supported in part by NSF grant DMS-1611851 and CAREER Award DMS-1654114. \n\nSupported in part by NSF grant DMS-0968762, NSF CAREER Award\nDMS-0954606 and BSF grant 2008274. \n\nThis work was supported by a grant from the Simons Foundation\n(#419427, Omer Tamuz).\n\n

Submitted - 1703.01282.pdf

", "abstract": "We study invariant random subgroups (IRSs) of semidirect products G = A\u22ca\u0393. In particular, we characterize all IRSs of parabolic subgroups of SL_d(R), and show that all ergodic IRSs of R^d\u22caSL_d(R) are either of the form R^d\u22caK for some IRS of SL_d(R), or are induced from IRSs of \u039b\u22caSL(\u039b), where \u039b < R^d is a lattice.", "date": "2017-03-03", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20200124-092403035", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200124-092403035", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1611851" }, { "agency": "NSF", "grant_number": "DMS-1654114" }, { "agency": "NSF", "grant_number": "DMS-0968762" }, { "agency": "NSF", "grant_number": "DMS-0954606" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2008274" }, { "agency": "Simons Foundation", "grant_number": "419427" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1703.01282", "primary_object": { "basename": "1703.01282.pdf", "url": "https://authors.library.caltech.edu/records/njjv8-zag17/files/1703.01282.pdf" }, "pub_year": "2017", "author_list": "Biringer, Ian; Bowen, Lewis; et al." }, { "id": "https://authors.library.caltech.edu/records/qz0qv-vck39", "eprint_id": 98256, "eprint_status": "archive", "datestamp": "2023-08-20 13:31:28", "lastmod": "2026-03-18 04:20:05", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Wales-D-B", "name": { "family": "Wales", "given": "David B." } } ] }, "title": "Classification of Imprimitive Irreducible Finite Sugroups of O(7)", "ispublished": "unpub", "full_text_status": "public", "keywords": "Orthogonal group, imprimitive linear group, quasiprimitve linear\ngroup", "note": "

Submitted - 1608.07543.pdf

", "abstract": "This work gives a classification of imprimitive irreducible finite subgroups of the orthogonal group O(7) plus the number of conjugate classes for each group.", "date": "2016-08-26", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20190827-080735174", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190827-080735174", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" }, { "id": "Physics-Department" } ] }, "doi": "10.48550/arXiv.1608.07543", "primary_object": { "basename": "1608.07543.pdf", "url": "https://authors.library.caltech.edu/records/qz0qv-vck39/files/1608.07543.pdf" }, "pub_year": "2016", "author_list": "Wales, David B." }, { "id": "https://authors.library.caltech.edu/records/9deyp-p1157", "eprint_id": 69672, "eprint_status": "archive", "datestamp": "2023-08-20 13:18:31", "lastmod": "2026-04-02 02:24:05", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Andersen-J-E", "name": { "family": "Andersen", "given": "J\u00f8rgen Ellegaard" }, "orcid": "0000-0001-9721-0722" }, { "id": "Gukov-S", "name": { "family": "Gukov", "given": "Sergei" }, "orcid": "0000-0002-9486-1762" }, { "id": "Pei-Du", "name": { "family": "Pei", "given": "Du" }, "orcid": "0000-0001-5587-6905" } ] }, "title": "The Verlinde formula for Higgs bundles", "ispublished": "unpub", "full_text_status": "public", "note": "Date: August 8, 2016. \n\nSupported in part by the center of excellence grant \"Center for Quantum Geometry of Moduli Space\" from the Danish National Research Foundation (DNRF95), by the Walter Burke Institute for Theoretical Physics, and by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number de-sc0011632.\n\n

Submitted - 1608.01761v1.pdf

", "abstract": "We propose and prove the Verlinde formula for the quantization of the Higgs\nbundle moduli spaces and stacks for any simple and simply-connected group. This\ngeneralizes the equivariant Verlinde formula for the case of SU(n) proposed\npreviously by the second and third author. We further establish a Verlinde\nformula for the quantization of parabolic Higgs bundle moduli spaces and\nstacks.", "date": "2016-08-05", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20160816-123626732", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160816-123626732", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Danish National Research Foundation", "grant_number": "DNRF95" }, { "agency": "Walter Burke Institute for Theoretical Physics, Caltech" }, { "agency": "Department of Energy (DOE)", "grant_number": "DE-SC0011632" } ] }, "other_numbering_system": { "items": [ { "id": "2016-020", "name": "CALT-TH" } ] }, "local_group": { "items": [ { "id": "Walter-Burke-Institute-for-Theoretical-Physics" }, { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1608.01761", "primary_object": { "basename": "1608.01761v1.pdf", "url": "https://authors.library.caltech.edu/records/9deyp-p1157/files/1608.01761v1.pdf" }, "pub_year": "2016", "author_list": "Andersen, J\u00f8rgen Ellegaard; Gukov, Sergei; et al." }, { "id": "https://authors.library.caltech.edu/records/ed3w4-wkj58", "eprint_id": 81757, "eprint_status": "archive", "datestamp": "2023-08-20 13:02:07", "lastmod": "2026-03-09 22:10:55", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Rains-E-M", "name": { "family": "Rains", "given": "Eric M." }, "orcid": "0000-0002-9915-0919" } ] }, "title": "The noncommutative geometry of elliptic difference equations", "ispublished": "unpub", "full_text_status": "public", "note": "The author would like to thank A. Borodin, P. Etingof, T. Graber, A. Okounkov, M. Van den Bergh, and X. Zhu for helpful conversations. This work was partially supported by grants from the National Science Foundation, DMS-1001645 and DMS-1500806.\n\n

Submitted - 1607.08876.pdf

", "abstract": "We give a new construction of noncommutative surfaces via elliptic difference operators, attaching a 1-parameter noncommutative deformation to any projective rational surface with smooth anticanonical curve. The construction agrees with one implicit in work of Van den Bergh (iterated blowups of noncommutative Hirzebruch surfaces), but the construction enables one to prove a number of new facts about these surfaces. We show that they are noncommutative smooth proper surfaces in the sense of Chan and Nyman, with projective Quot schemes, that moduli spaces of simple sheaves are Poisson and that moduli spaces classifying semistable sheaves of rank 0 or 1 are projective. We further show that the action of SL_2(Z) as derived autoequivalences of rational elliptic surfaces extends to an action as derived equivalences of surfaces in our family with K^2=0. \nWe also discuss applications to the theory of special functions arising by interpreting moduli spaces of 1-dimensional sheaves as moduli spaces of difference equations. When the moduli space is a single point, the equation is rigid, and we give an integral representation for the solutions. More generally, twisting by line bundles corresponds to isomonodromy deformations, so this gives rise to Lax pairs. When the moduli space is 2-dimensional, one obtains Lax pairs for the elliptic Painlev\u00e9 equation; this associates a Lax pair to any rational number, of order twice the denominator. There is also an elliptic analogue of the Riemann-Hilbert correspondence: an analytic equivalence between categories of elliptic difference equations, swapping the role of the shift of the equation and the nome of the curve.", "date": "2016-07-29", "date_type": "published", "id_number": "CaltechAUTHORS:20170922-135420800", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170922-135420800", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1001645" }, { "agency": "NSF", "grant_number": "DMS-1500806" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1607.08876", "primary_object": { "basename": "1607.08876.pdf", "url": "https://authors.library.caltech.edu/records/ed3w4-wkj58/files/1607.08876.pdf" }, "pub_year": "2016", "author_list": "Rains, Eric M." }, { "id": "https://authors.library.caltech.edu/records/wb14a-90h62", "eprint_id": 77075, "eprint_status": "archive", "datestamp": "2023-08-20 13:00:27", "lastmod": "2026-04-02 06:07:09", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Lieb-E-H", "name": { "family": "Lieb", "given": "Elliott H." }, "orcid": "0000-0001-5843-3587" } ] }, "title": "A 'liquid-solid' phase transition in a simple model for swarming, based on the 'no flat-spots' theorem for subharmonic functions", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 1607.07971.pdf

", "abstract": "We consider a non-local shape optimization problem, which is motivated by a simple model for swarming and other self-assembly/aggregation models, and prove the existence of different phases. A technical key ingredient, which we establish, is that a strictly subharmonic function cannot be constant on a set of positive measure.", "date": "2016-07-27", "date_type": "published", "id_number": "CaltechAUTHORS:20170428-161302774", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170428-161302774", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1607.07971", "primary_object": { "basename": "1607.07971.pdf", "url": "https://authors.library.caltech.edu/records/wb14a-90h62/files/1607.07971.pdf" }, "pub_year": "2016", "author_list": "Frank, Rupert L. and Lieb, Elliott H." }, { "id": "https://authors.library.caltech.edu/records/qzvbt-fjm79", "eprint_id": 79007, "eprint_status": "archive", "datestamp": "2023-09-15 05:45:13", "lastmod": "2026-04-02 16:03:21", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" }, { "id": "Zolman-N", "name": { "family": "Zolman", "given": "Nick" } } ] }, "title": "Adinkras, Dessins, Origami, and Supersymmetry Spectral Triples", "ispublished": "unpub", "full_text_status": "public", "note": "The first author is supported by NSF grants DMS-1201512 and PHY-1205440. The second author is supported by a Summer Undergraduate Research Fellowship at Caltech. We thank Kevin Iga for a careful reading of the manuscript and for very useful comments and suggestions.\n\n

Submitted - 1606.04463.pdf

", "abstract": "We investigate the spectral geometry and spectral action functionals associated to 1D Supersymmetry Algebras, using the classification of these superalgebras in terms of Adinkra graphs and the construction of associated dessin d'enfant and origami curves. The resulting spectral action functionals are computed in terms of the Selberg (super) trace formula.", "date": "2016-06-14", "date_type": "published", "id_number": "CaltechAUTHORS:20170712-101818792", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-101818792", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1201512" }, { "agency": "NSF", "grant_number": "PHY-1205440" }, { "agency": "Caltech Summer Undergraduate Research Fellowship (SURF)" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1606.04463", "primary_object": { "basename": "1606.04463.pdf", "url": "https://authors.library.caltech.edu/records/qzvbt-fjm79/files/1606.04463.pdf" }, "pub_year": "2016", "author_list": "Marcolli, Matilde and Zolman, Nick" }, { "id": "https://authors.library.caltech.edu/records/0xpnc-07837", "eprint_id": 68894, "eprint_status": "archive", "datestamp": "2023-08-20 11:43:13", "lastmod": "2026-04-02 14:47:03", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Gukov-S", "name": { "family": "Gukov", "given": "Sergei" }, "orcid": "0000-0002-9486-1762" }, { "id": "Marino-M", "name": { "family": "Marino", "given": "Marcos" } }, { "id": "Putrov-P", "name": { "family": "Putrov", "given": "Pavel" } } ] }, "title": "Resurgence in complex Chern-Simons theory", "ispublished": "unpub", "full_text_status": "public", "note": "Submitted on 24 May 2016. \n\nWe would like to thank Miranda Cheng, Mikhail Kapranov, Amir Kashani-Poor, Albrecht Klemm, Maxim Kontsevich, Cumrun Vafa, Edward Witten, Masahito Yamazaki for useful comments and discussions. The work of S.G. is funded in part by the DOE Grant DE-SC0011632 and the Walter Burke Institute for Theoretical Physics. The work of M.M. is supported in part by the Swiss National Science Foundation, subsidies 200020-149226, 200021-156995, and by the NCCR 51NF40-141869 \"The Mathematics of Physics\" (SwissMAP). P.P. gratefully acknowledges support from the Institute for Advanced Study. Opinions and conclusions expressed here are those of the authors and do not necessarily reflect the views of funding agencies.\n\n

Submitted - 1605.07615v1.pdf

", "abstract": "We study resurgence properties of partition function of SU(2) Chern-Simons\ntheory (WRT invariant) on closed three-manifolds. We check explicitly that in\nvarious examples Borel transforms of asymptotic expansions posses expected\nanalytic properties. In examples that we study we observe that contribution of\nirreducible flat connections to the path integral can be recovered from\nasymptotic expansions around abelian flat connections. We also discuss\nconnection to Floer instanton moduli spaces, disk instantons in 2d sigma models, and length spectra of \"complex geodesics\" on the A-polynomial curve.", "date": "2016-05-24", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20160707-134251692", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160707-134251692", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-SC0011632" }, { "agency": "Swiss National Science Foundation (SNSF)", "grant_number": "200020-149226" }, { "agency": "Swiss National Science Foundation (SNSF)", "grant_number": "200021-156995" }, { "agency": "Institute for Advanced Study" }, { "agency": "Walter Burke Institute for Theoretical Physics, Caltech" } ] }, "other_numbering_system": { "items": [ { "id": "2016-011", "name": "CALT-TH" } ] }, "local_group": { "items": [ { "id": "Walter-Burke-Institute-for-Theoretical-Physics" }, { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1605.07615", "primary_object": { "basename": "1605.07615v1.pdf", "url": "https://authors.library.caltech.edu/records/0xpnc-07837/files/1605.07615v1.pdf" }, "pub_year": "2016", "author_list": "Gukov, Sergei; Marino, Marcos; et al." }, { "id": "https://authors.library.caltech.edu/records/1fbtx-nyk67", "eprint_id": 71952, "eprint_status": "archive", "datestamp": "2023-08-20 10:55:59", "lastmod": "2026-04-03 00:01:26", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Burton-P", "name": { "family": "Burton", "given": "Peter" } }, { "id": "Lupini-M", "name": { "family": "Lupini", "given": "Martino" }, "orcid": "0000-0003-1588-7057" }, { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "Omer" }, "orcid": "0000-0002-0111-0418" } ] }, "title": "Weak equivalence of stationary actions and the entropy realization problem", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 1603.05013.pdf

", "abstract": "We introduce the notion of weak containment for stationary actions of a countable group and define a natural topology on the space of weak equivalence classes. We prove that Furstenberg entropy is an invariant of weak equivalence, and moreover that it descends to a continuous function on the space of weak equivalence classes.", "date": "2016-03-16", "date_type": "published", "id_number": "CaltechAUTHORS:20161111-122744051", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161111-122744051", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1603.05013", "primary_object": { "basename": "1603.05013.pdf", "url": "https://authors.library.caltech.edu/records/1fbtx-nyk67/files/1603.05013.pdf" }, "pub_year": "2016", "author_list": "Burton, Peter; Lupini, Martino; et al." }, { "id": "https://authors.library.caltech.edu/records/fn6wv-r1370", "eprint_id": 98023, "eprint_status": "archive", "datestamp": "2023-08-20 09:00:40", "lastmod": "2026-04-04 20:26:42", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" }, { "id": "Fox-J", "name": { "family": "Fox", "given": "Jacob" } }, { "id": "R\u00f6dl-V", "name": { "family": "R\u00f6dl", "given": "Vojt\u011bch" } } ] }, "title": "Hedgehogs are not colour blind", "ispublished": "unpub", "full_text_status": "public", "note": "Conlon research supported by a Royal Society University Research Fellowship. Fox research supported by a Packard Fellowship, by NSF Career Award DMS-1352121 and by an Alfred P. Sloan Fellowship. R\u00f6dl research partially supported by NSF grants DMS-1102086 and DMS-1301698.\n\n

Submitted - 1511.00563.pdf

", "abstract": "We exhibit a family of 3-uniform hypergraphs with the property that their 2-colour Ramsey numbers grow polynomially in the number of vertices, while their 4-colour Ramsey numbers grow exponentially. This is the first example of a class of hypergraphs whose Ramsey numbers show a strong dependence on the number of colours.", "date": "2015-11-02", "date_type": "published", "id_number": "CaltechAUTHORS:20190819-170900578", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170900578", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Royal Society" }, { "agency": "David and Lucile Packard Foundation" }, { "agency": "NSF", "grant_number": "DMS-1352121" }, { "agency": "Alfred P. Sloan Foundation" }, { "agency": "NSF", "grant_number": "DMS-1102086" }, { "agency": "NSF", "grant_number": "DMS-1301698" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1511.00563", "primary_object": { "basename": "1511.00563.pdf", "url": "https://authors.library.caltech.edu/records/fn6wv-r1370/files/1511.00563.pdf" }, "pub_year": "2015", "author_list": "Conlon, David; Fox, Jacob; et al." }, { "id": "https://authors.library.caltech.edu/records/amw51-r9r61", "eprint_id": 77225, "eprint_status": "archive", "datestamp": "2023-08-20 08:18:20", "lastmod": "2026-04-04 07:00:19", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "R. L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Hainzi-C", "name": { "family": "Hainzi", "given": "C." } }, { "id": "Seiringer-R", "name": { "family": "Seiringer", "given": "R." } }, { "id": "Solovej-J-P", "name": { "family": "Solovej", "given": "J. P." } } ] }, "title": "Microscopic Derivation of the Ginzburg-Landau Model", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2012 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\nLast revised 25 Sep 2015 (this version, v2) \n\nFinancial support via U.S. NSF grant PHY-1068285 (R.F.), NSERC (R.S.) and a grant from the Danish council for independent research (J.P.S.) is gratefully acknowledged.\n\n

Submitted - 1209.1080.pdf

", "abstract": "We present a summary of our recent rigorous derivation of the celebrated Ginzburg\u2013Landau (GL) theory, starting from the microscopic Bardeen\u2013Cooper\u2013Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof.", "date": "2015-09-25", "date_type": "published", "id_number": "CaltechAUTHORS:20170505-112911639", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170505-112911639", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "PHY-1068285" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)" }, { "agency": "Danish Council for Independent Research" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1209.1080", "primary_object": { "basename": "1209.1080.pdf", "url": "https://authors.library.caltech.edu/records/amw51-r9r61/files/1209.1080.pdf" }, "pub_year": "2015", "author_list": "Frank, R. L.; Hainzi, C.; et al." }, { "id": "https://authors.library.caltech.edu/records/25t19-ds637", "eprint_id": 62645, "eprint_status": "archive", "datestamp": "2023-08-20 07:22:50", "lastmod": "2026-04-04 07:28:32", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Chun-Sungbong", "name": { "family": "Chun", "given": "Sungbong" } }, { "id": "Gukov-S", "name": { "family": "Gukov", "given": "Sergei" }, "orcid": "0000-0002-9486-1762" }, { "id": "Roggenkamp-D", "name": { "family": "Roggenkamp", "given": "Daniel" } } ] }, "title": "Junctions of surface operators and categorification of quantum groups", "ispublished": "unpub", "full_text_status": "public", "note": "We thank A. Lauda and D. Rose for many patient and very helpful explanations, as well\nas J. Kamnitzer, M. Khovanov, A. Lobb, M. Mackaay, L. Rozansky, M. Stosic, C. Stroppel,\nC. Teleman, B. Webster, and P. Wedrich for helpful comments and advice. We also thank\nparticipants of the \"Knot homologies, BPS states, and SUSY gauge theories\" program at the\nSimons Center for Geometry and Physics for useful discussions.\nThe work of S.G. is funded in part by the DOE Grant DE-SC0011632 and the Walter\nBurke Institute for Theoretical Physics.\n\n

Submitted - 1507.06318v1.pdf

", "abstract": "We show how networks of Wilson lines realize quantum groups U_q(sl_m), for arbitrary m, in 3d SU(N) Chern-Simons theory. Lifting this construction to foams of surface operators in 4d theory we find that rich structure of junctions is encoded in combinatorics of planar diagrams. For a particular choice of surface operators we reproduce known mathematical constructions of categorical representations and categorified quantum groups.", "date": "2015-07-22", "date_type": "published", "id_number": "CaltechAUTHORS:20151207-102030601", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20151207-102030601", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-SC0011632" }, { "agency": "Walter Burke Institute for Theoretical Physics, Caltech" } ] }, "local_group": { "items": [ { "id": "Walter-Burke-Institute-for-Theoretical-Physics" }, { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1507.06318", "primary_object": { "basename": "1507.06318v1.pdf", "url": "https://authors.library.caltech.edu/records/25t19-ds637/files/1507.06318v1.pdf" }, "pub_year": "2015", "author_list": "Chun, Sungbong; Gukov, Sergei; et al." }, { "id": "https://authors.library.caltech.edu/records/kt2g3-psq55", "eprint_id": 81762, "eprint_status": "archive", "datestamp": "2023-08-20 06:51:42", "lastmod": "2026-04-04 05:52:48", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Rains-E-M", "name": { "family": "Rains", "given": "Eric M." }, "orcid": "0000-0002-9915-0919" }, { "id": "Warnaar-S-O", "name": { "family": "Warnaar", "given": "S. Ole" }, "orcid": "0000-0002-9786-0175" } ] }, "title": "Bounded Littlewood identities", "ispublished": "unpub", "full_text_status": "public", "note": "Work supported by the National Science Foundation (grant number DMS-1001645) and the Australian Research Council. \n\nWe thank Michael Schlosser, Hjalmar Rosengren and Jasper Stokman for helpful discussions on hypergeometric function, Macdonald identities and Macdonald\u2013Koornwinder polynomials. We thank Richard Stanley for pointing out the paper [86] by Schur.\n\n

Submitted - 1506.02755.pdf

", "abstract": "We describe a method, based on the theory of Macdonald-Koornwinder polynomials, for proving bounded Littlewood identities. Our approach provides an alternative to Macdonald's partial fraction technique and results in the first examples of bounded Littlewood identities for Macdonald polynomials. These identities, which take the form of decomposition formulas for Macdonald polynomials of type (R,S) in terms Macdonald polynomials of type A, are q,t-analogues of known branching formulas for characters of the symplectic, orthogonal and special orthogonal groups, important in the theory of plane partitions. \nAs applications of our results we obtain combinatorial formulas for characters of affine Lie algebras, Rogers-Ramanujan identities for such algebras complementing recent results of Griffin et al., and transformation formulas for Kaneko-Macdonald-type hypergeometric series.", "date": "2015-06-09", "date_type": "published", "id_number": "CaltechAUTHORS:20170922-141021522", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170922-141021522", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1001645" }, { "agency": "Australian Research Council" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1506.02755", "primary_object": { "basename": "1506.02755.pdf", "url": "https://authors.library.caltech.edu/records/kt2g3-psq55/files/1506.02755.pdf" }, "pub_year": "2015", "author_list": "Rains, Eric M. and Warnaar, S. Ole" }, { "id": "https://authors.library.caltech.edu/records/ntgs9-nhh54", "eprint_id": 79013, "eprint_status": "archive", "datestamp": "2023-08-20 05:49:36", "lastmod": "2026-04-04 21:17:53", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Manin-Y-I", "name": { "family": "Manin", "given": "Yuri I." } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 1504.04005.pdf

", "abstract": "It is well known that the so called Bianchi IX spacetimes with SO(3)-symmetry in a neighbourhood of the Big Bang exhibit a chaotic behaviour of typical trajectories in the backward movement of time. This behaviour (Mixmaster Model of the Universe) can be encoded by the shift of two-sided continued fractions. Exactly the same shift encodes the sequences of intersections of hyperbolic geodesics with purely imaginary axis in the upper complex half-plane, that is geodesic flow on an appropriate modular surface. A physical interpretation of this coincidence was suggested in arXiv:1402.2158: namely, that Mixmaster chaos is an approximate description of the passage from a hot quantum Universe at the Big Bang moment to the cooling classical Universe. Here we discuss and elaborate this suggestion, looking at the Mixmaster Model from the perspective of the second class of Bianchi IX spacetimes: those with SU(2)-symmetry (self-dual Einstein metrics). We also extend it to the more general context related to Painlev\u00e9 VI equations.", "date": "2015-04-15", "date_type": "published", "id_number": "CaltechAUTHORS:20170712-124502018", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-124502018", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1504.04005", "primary_object": { "basename": "1504.04005.pdf", "url": "https://authors.library.caltech.edu/records/ntgs9-nhh54/files/1504.04005.pdf" }, "pub_year": "2015", "author_list": "Manin, Yuri I. and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/0qwck-ktg64", "eprint_id": 77083, "eprint_status": "archive", "datestamp": "2023-08-20 05:31:01", "lastmod": "2026-04-04 21:18:34", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Lenzmann-E", "name": { "family": "Lenzmann", "given": "Enno" } } ] }, "title": "Uniqueness and Nondegeneracy of Ground States for (\u2212\u0394)^sQ+Q\u2212Q^(\u03b1+1)=0 in R", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 21 Sep 2010 (v1), last revised 23 Mar 2015 (this version, v2)) \n\n\nR. F. acknowledges support from NSF grant PHY-0652854. \n\nE. L. was supported by a Steno fellowship from the Danish science research council, and he also gratefully acknowledges partial support from NSF grant DMS-0702492.\n\n

Submitted - 1009.4042.pdf

", "abstract": "We prove uniqueness of ground state solutions Q = Q(|x|)\u22650 for the nonlinear equation (\u2212\u0394)^sQ + Q \u2212 Q^(\u03b1+)1 = 0 in R, where 0 < s < 1 and 0 < \u03b1 < _(4s) ^(1\u22122s) for s < 1/2 and 0 < \u03b1 < \u221e for s \u2265 1/2. Here (\u2212\u0394)^s denotes the fractional Laplacian in one dimension. In particular, we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for s = 1/2 and \u03b1 = 1 in [Acta Math.,167 (1991), 107-126]. As a technical key result in this paper, we show that the associated linearized operator L_+ = (\u2212\u0394)^s + 1\u2212 (\u03b1+1)Q^\u03b1 is nondegenerate; i.,e., its kernel satisfies ker L_+ = span {Q\u2032}. This result about L_+ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for nonlinear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.", "date": "2015-03-23", "date_type": "published", "id_number": "CaltechAUTHORS:20170501-072727175", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170501-072727175", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "PHY-0652854" }, { "agency": "Danish Natural Science Research Council" }, { "agency": "NSF", "grant_number": "DMS-0702492" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1009.4042", "primary_object": { "basename": "1009.4042.pdf", "url": "https://authors.library.caltech.edu/records/0qwck-ktg64/files/1009.4042.pdf" }, "pub_year": "2015", "author_list": "Frank, Rupert L. and Lenzmann, Enno" }, { "id": "https://authors.library.caltech.edu/records/tkvca-n5335", "eprint_id": 71956, "eprint_status": "archive", "datestamp": "2023-08-20 05:30:47", "lastmod": "2026-04-04 06:51:24", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frisch-J", "name": { "family": "Frisch", "given": "Joshua" } }, { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "Omer" }, "orcid": "0000-0002-0111-0418" } ] }, "title": "Symbolic dynamics on amenable groups: the entropy of generic shifts", "ispublished": "unpub", "full_text_status": "public", "note": "J. Frisch was supported by MIT's Undergraduate Research Opportunities Program. This research was partially conducted at Microsoft Research, New England.\n\n

Submitted - 1503.06251.pdf

", "abstract": "Let G be a finitely generated amenable group. We study the space of shifts on G over a given finite alphabet A. We show that the zero entropy shifts are generic in this space, and that more generally the shifts of entropy c are generic in the space of shifts with entropy at least c. The same is shown to hold for the space of transitive shifts and for the space of weakly mixing shifts. \nAs applications of this result, we show that for every entropy value c\u2208[0,log|A|] there is a weakly mixing subshift of A^G with entropy c. We also show that the set of strongly irreducible shifts does not form a G_\u03b4 in the space of shifts, and that all non-trivial, strongly irreducible shifts are non-isolated points in this space.", "date": "2015-03-21", "date_type": "published", "id_number": "CaltechAUTHORS:20161111-134938105", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161111-134938105", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Massachusetts Institute of Technology (MIT)" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1503.06251", "primary_object": { "basename": "1503.06251.pdf", "url": "https://authors.library.caltech.edu/records/tkvca-n5335/files/1503.06251.pdf" }, "pub_year": "2015", "author_list": "Frisch, Joshua and Tamuz, Omer" }, { "id": "https://authors.library.caltech.edu/records/t1k1c-yp061", "eprint_id": 117602, "eprint_status": "archive", "datestamp": "2023-08-20 04:02:37", "lastmod": "2026-03-09 23:57:12", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Song-Antoine", "name": { "family": "Song", "given": "Antoine" } } ] }, "title": "A maximum principle for self-shrinkers and some consequences", "ispublished": "unpub", "full_text_status": "public", "note": "I would like to thank Niels M\u00f8ller for proofreading a preliminary version of this note and for his numerous suggestions. I also wish to thank my professors Fernando C. Marques, Olivier Biquard and Laurent Hauswirth for their guidance.", "abstract": "Using a maximum principle for self-shrinkers of the mean curvature flow, we give new proofs of a rigidity theorem for rotationally symmetric compact self-shrinkers and a result about the asymptotic behavior of self-shrinkers. This comparison argument also implies a linear bound for the second fundamental form of self-shrinking surfaces under natural assumptions. As a consequence, translating solitons can be related to these self-shrinkers.", "date": "2014-12-15", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20221026-539161000.17", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221026-539161000.17", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1412.4755", "pub_year": "2014", "author_list": "Song, Antoine" }, { "id": "https://authors.library.caltech.edu/records/w5562-59k87", "eprint_id": 66656, "eprint_status": "archive", "datestamp": "2023-08-20 03:41:36", "lastmod": "2026-03-09 02:39:05", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" } ] }, "title": "Bosonic Topological Insulators and Paramagnets: a view from cobordisms", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 26 Apr 2014 (v1), last revised 14 Nov 2014 (this version, v2)). November 17, 2014. \n\nI am grateful to Ryan Thorngren for discussions and to Ashvin Vishwanath for patiently answering my questions regarding Symmetry Protected Topological Phases. I also would like to acknowledge discussions with Xiao-Gang Wen which helped me to find an error in the first version of the paper. This work was supported in part by the DOE grant DE-FG02-92ER40701.\n\n

Submitted - 1404.6659.pdf

", "abstract": "We classify Bosonic Topological Insulators and Paramagnets in D \u2264 4 spatial dimensions using the cobordism approach. For D < 4 we confirm that the only such phase which does not fit into the group cohomology classification is the 3D Bosonic Topological Insulator protected by time-reversal symmetry whose surface admits an all-fermion topologically ordered state. For D = 4 there is a unique \"beyond group\ncohomology\" phase. It is protected by gravitational anomalies of the boundary theory and is stable without any additional symmetry.", "date": "2014-11-14", "date_type": "published", "id_number": "CaltechAUTHORS:20160504-111658328", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160504-111658328", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-92ER40701" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1404.6659", "primary_object": { "basename": "1404.6659.pdf", "url": "https://authors.library.caltech.edu/records/w5562-59k87/files/1404.6659.pdf" }, "pub_year": "2014", "author_list": "Kapustin, Anton" }, { "id": "https://authors.library.caltech.edu/records/nfrxw-wd767", "eprint_id": 79027, "eprint_status": "archive", "datestamp": "2023-08-20 02:00:24", "lastmod": "2026-04-05 14:55:58", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Principles and Parameters: a coding theory perspective", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 26 Jul 2014).\n\n

Submitted - 1407.7169.pdf

", "abstract": "We propose an approach to Longobardi's parametric comparison method (PCM) via the theory of error-correcting codes. One associates to a collection of languages to be analyzed with the PCM a binary (or ternary) code with one code words for each language in the family and each word consisting of the binary values of the syntactic parameters of the language, with the ternary case allowing for an additional parameter state that takes into account phenomena of entailment of parameters. The code parameters of the resulting code can be compared with some classical bounds in coding theory: the asymptotic bound, the Gilbert\u2013Varshamov bound, etc. The position of the code parameters with respect to some of these bounds provides quantitative information on the variability of syntactic parameters within and across historical-linguistic families. While computations carried out for languages belonging to the same family yield codes below the GV curve, comparisons across different historical families can give examples of isolated codes lying above the asymptotic bound.", "date": "2014-07-26", "date_type": "published", "id_number": "CaltechAUTHORS:20170712-150210854", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-150210854", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1407.7169", "primary_object": { "basename": "1407.7169.pdf", "url": "https://authors.library.caltech.edu/records/nfrxw-wd767/files/1407.7169.pdf" }, "pub_year": "2014", "author_list": "Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/dnntz-k8493", "eprint_id": 56818, "eprint_status": "archive", "datestamp": "2023-08-20 01:46:12", "lastmod": "2026-04-05 17:14:24", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Ni-Yi", "name": { "family": "Ni", "given": "Yi" } }, { "id": "Zhang-Xingru", "name": { "family": "Zhang", "given": "Xingru" } } ] }, "title": "Dehn surgery on knots in S^3 producing Nil Seifert fibred spaces", "ispublished": "unpub", "full_text_status": "public", "note": "The first author was partially supported by NSF grant numbers DMS-1103976, DMS-1252992, and an Alfred P. Sloan Research Fellowship.\n\n

Submitted - 1407.0648.pdf

", "abstract": "We prove that there are exactly 6 Nil Seifert fibred spaces which can be\nobtained by Dehn surgeries on non-trefoil knots in S^3, with {60, 144, 156,\n288, 300} as the exact set of all such surgery slopes up to taking the mirror\nimages of the knots. We conjecture that there are exactly 4 specific\nhyperbolic knots in S^3 which admit Nil Seifert fibred surgery. We also give\nsome more general results and a more general conjecture concerning Seifert\nfibred surgeries on hyperbolic knots in S^3.", "date": "2014-07-02", "date_type": "published", "id_number": "CaltechAUTHORS:20150421-115833976", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150421-115833976", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1103976" }, { "agency": "NSF", "grant_number": "DMS-1252992" }, { "agency": "Alfred P. Sloan Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1407.0648", "primary_object": { "basename": "1407.0648.pdf", "url": "https://authors.library.caltech.edu/records/dnntz-k8493/files/1407.0648.pdf" }, "pub_year": "2014", "author_list": "Ni, Yi and Zhang, Xingru" }, { "id": "https://authors.library.caltech.edu/records/av4tc-q2055", "eprint_id": 71961, "eprint_status": "archive", "datestamp": "2023-08-20 00:55:04", "lastmod": "2026-04-06 01:23:39", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Benjamini-I", "name": { "family": "Benjamini", "given": "Itai" } }, { "id": "Chan-Siu-On", "name": { "family": "Chan", "given": "Siu-On" } }, { "id": "O'Donnell-R", "name": { "family": "O'Donnell", "given": "Ryan" } }, { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "Omer" }, "orcid": "0000-0002-0111-0418" }, { "id": "Tan-Li-Yang", "name": { "family": "Tan", "given": "Li-Yang" } } ] }, "title": "Convergence, unanimity and disagreement in majority dynamics on unimodular graphs and random graphs", "ispublished": "unpub", "full_text_status": "public", "note": "R. O'Donnell is supported by NSF grants CCF-1319743 and CCF-1116594.\n\n

Submitted - 1405.2486.pdf

", "abstract": "In majority dynamics, agents located at the vertices of an undirected simple graph update their binary opinions synchronously by adopting those of the majority of their neighbors. \nOn infinite unimodular transitive graphs (e.g., Cayley graphs), when initial opinions are chosen from a distribution that is invariant with respect to the graph automorphism group, we show that the opinion of each agent almost surely either converges, or else eventually oscillates with period two; this is known to hold for finite graphs, but not for all infinite graphs. \nOn Erd\u0151s-R\u00e9nyi random graphs with degrees \u03a9(n\u221a), we show that when initial opinions are chosen i.i.d. then agents all converge to the initial majority opinion, with constant probability. Conversely, on random 4-regular finite graphs, we show that with high probability different agents converge to different opinions.", "date": "2014-05-11", "date_type": "published", "id_number": "CaltechAUTHORS:20161111-145123837", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161111-145123837", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCF-1319743" }, { "agency": "NSF", "grant_number": "CCF-1116594" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1405.2486", "primary_object": { "basename": "1405.2486.pdf", "url": "https://authors.library.caltech.edu/records/av4tc-q2055/files/1405.2486.pdf" }, "pub_year": "2014", "author_list": "Benjamini, Itai; Chan, Siu-On; et al." }, { "id": "https://authors.library.caltech.edu/records/qa0fj-1sh90", "eprint_id": 66638, "eprint_status": "archive", "datestamp": "2023-08-20 00:24:52", "lastmod": "2026-03-09 02:41:51", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" } ] }, "title": "Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 6 Mar 2014 (v1), last revised 14 Apr 2014 (this version, v3)). April 16, 2014. \n\nI am would like to thank Ryan Thorngren for a collaboration on a related project, Dan Freed for drawing my attention to cobordism groups, and John Morgan and Michael Hopkins for advice. I am especially grateful to Alexei Kitaev for pointing out a number of erroneous statements in the first version of the manuscript. This work was supported in part by the DOE grant DE-FG02-92ER40701.\n\n

Submitted - 1403.1467.pdf

", "abstract": "We propose that Symmetry Protected Topological Phases with a\nfinite symmetry group G are classified by cobordism groups of the\nclassifying space of G. This provides an explanation for the recent\ndiscovery of bosonic SPT phases which do not fit into the group cohomology\nclassification. We discuss the connection of the cobordism\nclassification of SPT phases to gauge and gravitational anomalies in\nvarious dimensions.", "date": "2014-04-14", "date_type": "published", "id_number": "CaltechAUTHORS:20160504-095515747", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160504-095515747", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-92ER40701" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1403.1467", "primary_object": { "basename": "1403.1467.pdf", "url": "https://authors.library.caltech.edu/records/qa0fj-1sh90/files/1403.1467.pdf" }, "pub_year": "2014", "author_list": "Kapustin, Anton" }, { "id": "https://authors.library.caltech.edu/records/v4xvz-91c74", "eprint_id": 66610, "eprint_status": "archive", "datestamp": "2023-08-20 00:22:59", "lastmod": "2026-04-05 14:32:08", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Gadde-A", "name": { "family": "Gadde", "given": "Abhijit" } }, { "id": "Gukov-S", "name": { "family": "Gukov", "given": "Sergei" }, "orcid": "0000-0002-9486-1762" }, { "id": "Putrov-P", "name": { "family": "Putrov", "given": "Pavel" } } ] }, "title": "Duality Defects", "ispublished": "unpub", "full_text_status": "public", "note": "We are grateful to C. Bachas, O. Ganor and I. Runkel for useful discussions. The work of A.G. is supported in part by the John A. McCone fellowship and by DOE Grant DE-FG02-92-ER40701. The work of S.G. is supported in part by DOE Grant DE-FG02-92ER40701. The work of P.P. is supported in part by the Sherman Fairchild scholarship and by NSF Grant PHY-1050729.\n\n

Submitted - 1404.2929.pdf

", "abstract": "We propose a unified approach to a general class of codimension-2 defects in field theories with non-trivial duality symmetries and discuss various constructions of such \"duality defects\" in diverse dimensions. In particular, in d=4 we propose a new interpretation of the Seiberg-Witten u-plane by \"embedding\" it in the physical space-time: we argue that it describes a BPS configuration of two duality defects (at the monopole/dyon points) and propose its vast generalization based on Lefschetz fibrations of 4-manifolds.", "date": "2014-04-10", "date_type": "published", "id_number": "CaltechAUTHORS:20160503-090914800", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160503-090914800", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "John A. McCone Fellowship" }, { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-92-ER40701" }, { "agency": "Sherman Fairchild Foundation" }, { "agency": "NSF", "grant_number": "PHY-1050729" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1404.2929", "primary_object": { "basename": "1404.2929.pdf", "url": "https://authors.library.caltech.edu/records/v4xvz-91c74/files/1404.2929.pdf" }, "pub_year": "2014", "author_list": "Gadde, Abhijit; Gukov, Sergei; et al." }, { "id": "https://authors.library.caltech.edu/records/e67ha-vhb23", "eprint_id": 45240, "eprint_status": "archive", "datestamp": "2023-08-20 00:04:52", "lastmod": "2026-04-23 16:16:22", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Dimitrov-Mladen", "name": { "family": "Dimitrov", "given": "Mladen" }, "orcid": "0000-0002-7228-9136" }, { "id": "Ramakrishnan-D", "name": { "family": "Ramakrishnan", "given": "Dinakar" }, "orcid": "0000-0002-0159-087X" } ] }, "title": "Compact arithmetic quotients of the complex 2-ball and a conjecture of Lang", "ispublished": "unpub", "full_text_status": "public", "note": "

We would like to thank Don Blasius, Laurent Clozel, Najmuddin Fakhruddin, Dick Gross, Haruzo Hida, Barry Mazur, David Rohrlich, Matthew Stover and Shing-Tung Yau for helpful conversations. In fact it was Fakhruddin who suggested our use of Lang's conjecture for abelian varieties. Needless to say, this Note owes much to the deep results of Faltings. Thanks are also due to Serge Lang (posthumously), and to John Tate, for getting one of us interested in the conjectural Mordellic property of hyperbolic varieties. Finally, we are also happy to acknowledge partial support the following sources: the Agence Nationale de la Recherche grants ANR-10-BLAN-0114 and ANR-11-LABX-0007-01 for the first author (M.D.), and from the NSF grant DMS-1001916 for the second author (D.R.).

\n\n

Submitted - 1401.1628v2.pdf

", "abstract": "

Let X be a compact quotient of the unit ball in \u2102^2 by an arithmetic subgroup Γ of a unitary group defined by an anisotropic hermitian form on a three dimensional vector space over a CM field with signature (2,1) at one archimedean place and (3,0) at the others. We prove that if all the torsion elements of Γ are scalar, then X is Mordellic, meaning that for any number field k containing the field of definition of X, the set X(k) of k-rational points of X is finite. The proof applies and combines certain key results of Faltings with the work of Rogawski and the hyperbolicity of X.

", "date": "2014-03-31", "date_type": "published", "publisher": "ArXiv", "id_number": "CaltechAUTHORS:20140428-101450443", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140428-101450443", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Agence Nationale de la Recherche (ANR)", "grant_number": "ANR-10-BLAN-0114" }, { "agency": "Agence Nationale de la Recherche (ANR)", "grant_number": "ANR-11-LABX-0007-01" }, { "agency": "NSF", "grant_number": "DMS-1001916" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" }, { "id": "Division-of-Physics-Mathematics-and-Astronomy" } ] }, "primary_object": { "basename": "1401.1628v2.pdf", "url": "https://authors.library.caltech.edu/records/e67ha-vhb23/files/1401.1628v2.pdf" }, "pub_year": "2014", "author_list": "Dimitrov, Mladen and Ramakrishnan, Dinakar" }, { "id": "https://authors.library.caltech.edu/records/ptnxn-m7r48", "eprint_id": 98020, "eprint_status": "archive", "datestamp": "2023-08-19 23:13:45", "lastmod": "2026-04-06 01:18:47", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" }, { "id": "Fox-J", "name": { "family": "Fox", "given": "Jacob" } }, { "id": "Sudakov-B", "name": { "family": "Sudakov", "given": "Benny" } } ] }, "title": "Large subgraphs without complete bipartite graphs", "ispublished": "unpub", "full_text_status": "public", "note": "We would like to thank M. Krivelevich for bringing this problem to our attention and for sharing with us preprint 'Large subgraphs without short cycles' by F. Foucaud, M. Krivelevich and G. Perarnau.\n\n

Submitted - 1401.6711.pdf

", "abstract": "In this note, we answer the following question of Foucaud, Krivelevich and Perarnau. What is the size of the largest K_(r,s)-free subgraph one can guarantee in every graph G with m edges? We also discuss the analogous problem for hypergraphs.", "date": "2014-01-27", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20190819-170849884", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170849884", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1401.6711", "primary_object": { "basename": "1401.6711.pdf", "url": "https://authors.library.caltech.edu/records/ptnxn-m7r48/files/1401.6711.pdf" }, "pub_year": "2014", "author_list": "Conlon, David; Fox, Jacob; et al." }, { "id": "https://authors.library.caltech.edu/records/y8f5w-70x05", "eprint_id": 56819, "eprint_status": "archive", "datestamp": "2023-08-19 21:50:38", "lastmod": "2026-03-09 21:52:41", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Li-Eileen", "name": { "family": "Li", "given": "Eileen" } }, { "id": "Ni-Yi", "name": { "family": "Ni", "given": "Yi" } } ] }, "title": "Half-integral finite surgeries on knots in S^3", "ispublished": "unpub", "full_text_status": "public", "note": "The second author wishes to thank Xingru Zhang for asking the question about half-integral finite surgery and explaining the background. The second author is also grateful to Liling Gu, whose work [7] benefits our paper a lot. The first author was supported by Caltech's Summer Undergraduate Research Fellowships program. The second author was partially supported by an AIM Five-Year Fellowship, NSF grant numbers DMS-1103976, DMS-1252992, and an Alfred P. Sloan Research Fellowship.\n\n

Submitted - 1310.1346.pdf

", "abstract": "Suppose that a hyperbolic knot in S^3 admits a finite surgery, Boyer and\nZhang proved that the surgery slope must be either integral or half-integral, and they conjectured that the latter case does not happen. Using the correction terms in Heegaard Floer homology, we prove that if a hyperbolic knot in S^3 admits a half-integral finite surgery, then the knot must have the same knot Floer homology as one of eight non-hyperbolic knots which are known to admit such surgeries, and the resulting manifold must be one of ten spherical space forms. As knot Floer homology carries a lot of information about the knot, this gives a strong evidence to Boyer-Zhang's conjecture.", "date": "2013-10-04", "date_type": "published", "id_number": "CaltechAUTHORS:20150421-115837671", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150421-115837671", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Caltech Summer Undergraduate Research Fellowship (SURF)" }, { "agency": "AIM Five-Year Fellowship" }, { "agency": "NSF", "grant_number": "DMS-1103976" }, { "agency": "NSF", "grant_number": "DMS-1252992" }, { "agency": "Alfred P. Sloan Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1310.1346", "primary_object": { "basename": "1310.1346.pdf", "url": "https://authors.library.caltech.edu/records/y8f5w-70x05/files/1310.1346.pdf" }, "pub_year": "2013", "author_list": "Li, Eileen and Ni, Yi" }, { "id": "https://authors.library.caltech.edu/records/vtetj-feb46", "eprint_id": 66613, "eprint_status": "archive", "datestamp": "2023-08-19 20:47:51", "lastmod": "2026-03-09 02:20:14", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Gukov-S", "name": { "family": "Gukov", "given": "Sergei" }, "orcid": "0000-0002-9486-1762" }, { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" } ] }, "title": "Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories", "ispublished": "unpub", "full_text_status": "public", "note": "We are grateful to Nathan Seiberg for a collaboration during various stages of this project and to Gregory Moore for a discussion. This work was supported in part by the DOE grant DE-FG02-92ER40701 and by the National Science Foundation under Grant No. PHYS-1066293 and the hospitality of the Aspen Center for Physics.\n\n

Submitted - 1307.4793.pdf

", "abstract": "We revisit the role of loop and surface operators as order parameters for gapped phases of four-dimensional gauge theories. We show that in some cases surface operators are confined, and that this fact can be used to distinguish phases which are not distinguished by the Wilson-'t Hooft criterion. The long-distance behavior of loop and surface operators which are neither confined nor screened is controlled by a 4d TQFT. We construct these TQFTs for phases which are characterized by the presence of electrically and/or magnetically charged condensates. Interestingly, the TQFT describing a phase with a nonabelian monopole condensate is based on the theory of nonabelian gerbes. We also show that in phases with a dyonic condensate the low-energy theta-angle is quantized.", "date": "2013-07-17", "date_type": "published", "id_number": "CaltechAUTHORS:20160503-094901740", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160503-094901740", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-92ER40701" }, { "agency": "NSF", "grant_number": "PHY-1066293" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1307.4793", "primary_object": { "basename": "1307.4793.pdf", "url": "https://authors.library.caltech.edu/records/vtetj-feb46/files/1307.4793.pdf" }, "pub_year": "2013", "author_list": "Gukov, Sergei and Kapustin, Anton" }, { "id": "https://authors.library.caltech.edu/records/a92w2-1va55", "eprint_id": 81767, "eprint_status": "archive", "datestamp": "2023-08-19 20:46:37", "lastmod": "2026-03-09 22:49:02", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Rains-E-M", "name": { "family": "Rains", "given": "Eric M." }, "orcid": "0000-0002-9915-0919" } ] }, "title": "Generalized Hitchin systems on rational surfaces", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 1307.4033.pdf

", "abstract": "By analogy with work of Hitchin on integrable systems, we construct natural relaxations of several kinds of moduli spaces of difference equations, with special attention to a particular class of difference equations on an elliptic curve (arising in the theory of elliptic special functions). The common feature of the relaxations is that they can be identified with moduli spaces of sheaves on rational surfaces. Not only does this make various natural questions become purely geometric (rigid equations correspond to -2-curves), it also establishes a number of nontrivial correspondences between different moduli spaces, since a given moduli space of sheaves is typically the relaxation of infinitely many moduli spaces of equations. In the process of understanding this, we also consider a number of purely geometric questions about rational surfaces with anticanonical curves; e.g., we give an essentially combinatorial algorithm for testing whether a given divisor is the class of a -2-curve or is effective with generically integral representative.", "date": "2013-07-15", "date_type": "published", "id_number": "CaltechAUTHORS:20170922-142713387", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170922-142713387", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1307.4033", "primary_object": { "basename": "1307.4033.pdf", "url": "https://authors.library.caltech.edu/records/a92w2-1va55/files/1307.4033.pdf" }, "pub_year": "2013", "author_list": "Rains, Eric M." }, { "id": "https://authors.library.caltech.edu/records/0fj52-92r24", "eprint_id": 81770, "eprint_status": "archive", "datestamp": "2023-08-19 20:46:42", "lastmod": "2026-03-09 22:11:20", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Rains-E-M", "name": { "family": "Rains", "given": "Eric M." }, "orcid": "0000-0002-9915-0919" } ] }, "title": "Birational morphisms and Poisson moduli spaces", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 1307.4032.pdf

", "abstract": "We study birational morphisms between smooth projective surfaces that respect a given Poisson structure, with particular attention to induced birational maps between the (Poisson) moduli spaces of sheaves on those surfaces. In particular, to any birational morphism, we associate a corresponding \"minimal lift\" operation on sheaves of homological dimension \u2264 1, and study its properties. In particular, we show that minimal lift induces a stratification of the moduli space of simple sheaves on the codomain by open subspaces of the moduli space of simple sheaves on the domain, compatibly with the induced Poisson structures.", "date": "2013-07-15", "date_type": "published", "id_number": "CaltechAUTHORS:20170922-144027436", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170922-144027436", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1307.4032", "primary_object": { "basename": "1307.4032.pdf", "url": "https://authors.library.caltech.edu/records/0fj52-92r24/files/1307.4032.pdf" }, "pub_year": "2013", "author_list": "Rains, Eric M." }, { "id": "https://authors.library.caltech.edu/records/ahedw-srw71", "eprint_id": 66634, "eprint_status": "archive", "datestamp": "2023-08-19 20:17:51", "lastmod": "2026-03-09 02:38:42", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" } ] }, "title": "Is there life beyond Quantum Mechanics?", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 27 Mar 2013 (v1), last revised 3 Jun 2013 (this version, v4)). \n\nThis work was supported in part by the Department of Energy grant DE-FG02-92ER40701.\n\n

Submitted - 1303.6917.pdf

", "abstract": "We formulate physically-motivated axioms for a physical theory which for systems with a finite number of degrees of freedom uniquely lead to Quantum Mechanics as the only non-\ntrivial consistent theory. Complex numbers and the existence of the Planck constant common to all systems arise naturally in this approach. The axioms are divided into two groups covering kinematics and basic measurement theory respectively. We show that even if the second group of axioms is dropped, there are no deformations of Quantum Mechanics which preserve the kinematic axioms. Thus any theory going beyond Quantum Mechanics must represent a radical departure from the usual a priori assumptions\nabout the laws of Nature.", "date": "2013-06", "date_type": "published", "id_number": "CaltechAUTHORS:20160504-083212672", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160504-083212672", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-92ER40701" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1303.6917", "primary_object": { "basename": "1303.6917.pdf", "url": "https://authors.library.caltech.edu/records/ahedw-srw71/files/1303.6917.pdf" }, "pub_year": "2013", "author_list": "Kapustin, Anton" }, { "id": "https://authors.library.caltech.edu/records/zsmzm-xth94", "eprint_id": 98019, "eprint_status": "archive", "datestamp": "2023-08-19 20:00:27", "lastmod": "2026-03-08 03:20:38", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" }, { "id": "Fox-J", "name": { "family": "Fox", "given": "Jacob" } }, { "id": "Zhao-Yufei", "name": { "family": "Zhao", "given": "Yufei" } } ] }, "title": "Linear forms from the Gowers uniformity norm", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 1305.5565.pdf

", "abstract": "This is a companion note to our paper 'A relative Szemer\u00e9di theorem', elaborating on a concluding remark. In that paper, we showed how to prove a relative Szemer\u00e9di theorem for (r + 1)-term arithmetic progressions assuming a linear forms condition. Here we show how to replace this condition with an assumption about the Gowers uniformity norm U^r.", "date": "2013-05-23", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20190819-170846378", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170846378", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1305.5565", "primary_object": { "basename": "1305.5565.pdf", "url": "https://authors.library.caltech.edu/records/zsmzm-xth94/files/1305.5565.pdf" }, "pub_year": "2013", "author_list": "Conlon, David; Fox, Jacob; et al." }, { "id": "https://authors.library.caltech.edu/records/93wjr-saq39", "eprint_id": 79016, "eprint_status": "archive", "datestamp": "2023-08-19 20:00:15", "lastmod": "2026-04-09 21:01:41", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Greenfield-Mark", "name": { "family": "Greenfield", "given": "Mark" } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" }, { "id": "Teh-Kevin", "name": { "family": "Teh", "given": "Kevin" } } ] }, "title": "Type III \u03c3-spectral triples and quantum statistical mechanical systems", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 23 May 2013). \n\nThe first author was supported for this work by a Summer Undergraduate Research Fellowship at Caltech. The second author is partially supported by NSF grants DMS-0901221, DMS-1007207, DMS-1201512, and PHY-1205440. The second author thanks MSRI for hospitality and support.\n\n

Submitted - 1305.5492.pdf

", "abstract": "Spectral triples and quantum statistical mechanical systems are two important constructions in noncommutative geometry. In particular, both lead to interesting reconstruction theorems for a broad range of geometric objects, including number fields, spin manifolds, graphs. \n\nThere are similarities between the two structures, and we show that the notion of type III \u03c3-spectral triple, introduced recently by Connes and Moscovici, provides a natural bridge between them. We investigate explicit examples, related to the Bost\u2013Connes quantum statistical mechanical system and to Riemann surfaces and graphs.", "date": "2013-05-23", "date_type": "published", "id_number": "CaltechAUTHORS:20170712-132224908", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-132224908", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Caltech Summer Undergraduate Research Fellowship (SURF)" }, { "agency": "NSF", "grant_number": "DMS-0901221" }, { "agency": "NSF", "grant_number": "DMS-1007207" }, { "agency": "NSF", "grant_number": "DMS-1201512" }, { "agency": "NSF", "grant_number": "PHY-1205440" }, { "agency": "Mathematical Sciences Research Institute (MSRI)" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1305.5492", "primary_object": { "basename": "1305.5492.pdf", "url": "https://authors.library.caltech.edu/records/93wjr-saq39/files/1305.5492.pdf" }, "pub_year": "2013", "author_list": "Greenfield, Mark; Marcolli, Matilde; et al." }, { "id": "https://authors.library.caltech.edu/records/bwyq7-kyt42", "eprint_id": 56821, "eprint_status": "archive", "datestamp": "2023-08-19 19:52:30", "lastmod": "2026-03-09 21:52:59", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Greene-J-E", "name": { "family": "Greene", "given": "Joshua Evan" } }, { "id": "Ni-Yi", "name": { "family": "Ni", "given": "Yi" } } ] }, "title": "Non-simple genus minimizers in lens spaces", "ispublished": "unpub", "full_text_status": "public", "note": "We thank Ian Agol, Ken Baker, Josh Batson, Cameron Gordon, Adam Levine, Danny Ruberman, and Sucharit Sarkar for helpful discussions. We especially thank Eli Grigsby for providing her computer program to compute knot Floer homology of knots in lens spaces. We also thank the Simons Center for Geometry and Physics and the organizers of the workshop \"Symplectic and Low Dimensional Topologies in Interaction\" for providing an ideal place for us to collaborate. The first author was partially supported by NSF grant number DMS-1207812. The second author was partially supported by NSF grant number DMS-1103976 and an Alfred P. Sloan Research Fellowship.\n\n

Submitted - 1305.0517.pdf

", "abstract": "Given a one-dimensional homology class in a lens space, a question related to\nthe Berge conjecture on lens space surgeries is to determine all knots\nrealizing the minimal rational genus of all knots in this homology class. It is\nknown that simple knots are rational genus minimizers. In this paper, we\nconstruct many non-simple genus minimizers. This negatively answers a question\nof Rasmussen.", "date": "2013-05-02", "date_type": "published", "id_number": "CaltechAUTHORS:20150421-115844735", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150421-115844735", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1207812" }, { "agency": "NSF", "grant_number": "DMS-1103976" }, { "agency": "Alfred P. Sloan Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1305.0517", "primary_object": { "basename": "1305.0517.pdf", "url": "https://authors.library.caltech.edu/records/bwyq7-kyt42/files/1305.0517.pdf" }, "pub_year": "2013", "author_list": "Greene, Joshua Evan and Ni, Yi" }, { "id": "https://authors.library.caltech.edu/records/ghmd9-fzb84", "eprint_id": 66635, "eprint_status": "archive", "datestamp": "2023-08-19 14:39:14", "lastmod": "2026-03-09 02:37:59", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" }, { "id": "Willett-B", "name": { "family": "Willett", "given": "Brian" } } ] }, "title": "Wilson loops in supersymmetric Chern-Simons-matter theories and duality", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 8 Feb 2013). \n\nA.K. would like to thank Kentaro Hori for a useful discussion which contributed to our understanding of section 7. The work of A.K. was supported in part by the DOE grant DE-FG02-92ER40701, and that of B.W. was supported by DOE grant DE-FG02-90ER40542.\n\n

Submitted - 1302.2164.pdf

", "abstract": "We study the algebra of BPSWilson loops in 3d gauge theories with N = 2 supersymmetry\nand Chern-Simons terms. We argue that new relations appear on the quantum level, and that in many\ncases this makes the algebra finite-dimensional. We use our results to propose the mapping of Wilson\nloops under Seiberg-like dualities and verify that the proposed map agrees with the exact results for\nexpectation values of circular Wilson loops. In some cases we also relate the algebra of Wilson loops to\nthe equivariant quantum K-ring of certain quasi projective varieties. This generalizes the connection\nbetween the Verlinde algebra and the quantum cohomology of the Grassmannian found by Witten.", "date": "2013-02-08", "date_type": "published", "id_number": "CaltechAUTHORS:20160504-094028479", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160504-094028479", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-92ER40701" }, { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-90ER40542" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1302.2164", "primary_object": { "basename": "1302.2164.pdf", "url": "https://authors.library.caltech.edu/records/ghmd9-fzb84/files/1302.2164.pdf" }, "pub_year": "2013", "author_list": "Kapustin, Anton and Willett, Brian" }, { "id": "https://authors.library.caltech.edu/records/f9txd-6jq82", "eprint_id": 43591, "eprint_status": "archive", "datestamp": "2023-08-19 14:18:46", "lastmod": "2026-06-01 22:23:03", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Arad-Itai", "name": { "family": "Arad", "given": "Itai" } }, { "id": "Kitaev-A", "name": { "family": "Kitaev", "given": "Alexei" }, "orcid": "0000-0002-5777-642X" }, { "id": "Landau-Zeph", "name": { "family": "Landau", "given": "Zeph" } }, { "id": "Vazirani-Umesh-V", "name": { "family": "Vazirani", "given": "Umesh" } } ] }, "title": "An area law and sub-exponential algorithm for 1D systems", "ispublished": "unpub", "full_text_status": "public", "note": "We are grateful to Dorit Aharonov, Fernando Brandao, and Matt Hastings for inspiring discussions\nabout the above and related topics.\n\n

Submitted - 1301.1162v1.pdf

", "abstract": "We give a new proof for the area law for general 1D gapped systems, which exponentially improves Hastings' famous result [1]. Specifically, we show that for a chain of d-dimensional spins, governed by a 1D local Hamiltonian with a spectral gap \u03b5 > 0, the entanglement entropy of the ground state with respect to any cut in the chain is upper bounded by O(log^3 d/\u03b5 ). Our approach uses the framework of Refs. [2, 3] to construct a Chebyshev-based AGSP (Approximate Ground Space Projection) with favorable factors. However, our construction uses the Hamiltonian directly, instead of using the Detectability lemma, which allows us to work with general (frustrated) Hamiltonians, as well as slightly improving the 1/\u03b5 dependence of the bound in Ref. [3]. To achieve that, we establish a new, \"random-walk like\", bound on the entanglement rank of an arbitrary power of a 1D Hamiltonian, which might be of independent interest: ER(H^\u2113) \u2264 (\u2113d)O(\u221a\u2113). Finally, treating d as a constant, our AGSP shows that the ground state is well approximated by a matrix product state with a sublinear bond dimension B = \u03b5 ^O(log^(3/4) n/\u03b5^(1/4)). Using this in conjunction with known dynamical programing algorithms, yields an algorithm for a 1=poly(n) approximation of the ground energy with a subexponential running time T \u2264 exp (\u03b5O(log^(3/4) n/\u03b5^(1/4))).", "date": "2013-01-07", "date_type": "published", "id_number": "CaltechAUTHORS:20140130-142058060", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140130-142058060", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "IQIM" }, { "id": "Mathematics-Department" }, { "id": "Physics-Department" } ] }, "doi": "10.48550/arXiv.1301.1162", "primary_object": { "basename": "1301.1162v1.pdf", "url": "https://authors.library.caltech.edu/records/f9txd-6jq82/files/1301.1162v1.pdf" }, "pub_year": "2013", "author_list": "Arad, Itai; Kitaev, Alexei; et al." }, { "id": "https://authors.library.caltech.edu/records/mymxh-x2t29", "eprint_id": 79003, "eprint_status": "archive", "datestamp": "2023-08-19 13:15:04", "lastmod": "2026-04-10 17:24:28", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Ceyhan-\u00d6", "name": { "family": "Ceyhan", "given": "\u00d6zg\u00fcr" } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Feynman integrals and periods in configuration spaces", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 15 Jul 2012 (v1), last revised 26 Oct 2012 (this version, v2). \n\nParts of this work have been carried out during visits\nof the first author to the California Institute of Technology, the Institut des Hautes \u00c9tudes Scientifiques and the Max Planck Institut f\u00fcr Mathematik. We thank these institutions for their support. The second author acknowledges support from NSF grants DMS-0901221, DMS-1007207, DMS-1201512, and PHY-1205440. The authors thank Paolo Aluffi and Spencer Bloch for many useful conversations. The first author's son, Uzay, was diagnosed with neuroblastoma, at the time when we were in the early stages of this project. His doctor, Tanju Ba\u015farir \u00d6zkan, not only saved Uzay with her exceptional professional skills, but also gave constant personal support, so that \u00d6.C. could return to work and continue the project. This paper is dedicated to her, to express the author's deepest gratitude: Ellerin dert g\u00f6rmesin Tanju abla.\n\n

Submitted - 1207.3544.pdf

", "abstract": "We describe differential forms representing Feynman amplitudes in configuration spaces of Feynman graphs, and regularization and evaluation techniques, for suitable chains of integration, that give rise to periods of mixed Tate motives.", "date": "2012-10-26", "date_type": "published", "id_number": "CaltechAUTHORS:20170712-093539099", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-093539099", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Caltech" }, { "agency": "Institut des Hautes \u00c9tudes Scientifiques" }, { "agency": "Max Planck Institut f\u00fcr Mathematik" }, { "agency": "NSF", "grant_number": "DMS-0901221" }, { "agency": "NSF", "grant_number": "DMS-1007207" }, { "agency": "NSF", "grant_number": "DMS-1201512" }, { "agency": "NSF", "grant_number": "PHY-1205440" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1207.3544", "primary_object": { "basename": "1207.3544.pdf", "url": "https://authors.library.caltech.edu/records/mymxh-x2t29/files/1207.3544.pdf" }, "pub_year": "2012", "author_list": "Ceyhan, \u00d6zg\u00fcr and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/z6wkq-brk71", "eprint_id": 71974, "eprint_status": "archive", "datestamp": "2023-08-19 12:47:41", "lastmod": "2026-04-10 01:53:05", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Menon-A-K", "name": { "family": "Menon", "given": "Aditya Krishna" } }, { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "Omer" }, "orcid": "0000-0002-0111-0418" }, { "id": "Gulwani-S", "name": { "family": "Gulwani", "given": "Sumit" } }, { "id": "Lampson-B", "name": { "family": "Lampson", "given": "Butler" } }, { "id": "Kalai-A-T", "name": { "family": "Kalai", "given": "Adam Tauman" } } ] }, "title": "Textual Features for Programming by Example", "ispublished": "unpub", "full_text_status": "public", "note": "Submitted on 17 Sep 2012. \n\nMay 28, 2013.\n\n

Submitted - 1209.3811.pdf

", "abstract": "In Programming by Example, a system attempts to infer a program from input and output examples, generally by searching for a composition of certain base functions. Performing a na\u0457ve brute force search is infeasible for even mildly involved tasks. We note that the examples themselves often present clues as to which\nfunctions to compose, and how to rank the resulting programs. In text processing, which is our domain of interest, clues arise from simple textual features: for example, if parts of the input and output strings are permutations of one another, this suggests that sorting may be useful. We describe a system that learns the reliability of such clues, allowing for faster search and a principled ranking over programs. Experiments on a prototype of this system show that this learning scheme facilitates efficient inference on a range of text processing tasks.", "date": "2012-09-17", "date_type": "published", "id_number": "CaltechAUTHORS:20161114-080916122", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161114-080916122", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1209.3811", "primary_object": { "basename": "1209.3811.pdf", "url": "https://authors.library.caltech.edu/records/z6wkq-brk71/files/1209.3811.pdf" }, "pub_year": "2012", "author_list": "Menon, Aditya Krishna; Tamuz, Omer; et al." }, { "id": "https://authors.library.caltech.edu/records/wz7k9-3km37", "eprint_id": 77222, "eprint_status": "archive", "datestamp": "2023-08-19 12:47:46", "lastmod": "2026-04-10 14:54:09", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Lieb-E-H", "name": { "family": "Lieb", "given": "Elliott H." }, "orcid": "0000-0001-5843-3587" }, { "id": "Seiringer-R", "name": { "family": "Seiringer", "given": "Robert" } }, { "id": "Thomas-L-E", "name": { "family": "Thomas", "given": "Lawrence E." } } ] }, "title": "Ground state properties of multi-polaron systems", "ispublished": "unpub", "full_text_status": "public", "keywords": "Polaron, binding energies, stability, Coulomb system", "note": "\u00a9 2012 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\nSubmitted on 17 Sep 2012. \n\nPartial financial support from the U.S. National Science Foundation through grants PHY-1068285 (R.F.), PHY-0965859 (E.L.), the Simons Foundation (# 230207, E.L.) and the NSERC (R.S.) is acknowledged. L.T. would like to thank the PIMS Institute, University of British Columbia, for their hospitality and support. We are grateful to Herbert Spohn for stimulating our interest.\n\n

Submitted - 1209.3717.pdf

", "abstract": "We summarize our recent results on the ground state energy of multi-polaron systems. In particular, we discuss stability and existence of the thermodynamic limit, and we discuss the absence of binding in the case of large Coulomb repulsion and the corresponding binding\u2013unbinding transition. We also consider the Pekar-Tomasevich approximation to the ground state energy and we study radial symmetry of the ground state density.", "date": "2012-09-17", "date_type": "published", "id_number": "CaltechAUTHORS:20170505-111204119", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170505-111204119", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "PHY-1068285" }, { "agency": "NSF", "grant_number": "PHY-0965859" }, { "agency": "Simons Foundation", "grant_number": "230207" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)" }, { "agency": "University of British Columbia" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1209.3717", "primary_object": { "basename": "1209.3717.pdf", "url": "https://authors.library.caltech.edu/records/wz7k9-3km37/files/1209.3717.pdf" }, "pub_year": "2012", "author_list": "Frank, Rupert L.; Lieb, Elliott H.; et al." }, { "id": "https://authors.library.caltech.edu/records/cs1hz-ws508", "eprint_id": 98018, "eprint_status": "archive", "datestamp": "2023-08-19 12:37:34", "lastmod": "2026-04-10 00:06:47", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" }, { "id": "Fox-J", "name": { "family": "Fox", "given": "Jacob" } }, { "id": "Sudakov-B", "name": { "family": "Sudakov", "given": "Benny" } } ] }, "title": "Sidorenko's conjecture for a class of graphs: an exposition", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 1209.0184.pdf

", "abstract": "A famous conjecture of Sidorenko and Erd\u0151s-Simonovits states that if H is a bipartite graph then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. The goal of this expository note is to give a short self-contained proof (suitable for teaching in class) of the conjecture if H has a vertex complete to all vertices in the other part.", "date": "2012-09-02", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20190819-170842963", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170842963", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1209.0184", "primary_object": { "basename": "1209.0184.pdf", "url": "https://authors.library.caltech.edu/records/cs1hz-ws508/files/1209.0184.pdf" }, "pub_year": "2012", "author_list": "Conlon, David; Fox, Jacob; et al." }, { "id": "https://authors.library.caltech.edu/records/98738-m0857", "eprint_id": 32454, "eprint_status": "archive", "datestamp": "2023-08-19 11:40:21", "lastmod": "2026-03-09 02:38:14", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" } ] }, "title": "Remarks on nonrelativistic Goldstone bosons", "ispublished": "unpub", "full_text_status": "public", "note": "I would like to thank Ira Rothstein for discussions and Hiroshi Ooguri for\ndrawing my attention to Ref. [7]. This work was supported in part by the\nDOE grant DE-FG02-92ER40701.\n\n

Submitted - 1207.0457v2.pdf

", "abstract": "We discuss excitations in nonrelativistic field theories with spontaneous breaking of a continuous global symmetry. It is known that in such systems there are two types of Goldstone bosons (Type A and Type B) whose dispersion law is generically linear or quadratic, respectively. We show that Type B Goldstone bosons may have gapped partners which we call almost-Goldstone bosons. With some nondegeneracy assumption about the low-energy effective action, the total number of Goldstone and almost-Goldstone bosons adds up to the number of broken symmetry generators. We propose that deviations of the dispersion law of Goldstone bosons from linearity at small momenta may serve as a signature of small breaking of time-reversal symmetry.", "date": "2012-07-05", "date_type": "published", "id_number": "CaltechAUTHORS:20120716-084258892", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120716-084258892", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-92ER40701" } ] }, "local_group": { "items": [ { "id": "Caltech-Theory" }, { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1207.0457", "primary_object": { "basename": "1207.0457v2.pdf", "url": "https://authors.library.caltech.edu/records/98738-m0857/files/1207.0457v2.pdf" }, "pub_year": "2012", "author_list": "Kapustin, Anton" }, { "id": "https://authors.library.caltech.edu/records/pq41f-zt672", "eprint_id": 77132, "eprint_status": "archive", "datestamp": "2023-08-19 11:02:31", "lastmod": "2026-04-10 01:19:13", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Chen-Shibing", "name": { "family": "Chen", "given": "Shibing" } }, { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Weth-T", "name": { "family": "Weth", "given": "Tobias" } } ] }, "title": "Remainder terms in the fractional Sobolev inequality", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 25 May 2012) \n\nU.S. National Science Foundation grant PHY-1068285 (R.F.) and German Science Foundation (DFG) grant WE 2821/4-1 (T.W.) is acknowledged. Shibing Chen wants to thank Robert McCann for helpful discussions.\n\n

Submitted - 1205.5666.pdf

", "abstract": "We show that the fractional Sobolev inequality for the embedding H^(s/2)(R^N),\u21aa L^(2N)/_(N-s) (R^N) s \u2208 (0,N) can be sharpened by adding a remainder term proportional to the distance to the set of optimizers. As a corollary, we derive the existence of a remainder term in the weak L^N/_(N\u2212s)-norm for functions supported in a domain of finite measure. Our results generalize earlier work for the non-fractional case where s is an even integer.", "date": "2012-05-25", "date_type": "published", "id_number": "CaltechAUTHORS:20170502-150250004", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170502-150250004", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "PHY-1068285" }, { "agency": "Deutsche Forschungsgemeinschaft (DFG)", "grant_number": "WE 2821/4-1" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1205.5666", "primary_object": { "basename": "1205.5666.pdf", "url": "https://authors.library.caltech.edu/records/pq41f-zt672/files/1205.5666.pdf" }, "pub_year": "2012", "author_list": "Chen, Shibing; Frank, Rupert L.; et al." }, { "id": "https://authors.library.caltech.edu/records/tf8r6-qs172", "eprint_id": 32421, "eprint_status": "archive", "datestamp": "2023-08-19 10:37:37", "lastmod": "2026-04-16 01:40:12", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kitaev-A", "name": { "family": "Kitaev", "given": "Alexei" }, "orcid": "0000-0002-5777-642X" }, { "id": "Wang-Zhenghan", "name": { "family": "Wang", "given": "Zhenghan" }, "orcid": "0000-0002-5253-6400" } ] }, "title": "Solutions to generalized Yang-Baxter equations via ribbon fusion categories", "ispublished": "unpub", "full_text_status": "public", "note": "The second author is partially supported by NSF DMS 1108736 and would like to thank E.\nRowell for observing (3) of Thm. 2.5, S. Hong for helping on 6j symbols, and R. Chen for\nnumerically testing the solutions.\n\n

Submitted - 1203.1063v2.pdf

", "abstract": "Inspired by quantum information theory, we look for representations of the braid groups B_n on V^(\u2297(n+m\u22122)) for some fixed vector space V such\nthat each braid generator \u03c3_i, i = 1, ..., n\u22121, acts on m consecutive tensor factors\nfrom i through i +m\u22121. The braid relation for m = 2 is essentially the Yang-Baxter equation, and the cases for m > 2 are called generalized Yang-Baxter\nequations. We observe that certain objects in ribbon fusion categories naturally give rise to such representations for the case m = 3. Examples are given\nfrom the Ising theory (or the closely related SU(2)_2), SO(N)_2 for N odd, and\nSU(3)_3. The solution from the Jones-Kauffman theory at a 6th root of unity,\nwhich is closely related to SO(3)_2 or SU(2)_4, is explicitly described in the end.", "date": "2012-04-09", "date_type": "published", "id_number": "CaltechAUTHORS:20120713-102318475", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120713-102318475", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1108736" } ] }, "local_group": { "items": [ { "id": "IQIM" }, { "id": "Mathematics-Department" }, { "id": "Physics-Department" } ] }, "doi": "10.48550/arXiv.1203.1063", "primary_object": { "basename": "1203.1063v2.pdf", "url": "https://authors.library.caltech.edu/records/tf8r6-qs172/files/1203.1063v2.pdf" }, "pub_year": "2012", "author_list": "Kitaev, Alexei and Wang, Zhenghan" }, { "id": "https://authors.library.caltech.edu/records/1248d-kht92", "eprint_id": 71918, "eprint_status": "archive", "datestamp": "2023-08-19 10:36:19", "lastmod": "2026-04-10 01:17:38", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Mossel-E", "name": { "family": "Mossel", "given": "Elchanan" }, "orcid": "0000-0001-7812-7886" }, { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "Omer" }, "orcid": "0000-0002-0111-0418" } ] }, "title": "Bundling Customers: How to Exploit Trust Among Customers to Maximize Seller Profit", "ispublished": "unpub", "full_text_status": "public", "note": "Submitted on 5 Feb 2012 (v1), last revised 4 Apr 2012 (this version, v2) October 28, 2013. \n\nSupported by a Sloan fellowship in Mathematics, by BSF grant 2004105, by NSF Career Award (DMS 054829) by ONR award N00014-07-1-0506 and by ISF grant 1300/08 Weizmann Institute. Supported by ISF grant 1300/08. Omer Tamuz is a recipient of the Google Europe Fellowship in Social Computing, and this research is supported in part by this Google Fellowship.\n\n

Submitted - 1202.0969.pdf

", "abstract": "We consider an auction of identical digital goods to customers whose valuations are drawn independently from known distributions. Myerson's classic result identifies the truthful mechanism that maximizes the seller's expected profit. Under the assumption that in small groups customers can learn each others' valuations, we show how Myerson's result can be improved to yield a higher payoff to the seller using a mechanism that offers groups of customers to buy bundles of items.", "date": "2012-04-04", "date_type": "published", "id_number": "CaltechAUTHORS:20161110-143929904", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161110-143929904", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Alfred P. Sloan Foundation" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2004105" }, { "agency": "NSF", "grant_number": "DMS 054829" }, { "agency": "Office of Naval Research (ONR)", "grant_number": "N00014-07-1-0506" }, { "agency": "Israel Science Foundation", "grant_number": "1300/08" }, { "agency": "Google" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1202.0969", "primary_object": { "basename": "1202.0969.pdf", "url": "https://authors.library.caltech.edu/records/1248d-kht92/files/1202.0969.pdf" }, "pub_year": "2012", "author_list": "Mossel, Elchanan and Tamuz, Omer" }, { "id": "https://authors.library.caltech.edu/records/791z2-f3f14", "eprint_id": 81775, "eprint_status": "archive", "datestamp": "2023-08-19 08:20:03", "lastmod": "2026-04-10 21:08:57", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "van-de-Bult-F-J", "name": { "family": "van de Bult", "given": "Fokko J." } }, { "id": "Rains-E-M", "name": { "family": "Rains", "given": "Eric M." }, "orcid": "0000-0002-9915-0919" } ] }, "title": "Limits of multivariate elliptic hypergeometric biorthogonal functions", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 1110.1458.pdf

", "abstract": "In this article we extend the results of our article \"Limits of elliptic hypergeometric biorthogonal functions\" to the multivariate setting. In that article we determined which families of biorthogonal functions arise as limits from the elliptic hypergeometric biorthogonal functions from Spiridonov when p \u2192 0. Here we show that the classification of the possible limits of the BC_n type multivariate biorthogonal functions previously introduced by the second author is identical to the univariate classification. That is, for each univariate limit family there exists a multivariate extension, and in particular we obtain multivariate versions for all elements of the q-Askey scheme. For the Askey-Wilson polynomials these are the Koornwinder polynomials, and the multivariate versions of the Pastro polynomials form a two-parameter family which include the Macdonald polynomials.", "date": "2011-10-07", "date_type": "published", "id_number": "CaltechAUTHORS:20170922-152223616", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170922-152223616", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1110.1458", "primary_object": { "basename": "1110.1458.pdf", "url": "https://authors.library.caltech.edu/records/791z2-f3f14/files/1110.1458.pdf" }, "pub_year": "2011", "author_list": "van de Bult, Fokko J. and Rains, Eric M." }, { "id": "https://authors.library.caltech.edu/records/r6h0m-sqs92", "eprint_id": 81771, "eprint_status": "archive", "datestamp": "2023-08-19 08:19:59", "lastmod": "2026-04-10 21:19:21", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "van-de-Bult-F-J", "name": { "family": "van de Bult", "given": "Fokko J." } }, { "id": "Rains-E-M", "name": { "family": "Rains", "given": "Eric M." }, "orcid": "0000-0002-9915-0919" } ] }, "title": "Limits of multivariate elliptic beta integrals and related bilinear forms", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 1110.1460.pdf

", "abstract": "In this article we consider the elliptic Selberg integral, which is a BC_n symmetric multivariate extension of the elliptic beta integral. We categorize the limits that are obtained as p \u2192 0, for given behavior of the parameters as p \u2192 0. This article is therefore the multivariate version of our earlier paper \"Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions\". The integrand of the elliptic Selberg integral is the measure for the BC_n symmetric biorthogonal functions introduced by the second author, so we also consider the limits of the associated bilinear form. We also provide the limits for the discrete version of this bilinear form, which is related to a multivariate extension of the Frenkel-Turaev summation.", "date": "2011-10-07", "date_type": "published", "id_number": "CaltechAUTHORS:20170922-144400917", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170922-144400917", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1110.1460", "primary_object": { "basename": "1110.1460.pdf", "url": "https://authors.library.caltech.edu/records/r6h0m-sqs92/files/1110.1460.pdf" }, "pub_year": "2011", "author_list": "van de Bult, Fokko J. and Rains, Eric M." }, { "id": "https://authors.library.caltech.edu/records/n4ywj-7m705", "eprint_id": 77099, "eprint_status": "archive", "datestamp": "2023-08-19 08:10:52", "lastmod": "2026-04-10 23:27:37", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Dyda-B", "name": { "family": "Dyda", "given": "Bartlomiej" } }, { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" } ] }, "title": "Fractional Hardy-Sobolev-Maz'ya inequality for domains", "ispublished": "unpub", "full_text_status": "public", "keywords": "fractional Hardy-Sobolev-Maz'ya inequality, fractional Hardy\ninequality", "note": "(Submitted on 29 Sep 2011) \n\nWork supported by the DFG through SFB-701 'Spectral Structures and Topological Methods in Mathematics' and by grant N N201 397137, MNiSW (B.D.) and by U.S. NSF grant PHY1068285 (R.L.F.)\n\n

Submitted - 1109.6570.pdf

", "abstract": "We prove a fractional version of the Hardy\u2013Sobolev\u2013Maz'ya inequality for arbitrary domains and Lp norms with p \u2265 2. This inequality combines the fractional Sobolev and the fractional Hardy inequality into a single inequality, while keeping the sharp constant in the Hardy inequality.", "date": "2011-09-29", "date_type": "published", "id_number": "CaltechAUTHORS:20170501-094456500", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170501-094456500", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Deutsche Forschungsgemeinschaft (DFG)", "grant_number": "SFB-701" }, { "agency": "Ministerstwo Nauki i Szkolnictwa Wy\u017cszego (MNiSW)", "grant_number": "N201 397137" }, { "agency": "NSF", "grant_number": "PHY-1068285" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1109.6570", "primary_object": { "basename": "1109.6570.pdf", "url": "https://authors.library.caltech.edu/records/n4ywj-7m705/files/1109.6570.pdf" }, "pub_year": "2011", "author_list": "Dyda, Bartlomiej and Frank, Rupert L." }, { "id": "https://authors.library.caltech.edu/records/dq1x2-78f66", "eprint_id": 77101, "eprint_status": "archive", "datestamp": "2023-08-19 07:40:58", "lastmod": "2026-04-10 21:26:32", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Cornean-H-D", "name": { "family": "Cornean", "given": "Horia D." } }, { "id": "Fournais-S", "name": { "family": "Fournais", "given": "S\u00f8ren" } }, { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Helffer-B", "name": { "family": "Helffer", "given": "Bernard" } } ] }, "title": "Sharp trace asymptotics for a class of 2D-magnetic operators", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 3 Aug 2011) \n\nH.C. acknowledges support from the Danish F.N.U. grant Mathematical Physics. S.F. was supported by the Lundbeck Foundation, the Danish Natural Science Research Council and by the European Research Council under the European Community's Seventh Framework Program (FP7/2007\u20132013)/ERC grant agreement 202859.\n\n

Submitted - 1108.0777.pdf

", "abstract": "In this paper we prove a two-term asymptotic formula for for the spectral counting function for a 2D magnetic Schr\u00f6dinger operator on a domain (with Dirichlet boundary conditions) in a semiclassical limit and with strong magnetic field. By scaling, this is equivalent to a thermodynamic limit of a 2D Fermi gas submitted to a constant external magnetic field. The original motivation comes from a paper by H. Kunz in which he studied, among other things, the boundary correction for the grand-canonical pressure and density of such a Fermi gas. Our main theorem yields a rigorous proof of the formulas announced by Kunz. Moreover, the same theorem provides several other results on the integrated density of states for operators of the type (\u2212ih\u2207\u2212\u03bcA)^2 in L^2(\u03a9) with Dirichlet boundary conditions.", "date": "2011-08-03", "date_type": "published", "id_number": "CaltechAUTHORS:20170501-100604320", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170501-100604320", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Lundbeck Foundation" }, { "agency": "Danish Natural Science Research Council" }, { "agency": "European Research Council (ERC)", "grant_number": "202859" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1108.0777", "primary_object": { "basename": "1108.0777.pdf", "url": "https://authors.library.caltech.edu/records/dq1x2-78f66/files/1108.0777.pdf" }, "pub_year": "2011", "author_list": "Cornean, Horia D.; Fournais, S\u00f8ren; et al." }, { "id": "https://authors.library.caltech.edu/records/saq03-48z82", "eprint_id": 66639, "eprint_status": "archive", "datestamp": "2023-08-19 07:09:26", "lastmod": "2026-03-09 02:36:52", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" }, { "id": "Willett-B", "name": { "family": "Willett", "given": "Brian" } } ] }, "title": "Generalized Superconformal Index for Three Dimensional Field Theories", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 13 Jun 2011). \n\nThis work was supported in part by the DOE grant DE-FG02-92ER40701.\n\n

Submitted - 1106.2484.pdf

", "abstract": "We introduce a generalization of the S^2 \u00d7 S^1 superconformal index where background gauge fields with magnetic flux are coupled to the global symmetries of the\ntheory. This allows one to gauge a global symmetry at the level of the index, which we use to show the matching of the superconformal index for N = 2 SQED with N_f flavors and\nits mirror dual.", "date": "2011-06-13", "date_type": "published", "id_number": "CaltechAUTHORS:20160504-095845547", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160504-095845547", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-92ER40701" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1106.2484", "primary_object": { "basename": "1106.2484.pdf", "url": "https://authors.library.caltech.edu/records/saq03-48z82/files/1106.2484.pdf" }, "pub_year": "2011", "author_list": "Kapustin, Anton and Willett, Brian" }, { "id": "https://authors.library.caltech.edu/records/6jyst-gsc52", "eprint_id": 77095, "eprint_status": "archive", "datestamp": "2023-08-19 07:05:33", "lastmod": "2026-04-10 22:23:20", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Lieb-E-H", "name": { "family": "Lieb", "given": "Elliott H." }, "orcid": "0000-0001-5843-3587" }, { "id": "Seiringer-R", "name": { "family": "Seiringer", "given": "Robert" } } ] }, "title": "Binding of Polarons and Atoms at Threshold", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2011 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\n(Submitted on 3 Jun 2011) \n\nWe are grateful to Herbert Spohn for making us aware of\nthis problem. Partial financial support from the U.S. National Science Foundation through grant PHY-0965859 (E.L.) and the NSERC (R.S.) is gratefully acknowledged.\n\n

Submitted - 1106.0729.pdf

", "abstract": "If the polaron coupling constant \u03b1 is large enough, bipolarons or multi-polarons will form. When passing through the critical \u03b1_c from above, does the radius of the system simply get arbitrarily large or does it reach a maximum and then explodes? We prove that it is always the latter. We also prove the analogous statement for the Pekar-Tomasevich (PT) approximation to the energy, in which case there is a solution to the PT equation at \u03b1_c. Similarly, we show that the same phenomenon occurs for atoms, e.g., helium, at the critical value of the nuclear charge. Our proofs rely only on energy estimates, not on a detailed analysis of the Schr\u00f6dinger equation, and are very general. They use the fact that the Coulomb repulsion decays like 1/r, while 'uncertainty principle' localization energies decay more rapidly, as 1/r^2.", "date": "2011-06-03", "date_type": "published", "id_number": "CaltechAUTHORS:20170501-091837793", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170501-091837793", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "PHY-0965859" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1106.0729", "primary_object": { "basename": "1106.0729.pdf", "url": "https://authors.library.caltech.edu/records/6jyst-gsc52/files/1106.0729.pdf" }, "pub_year": "2011", "author_list": "Frank, Rupert L.; Lieb, Elliott H.; et al." }, { "id": "https://authors.library.caltech.edu/records/5h0vx-dv976", "eprint_id": 77089, "eprint_status": "archive", "datestamp": "2023-08-19 06:28:19", "lastmod": "2026-04-10 21:45:05", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Lieb-E-H", "name": { "family": "Lieb", "given": "Elliott H." }, "orcid": "0000-0001-5843-3587" } ] }, "title": "Possible Lattice Distortions in the Hubbard Model for Graphene", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2011 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\n(Submitted on 3 May 2011) \n\nWe are indebted to G. Dunne, D. Abanin and A. Giuliani\nfor illuminating discussions and G. Holzegel for technical\nhelp. A grant from the U.S. National Science Foundation\nis acknowledged: PHY-0965859 (E.L.).\n\n

Submitted - 1105.0693.pdf

", "abstract": "The Hubbard model on the honeycomb lattice is a well known model for graphene. Equally well known is the Peierls type of instability of the lattice bond lengths. In the context of these two approximations we ask and answer the question of the possible lattice distortions for graphene in zero magnetic field. The answer is that in the thermodynamic limit only periodic, reflection-symmetric distortions are allowed and these have at most six atoms per unit cell as compared to two atoms for the undistorted lattice.", "date": "2011-05-03", "date_type": "published", "id_number": "CaltechAUTHORS:20170501-082647988", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170501-082647988", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "PHY-0965859" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1105.0693", "primary_object": { "basename": "1105.0693.pdf", "url": "https://authors.library.caltech.edu/records/5h0vx-dv976/files/1105.0693.pdf" }, "pub_year": "2011", "author_list": "Frank, Rupert L. and Lieb, Elliott H." }, { "id": "https://authors.library.caltech.edu/records/abm9v-aza09", "eprint_id": 66636, "eprint_status": "archive", "datestamp": "2023-08-19 06:10:02", "lastmod": "2026-03-09 02:38:37", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" } ] }, "title": "Seiberg-like duality in three dimensions for orthogonal gauge groups", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 4 Apr 2011). April 5, 2011. \n\nI would like to thank Denis Bashkirov for discussions and the Simons Center for Geometry and Physics for hospitality. This work was supported in part by the DOE grant DE-FG02-92ER40701.\n\n

Submitted - 1104.0466.pdf

", "abstract": "We propose a duality for N = 2 d = 3 Chern-Simons gauge theories with orthogonal\ngauge groups and matter in the vector representation. This duality generalizes level-\nrank duality for pure Chern-Simons gauge theories with orthogonal gauge groups and\nis reminiscent of Seiberg duality in four dimensions. We perform extensive checks by\ncomparing partition functions of theories related by dualities. We also determine the\nconformal dimensions of fields using Z-extremization.", "date": "2011-04-04", "date_type": "published", "id_number": "CaltechAUTHORS:20160504-094942955", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160504-094942955", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-92ER40701" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1104.0466", "primary_object": { "basename": "1104.0466.pdf", "url": "https://authors.library.caltech.edu/records/abm9v-aza09/files/1104.0466.pdf" }, "pub_year": "2011", "author_list": "Kapustin, Anton" }, { "id": "https://authors.library.caltech.edu/records/cazf0-pdg89", "eprint_id": 71911, "eprint_status": "archive", "datestamp": "2023-08-19 05:26:30", "lastmod": "2026-04-11 00:08:42", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kanoria-Y", "name": { "family": "Kanoria", "given": "Yashodhan" } }, { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "Omer" }, "orcid": "0000-0002-0111-0418" } ] }, "title": "Efficient Bayesian Social Learning on Trees", "ispublished": "unpub", "full_text_status": "public", "note": "Submitted on 7 Feb 2011. October 29, 2013. \n\nSupported by 3Com Corporation Stanford Graduate Fellowship. Supported by ISF grant 1300/08.\n\nWe would like to thank Andrea Montanari, Elchanan Mossel and Allan Sly for valuable discussions.\n\n

Submitted - 1102.1398.pdf

", "abstract": "We consider a set of agents who are attempting to\niteratively learn the 'state of the world' from their\nneighbors in a social network. Each agent initially\nreceives a noisy observation of the true state of the\nworld. The agents then repeatedly 'vote' and observe the votes of some of their peers, from which they gain more information. The agents' calculations are Bayesian and aim to myopically maximize the expected utility at each iteration. \nThis model, introduced by Gale and Kariv (2003),\nis a natural approach to learning on networks. However, it has been criticized, chiefly because the agents' decision rule appears to become computationally intractable as the number of iterations advances. For instance, a dynamic programming approach (part of this work) has running time that is exponentially large in min(n; (d - 1)^t), where n is the number of agents. \nWe provide a new algorithm to perform the agents' computations on locally tree-like graphs. Our algorithm uses the dynamic cavity method to drastically reduce computational effort. Let d be the maximum degree and t be the iteration number. The computational effort needed per agent is exponential only in O(td) (note that the number of possible information sets of a neighbor at time t is itself exponential in td).\nUnder appropriate assumptions on the rate of convergence, we deduce that each agent is only required to spend polylogarithmic (in 1=\u0454) computational effort to approximately learn the true state of the world with error probability \u0454, on regular trees of degree at least five. We provide numerical and other evidence to justify our assumption on convergence rate.\nWe extend our results in various directions, including loopy graphs. Our results indicate efficiency of iterative Bayesian social learning in a wide range of situations, contrary to widely held beliefs.", "date": "2011-02", "date_type": "published", "id_number": "CaltechAUTHORS:20161110-083842987", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161110-083842987", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Israel Science Foundation", "grant_number": "1300/08" }, { "agency": "3Com Corporation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1102.1398", "primary_object": { "basename": "1102.1398.pdf", "url": "https://authors.library.caltech.edu/records/cazf0-pdg89/files/1102.1398.pdf" }, "pub_year": "2011", "author_list": "Kanoria, Yashodhan and Tamuz, Omer" }, { "id": "https://authors.library.caltech.edu/records/67yzy-bm247", "eprint_id": 77810, "eprint_status": "archive", "datestamp": "2023-08-19 04:11:17", "lastmod": "2026-04-11 17:43:12", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Lenzmann-E", "name": { "family": "Lenzmann", "given": "Enno" } } ] }, "title": "On ground states for the L^2-critical boson star equation", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2010 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\n(Submitted on 14 Oct 2009 (v1), last revised 26 Oct 2010 (this version, v2) \n\nR. F. gratefully acknowledges support through DFG grant\nFR 2664/1-1 and NSF grant PHY 06 52854. E. L. is supported by a Steno fellowship of the Danish research council and NSF grant DMS-0702492.\n\n

Submitted - 0910.2721.pdf

", "abstract": "We consider ground state solutions u \u2a7e 0 for the L^2-critical boson star equation \u221a\u2212u \u2212 (|x|^(\u22121) \u2217 |u|^2)u = \u2212u in R^3. We prove analyticity and radial symmetry of u.\nIn a previous version of this paper, we also stated uniqueness and nondegeneracy of ground states for the L^2-critical boson star equation in R^3, but the arguments given there contained a gap. However, we refer to our recent preprint [FraLe] in arXiv:1009.4042, where we prove a general uniqueness and nondegeneracy result for ground states of nonlinear equations with fractional Laplacians in d = 1 space dimension.", "date": "2010-10-26", "date_type": "published", "id_number": "CaltechAUTHORS:20170526-101905288", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170526-101905288", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Deutsche Forschungsgemeinschaft (DFG)", "grant_number": "FR 2664/1-1" }, { "agency": "NSF", "grant_number": "PHY-0652854" }, { "agency": "Danish Research Council" }, { "agency": "NSF", "grant_number": "DMS-0702492" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0910.2721", "primary_object": { "basename": "0910.2721.pdf", "url": "https://authors.library.caltech.edu/records/67yzy-bm247/files/0910.2721.pdf" }, "pub_year": "2010", "author_list": "Frank, Rupert L. and Lenzmann, Enno" }, { "id": "https://authors.library.caltech.edu/records/sksr0-s9f09", "eprint_id": 66626, "eprint_status": "archive", "datestamp": "2023-08-19 03:54:22", "lastmod": "2026-03-09 02:38:10", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" }, { "id": "Setter-K", "name": { "family": "Setter", "given": "Kevin" } } ] }, "title": "Geometry of Topological Defects of Two-dimensional Sigma Models", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 29 Sep 2010) October 1, 2010. \n\nK.S. thanks Ketan Vyas for useful discussions. This work was supported in part by the DOE grant DE-FG02-92ER-40701.\n\n

Submitted - 1009.5999.pdf

", "abstract": "A topological defect separating a pair of two-dimensional CFTs is a codimension one interface along which all components of the stress-energy tensor glue continuously. We\nstudy topological defects of the bosonic, (0,1)- and (0,2)-supersymmetric sigma models in two dimensions. We find a geometric classification of such defects closely analogous to that of A-branes of symplectic manifolds, with the role of symplectic form played instead by a neutral signature metric. Alternatively, we find a compact description in terms of a generalized metric on the product of the targets. In the (0,1) case, we describe the target\nspace geometry of a bundle in which the fermions along the defect take values. In the (0,2) case, we describe the defects as being simultaneously A-branes and B-branes.", "date": "2010-09-29", "date_type": "published", "id_number": "CaltechAUTHORS:20160503-151908685", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160503-151908685", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-92ER-40701" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1009.5999", "primary_object": { "basename": "1009.5999.pdf", "url": "https://authors.library.caltech.edu/records/sksr0-s9f09/files/1009.5999.pdf" }, "pub_year": "2010", "author_list": "Kapustin, Anton and Setter, Kevin" }, { "id": "https://authors.library.caltech.edu/records/aedv2-pn049", "eprint_id": 79001, "eprint_status": "archive", "datestamp": "2023-08-19 03:36:43", "lastmod": "2026-04-11 19:27:15", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Bellissard-J-V", "name": { "family": "Bellissard", "given": "Jean V." } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" }, { "id": "Reihani-K", "name": { "family": "Reihani", "given": "Kamran" } } ] }, "title": "Dynamical Systems on Spectral Metric Spaces", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 26 Aug 2010) \n\nWork supported in part by NSF Grants No. 0901514, DMS-0651925 and DMS-1007207.\n\n

Submitted - 1008.4617.pdf

", "abstract": "Let (A,H,D) be a spectral triple, namely: A is a C^*-algebra, H is a Hilbert space on which A acts and D is a selfadjoint operator with compact resolvent such that the set of elements of A having a bounded commutator with D is dense. A spectral metric space, the noncommutative analog of a complete metric space, is a spectral triple (A,H,D) with additional properties which guaranty that the Connes metric induces the weak^*-topology on the state space of A. A ^*-automorphism respecting the metric defined a dynamical system. This article gives various answers to the question: is there a canonical spectral triple based upon the crossed product algebra Ax_\u03b1Z, characterizing the metric properties of the dynamical system ? If \u03b1 is the noncommutative analog of an isometry the answer is yes. Otherwise, the metric bundle construction of Connes and Moscovici is used to replace (A,\u03b1) by an equivalent dynamical system acting isometrically. The difficulties relating to the non compactness of this new system are discussed. Applications, in number theory, in coding theory are given at the end.", "date": "2010-08-26", "date_type": "published", "id_number": "CaltechAUTHORS:20170712-092526227", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-092526227", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0901514" }, { "agency": "NSF", "grant_number": "DMS-0651925" }, { "agency": "NSF", "grant_number": "DMS-1007207" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1008.4617", "primary_object": { "basename": "1008.4617.pdf", "url": "https://authors.library.caltech.edu/records/aedv2-pn049/files/1008.4617.pdf" }, "pub_year": "2010", "author_list": "Bellissard, Jean V.; Marcolli, Matilde; et al." }, { "id": "https://authors.library.caltech.edu/records/e1744-79007", "eprint_id": 79018, "eprint_status": "archive", "datestamp": "2023-08-19 01:59:46", "lastmod": "2026-04-11 23:12:45", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Noncommutative Geometry and Arithmetic", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 18 Mar 2010).\n\n

Submitted - 1003.3662.pdf

", "abstract": "This is an overview of recent results aimed at developing a geometry of noncommutative tori with real multiplication, with the purpose of providing a parallel, for real quadratic fields, of the classical theory of elliptic curves with complex multiplication for imaginary quadratic fields. This talk concentrates on two main aspects: the relation of Stark numbers to the geometry of noncommutative tori with real multiplication, and the shadows of modular forms on the noncommutative boundary of modular curves, that is, the moduli space of on commutative tori.", "date": "2010-03-18", "date_type": "published", "id_number": "CaltechAUTHORS:20170712-133251099", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-133251099", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1003.3662", "primary_object": { "basename": "1003.3662.pdf", "url": "https://authors.library.caltech.edu/records/e1744-79007/files/1003.3662.pdf" }, "pub_year": "2010", "author_list": "Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/1kac7-tv204", "eprint_id": 66623, "eprint_status": "archive", "datestamp": "2023-08-19 01:42:17", "lastmod": "2026-03-09 02:40:35", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" }, { "id": "Vyas-K", "name": { "family": "Vyas", "given": "Ketan" } } ] }, "title": "A-models in three and four dimensions", "ispublished": "unpub", "full_text_status": "public", "note": "Submitted on 23 Feb 2010. \n\nA.K. would like to thank Lev Rozansky for useful discussions. This work was supported in part by the DOE grant DE-FG02-92ER40701.\n\n

Submitted - 1002.4241.pdf

", "abstract": "We introduce and study a new 3d Topological Field Theory which can be associated to any compact real manifold X. This TFT is analogous to the 2d A-model and reduces\nto it upon compactification on an interval with suitable boundary conditions. It plays a role in 3d mirror symmetry as well as in the physical approach to the geometric Langlands duality. A similar TFT can be defined in four dimensions.", "date": "2010-02-23", "date_type": "published", "id_number": "CaltechAUTHORS:20160503-143950612", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160503-143950612", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-92ER40701" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1002.4241", "primary_object": { "basename": "1002.4241.pdf", "url": "https://authors.library.caltech.edu/records/1kac7-tv204/files/1002.4241.pdf" }, "pub_year": "2010", "author_list": "Kapustin, Anton and Vyas, Ketan" }, { "id": "https://authors.library.caltech.edu/records/qhfba-bb842", "eprint_id": 66624, "eprint_status": "archive", "datestamp": "2023-08-19 01:35:53", "lastmod": "2026-03-09 02:37:13", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" }, { "id": "Setter-K", "name": { "family": "Setter", "given": "Kevin" } }, { "id": "Vyas-K", "name": { "family": "Vyas", "given": "Ketan" } } ] }, "title": "Surface operators in four-dimensional topological gauge\n theory and Langlands duality", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 2 Feb 2010) \n\nA.K. would like to thank R. Bezrukavnikov, A. Braverman, D. Orlov, V. Lunts and especially L. Rozansky for useful discussions. K.S. acknowledges the support of the Jack Kent Cooke Foundation. This work was supported in part by the DOE grant DE-FG02-92ER40701.\n\n

Submitted - 1002.0385.pdf

", "abstract": "We study surface and line operators in the GL-twisted N = 4 gauge theory in four dimensions. Their properties depend on the parameter t which determines the BRST operator of theory. For t = i we propose a complete description of the 2-category of surface operators in terms of module categories. We also determine the monoidal category of line\noperators which includes Wilson lines as special objects. For t = 1 and t = 0 we only discuss surface and line operators in the abelian case. Applications to the categorification of the local geometric Langlands duality and its quantum version are briefly described. In the appendices we discuss several 3d and 2d topological field theories with gauge fields. In particular, we explain a relationship between the category of branes in the gauged\nB-model and the equivariant derived category of coherent sheaves.", "date": "2010-02-02", "date_type": "published", "id_number": "CaltechAUTHORS:20160503-145417819", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160503-145417819", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Jack Kent Cooke Foundation" }, { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-92ER40701" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.1002.0385", "primary_object": { "basename": "1002.0385.pdf", "url": "https://authors.library.caltech.edu/records/qhfba-bb842/files/1002.0385.pdf" }, "pub_year": "2010", "author_list": "Kapustin, Anton; Setter, Kevin; et al." }, { "id": "https://authors.library.caltech.edu/records/n6c21-pkf92", "eprint_id": 78996, "eprint_status": "archive", "datestamp": "2023-08-19 00:16:17", "lastmod": "2026-04-12 21:00:06", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Rej-A", "name": { "family": "Rej", "given": "Abhijnan" } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Motives: an introductory survey for physicists", "ispublished": "unpub", "full_text_status": "public", "keywords": "motives, K-theory, Grothendieck ring, algebraic cycles, motivic zeta functions, Feynman graphs, tensor categories, gauge theory, dualities, mirror symmetry", "note": "(Submitted on 23 Jul 2009 (v1), last revised 1 Oct 2009 (this version, v2). \n\nPart of this work was done while the author was employed by the Clay Mathematics Institute as a Junior Research Scholar.\n\n

Submitted - 0907.4046.pdf

", "abstract": "We survey certain accessible aspects of Grothendieck's theory of motives in arithmetic algebraic geometry for mathematical physicists, focussing on areas that have\nrecently found applications in quantum field theory. An appendix (by Matilde Marcolli) sketches further connections between motivic theory and theoretical physics.", "date": "2009-10-01", "date_type": "published", "id_number": "CaltechAUTHORS:20170712-090534368", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-090534368", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0907.4046", "primary_object": { "basename": "0907.4046.pdf", "url": "https://authors.library.caltech.edu/records/n6c21-pkf92/files/0907.4046.pdf" }, "pub_year": "2009", "author_list": "Rej, Abhijnan and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/5jvje-z1e26", "eprint_id": 77333, "eprint_status": "archive", "datestamp": "2023-08-19 00:06:22", "lastmod": "2026-04-12 05:19:18", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Lieb-E-H", "name": { "family": "Lieb", "given": "Elliott H." }, "orcid": "0000-0001-5843-3587" }, { "id": "Seiringer-R", "name": { "family": "Seiringer", "given": "Robert" } }, { "id": "Siedentop-H", "name": { "family": "Siedentop", "given": "Heinz" } } ] }, "title": "M\u00fceller's Exchange-Correlation Energy in Density-Matrix-Functional Theory", "ispublished": "unpub", "full_text_status": "public", "note": "Dated: September 28, 2009. (Submitted on 11 May 2007 (v1), last revised 29 Sep 2009 (this version, v3)). \n\nRupert Frank and Heinz Siedentop thank the Departments of Mathematics and Physics at Princeton University for hospitality while this work was done. The following partial support is gratefully acknowledged: The Swedish Foundation for International Cooperation in Research and Higher Education (STINT) (R.F.); U.S. National Science Foundation, grants PHY 01 39984 (E.H.L and H.S.) and PHY 03 53181 (R.S.); an A.P. Sloan Fellowship (R.S.); Deutsche Forschungsgemeinschaft, grant SI 348/13-1 (H.S.).\n\n

Submitted - 0705.1587.pdf

", "abstract": "The increasing interest in the M\u00fcller density-matrix-functional theory has led us to a systematic mathematical investigation of its properties. This functional is similar to the Hartree-Fock functional, but with a modified exchange term in which the square of the density matrix (x, x\u2032) is replaced by the square of y^(1/2)(x,x\u2032). After an extensive introductory discussion of densitymatrix-functional theory we show, among other things, that this functional is convex (unlike the HF functional) and that energy minimizing y's have unique densities p(r), which is a physically desirable property often absent in HF theory. We show that minimizers exist if N \u2264 Z, and derive various properties of the minimal energy and the corresponding minimizers. We also give a precise statement about the equation for the orbitals of y, which is more complex than for HF theory. We state some open mathematical questions about the theory together with conjectured solutions.", "date": "2009-09-20", "date_type": "published", "id_number": "CaltechAUTHORS:20170510-091442211", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170510-091442211", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Swedish Foundation for International Cooperation in Research and Higher Education (STINT)" }, { "agency": "NSF", "grant_number": "PHY-0139984" }, { "agency": "NSF", "grant_number": "PHY-0353181" }, { "agency": "Alfred P. Sloan Foundation" }, { "agency": "Deutsche Forschungsgemeinschaft (DFG)", "grant_number": "SI 348/13-1" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0705.1587", "primary_object": { "basename": "0705.1587.pdf", "url": "https://authors.library.caltech.edu/records/5jvje-z1e26/files/0705.1587.pdf" }, "pub_year": "2009", "author_list": "Frank, Rupert L.; Lieb, Elliott H.; et al." }, { "id": "https://authors.library.caltech.edu/records/3qgja-5pe21", "eprint_id": 77803, "eprint_status": "archive", "datestamp": "2023-08-20 01:44:40", "lastmod": "2026-04-12 05:12:46", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Lenzmann-E", "name": { "family": "Lenzmann", "given": "Enno" } } ] }, "title": "Uniqueness of ground states for the L^2-critical boson star equation", "ispublished": "unpub", "full_text_status": "public", "note": "Submitted on 19 May 2009. \n\nR.L. acknowledges support through DFG grant FR 2664/1-1, NSF grant PHY 06 52854; and E.L. is partially supported by NSF grant DMS-070249 and CTS at ETH Z\u00fcrich. We thank Joachim Krieger, Mathieu Lewin and Pierre Rapha\u00ebl for valuable discussions.\n\n

Submitted - 0905.3105.pdf

", "abstract": "We establish uniqueness of ground states u(x)\u22650 for the L^2-critical boson star equation \u221a\u2212\u0394u\u2212(|x|^(\u22121)\u2217|u|2)u=\u2212u in R^3. The proof blends variational arguments with the harmonic extension to the halfspace R^4_+. Apart from uniqueness, we also show radiality of ground states (up to translations) and the nondegeneracy of the linearization. Our results provide an indispensable basis for the blowup analysis of the time-dependent L^2-critical boson star equation. The uniqueness proof can be generalized to different fractional Laplacians (\u2212\u0394)^s and space dimensions.", "date": "2009-05-19", "date_type": "published", "id_number": "CaltechAUTHORS:20170526-093631237", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170526-093631237", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Deutsche Forschungsgemeinschaft (DFG)", "grant_number": "FR 2664/1-1" }, { "agency": "NSF", "grant_number": "PHY-0652854" }, { "agency": "NSF", "grant_number": "DMS-070249" }, { "agency": "ETH Zurich" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0905.3105", "primary_object": { "basename": "0905.3105.pdf", "url": "https://authors.library.caltech.edu/records/3qgja-5pe21/files/0905.3105.pdf" }, "pub_year": "2009", "author_list": "Frank, Rupert L. and Lenzmann, Enno" }, { "id": "https://authors.library.caltech.edu/records/k79gw-ywz71", "eprint_id": 77802, "eprint_status": "archive", "datestamp": "2023-08-20 01:07:21", "lastmod": "2026-03-09 00:47:32", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" } ] }, "title": "Remarks on eigenvalue estimates and semigroup domination", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2009 by the author. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\nSubmitted on 3 Mar 2009. \n\nSupport through DFG grant FR 2664/1-1 and U.S. National Science Foundation grant PHY 06 52854 is gratefully acknowledged.\n\n

Submitted - 0903.0561.pdf

", "abstract": "We present an overview over recent results concerning semi-classical spectral estimates for magnetic Schroedinger operators. We discuss how the constants in magnetic and non-magnetic eigenvalue bounds are related and we prove, in an abstract setting, that any non-magnetic Lieb-Thirring-type inequality implies a magnetic Lieb-Thirring-type inequality with possibly a larger constant.", "date": "2009-03-03", "date_type": "published", "id_number": "CaltechAUTHORS:20170526-092635990", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170526-092635990", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Deutsche Forschungsgemeinschaft (DFG)", "grant_number": "FR 2664/1-1" }, { "agency": "NSF", "grant_number": "PHY-0652854" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0903.0561", "primary_object": { "basename": "0903.0561.pdf", "url": "https://authors.library.caltech.edu/records/k79gw-ywz71/files/0903.0561.pdf" }, "pub_year": "2009", "author_list": "Frank, Rupert L." }, { "id": "https://authors.library.caltech.edu/records/d613w-v2711", "eprint_id": 77400, "eprint_status": "archive", "datestamp": "2023-08-19 23:18:29", "lastmod": "2026-04-12 20:44:34", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Siedentop-H", "name": { "family": "Siedentop", "given": "Heinz" } }, { "id": "Warzel-S", "name": { "family": "Warzel", "given": "Simone" } } ] }, "title": "The Energy of Heavy Atoms According to Brown and Ravenhall: The Scott Correction", "ispublished": "unpub", "full_text_status": "public", "keywords": "Heavy atoms, ground state energy, relativistic Coulomb system, Scott correction, Brown-Ravenhall operator.", "note": "Last revised 23 Jul 2008 (this version, v2) \n\nWe thank Elliott Lieb and Robert Seiringer for supportive\ndiscussions. H.S. thanks the Departments of Mathematics and Physics of Princeton University and R.F. and S.W. thank the Department of Mathematics of LMU Munich for hospitality while parts of this work were done. We also thank Volker Bach for stimulating questions resulting in several improvements. The work has been partially supported by the Deutscher Akademischer Austauschdienst, grant D/06/49117 (R.F.), the U.S. National Science Foundation, grant PHY 01 39984 (H.S.), the Deutsche Forschungsgemeinschaft, grant SI 348/13-1 (H.S.), and a Sloan Fellowship (S.W.).\n\n

Submitted - 0805.4441.pdf

", "abstract": "We consider relativistic many-particle operators which \u2013 according to Brown and Ravenhall \u2013 describe the electronic states of heavy atoms. Their ground state energy is investigated in the limit of large nuclear charge and velocity of light. We show that the leading quasi-classical behavior given by the Thomas-Fermi theory is raised by a subleading correction, the Scott correction. Our result is valid for the maximal range of coupling constants, including the critical one. As a technical tool, a Sobolev-Gagliardo-Nirenberg type inequality is established for the critical atomic Brown-Ravenhall operator. Moreover, we prove sharp upper and lower bound on the eigenvalues of the hydrogenic Brown-Ravenhall operator up to and including the critical coupling constant.", "date": "2008-07-23", "date_type": "published", "id_number": "CaltechAUTHORS:20170512-100456574", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170512-100456574", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Deutscher Akademischer Austauschdienst (DAAD)", "grant_number": "D/06/49117" }, { "agency": "NSF", "grant_number": "PHY-0139984" }, { "agency": "Deutsche Forschungsgemeinschaft (DFG)", "grant_number": "SI 348/13-1" }, { "agency": "Alfred P. Sloan Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0805.4441", "primary_object": { "basename": "0805.4441.pdf", "url": "https://authors.library.caltech.edu/records/d613w-v2711/files/0805.4441.pdf" }, "pub_year": "2008", "author_list": "Frank, Rupert L.; Siedentop, Heinz; et al." }, { "id": "https://authors.library.caltech.edu/records/4sv4q-5rb14", "eprint_id": 78997, "eprint_status": "archive", "datestamp": "2023-08-19 23:15:11", "lastmod": "2026-04-12 23:54:14", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aluffi-P", "name": { "family": "Aluffi", "given": "Paolo" } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Feynman motives of banana graphs", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 10 Jul 2008 (v1), last revised 16 Jul 2008 (this version, v2).\n\n

Submitted - 0807.1690.pdf

", "abstract": "We consider the infinite family of Feynman graphs known as the \"banana graphs\" and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern\u2013Schwartz\u2013MacPherson classes, using the classical Cremona transformation and the dual graph, and a blowup formula for characteristic classes. We outline the interesting similarities between these operations and we give formulae for cones obtained by simple operations on graphs. We formulate a positivity conjecture for characteristic classes of graph hypersurfaces and discuss briefly the effect of passing to noncommutative spacetime.", "date": "2008-07-16", "date_type": "published", "id_number": "CaltechAUTHORS:20170712-091134506", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-091134506", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0807.1690", "primary_object": { "basename": "0807.1690.pdf", "url": "https://authors.library.caltech.edu/records/4sv4q-5rb14/files/0807.1690.pdf" }, "pub_year": "2008", "author_list": "Aluffi, Paolo and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/pmp7m-c5w41", "eprint_id": 88816, "eprint_status": "archive", "datestamp": "2023-08-19 22:54:18", "lastmod": "2026-03-18 00:07:36", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Simon-B", "name": { "family": "Simon", "given": "Barry" }, "orcid": "0000-0003-2561-8539" } ] }, "title": "The Christoffel-Darboux Kernel", "ispublished": "unpub", "full_text_status": "public", "keywords": "Orthogonal polynomials, spectral theory", "note": "This work was supported in part by NSF grant DMS-0652919 and U.S.\u2013Israel Binational Science Foundation (BSF) Grant No. 2002068.\n\n

Submitted - 0806.1528.pdf

", "abstract": "A review of the uses of the CD kernel in the spectral theory of orthogonal polynomials, concentrating on recent results.", "date": "2008-06-09", "date_type": "published", "id_number": "CaltechAUTHORS:20180815-095937666", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180815-095937666", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0652919" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2002068" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" }, { "id": "Physics-Department" } ] }, "doi": "10.48550/arXiv.0806.1528", "primary_object": { "basename": "0806.1528.pdf", "url": "https://authors.library.caltech.edu/records/pmp7m-c5w41/files/0806.1528.pdf" }, "pub_year": "2008", "author_list": "Simon, Barry" }, { "id": "https://authors.library.caltech.edu/records/x836g-v1m96", "eprint_id": 72926, "eprint_status": "archive", "datestamp": "2023-08-19 21:40:43", "lastmod": "2026-04-13 14:37:38", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marsano-J", "name": { "family": "Marsano", "given": "Joseph" } }, { "id": "Ooguri-H", "name": { "family": "Ooguri", "given": "Hirosi" }, "orcid": "0000-0001-6021-3778" }, { "id": "Ookouchi-Yutaka", "name": { "family": "Ookouchi", "given": "Yutaka" } }, { "id": "Park-Chang-Soon", "name": { "family": "Park", "given": "Chang-Soon" } } ] }, "title": "Metastable Vacua in Perturbed Seiberg-Witten Theories, Part 2: Fayet-Iliopoulos Terms and K\u00e4hler Normal Coordinates", "ispublished": "unpub", "full_text_status": "public", "note": "We would like to thank S. Ferrara, S. Kachru, S. Ketov, M. Nitta, M. Shigemori, and S. Trivedi for discussions. This research is supported in part by DOE grant DE-FG03-92-ER40701. J.M. is also supported by a John A McCone postdoctoral fellowship. The research of H.O. is also\nsupported in part by the NSF grant OISE-0403366, by the 21st Century COE Program at the University of Tokyo, and by the Kavli Foundation. Y.O. is also supported in part by the JSPS Fellowship for Research Abroad. C.P. is also supported in part by Samsung Scholarship. H.O. thanks the Aspen Center for Physics for the hospitality at the initial stage of this work.\n\n

Submitted - 0712.3305.pdf

", "abstract": "We show that the perturbation of an N=2 supersymmetric gauge theory by a superpotential linear in the Kahler normal coordinates of the Coulomb branch, discussed in arXiv:0704.3613, is equivalent to the perturbation by Fayet-Iliopoulos terms. It follows that the would-be meta-stable vacuum at the origin of the normal coordinates in fact preserves N=1 supersymmetry unless the superpotential is truncated to a finite-degree polynomial of the adjoint scalar fields. We examine the criteria for supersymmetry breaking under a perturbation by Fayet-Iliopoulos terms and present a general classification of non-supersymmetric critical points. In some explicit examples, we are also able to study local stability of these points and demonstrate that, if the perturbation is chosen appropriately, they indeed correspond to supersymmetry-breaking vacua. Relations of these constructions to flux compactifications and geometric meta-stability are also discussed.", "date": "2007-12-20", "date_type": "published", "id_number": "CaltechAUTHORS:20161219-090406382", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161219-090406382", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG03-92-ER40701" }, { "agency": "John A. McCone Postdoctoral Fellowship" }, { "agency": "NSF", "grant_number": "OISE-0403366" }, { "agency": "University of Tokyo" }, { "agency": "Kavli Foundation" }, { "agency": "Japan Society for the Promotion of Science (JSPS)" }, { "agency": "Samsung Scholarship" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0712.3305", "primary_object": { "basename": "0712.3305.pdf", "url": "https://authors.library.caltech.edu/records/x836g-v1m96/files/0712.3305.pdf" }, "pub_year": "2007", "author_list": "Marsano, Joseph; Ooguri, Hirosi; et al." }, { "id": "https://authors.library.caltech.edu/records/s7d7k-cwd29", "eprint_id": 98015, "eprint_status": "archive", "datestamp": "2023-08-19 21:31:40", "lastmod": "2026-03-08 03:39:59", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Conlon-David", "name": { "family": "Conlon", "given": "David" }, "orcid": "0000-0001-5899-1829" } ] }, "title": "A note on lower bounds for hypergraph Ramsey numbers", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 0711.5004.pdf

", "abstract": "We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that \n\nr_(3)(l, l, l) \u2265 2^(l^(c log log l)).\n\nThe old bound, due to Erd\u0151s and Hajnal, was \n\nr_(3)(l, l, l) \u2265 2^(cl^(2) log^(2) l).", "date": "2007-11-30", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20190819-170832267", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170832267", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0711.5004", "primary_object": { "basename": "0711.5004.pdf", "url": "https://authors.library.caltech.edu/records/s7d7k-cwd29/files/0711.5004.pdf" }, "pub_year": "2007", "author_list": "Conlon, David" }, { "id": "https://authors.library.caltech.edu/records/t7f16-spd84", "eprint_id": 77399, "eprint_status": "archive", "datestamp": "2023-08-19 21:30:25", "lastmod": "2026-03-08 20:41:24", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Laptev-A", "name": { "family": "Laptev", "given": "Ari" } } ] }, "title": "Spectral inequalities for Schr\u00f6dinger operators with surface potentials", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2007 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\n(Submitted on 21 Nov 2007) \n\nThis work has been supported by DAAD grant D/06/49117 (R. F.).\n\n

Submitted - 0711.3473.pdf

", "abstract": "We prove sharp Lieb-Thirring inequalities for Schr\u00f6dinger operators with potentials supported on a hyperplane and we show how these estimates are related to Lieb-Thirring inequalities for relativistic Schr\u00f6dinger operators.", "date": "2007-11-21", "date_type": "published", "id_number": "CaltechAUTHORS:20170512-095723566", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170512-095723566", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Deutscher Akademischer Austauschdienst (DAAD)", "grant_number": "D/06/49117" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0711.3473", "primary_object": { "basename": "0711.3473.pdf", "url": "https://authors.library.caltech.edu/records/t7f16-spd84/files/0711.3473.pdf" }, "pub_year": "2007", "author_list": "Frank, Rupert L. and Laptev, Ari" }, { "id": "https://authors.library.caltech.edu/records/wy6jr-ghn11", "eprint_id": 77394, "eprint_status": "archive", "datestamp": "2023-08-19 21:29:26", "lastmod": "2026-03-18 00:06:48", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Simon-B", "name": { "family": "Simon", "given": "Barry" }, "orcid": "0000-0003-2561-8539" } ] }, "title": "Equilibrium measures and capacities in spectral theory", "ispublished": "unpub", "full_text_status": "public", "keywords": "Potential theory, spectral theory, regular orthogonal polynomials", "note": "August 23, 2007. \n\n(Submitted on 16 Nov 2007) \n\nSupported in part by NSF grants DMS-0140592 and DMS-0652919 and U.S.\u2013Israel Binational Science Foundation (BSF) Grant No.2002068.\n\n

Submitted - 0711.2700.pdf

", "abstract": "This is a comprehensive review of the uses of potential\ntheory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl\u2013Totik theory of regular measures, especially the case of OPRL and OPUC. Links are made to the study of ergodic Schr\u00a8odinger operators where one of our new results implies that, in complete generality, the spectral measure is supported on a set of zero Hausdorff dimension (indeed, of capacity zero) in the region of strictly positive Lyapunov exponent. There are many examples and some new conjectures and indications of new research directions. Included are appendices on potential theory and on Fekete\u2013Szeg\u0151 theory.", "date": "2007-11-16", "date_type": "published", "id_number": "CaltechAUTHORS:20170512-091544520", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170512-091544520", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0140592" }, { "agency": "NSF", "grant_number": "DMS-0652919" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2002068" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" }, { "id": "Physics-Department" } ] }, "doi": "10.48550/arXiv.0711.2700", "primary_object": { "basename": "0711.2700.pdf", "url": "https://authors.library.caltech.edu/records/wy6jr-ghn11/files/0711.2700.pdf" }, "pub_year": "2007", "author_list": "Simon, Barry" }, { "id": "https://authors.library.caltech.edu/records/9pjwb-3em17", "eprint_id": 78993, "eprint_status": "archive", "datestamp": "2023-08-19 21:28:10", "lastmod": "2026-04-13 15:06:06", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Solvmanifolds and noncommutative tori with real multiplication", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 13 Nov 2007)\n\n

Submitted - 0711.2036.pdf

", "abstract": "We prove that the Shimizu L-function of a real quadratic field is obtained from a (Lorentzian) spectral triple on a noncommutative torus with real multiplication, as an adiabatic limit of the Dirac operator on a 3-dimensional solvmanifold. The Dirac operator on this 3-dimensional geometry gives, via the Connes-Landi isospectral deformations, a spectral triple for the noncommutative tori obtained by deforming the fiber tori to noncommutative spaces. The 3-dimensional solvmanifold is the homotopy quotient in the sense of Baum--Connes of the noncommutative space obtained as the crossed product of the noncommutative torus by the action of the units of the real quadratic field. This noncommutative space is identified with the twisted group C*-algebra of the fundamental group of the 3-manifold. The twisting can be interpreted as the cocycle arising from a magnetic field, as in the theory of the quantum Hall effect. We prove a twisted index theorem that computes the range of the trace on the K-theory of this noncommutative space and gives an estimate on the gaps in the spectrum of the associated Harper operator.", "date": "2007-11-13", "date_type": "published", "id_number": "CaltechAUTHORS:20170712-090024525", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-090024525", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0711.2036", "primary_object": { "basename": "0711.2036.pdf", "url": "https://authors.library.caltech.edu/records/9pjwb-3em17/files/0711.2036.pdf" }, "pub_year": "2007", "author_list": "Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/89fkm-jg794", "eprint_id": 77393, "eprint_status": "archive", "datestamp": "2023-08-19 21:28:05", "lastmod": "2026-03-09 02:11:12", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Ekholm-T", "name": { "family": "Ekholm", "given": "Tomas" } }, { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Kova\u0159\u00edk-H", "name": { "family": "Kova\u0159\u00edk", "given": "Hynek" } } ] }, "title": "Remarks about Hardy inequalities on metric trees", "ispublished": "unpub", "full_text_status": "public", "keywords": "Laplace operator, metric tree, Hardy inequality", "note": "\u00a9 2007 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\n(Submitted on 13 Nov 2007) \n\nThe authors are grateful to Timo Weidl for several useful discussions, and to the organizers of the workshop 'Analysis on Graphs' at the Isaac Newton Institute in Cambridge for their kind invitation. This work has been supported by FCT grant SFRH/BPD/23820/2005 (T.E.) and DAAD grant D/06/49117 (R.F.). Partial support by the ESF programme SPECT (T.E. and H.K.) and the DAAD-STINT PPP programme (R.F.) is gratefully acknowledged.\n\n

Submitted - 0711.1943.pdf

", "abstract": "We find sharp conditions on the growth of a rooted regular metric tree such that the Neumann Laplacian on the tree satisfies a Hardy inequality. In particular, we consider homogeneous metric trees. Moreover, we show that a non-trivial Aharonov-Bohm magnetic field leads to a Hardy inequality on a loop graph.", "date": "2007-11-13", "date_type": "published", "publisher": "Caltech Library", "id_number": "CaltechAUTHORS:20170512-090412324", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170512-090412324", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia (FCT)", "grant_number": "SFRH/BPD/23820/2005" }, { "agency": "Deutscher Akademischer Austauschdienst (DAAD)", "grant_number": "D/06/49117" }, { "agency": "European Science Foundation" }, { "agency": "Swedish Foundation for International Cooperation in Research and Higher Education (STINT)" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "primary_object": { "basename": "0711.1943.pdf", "url": "https://authors.library.caltech.edu/records/89fkm-jg794/files/0711.1943.pdf" }, "pub_year": "2007", "author_list": "Ekholm, Tomas; Frank, Rupert L.; et al." }, { "id": "https://authors.library.caltech.edu/records/xd4kg-e6671", "eprint_id": 77326, "eprint_status": "archive", "datestamp": "2023-08-19 21:06:01", "lastmod": "2026-04-14 03:19:31", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Loss-M", "name": { "family": "Loss", "given": "Michael" } }, { "id": "Weidl-T", "name": { "family": "Weidl", "given": "Timo" } } ] }, "title": "P\u03cclya's conjecture in the presence of a constant magnetic field", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2007 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\n(Submitted on 4 Oct 2007) \n\nThis work had its gestation at the workshop 'Low eigenvalues of Laplace and Schr\u00f6dinger operators' which was held at AIM in May 2006. The support of AIM is gratefully acknowledged. This work has been partially supported by DAAD grant D/06/49117 (R. F.), NSF grant DMS 0600037 (M. L.) and DFG grant WE-1964/2-1 (T.W.), as well as by the DAAD-STINT PPP program (R. F. and T. W.).\n\n

Submitted - 0710.1078.pdf

", "abstract": "We consider the Dirichlet Laplacian with a constant magnetic field in a two-dimensional domain of finite measure. We determine the sharp constants in semi-classical eigenvalue estimates and show, in particular, that P\u03cclya's conjecture is not true in the presence of a magnetic field.", "date": "2007-10-04", "date_type": "published", "id_number": "CaltechAUTHORS:20170510-073002205", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170510-073002205", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Deutscher Akademischer Austauschdienst (DAAD)", "grant_number": "D/06/49117" }, { "agency": "NSF", "grant_number": "DMS 0600037" }, { "agency": "Deutsche Forschungsgemeinschaft (DFG)", "grant_number": "WE-1964/2-1" }, { "agency": "Swedish Foundation for International Cooperation in Research and Higher Education (STINT)" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0710.1078", "primary_object": { "basename": "0710.1078.pdf", "url": "https://authors.library.caltech.edu/records/xd4kg-e6671/files/0710.1078.pdf" }, "pub_year": "2007", "author_list": "Frank, Rupert L.; Loss, Michael; et al." }, { "id": "https://authors.library.caltech.edu/records/y49sy-e7r68", "eprint_id": 77332, "eprint_status": "archive", "datestamp": "2023-08-19 21:06:06", "lastmod": "2026-03-09 02:13:21", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Hansson-A", "name": { "family": "Hansson", "given": "Anders" } } ] }, "title": "Eigenvalue estimates for the Aharonov-Bohm operator in a domain", "ispublished": "unpub", "full_text_status": "public", "note": "Received by the editors February 5, 2007. (Submitted on 4 Oct 2007) \n\nThe authors would like to thank A. Laptev for the setting of the problem and helpful remarks. The first author is grateful to E. H. Lieb and R. Seiringer for their hospitality at Princeton University and thanks them, H. Kalf and M. Loss for fruitful discussions.\n\n

Submitted - 0710.1089.pdf

", "abstract": "We prove semi-classical estimates on moments of eigenvalues of the Aharonov-Bohm operator in bounded two-dimensional domains. Moreover, we present a counterexample to the generalized diamagnetic inequality which was proposed by Erdos, Loss and Vougalter. Numerical studies complement these results.", "date": "2007-10-04", "date_type": "published", "id_number": "CaltechAUTHORS:20170510-090747676", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170510-090747676", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0710.1089", "primary_object": { "basename": "0710.1089.pdf", "url": "https://authors.library.caltech.edu/records/y49sy-e7r68/files/0710.1089.pdf" }, "pub_year": "2007", "author_list": "Frank, Rupert L. and Hansson, Anders" }, { "id": "https://authors.library.caltech.edu/records/rb2w2-4e432", "eprint_id": 77362, "eprint_status": "archive", "datestamp": "2023-08-19 20:14:09", "lastmod": "2026-04-13 23:16:25", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Benguria-R-D", "name": { "family": "Benguria", "given": "Rafael D." } }, { "id": "Frank-R-L", "name": { "family": "Frank", "given": "Rupert L." }, "orcid": "0000-0001-7973-4688" }, { "id": "Loss-M", "name": { "family": "Loss", "given": "Michael" } } ] }, "title": "The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2007 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \n\nMay 7, 2007 \n\n(Submitted on 25 May 2007) \n\nWork partially supported by Fondecyt (CHILE) projects 106\u20130651 and 706\u20130200, and CONICYT/PBCT Proyecto Anillo de Investigaci\u03ccn en Ciencia y Tecnolog\u00eda ACT30/2006. Work partially supported by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT). Work partially supported by NSF-grant DMS-0600037.\n\n

Submitted - 0705.3833.pdf

", "abstract": "It is shown that the sharp constant in the Hardy-Sobolev-Maz'ya inequality on the upper half space H^3 \u2282 R^3 is given by the Sobolev constant. This is achieved by a duality argument relating the problem to a Hardy-Littlewood-Sobolev type inequality whose sharp constant is determined as well.", "date": "2007-05-25", "date_type": "published", "id_number": "CaltechAUTHORS:20170511-064913855", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170511-064913855", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Fondo Nacional de Desarrollo Cient\u00edfico y Tecnol\u00f3gico (FONDECYT) Chile", "grant_number": "106\u20130651" }, { "agency": "Fondo Nacional de Desarrollo Cient\u00edfico y Tecnol\u00f3gico (FONDECYT) Chile", "grant_number": "706\u20130200" }, { "agency": "Comisi\u00f3n Nacional de Investigaci\u00f3n Cient\u00edfica y Tecnol\u00f3gica (CONICYT)", "grant_number": "ACT30/2006" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0705.3833", "primary_object": { "basename": "0705.3833.pdf", "url": "https://authors.library.caltech.edu/records/rb2w2-4e432/files/0705.3833.pdf" }, "pub_year": "2007", "author_list": "Benguria, Rafael D.; Frank, Rupert L.; et al." }, { "id": "https://authors.library.caltech.edu/records/e707h-hty72", "eprint_id": 72021, "eprint_status": "archive", "datestamp": "2023-08-19 19:05:24", "lastmod": "2026-03-09 23:58:02", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Mazeh-T", "name": { "family": "Mazeh", "given": "T." } }, { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "O." }, "orcid": "0000-0002-0111-0418" }, { "id": "Zucker-S", "name": { "family": "Zucker", "given": "S." } } ] }, "title": "The Sys-Rem Detrending Algorithm: Implementation and Testing", "ispublished": "unpub", "full_text_status": "public", "note": "We wish to thank A. Udalski for letting us use the OGLE data. This work was supported by the German-Israeli Foundation for Scientific Research and Development.\n\n

Submitted - 0612418.pdf

", "abstract": "Sys-Rem (Tamuz, Mazeh & Zucker 2005) is a detrending algorithm designed to remove systematic effects in a large set of lightcurves obtained by a photometric survey. The algorithm works without any prior knowledge of the effects, as long as they appear in many stars of the sample. This paper presents the basic principles of Sys-Rem and discusses a parameterization used to determine the number of effects removed. We assess the performance of Sys-Rem on simulated transits injected into WHAT survey data. This test is proposed as a general scheme to assess the effectiveness of detrending algorithms. Application of Sys-Rem to the OGLE dataset demonstrates the power of the algorithm. We offer a coded implementation of Sys-Rem to the community.", "date": "2006-12-15", "date_type": "published", "id_number": "CaltechAUTHORS:20161115-090420610", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161115-090420610", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "German-Israeli Foundation for Scientific Research and Development" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0612418", "primary_object": { "basename": "0612418.pdf", "url": "https://authors.library.caltech.edu/records/e707h-hty72/files/0612418.pdf" }, "pub_year": "2006", "author_list": "Mazeh, T.; Tamuz, O.; et al." }, { "id": "https://authors.library.caltech.edu/records/rzdn9-tcr40", "eprint_id": 66983, "eprint_status": "archive", "datestamp": "2023-08-19 19:04:32", "lastmod": "2026-03-09 02:38:26", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" } ] }, "title": "Holomorphic reduction of N = 2 gauge theories, Wilson-'t Hooft operators, and S-duality", "ispublished": "unpub", "full_text_status": "public", "note": "I would like to thank D. Arinkin, R. Bezrukavnikov, I. Mirkovic, and N. Hitchin for explanations and discussions, and S. Gukov, N. Hitchin, and N. Saulina for comments on the preliminary draft of the paper. I am also grateful to Ketan Vyas for performing some computations related to the discussion of OPE of Wilson and 't Hooft operators in section 4.2. This work was supported in part by the DOE grant DE-FG03-92-ER40701.\n\n

Submitted - 0612119.pdf

", "abstract": "We study twisted N=2 superconformal gauge theory on a product of two Riemann surfaces Sigma and C. The twisted theory is topological along C and holomorphic along Sigma and does not depend on the gauge coupling or theta-angle. Upon Kaluza-Klein reduction along Sigma, it becomes equivalent to a topological B-model on C whose target is the moduli space MV of nonabelian vortex equations on Sigma. The N=2 S-duality conjecture implies that the duality group acts by autoequivalences on the derived category of MV. This statement can be regarded as an N=2 counterpart of the geometric Langlands duality. We show that the twisted theory admits Wilson-'t Hooft loop operators labelled by both electric and magnetic weights. Correlators of these loop operators depend holomorphically on coordinates and are independent of the gauge coupling. Thus the twisted theory provides a convenient framework for studying the Operator Product Expansion of general Wilson-'t Hooft loop operators.", "date": "2006-12-12", "date_type": "published", "id_number": "CaltechAUTHORS:20160511-100705968", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160511-100705968", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG03-92-ER40701" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0612119", "primary_object": { "basename": "0612119.pdf", "url": "https://authors.library.caltech.edu/records/rzdn9-tcr40/files/0612119.pdf" }, "pub_year": "2006", "author_list": "Kapustin, Anton" }, { "id": "https://authors.library.caltech.edu/records/2sd25-3ax57", "eprint_id": 72020, "eprint_status": "archive", "datestamp": "2023-08-19 18:53:37", "lastmod": "2026-03-09 23:58:48", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Mazeh-T", "name": { "family": "Mazeh", "given": "Tsevi" } }, { "id": "Tamuz-O", "name": { "family": "Tamuz", "given": "Omer" }, "orcid": "0000-0002-0111-0418" }, { "id": "North-P", "name": { "family": "North", "given": "Pierre" } } ] }, "title": "Analysis of the eclipsing binaries in the LMC discovered by OGLE: period distribution and frequency of the short-period binaries", "ispublished": "unpub", "full_text_status": "public", "keywords": "methods: data analysis, binaries: eclipsing, Magellanic Clouds", "note": "We are grateful to the OGLE team, and to L. Wyrzykowski in particular, for the photometric data set and for the eclipsing binary analysis. This work was supported by the Israeli Science Foundation through grant no. 03/233.\n\n

Submitted - 0611482.pdf

", "abstract": "We review the results of our analysis of the OGLE LMC eclipsing binaries (Mazeh, Tamuz & North 2006), using EBAS -- Eclipsing Binary Automated Solver, an automated algorithm to fit lightcurves of eclipsing binaries (Tamuz, Mazeh & North 2006). \nAfter being corrected for observational selection effects, the set of detected eclipsing binaries yielded the period distribution and the frequency of all LMC short-period binaries, and not just the eclipsing systems. \nSomewhat surprisingly, the period distribution is consistent with a flat distribution in log P between 2 and 10 days. The total number of binaries with periods shorter than 10 days in the LMC was estimated to be about 5000. This figure led us to suggest that (0.7 \u00b1 0.4)% of the main-sequence A- and B-type stars are found in binaries with periods shorter than 10 days. This frequency is substantially smaller than the fraction of binaries found by small Galactic radial-velocity surveys of B stars.", "date": "2006-11-15", "date_type": "published", "id_number": "CaltechAUTHORS:20161115-085954023", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161115-085954023", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Israel Science Foundation", "grant_number": "03/233" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0611482", "primary_object": { "basename": "0611482.pdf", "url": "https://authors.library.caltech.edu/records/2sd25-3ax57/files/0611482.pdf" }, "pub_year": "2006", "author_list": "Mazeh, Tsevi; Tamuz, Omer; et al." }, { "id": "https://authors.library.caltech.edu/records/m344c-z4214", "eprint_id": 82255, "eprint_status": "archive", "datestamp": "2023-08-19 18:01:05", "lastmod": "2026-03-09 23:03:34", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Rains-E-M", "name": { "family": "Rains", "given": "Eric M." }, "orcid": "0000-0002-9915-0919" }, { "id": "Vazirani-Monica-J", "name": { "family": "Vazirani", "given": "Monica J." } } ] }, "title": "Quadratic transformations of Macdonald and Koornwinder polynomials", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 0606204.pdf

", "abstract": "When one expands a Schur function in terms of the irreducible characters of the symplectic (or orthogonal) group, the coefficient of the trivial character is 0 unless the indexing partition has an appropriate form. A number of q-analogues of this fact were conjectured in [8]; the present paper proves most of those conjectures, as well as some new identities suggested by the proof technique. The proof involves showing that a nonsymmetric version of the relevant integral is annihilated by a suitable ideal of the affine Hecke algebra, and that any such annihilated functional satisfies the desired vanishing property. This does not, however, give rise to vanishing identities for the standard nonsymmetric Macdonald and Koornwinder polynomials; we discuss the required modification to these polynomials to support such results.", "date": "2006-06-09", "date_type": "published", "id_number": "CaltechAUTHORS:20171010-113111196", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20171010-113111196", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0606204", "primary_object": { "basename": "0606204.pdf", "url": "https://authors.library.caltech.edu/records/m344c-z4214/files/0606204.pdf" }, "pub_year": "2006", "author_list": "Rains, Eric M. and Vazirani, Monica J." }, { "id": "https://authors.library.caltech.edu/records/y0xg2-pe539", "eprint_id": 66980, "eprint_status": "archive", "datestamp": "2023-08-19 17:43:06", "lastmod": "2026-03-09 02:41:56", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" }, { "id": "Witten-E", "name": { "family": "Witten", "given": "Edward" }, "orcid": "0000-0002-7752-6073" } ] }, "title": "Electric-Magnetic Duality And The Geometric Langlands Program", "ispublished": "unpub", "full_text_status": "public", "note": "A.K. would like to thank D. Arinkin, R. Bezrukavnikov, and D. Orlov for useful conversations. In particular, Orlov's explanations in 2002 about the abelian case of the geometric Langlands program partially motivated the paper [41], which will enter our story in section 11. \n\nE.W. would like to thank the many mathematicians who over the years have explained matters relevant to the Langlands program, including A. Beilinson, P. Deligne, V. Drinfeld, and K. Vilonen, and especially M. F. Atiyah, D. Ben-Zvi, R. Donagi, E. Frenkel, and D. Kazhdan, and most recently M. Goresky and R. MacPherson. In addition, among others, T. Hausel, N. Hitchin, M. Hopkins, P. Kronheimer, L. Jeffrey, J. Morgan, G. Moore, D. Morrison, N. Nekrasov, M. Thaddeus, C. Vafa, and E. J. Weinberg clarified some points relevant to the present paper, and many of the physicists at the IAS, including S. Hellerman, K. Intriligator, J. Maldacena, N. Seiberg, and J. Walcher, made helpful comments.\n\n

Submitted - 0604151.pdf

", "abstract": "The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and 't Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke eigensheaves and D-modules, arise naturally from the physics.", "date": "2006-04-21", "date_type": "published", "id_number": "CaltechAUTHORS:20160511-093906187", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160511-093906187", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0604151", "primary_object": { "basename": "0604151.pdf", "url": "https://authors.library.caltech.edu/records/y0xg2-pe539/files/0604151.pdf" }, "pub_year": "2006", "author_list": "Kapustin, Anton and Witten, Edward" }, { "id": "https://authors.library.caltech.edu/records/wzfhb-ngd47", "eprint_id": 79063, "eprint_status": "archive", "datestamp": "2023-08-19 17:41:03", "lastmod": "2026-03-09 21:38:39", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Cornelissen-G", "name": { "family": "Cornelissen", "given": "Gunther" } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" }, { "id": "Reihani-K", "name": { "family": "Reihani", "given": "Kamran" } }, { "id": "Vdovina-A", "name": { "family": "Vdovina", "given": "Alina" } } ] }, "title": "Noncommutative geometry on trees and buildings", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 0604114.pdf

", "abstract": "We describe the construction of theta summable and finitely summable spectral triples associated to Mumford curves and some classes of higher dimensional buildings. The finitely summable case is constructed by considering the stabilization of the algebra of the dual graph of the special fiber of the Mumford curve and a variant of the Antonescu-Christensen spectral geometries for AF algebras. The information on the Schottky uniformization is encoded in the spectral geometry through the Patterson-Sullivan measure on the limit set. Some higher rank cases are obtained by adapting the construction for trees.", "date": "2006-04-05", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-083009140", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-083009140", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0604114", "primary_object": { "basename": "0604114.pdf", "url": "https://authors.library.caltech.edu/records/wzfhb-ngd47/files/0604114.pdf" }, "pub_year": "2006", "author_list": "Cornelissen, Gunther; Marcolli, Matilde; et al." }, { "id": "https://authors.library.caltech.edu/records/jmgdc-tn948", "eprint_id": 79060, "eprint_status": "archive", "datestamp": "2023-08-19 17:12:36", "lastmod": "2026-03-09 21:42:52", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Connes-A", "name": { "family": "Connes", "given": "Alain" } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "A walk in the noncommutative garden", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 0601054.pdf

", "abstract": "This text is written for the volume of the school/conference \"Noncommutative Geometry 2005\" held at IPM Tehran. It gives a survey of methods and results in noncommutative geometry, based on a discussion of significant examples of noncommutative spaces in geometry, number theory, and physics. The paper also contains an outline (the \"Tehran program\") of ongoing joint work with Consani on the noncommutative geometry of the adeles class space and its relation to number theoretic questions.", "date": "2006-01-03", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-081431578", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-081431578", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0601054", "primary_object": { "basename": "0601054.pdf", "url": "https://authors.library.caltech.edu/records/jmgdc-tn948/files/0601054.pdf" }, "pub_year": "2006", "author_list": "Connes, Alain and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/tffs2-npj53", "eprint_id": 66979, "eprint_status": "archive", "datestamp": "2023-08-19 16:52:20", "lastmod": "2026-03-09 02:20:05", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Gukov-S", "name": { "family": "Gukov", "given": "Sergei" }, "orcid": "0000-0002-9486-1762" }, { "id": "Walcher-J", "name": { "family": "Walcher", "given": "Johannes" } } ] }, "title": "Matrix Factorizations and Kauffman Homology", "ispublished": "unpub", "full_text_status": "public", "note": "We would like to thank M. Khovanov, M. Mari\u00f1o, C. Vafa, and E. Witten for valuable discussions. We are grateful to the KITP, Santa Barbara for warm hospitality during the program \"Mathematical Structures in String Theory\", where part of this work was carried out. This work was conducted during the period S.G. served as a Clay Mathematics Institute Long-Term Prize Fellow. This work was also supported in part by the DOE under grant number DE-FG02-90ER40542, in part by RFBR grant 04-02-16880 and in part by the NSF under Grant No. PHY99-07949.\n\n

Submitted - 0512298.pdf

", "abstract": "The topological string interpretation of homological knot invariants has led to several insights into the structure of the theory in the case of sl(N). We study possible extensions of the matrix factorization approach to knot homology for other Lie groups and representations. In particular, we introduce a new triply graded theory categorifying the Kauffman polynomial, test it, and predict the Kauffman homology for several simple knots.", "date": "2005-12-22", "date_type": "published", "id_number": "CaltechAUTHORS:20160511-092422859", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160511-092422859", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-90ER40542" }, { "agency": "Russian Foundation for Basic Research", "grant_number": "04-02-16880" }, { "agency": "NSF", "grant_number": "PHY99-07949" }, { "agency": "Clay Mathematics Institute" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0512298", "primary_object": { "basename": "0512298.pdf", "url": "https://authors.library.caltech.edu/records/tffs2-npj53/files/0512298.pdf" }, "pub_year": "2005", "author_list": "Gukov, Sergei and Walcher, Johannes" }, { "id": "https://authors.library.caltech.edu/records/49pds-np505", "eprint_id": 82226, "eprint_status": "archive", "datestamp": "2023-08-19 15:38:30", "lastmod": "2026-03-09 22:11:04", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Rains-E-M", "name": { "family": "Rains", "given": "Eric M." }, "orcid": "0000-0002-9915-0919" } ] }, "title": "Recurrences for elliptic hypergeometric integrals", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 0504285.pdf

", "abstract": "In recent work (math.QA/0309252) on multivariate hypergeometric integrals, the author generalized a conjectural integral formula of van Diejen and Spiridonov to a ten parameter integral provably invariant under an action of the Weyl group E_7. In the present note, we consider the action of the affine Weyl group, or more precisely, the recurrences satisfied by special cases of the integral. These are of two flavors: linear recurrences that hold only up to dimension 6, and three families of bilinear recurrences that hold in arbitrary dimension, subject to a condition on the parameters. As a corollary, we find that a codimension one special case of the integral is a tau function for the elliptic Painlev\u00e9 equation.", "date": "2005-04-13", "date_type": "published", "id_number": "CaltechAUTHORS:20171009-142453684", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20171009-142453684", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0504285", "primary_object": { "basename": "0504285.pdf", "url": "https://authors.library.caltech.edu/records/49pds-np505/files/0504285.pdf" }, "pub_year": "2005", "author_list": "Rains, Eric M." }, { "id": "https://authors.library.caltech.edu/records/xfx4x-vvn29", "eprint_id": 66680, "eprint_status": "archive", "datestamp": "2023-08-19 15:38:01", "lastmod": "2026-03-09 02:43:29", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" } ] }, "title": "Chiral de Rham complex and the half-twisted sigma-model", "ispublished": "unpub", "full_text_status": "public", "note": "A.K. would like to thank Fyodor Malikov and Vassily Gorbounov for discussions.\nThis work was supported in part by the DOE grant DE-FG03-92-ER40701.\n\n

Submitted - 0504074.pdf

", "abstract": "On any Calabi-Yau manifold X one can define a certain sheaf of chiral N=2 superconformal field theories, known as the chiral de Rham complex of X. It depends only on the complex structure of X, and its local structure is described by a simple free field theory. We show that the cohomology of this sheaf can be identified with the infinite-volume limit of the half-twisted sigma-model defined by E. Witten more than a decade ago. We also show that the correlators of the half-twisted model are independent of the Kahler moduli to all orders in worldsheet perturbation theory, and that the relation to the chiral de Rham complex can be violated only by worldsheet instantons.", "date": "2005-04-08", "date_type": "published", "id_number": "CaltechAUTHORS:20160505-102220527", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160505-102220527", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG03-92-ER40701" } ] }, "other_numbering_system": { "items": [ { "id": "68-2547", "name": "CALT" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0504074", "primary_object": { "basename": "0504074.pdf", "url": "https://authors.library.caltech.edu/records/xfx4x-vvn29/files/0504074.pdf" }, "pub_year": "2005", "author_list": "Kapustin, Anton" }, { "id": "https://authors.library.caltech.edu/records/gz6cy-s6094", "eprint_id": 99130, "eprint_status": "archive", "datestamp": "2023-08-19 15:23:29", "lastmod": "2026-03-18 04:19:45", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Cohen-A-M", "name": { "family": "Cohen", "given": "Arjeh M." } }, { "id": "Gijsbers-D-A-H", "name": { "family": "Gijsbers", "given": "Di\u00e9 A. H." } }, { "id": "Wales-D-B", "name": { "family": "Wales", "given": "David B." } } ] }, "title": "A Poset Connected to Artin Monoids of Simply Laced Type", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 0502501.pdf

", "abstract": "Let W be a Weyl group whose type is a simply laced Dynkin diagram. On several W-orbits of sets of mutually commuting reflections, a poset is described which plays a role in linear representatons of the corresponding Artin group A. The poset generalizes many properties of the usual order on positive roots of W given by height. In this paper, a linear representation of the positive monoid of A is defined by use of the poset.", "date": "2005-02-24", "date_type": "published", "id_number": "CaltechAUTHORS:20191007-160434367", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191007-160434367", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" }, { "id": "Physics-Department" } ] }, "doi": "10.48550/arXiv.0502501", "primary_object": { "basename": "0502501.pdf", "url": "https://authors.library.caltech.edu/records/gz6cy-s6094/files/0502501.pdf" }, "pub_year": "2005", "author_list": "Cohen, Arjeh M.; Gijsbers, Di\u00e9 A. H.; et al." }, { "id": "https://authors.library.caltech.edu/records/3zgdb-eha56", "eprint_id": 72942, "eprint_status": "archive", "datestamp": "2023-08-19 15:23:24", "lastmod": "2026-03-09 22:05:26", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Ooguri-H", "name": { "family": "Ooguri", "given": "Hirosi" }, "orcid": "0000-0001-6021-3778" }, { "id": "Vafa-C", "name": { "family": "Vafa", "given": "Cumrun" } }, { "id": "Verlinde-E", "name": { "family": "Verlinde", "given": "Erik" } } ] }, "title": "Hartle-Hawking Wave-Function for Flux Compactifications", "ispublished": "unpub", "full_text_status": "public", "note": "We would like to thank F. Denef, R. Dijkgraaf, G. Gibbons, R. Gopakumar, S. Gukov, G. Mandal, S. Minwalla, L. Motl, A. Neitzke, J. Preskill, A. Strominger, and L. Susskind for useful discussions. The research of H.O. was supported in part by DOE grant DE-FG03-92-ER40701. The research of C.V. was supported in part by NSF grants PHY-0244821 and DMS-0244464.\n\n

Submitted - 0502211.pdf

", "abstract": "We argue that the topological string partition function, which has been known to correspond to a wave-function, can be interpreted as an exact ``wave-function of the universe'' in the mini-superspace sector of physical superstring theory. This realizes the idea of Hartle and Hawking in the context of string theory, including all loop quantum corrections. The mini-superspace approximation is justified as an exact description of BPS quantities. Moreover this proposal leads to a conceptual explanation of the recent observation that the black hole entropy is the square of the topological string wave-function. This wave-function can be interpreted in the context of flux compactification of all spatial dimensions as providing a physical probability distribution on the moduli space of string compactification. Euclidean time is realized holographically in this setup.", "date": "2005-02-23", "date_type": "published", "id_number": "CaltechAUTHORS:20161219-113555787", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161219-113555787", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG03-92-ER40701" }, { "agency": "NSF", "grant_number": "PHY-0244821" }, { "agency": "NSF", "grant_number": "DMS-0244464" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0502211v2", "primary_object": { "basename": "0502211.pdf", "url": "https://authors.library.caltech.edu/records/3zgdb-eha56/files/0502211.pdf" }, "pub_year": "2005", "author_list": "Ooguri, Hirosi; Vafa, Cumrun; et al." }, { "id": "https://authors.library.caltech.edu/records/jayzb-mpc59", "eprint_id": 66970, "eprint_status": "archive", "datestamp": "2023-08-19 15:23:19", "lastmod": "2026-03-09 02:37:55", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" } ] }, "title": "A-branes and Noncommutative Geometry", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 23 Feb 2005) February 1, 2008. \n\nI would like to thank Dima Orlov, Oren Ben-Bassat, Jonathan Block, Tony Pantev, and Marco Gualtieri for helpful discussions. I am also grateful to the organizers of the Workshop on Mirror Symmetry at the University of Miami for providing a stimulating atmosphere. This work was supported in part by the DOE grant DE-FG03-92-ER40701.\n\n

Submitted - 0502212.pdf

", "abstract": "We argue that for a certain class of symplectic manifolds the category\nof A-branes (which includes the Fukaya category as a full subcategory)\nis equivalent to a noncommutative deformation of the category\nof B-branes (which is equivalent to the derived category of coherent\nsheaves) on the same manifold. This equivalence is different from Mirror\nSymmetry and arises from the Seiberg-Witten transform which\nrelates gauge theories on commutative and noncommutative spaces.\nMore generally, we argue that for certain generalized complex manifolds\nthe category of generalized complex branes is equivalent to a noncommutative\ndeformation of the derived category of coherent sheaves\non the same manifold. We perform a simple test of our proposal in\nthe case when the manifold in question is a symplectic torus.", "date": "2005-02-23", "date_type": "published", "id_number": "CaltechAUTHORS:20160511-075947917", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160511-075947917", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG03-92-ER40701" } ] }, "other_numbering_system": { "items": [ { "id": "68-2544", "name": "CALT" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0502212", "primary_object": { "basename": "0502212.pdf", "url": "https://authors.library.caltech.edu/records/jayzb-mpc59/files/0502212.pdf" }, "pub_year": "2005", "author_list": "Kapustin, Anton" }, { "id": "https://authors.library.caltech.edu/records/fmnf8-fne39", "eprint_id": 79034, "eprint_status": "archive", "datestamp": "2023-08-19 15:22:23", "lastmod": "2026-03-09 21:41:27", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" }, { "id": "Mathai-V", "name": { "family": "Mathai", "given": "Varghese" } } ] }, "title": "Towards the fractional quantum Hall effect: a noncommutative geometry perspective", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 0502356.pdf

", "abstract": "In this paper we give a survey of some models of the integer and fractional quantum Hall effect based on noncommutative geometry. We begin by recalling some classical geometry of electrons in solids and the passage to noncommutative geometry produced by the presence of a magnetic field. We recall how one can obtain this way a single electron model of the integer quantum Hall effect. While in the case of the integer quantum Hall effect the underlying geometry is Euclidean, we then discuss a model of the fractional quantum Hall effect, which is based on hyperbolic geometry simulating the multi-electron interactions. We derive the fractional values of the Hall conductance as integer multiples of orbifold Euler characteristics. We compare the results with experimental data.", "date": "2005-02-15", "date_type": "published", "id_number": "CaltechAUTHORS:20170712-154917444", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-154917444", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0502356", "primary_object": { "basename": "0502356.pdf", "url": "https://authors.library.caltech.edu/records/fmnf8-fne39/files/0502356.pdf" }, "pub_year": "2005", "author_list": "Marcolli, Matilde and Mathai, Varghese" }, { "id": "https://authors.library.caltech.edu/records/7557s-69t47", "eprint_id": 79059, "eprint_status": "archive", "datestamp": "2023-08-19 14:40:49", "lastmod": "2026-03-09 21:42:17", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Connes-A", "name": { "family": "Connes", "given": "Alain" } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 11 Nov 2004) \n\nWe are very grateful to Jean\u2013Pierre Ramis for many useful\ncomments on an early draft of this paper, for the kind invitation to Toulouse, and for the many stimulating discussions we had there with him, Fr\u00e9d\u00e9ric Fauvet, and Laurent Stolovitch. We thank Fr\u00e9d\u00e9ric Menous and Giorgio Parisi for some useful correspondence. Many thanks go to Dirk Kreimer, whose joint work with AC on perturbative renormalization is a main topic of this Chapter.\n\n

Submitted - 0411114.pdf

", "abstract": "We give here a comprehensive treatment of the mathematical theory of perturbative renormalization (in the minimal subtraction scheme with dimensional regularization), in the framework of the Riemann\u2013Hilbert correspondence and\nmotivic Galois theory. We give a detailed overview of the work of Connes\u2013Kreimer [31], [32]. We also cover some background material on affine group schemes, Tannakian categories, the Riemann\u2013Hilbert problem in the regular singular and irregular case, and a brief introduction to motives and motivic Galois theory. We then give a complete account of our results on renormalization and motivic Galois theory announced in [35].", "date": "2004-11-11", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-080604764", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-080604764", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0411114", "primary_object": { "basename": "0411114.pdf", "url": "https://authors.library.caltech.edu/records/7557s-69t47/files/0411114.pdf" }, "pub_year": "2004", "author_list": "Connes, Alain and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/p1ywv-nbs42", "eprint_id": 66976, "eprint_status": "archive", "datestamp": "2023-08-19 14:35:07", "lastmod": "2026-03-09 02:36:56", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" } ] }, "title": "A remark on worldsheet fermions and double-scaled matrix models", "ispublished": "unpub", "full_text_status": "public", "note": "Submitted on 27 Oct 2004. January 1, 2014.\n\nThe author would like to thank Jongwon Park for useful discussions and Juan Maldacena for comments. This work was supported in part by the DOE grant DE-FG03-92-ER40701.\n\n

Submitted - 0410268.pdf

", "abstract": "We provide a heuristic explanation for the emergence of worldsheet\nfermions in the continuum limit of some matrix models. We also argue\nthat turning on Ramond-Ramond flux confines the fermionic degrees\nof freedom of the Ramond-Neveu-Schwarz formalism.", "date": "2004-10-27", "date_type": "published", "id_number": "CaltechAUTHORS:20160511-090400845", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160511-090400845", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG03-92-ER40701" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0410268", "primary_object": { "basename": "0410268.pdf", "url": "https://authors.library.caltech.edu/records/p1ywv-nbs42/files/0410268.pdf" }, "pub_year": "2004", "author_list": "Kapustin, Anton" }, { "id": "https://authors.library.caltech.edu/records/04m4t-1vr74", "eprint_id": 79064, "eprint_status": "archive", "datestamp": "2023-08-19 14:18:09", "lastmod": "2026-03-09 21:45:28", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Lectures on Arithmetic Noncommutative Geometry", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 27 Sep 2004)\n\n

Submitted - 0409520.pdf

", "abstract": "This is the text of a series of five lectures given by the author at the \"Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras\" held at Vanderbilt University in May 2004. It is meant as an overview of recent results illustrating the interplay between noncommutative geometry and arithmetic geometry/number theory.", "date": "2004-09-27", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-083048947", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-083048947", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0409520", "primary_object": { "basename": "0409520.pdf", "url": "https://authors.library.caltech.edu/records/04m4t-1vr74/files/0409520.pdf" }, "pub_year": "2004", "author_list": "Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/dejrb-xjx76", "eprint_id": 79066, "eprint_status": "archive", "datestamp": "2023-08-19 13:59:22", "lastmod": "2026-03-09 21:31:42", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Consani-C", "name": { "family": "Consani", "given": "Caterina" } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Archimedean cohomology revisited", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 28 Jul 2004) \n\nPartially supported by NSERC grants 72016789, 72024520. \n\nPartially supported by Humboldt Foundation Sofja Kovalevskaja Award.\n\n

Submitted - 0407480.pdf

", "abstract": "Archimedean cohomology provides a cohomological interpretation for the calculation of the local L-factors at archimedean places as zeta regularized determinant of a log of Frobenius. In this paper we investigate further the properties of the Lefschetz and log of monodromy operators on this cohomology. We use the Connes-Kreimer formalism of renormalization to obtain a fuchsian connection whose residue is the log of the monodromy. We also present a dictionary of analogies between the geometry of a tubular neighborhood of the \"fiber at arithmetic infinity\" of an arithmetic variety and the complex of nearby cycles in the geometry of a degeneration over a disk, and we recall Deninger's approach to the archimedean cohomology through an interpretation as global sections of a analytic Rees sheaf. We show that action of the Lefschetz, the log of monodromy and the log of Frobenius on the archimedean cohomology combine to determine a spectral triple in the sense of Connes. The archimedean part of the Hasse-Weil L-function appears as a zeta function of this spectral triple. We also outline some formal analogies between this cohomological theory at arithmetic infinity and Givental's homological geometry on loop spaces.", "date": "2004-07-28", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-085828455", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-085828455", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "72016789" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "72024520" }, { "agency": "Alexander von Humboldt Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0407480", "primary_object": { "basename": "0407480.pdf", "url": "https://authors.library.caltech.edu/records/dejrb-xjx76/files/0407480.pdf" }, "pub_year": "2004", "author_list": "Consani, Caterina and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/x8m9v-m7289", "eprint_id": 82252, "eprint_status": "archive", "datestamp": "2023-08-19 13:46:18", "lastmod": "2026-03-09 22:47:17", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Draper-T-G", "name": { "family": "Draper", "given": "Thomas G." } }, { "id": "Kutin-S-A", "name": { "family": "Kutin", "given": "Samuel A." } }, { "id": "Rains-E-M", "name": { "family": "Rains", "given": "Eric M." }, "orcid": "0000-0002-9915-0919" }, { "id": "Svore-K-M", "name": { "family": "Svore", "given": "Krysta M." } } ] }, "title": "A logarithmic-depth quantum carry-lookahead adder", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 0406142.pdf

", "abstract": "We present an efficient addition circuit, borrowing techniques from the classical carry-lookahead arithmetic circuit. Our quantum carry-lookahead (QCLA) adder accepts two n-bit numbers and adds them in O(log n) depth using O(n) ancillary qubits. We present both in-place and out-of-place versions, as well as versions that add modulo 2^n and modulo 2^n - 1. \nPreviously, the linear-depth ripple-carry addition circuit has been the method of choice. Our work reduces the cost of addition dramatically with only a slight increase in the number of required qubits. The QCLA adder can be used within current modular multiplication circuits to reduce substantially the run-time of Shor's algorithm.", "date": "2004-06-20", "date_type": "published", "id_number": "CaltechAUTHORS:20171010-104601338", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20171010-104601338", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0406142", "primary_object": { "basename": "0406142.pdf", "url": "https://authors.library.caltech.edu/records/x8m9v-m7289/files/0406142.pdf" }, "pub_year": "2004", "author_list": "Draper, Thomas G.; Kutin, Samuel A.; et al." }, { "id": "https://authors.library.caltech.edu/records/ggmcy-djp61", "eprint_id": 66972, "eprint_status": "archive", "datestamp": "2023-08-19 13:29:27", "lastmod": "2026-03-09 02:40:23", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" }, { "id": "Murugan-A", "name": { "family": "Murugan", "given": "Arvind" } } ] }, "title": "Fatgraph expansion for noncritical superstrings", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 29 Apr 2004) August 14, 2013. \n\nA.K. would like to thank Michael Douglas and Jaume Gomis for useful discussions. This work was supported in part by the DOE grant DE-FG03-92-ER40701\n\n

Submitted - 0404238.pdf

", "abstract": "We study the fatgraph expansion for the Complex Matrix Quan-\ntum Mechanics (CMQM) with a Chern-Simons coupling. In the double-scaling limit this model is believed to describe Type 0A superstrings in 1+1 dimensions in a Ramond-Ramond electric field. With Euclidean time compactified, we show that the RR electric field acts as a chemical potential for vortices living on the Feynman diagrams of the CMQM. We interpret it as evidence that the CMQM Feynman diagrams discretize the NSR formulation of the noncritical Type 0A\nsuperstring. We also study T-duality for the CMQM diagrams and propose that a certain complex matrix model is dual to the noncritical Type 0B superstring.", "date": "2004-04-29", "date_type": "published", "id_number": "CaltechAUTHORS:20160511-081843750", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160511-081843750", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG03-92-ER40701" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0404238", "primary_object": { "basename": "0404238.pdf", "url": "https://authors.library.caltech.edu/records/ggmcy-djp61/files/0404238.pdf" }, "pub_year": "2004", "author_list": "Kapustin, Anton and Murugan, Arvind" }, { "id": "https://authors.library.caltech.edu/records/t8va9-8hj75", "eprint_id": 79061, "eprint_status": "archive", "datestamp": "2023-08-19 13:25:56", "lastmod": "2026-03-09 21:43:53", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Connes-A", "name": { "family": "Connes", "given": "Alain" } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 6 Apr 2004)\n\nWe are very grateful to Niranjan Ramachandran for many extremely useful conversations on class field theory and KMS states, that motivated the GL_2 system described here, whose relation to the theory of complex multiplication is being investigated in [13]. We thank Marcelo Laca for giving us an extensive update on the further developments on [5]. We benefited from visits of the first author to MPI and of the second author to IHES and we thank both institutions for their hospitality. The second author is partially supported by a Sofja Kovalevskaya Award of the Humboldt Foundation and the German Government.\n\n

Submitted - 0404128.pdf

", "abstract": "Several recent results reveal a surprising connection between modular forms and noncommutative geometry. The first occurrence came from the classification of noncommutative three spheres, [C\u2013DuboisViolette-I] [C\u2013DuboisViolette-II]. Hard computations with the noncommutative analog of the Jacobian involving the ninth power of the Dedekind eta function were necessary in order to analyze the relation between such spheres and noncommutative nilmanifolds. Another occurrence can be seen in the computation of the explicit cyclic cohomology\nChern character of a spectral triple on SU_q(2) [C\u201302]. Another surprise came recently from a remarkable action of the Hopf algebra of transverse geometry of foliations of codimension one on the space of lattices modulo Hecke correspondences, described in the framework of noncommutative geometry, using a modular Hecke algebra obtained as the cross product of modular forms by the\naction of Hecke correspondences [C\u2013Moscovici-I] [C\u2013Moscovici-II]. This action determines a differentiable structure on this noncommutative space, related to the Rankin\u2013Cohen brackets of modular forms, and shows their compatibility with Hecke operators. Another instance where properties of modular forms can be recast in the context of noncommutative geometry can be found in the theory\nof modular symbols and Mellin transforms of cusp forms of weight two, which can be recovered from the geometry of the moduli space of Morita equivalence classes of noncommutative tori viewed as boundary of the modular curve [Manin\u2013M].", "date": "2004-04-06", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-081805785", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-081805785", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Alexander von Humboldt Foundation" }, { "agency": "Deutsche Forschungsgemeinschaft (DFG)" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0404128", "primary_object": { "basename": "0404128.pdf", "url": "https://authors.library.caltech.edu/records/t8va9-8hj75/files/0404128.pdf" }, "pub_year": "2004", "author_list": "Connes, Alain and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/hd8vz-wmw82", "eprint_id": 66686, "eprint_status": "archive", "datestamp": "2023-08-19 13:25:05", "lastmod": "2026-03-09 02:17:40", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Gukov-S", "name": { "family": "Gukov", "given": "Sergei" }, "orcid": "0000-0002-9486-1762" }, { "id": "Martinec-E", "name": { "family": "Martinec", "given": "Emil" } }, { "id": "Moore-G", "name": { "family": "Moore", "given": "Gregory" } }, { "id": "Strominger-A", "name": { "family": "Strominger", "given": "Andrew" } } ] }, "title": "An Index for 2D field theories with large N=4 superconformal symmetry", "ispublished": "unpub", "full_text_status": "public", "note": "G.M. would like to that the KITP for hospitality during the course\nof some of this work. This work was conducted during the period S.G. served as a Clay Mathematics Institute Long-Term Prize Fellow. The work of E.M. is supported in part by\nDOE grant DE-FG02-90ER40560, that of G.M. by DOE grant DE-FG02-96ER40949 and\nthat of A.S. and S.G. by DE-FG02-91ER40654.\n\n

Submitted - 0404023.pdf

", "abstract": "We consider families of theories with large N=4 superconformal symmetry. We define an index generalizing the elliptic genus of theories with N=2 symmetry. In contrast to the N=2 case, the new index constrains part of the non-BPS spectrum. Motivated by aspects of the AdS/CFT correspondence we study the index in the examples of symmetric product theories. We give a physical interpretation of the Hecke operators which appear in the expressions for partition functions of such theories. Finally, we compute the index for a nontrivial example of a symmetric product theory.", "date": "2004-04-02", "date_type": "published", "id_number": "CaltechAUTHORS:20160505-105657599", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160505-105657599", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Clay Mathematics Institute" }, { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-90ER40560" }, { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-96ER40949" }, { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-91ER40654" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0404023", "primary_object": { "basename": "0404023.pdf", "url": "https://authors.library.caltech.edu/records/hd8vz-wmw82/files/0404023.pdf" }, "pub_year": "2004", "author_list": "Gukov, Sergei; Martinec, Emil; et al." }, { "id": "https://authors.library.caltech.edu/records/6wa97-6gp76", "eprint_id": 66716, "eprint_status": "archive", "datestamp": "2023-08-19 13:17:06", "lastmod": "2026-03-09 02:16:44", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Gukov-S", "name": { "family": "Gukov", "given": "Sergei" }, "orcid": "0000-0002-9486-1762" }, { "id": "Martinec-E", "name": { "family": "Martinec", "given": "Emil" } }, { "id": "Moore-G", "name": { "family": "Moore", "given": "Gregory" } }, { "id": "Strominger-A", "name": { "family": "Strominger", "given": "Andrew" } } ] }, "title": "Chern-Simons Gauge Theory and the AdS_3/CFT_2 Correspondence", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 22 Mar 2004). March 19, 2004.\n\nWe would like to thank J. Maldacena and E. Witten for discussions and C. Schweigert and N. Read for correspondence. G.M. would like to that the KITP for hospitality during the course of some of this work. This work was conducted during the period S.G. served as a\nClay Mathematics Institute Long-Term Prize Fellow. The work of E.M. is supported in part by DOE grant DE-FG02-90ER40560, that of G.M. by DOE grant DE-FG02-96ER40949 and that of A.S. by DE-FG02-91ER40654.\n\n

Submitted - 0403225.pdf

", "abstract": "The bulk partition function of pure Chern-Simons theory on a three-manifold is a state in the space of conformal blocks of the dual boundary RCFT, and therefore transforms\nnon-trivially under the boundary modular group. In contrast the bulk partition function of AdS_3 string theory is the modular-invariant partition function of the dual CFT on the\nboundary. This is a puzzle because AdS_3 string theory formally reduces to pure Chern-Simons theory at long distances. We study this puzzle in the context of massive Chern-Simons theory. We show that the puzzle is resolved in this context by the appearance of a chiral \"spectator boson\" in the boundary CFT which restores modular invariance. It couples to the conformal metric but not to the gauge field on the boundary. Consequently, we find a generalization of the standard Chern-Simons/RCFT correspondence involving \"nonholomorphic conformal blocks\" and nonrational boundary CFTs. These generalizations\nappear in the long-distance limit of AdS_3 string theory, where the role of the spectator boson is played by other degrees of freedom in the theory.", "date": "2004-03-22", "date_type": "published", "id_number": "CaltechAUTHORS:20160506-145809044", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160506-145809044", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-90ER40560" }, { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-96ER40949" }, { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG02-91ER40654" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0403225", "primary_object": { "basename": "0403225.pdf", "url": "https://authors.library.caltech.edu/records/6wa97-6gp76/files/0403225.pdf" }, "pub_year": "2004", "author_list": "Gukov, Sergei; Martinec, Emil; et al." }, { "id": "https://authors.library.caltech.edu/records/qcvad-fpq10", "eprint_id": 79062, "eprint_status": "archive", "datestamp": "2023-08-19 12:39:17", "lastmod": "2026-03-09 21:44:31", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Modular curves, C^* algebras, and chaotic cosmology", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 11 Dec 2003)\n\n

Submitted - 0312035.pdf

", "abstract": "We make some brief remarks on the relation of the mixmaster universe model of chaotic cosmology to the geometry of modular curves and to noncommutative geometry. We show that the full dynamics of the mixmaster universe\nis equivalent to the geodesic flow on the modular curve X_(\u03930(2)). We then consider a special class of solutions, with bounded number of cycles in each Kasner era, and describe their dynamical properties (invariant density, Lyapunov exponent, topological pressure). We relate these properties to the noncommutative geometry of a moduli space of such solutions, which is given by a Cuntz\u2013Krieger C^\u2217-algebra.", "date": "2003-12-11", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-082602443", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-082602443", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0312035", "primary_object": { "basename": "0312035.pdf", "url": "https://authors.library.caltech.edu/records/qcvad-fpq10/files/0312035.pdf" }, "pub_year": "2003", "author_list": "Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/pf36s-zm641", "eprint_id": 66682, "eprint_status": "archive", "datestamp": "2023-08-19 12:31:15", "lastmod": "2026-03-09 02:39:56", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kapustin-A", "name": { "family": "Kapustin", "given": "Anton" }, "orcid": "0000-0003-3903-5158" }, { "id": "Li-Yi", "name": { "family": "Li", "given": "Yi" } } ] }, "title": "Stability Conditions For Topological D-branes: A Worldsheet Approach", "ispublished": "unpub", "full_text_status": "public", "note": "A. K. would like to thank Nigel Hitchin and Dmitri Orlov for discussions.\nThis work was supported in part by the DOE grant DE-FG03-92-ER40701.\n\n

Submitted - 0311101.pdf

", "abstract": "We study conditions on the topological D-branes of types A and B obtained by requiring a proper matching of the spectral flow operators on the boundary. These conditions ensure space-time supersymmetry and stability of D-branes. In most cases, we reproduce the results of Marino-Minasian-Moore-Strominger, who studied the same problem using the supersymmetric Born-Infeld action. In some other cases, corresponding to coisotropic A-branes, our stability condition is new. Our results enable us to define an analogue of the Maslov class and grading for coisotropic A-branes. We expect that they play a role in a conjectural generalization of the Floer homology.", "date": "2003-11-12", "date_type": "published", "id_number": "CaltechAUTHORS:20160505-102849577", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160505-102849577", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Department of Energy (DOE)", "grant_number": "DE-FG03-92-ER40701" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0311101", "primary_object": { "basename": "0311101.pdf", "url": "https://authors.library.caltech.edu/records/pf36s-zm641/files/0311101.pdf" }, "pub_year": "2003", "author_list": "Kapustin, Anton and Li, Yi" }, { "id": "https://authors.library.caltech.edu/records/4mq62-k2841", "eprint_id": 79055, "eprint_status": "archive", "datestamp": "2023-08-19 12:24:43", "lastmod": "2026-03-09 21:43:33", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Consani-C", "name": { "family": "Consani", "given": "Caterina" } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Non-commutative geometry, dynamics, and infinity-adic Arakelov geometry", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 29 May 2002 (v1), last revised 27 Oct 2003 (this version, v2) \n\nPartially supported by NSERC grant 72016789. \n\nPartially supported by Humboldt Foundation Sofja Kovalevskaja Award.\n\n

Submitted - 0205306.pdf

", "abstract": "In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the \"closed fibers at infinity\". Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic\nhandlebody endowed with a Schottky uniformization. In this paper we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus g \u2265 2. We use Connes' theory of spectral triples to relate the hyperbolic geometry of the handlebody to Deninger's Archimedean cohomology and the cohomology of the cone of the local monodromy N at arithmetic infinity as introduced by the first author of this paper. First, we consider derived (cohomological) spectral data (A,H\u00b7(X^\u2217),\u0424), where the algebra is obtained from the SL(2,R) action on the cohomology of the cone, induced by the presence of a polarized Lefschetz module structure, and its restriction to the group ring of a Fuchsian Schottky group. In this setting we recover the alternating product of the Archimedean factors from a zeta function of a spectral triple. Then, we introduce a different\nconstruction, which is related to Manin's description of the dual graph of the fiber at infinity. We provide a geometric model for the dual graph as the mapping torus of a dynamical system T on a Cantor set. We consider a noncommutative space which describes the action of the Schottky group on its limit set and parameterizes the \"components of the closed fiber at infinity\". This can be identified with a Cuntz\u2013Krieger algebra O_A associated to a subshift of finite type. We\nconstruct a spectral triple for this noncommutative space, via a representation on the cochains of a \"dynamical cohomology\", defined in terms of the tangle of bounded geodesics in the handlebody. In both constructions presented in the paper, the Dirac operator agrees with the grading operator \u0424, that represents the \"logarithm of a Frobenius\u2013type operator\" on the Archimedean cohomology. In fact, the Archimedean cohomology embeds in the dynamical cohomology, compatibly with the action of a real Frobenius F_\u221e, so that the local factor can again be recovered from these data. The\nduality isomorphism on the cohomology of the cone of N corresponds to the pairing of dynamical homology and cohomology. This suggests the existence of a duality between the monodromy N and the dynamical map 1\u2212T. Moreover, the \"reduction mod infinity\" is described in terms of the\nhomotopy quotient associated to the noncommutative space OA and the \u03bc-map of Baum\u2013Connes. The geometric model of the dual graph can also be described as a homotopy quotient.", "date": "2003-10-27", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-075105845", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-075105845", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "72016789" }, { "agency": "Alexander von Humboldt Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0205306", "primary_object": { "basename": "0205306.pdf", "url": "https://authors.library.caltech.edu/records/4mq62-k2841/files/0205306.pdf" }, "pub_year": "2003", "author_list": "Consani, Caterina and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/4m25m-vt918", "eprint_id": 99179, "eprint_status": "archive", "datestamp": "2023-09-15 06:24:16", "lastmod": "2026-03-18 04:19:40", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Cohen-A-M", "name": { "family": "Cohen", "given": "Arjeh M." } }, { "id": "Gijsbers-D-A-H", "name": { "family": "Gijsbers", "given": "Di\u00e9 A. H." } }, { "id": "Wales-D-B", "name": { "family": "Wales", "given": "David B." } } ] }, "title": "BMW algebras of simply laced type", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 0310011.pdf

", "abstract": "It is known that the recently discovered representations of the Artin groups of type A_n, the braid groups, can be constructed via BMW algebras. We introduce similar algebras of type D_n and E_n which also lead to the newly found faithful representations of the Artin groups of the corresponding types. We establish finite dimensionality of these algebras. Moreover, they have ideals I_1 and I_2 with I_2 contained in I_1 such that the quotient with respect to I_1 is the Hecke algebra and I_1/I_2 is a module for the corresponding Artin group generalizing the Lawrence-Krammer representation. Finally we give conjectures on the structure, the dimension and parabolic subalgebras of the BMW algebra, as well as on a generalization of deformations to Brauer algebras for simply laced spherical type other than A_n.", "date": "2003-10-01", "date_type": "published", "id_number": "CaltechAUTHORS:20191009-091742763", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191009-091742763", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" }, { "id": "Physics-Department" } ] }, "doi": "10.48550/arXiv.0310011", "primary_object": { "basename": "0310011.pdf", "url": "https://authors.library.caltech.edu/records/4m25m-vt918/files/0310011.pdf" }, "pub_year": "2003", "author_list": "Cohen, Arjeh M.; Gijsbers, Di\u00e9 A. H.; et al." }, { "id": "https://authors.library.caltech.edu/records/k6wbf-5k411", "eprint_id": 79058, "eprint_status": "archive", "datestamp": "2023-08-19 10:18:30", "lastmod": "2026-03-09 21:38:06", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" }, { "id": "Wang-Bai-Ling", "name": { "family": "Wang", "given": "Bai-Ling" } } ] }, "title": "Variants of equivariant Seiberg-Witten Floer homology", "ispublished": "unpub", "full_text_status": "public", "keywords": "rational homology 3-spheres, equivariant Seiberg-Witten Floer homology, Spinc structures, topological invariants", "note": "

Submitted - 0211238.pdf

", "abstract": "For a rational homology 3-sphere Y with a Spin^c structure s, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology in [5] lead to a collection of variants HF^(SW,\u2212)_(*,U(1))(Y,s), HF^(SW,\u221e)_(*,U(1)(Y,s) HF^(SW,+)_(*,U(1))(Y,s), HF^(SW)_*(Y,s) and HF^(SW)_(red,*)(Y,s) which are topological invariants. We establish a long exact sequence relating HF^(SW,\u00b1)_(*,U(1))(Y,s) and HF^(SW,\u221e) _(*,U(1))(Y,s). We show they satisfy a duality under orientation reversal, and we explain their relation to the equivariant Seiberg-Witten Floer (co)homologies introduced in [5]. We conjecture the equivalence of these versions of equivariant Seiberg-Witten Floer homology with the Heegaard Floer invariants introduced by Ozsv\u00e1th and Szab\u00f3.", "date": "2002-11-15", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-080540298", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-080540298", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0211238", "primary_object": { "basename": "0211238.pdf", "url": "https://authors.library.caltech.edu/records/k6wbf-5k411/files/0211238.pdf" }, "pub_year": "2002", "author_list": "Marcolli, Matilde and Wang, Bai-Ling" }, { "id": "https://authors.library.caltech.edu/records/3cw7z-prv14", "eprint_id": 79057, "eprint_status": "archive", "datestamp": "2023-08-19 10:12:05", "lastmod": "2026-03-09 21:41:02", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Consani-C", "name": { "family": "Consani", "given": "C." } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "M." }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Spectral triples from Mumford curves", "ispublished": "unpub", "full_text_status": "public", "note": "Part of this work was done during visits of the first author to the Max Planck Institute in Bonn and of the second author to Florida State University and University of Toronto. We thank these institutions for hospitality and support. We are very grateful to Christopher Deninger for a crucial remark about normalizations. This research has been partially supported by NSERC grant 72016789 and by Humboldt Foundation and the German Government (Sofja Kovalevskaya Award).\n\n

Submitted - 0210435.pdf

", "abstract": "We construct spectral triples associated to Schottky--Mumford curves, in such a way that the local Euler factor can be recovered from the zeta functions of such spectral triples. We propose a way of extending this construction to the case where the curve is not k-split degenerate.", "date": "2002-10-28", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-075847963", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-075847963", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "72016789" }, { "agency": "Alexander von Humboldt Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0210435", "primary_object": { "basename": "0210435.pdf", "url": "https://authors.library.caltech.edu/records/3cw7z-prv14/files/0210435.pdf" }, "pub_year": "2002", "author_list": "Consani, C. and Marcolli, M." }, { "id": "https://authors.library.caltech.edu/records/p2n5v-9e676", "eprint_id": 79056, "eprint_status": "archive", "datestamp": "2023-08-19 10:11:26", "lastmod": "2026-03-09 21:42:31", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Consani-C", "name": { "family": "Consani", "given": "Caterina" } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "New perspectives in Arakelov geometry", "ispublished": "unpub", "full_text_status": "public", "note": "This paper was partly written during visits of the second author to Florida State University and University of Toronto. We thank these institutions for the hospitality. The first author is partially supported by NSERC grant 72016789. The second author is partially supported by the Humboldt Foundation and the German Government (Sofja Kovalevskaya Award). We thank Alain Connes for many extremely helpful discussions.\n\n

Submitted - 0210357.pdf

", "abstract": "In this survey, written for the proceedings of the VII meeting of the CNTA held in May 2002 in Montreal, we describe how Connes' theory of spectral triples provides a unified view, via noncommutative geometry, of the archimedean and the totally split degenerate fibers of an arithmetic surface.", "date": "2002-10-23", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-075158422", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-075158422", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "72016789" }, { "agency": "Alexander von Humboldt Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0210357", "primary_object": { "basename": "0210357.pdf", "url": "https://authors.library.caltech.edu/records/p2n5v-9e676/files/0210357.pdf" }, "pub_year": "2002", "author_list": "Consani, Caterina and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/0h2cb-9e174", "eprint_id": 79051, "eprint_status": "archive", "datestamp": "2023-08-19 10:03:06", "lastmod": "2026-03-09 21:41:15", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Consani-C", "name": { "family": "Consani", "given": "Caterina" } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Triplets spectraux en g\u00e9om\u00e9trie d'Arakelov", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 15 Sep 2002) \n\nPartiellement support\u00e9e par la bourse du NSERC num\u00e9ro 72016789. \n\nPartiellement support\u00e9e par la bourse Sofja Kovalevskaja de l'Humboldt Foundation.\n\n

Submitted - 0209182.pdf

", "abstract": "In this note, we use Connes' theory of spectral triples to provide a connection between Manin's model of the dual graph of the fiber at infinity of an Arakelov surface and the cohomology of the mapping cone of the local monodromy.", "date": "2002-09-15", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-071539391", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-071539391", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)", "grant_number": "72016789" }, { "agency": "Alexander von Humboldt Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0209182", "primary_object": { "basename": "0209182.pdf", "url": "https://authors.library.caltech.edu/records/0h2cb-9e174/files/0209182.pdf" }, "pub_year": "2002", "author_list": "Consani, Caterina and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/9d42w-jf423", "eprint_id": 83432, "eprint_status": "archive", "datestamp": "2023-08-19 09:53:36", "lastmod": "2026-03-09 22:46:39", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Rains-E-M", "name": { "family": "Rains", "given": "E. M." }, "orcid": "0000-0002-9915-0919" }, { "id": "Sloane-N-J-A", "name": { "family": "Sloane", "given": "N. J. A." } } ] }, "title": "Self-Dual Codes", "ispublished": "unpub", "full_text_status": "public", "note": "Over the past 26 years NJAS has had the pleasure of collaborating with many of the people whose names are listed in the bibliography: he wishes to express his appreciation to all of them. We thank Eiichi Bannai, Dave Forney, Aaron Gulliver, Masaaki Harada, Cary Huffman, Michio Ozeki, Vera Pless and Patrick Sol\u00e9 for helpful comments on the manuscript of this chapter. We also thank Susan Pope for a superb typing job.\n\n

Submitted - 0208001.pdf

", "abstract": "Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.", "date": "2002-08-01", "date_type": "published", "id_number": "CaltechAUTHORS:20171122-102726979", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20171122-102726979", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0208001", "primary_object": { "basename": "0208001.pdf", "url": "https://authors.library.caltech.edu/records/9d42w-jf423/files/0208001.pdf" }, "pub_year": "2002", "author_list": "Rains, E. M. and Sloane, N. J. A." }, { "id": "https://authors.library.caltech.edu/records/tatr5-4kp41", "eprint_id": 79067, "eprint_status": "archive", "datestamp": "2023-08-19 09:14:41", "lastmod": "2026-03-09 21:43:09", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Carey-A-L", "name": { "family": "Carey", "given": "A. L." } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "M." }, "orcid": "0000-0002-2045-2907" }, { "id": "Wang-Bai-Ling", "name": { "family": "Wang", "given": "Bai-Ling" } } ] }, "title": "The geometric triangle for 3-dimensional Seiberg-Witten monopoles", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 11 Jul 1999 (v1), last revised 21 Mar 2002 (this version, v3) \n\nWe are very grateful to Ronnie Lee, Tom Mrowka, Vicente\nMunoz, and Peter Ozsvath for useful discussions and suggestions. We like to thank Cliff Taubes for providing the proof of Lemma 2.3 and Liviu Nicolaescu for the proof of Lemma 4.11. The three authors also thank the Max-Planck-Institut f\u00fcr Mathematik, Bonn for the kind hospitality and for support. AC and BW are partially supported by Australian Research Council. MM is partially supported by NSF grant DMS-9802480 and by Humboldt Foundation (Sofja Kovalevskaya Award).\n\n

Submitted - 9907065.pdf

", "abstract": "We establish a surgery formula for 3-dimensional Seiberg-Witten monopoles under (+1) Dehn surgery on a knot in a homology 3-sphere. (substantial revision)", "date": "2002-03-21", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-090503578", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-090503578", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Max-Planck-Institut f\u00fcr Mathematik" }, { "agency": "Australian Research Council" }, { "agency": "NSF", "grant_number": "DMS-9802480" }, { "agency": "Alexander von Humboldt Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.9907065", "primary_object": { "basename": "9907065.pdf", "url": "https://authors.library.caltech.edu/records/tatr5-4kp41/files/9907065.pdf" }, "pub_year": "2002", "author_list": "Carey, A. L.; Marcolli, M.; et al." }, { "id": "https://authors.library.caltech.edu/records/zc611-c2e58", "eprint_id": 79054, "eprint_status": "archive", "datestamp": "2023-08-19 08:57:27", "lastmod": "2026-03-09 21:41:49", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Manin-Y-I", "name": { "family": "Manin", "given": "Yuri I." } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Holography principle and arithmetic of algebraic curves", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 7 Jan 2002) \n\nWe are grateful to Alain Connes who suggested the authors to look at [Man2] from the perspective of the holography principle. We also thank Kirill Krasnov for several useful and encouraging comments.\n\n

Submitted - 0201036.pdf

", "abstract": "According to the holography principle (due to G. 't Hooft, L. Susskind, J. Maldacena, et al.), quantum gravity and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few mathematically exact results corroborating this exciting program are known. In this paper we interpret from this perspective several constructions which arose initially in the arithmetic geometry of algebraic curves. We show that the relation\nbetween hyperbolic geometry and Arakelov geometry at arithmetic infinity involves exactly the same geometric data as the Euclidean AdS_3 holography of black holes. Moreover, in the case of Euclidean AdS_2 holography, we present some results on bulk/boundary correspondence where the boundary is a non\u2013commutative space.", "date": "2002-01-07", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-074226681", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-074226681", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0201036", "primary_object": { "basename": "0201036.pdf", "url": "https://authors.library.caltech.edu/records/zc611-c2e58/files/0201036.pdf" }, "pub_year": "2002", "author_list": "Manin, Yuri I. and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/tdar8-qyj53", "eprint_id": 79052, "eprint_status": "archive", "datestamp": "2023-08-19 08:07:00", "lastmod": "2026-03-09 21:45:51", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Manin-Y-I", "name": { "family": "Manin", "given": "Yuri I." } }, { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" } ] }, "title": "Continued fractions, modular symbols, and non-commutative geometry", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 1 Feb 2001 (v1), last revised 7 Aug 2001 (this version, v2). \n\nWe thank Dieter Mayer, Victor Nistor, and Don Zagier for useful conversations. The second author is partially supported by Sofja Kovalevskaya Award.\n\n

Submitted - 0102006.pdf

", "abstract": "Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss\u2013Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols and show that Dirichlet series related to modular forms of weight 2 can be obtained by integrating certain functions on real axis defined in terms of continued fractions. We argue that the quotient PGL(2,Z) \\ P^1(R) should be considered as non\u2013commutative modular curve, and show that the modular complex can be seen as a sequence of K0\u2013groups of the related crossed\u2013product C^\u2217\u2013algebras.", "date": "2001-08-07", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-072319363", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-072319363", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Alexander von Humboldt Foundation" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0102006", "primary_object": { "basename": "0102006.pdf", "url": "https://authors.library.caltech.edu/records/tdar8-qyj53/files/0102006.pdf" }, "pub_year": "2001", "author_list": "Manin, Yuri I. and Marcolli, Matilde" }, { "id": "https://authors.library.caltech.edu/records/p9bdm-qw833", "eprint_id": 79053, "eprint_status": "archive", "datestamp": "2023-08-19 07:13:52", "lastmod": "2026-03-09 21:40:25", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" }, { "id": "Wang-Bai-Ling", "name": { "family": "Wang", "given": "Bai-Ling" } } ] }, "title": "Exact triangles in monopole homology and the Casson-Walker invariant", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 16 Jan 2001) \n\nBLW likes to acknowledge the paper of Ozsv\u00e1th and Szab\u00f3 [10] on the theta divisor and the Casson-Walker invariant which leads to his proof of the equivalence of SW_Y and the Casson-Walker invariant, hence proving the conjecture formulated in [10] on the equivalent between SW_Y and their \u03b8 invariant for all rational homology 3-sphere. BLW is partially supported by Australia Research Council Fellowship.\n\n

Submitted - 0101127.pdf

", "abstract": "The purpose of this paper is to give a general outline of the problem of the exact triangles in Seiberg\u2013Witten\u2013Floer theory. We present here the most general case, where the problem consists of producing a surgery formula relating the monopole homology of a compact oriented 3\u2013manifold Y with an embedded knot K, and the monopole homologies of some 3\u2013manifolds obtained by Dehn surgery on K.", "date": "2001-01-16", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-073152604", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-073152604", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Australian Research Council" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0101127", "primary_object": { "basename": "0101127.pdf", "url": "https://authors.library.caltech.edu/records/p9bdm-qw833/files/0101127.pdf" }, "pub_year": "2001", "author_list": "Marcolli, Matilde and Wang, Bai-Ling" }, { "id": "https://authors.library.caltech.edu/records/s0sjj-kfm53", "eprint_id": 79031, "eprint_status": "archive", "datestamp": "2023-08-19 06:20:10", "lastmod": "2026-03-09 21:38:44", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" }, { "id": "Wang-Bai-Ling", "name": { "family": "Wang", "given": "Bai-Ling" } } ] }, "title": "Exact triangles in Seiberg-Witten Floer theory. Part III: proof of exactness", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 15 Sep 2000). \n\nThe first author is partially supported by NSF grant DMS-9802480. The second author is partially supported by ARC Fellowship. \n\nWe thank the Max\u2013Planck\u2013Institut f\u00fcr Mathematik, where a large part of the work was done.\n\n

Submitted - 0009157.pdf

", "abstract": "This is the third part of the work on the exact triangles. We construct chain homomorphisms and show exactness of the resulting sequence.", "date": "2000-09-15", "date_type": "published", "id_number": "CaltechAUTHORS:20170712-152627045", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-152627045", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-9802480" }, { "agency": "Australian Research Council" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0009157v1", "primary_object": { "basename": "0009157.pdf", "url": "https://authors.library.caltech.edu/records/s0sjj-kfm53/files/0009157.pdf" }, "pub_year": "2000", "author_list": "Marcolli, Matilde and Wang, Bai-Ling" }, { "id": "https://authors.library.caltech.edu/records/zhvr5-v9796", "eprint_id": 79030, "eprint_status": "archive", "datestamp": "2023-08-19 06:20:06", "lastmod": "2026-03-09 21:47:58", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" }, { "id": "Wang-B-L", "name": { "family": "Wang", "given": "Bai-Ling" } } ] }, "title": "Exact triangles in Seiberg-Witten Floer theory. Part IV: Z-graded monopole homology", "ispublished": "unpub", "full_text_status": "public", "note": "Both the authors benefited of conversations with Ron Wang on some of the issues discussed in this paper. We thank Vicente Mu\u00f1oz for his help in formulating the gluing formulae. We also thank the referee for useful comments. We thank the Max Planck Institut f\u00fcr Mathematik for the kind hospitality and for support. The first author is partially supported by NSF grant DMS-9802480. The second author is partially supported by ARC Fellowship.\n\n

Submitted - 0009159.pdf

", "abstract": "Some aspects of the construction of SW Floer homology for manifolds with non-trivial rational homology are analyzed. In particular, the case of manifolds that are obtained as zero-surgery on a knot in a homology sphere, and for torsion spinc structures. We discuss relative invariants in the case of torsion spinc structures.", "date": "2000-09-15", "date_type": "published", "id_number": "CaltechAUTHORS:20170712-151923729", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-151923729", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-9802480" }, { "agency": "Australian Research Council" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.0009159", "primary_object": { "basename": "0009159.pdf", "url": "https://authors.library.caltech.edu/records/zhvr5-v9796/files/0009159.pdf" }, "pub_year": "2000", "author_list": "Marcolli, Matilde and Wang, Bai-Ling" }, { "id": "https://authors.library.caltech.edu/records/8cb8c-89p40", "eprint_id": 79073, "eprint_status": "archive", "datestamp": "2023-08-19 06:20:15", "lastmod": "2026-03-09 21:43:26", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" }, { "id": "Wang-Bai-Ling", "name": { "family": "Wang", "given": "Bai-Ling" } } ] }, "title": "Exact triangles in Seiberg-Witten Floer theory. Part II: geometric limits of flow lines", "ispublished": "unpub", "full_text_status": "public", "note": "(Submitted on 12 Jul 1999 (v1), last revised 15 Sep 2000 (this version, v2) \n\nWe are deeply grateful to Tom Mrowka who corrected several mistakes in our understanding of the convergence and gluing of flow lines, and referred us to the results of [5] and to their rescaling technique as a possible model for our problem. The first author benefited of several conversations with T.R. Ramadas, who suggested the use of the radial gauge limits. Part of this work was done during visits of the first author to the Tata Institute of Fundamental Research in Mumbai, and to the Max Planck Institut f\u00fcr Mathematik in Bonn. We thank these institutions for the kind hospitality and for support. The first author is partially supported by NSF grant DMS-9802480. The second author is supported by ARC Fellowship.\n\n

Submitted - 9907080.pdf

", "abstract": "This is the second part of the proof of the exact traiangles in Seiberg-Witten Floer theory. We analyse the splitting and gluing of flow lines of the Chern-Simons-Dirac functional when the underlying three-manifold splits along a torus. (two corrections added)", "date": "2000-09-15", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-094456115", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-094456115", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Max Planck Institut f\u00fcr Mathematik" }, { "agency": "NSF", "grant_number": "DMS-9802480" }, { "agency": "Australian Research Council" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.9907080", "primary_object": { "basename": "9907080.pdf", "url": "https://authors.library.caltech.edu/records/8cb8c-89p40/files/9907080.pdf" }, "pub_year": "2000", "author_list": "Marcolli, Matilde and Wang, Bai-Ling" }, { "id": "https://authors.library.caltech.edu/records/hgv8d-csh56", "eprint_id": 72939, "eprint_status": "archive", "datestamp": "2023-08-19 04:53:58", "lastmod": "2026-03-09 21:54:25", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Ooguri-H", "name": { "family": "Ooguri", "given": "Hirosi" }, "orcid": "0000-0001-6021-3778" }, { "id": "Robins-H", "name": { "family": "Robins", "given": "Harlan" } }, { "id": "Tannenhauser-J", "name": { "family": "Tannenhauser", "given": "Jonathan" } } ] }, "title": "Glueballs and Their Kaluza-Klein Cousins", "ispublished": "unpub", "full_text_status": "public", "note": "We thank Csaba Cs\u00e1ki, Aki Hashimoto, Yaron Oz, John Terning, and especially David Gross for useful discussions. We thank the Institute for Theoretical Physics at Santa Barbara for its hospitality. This work was supported in part by the NSF grant PHY-95-14797 and the DOE grant DE-AC03-76SF00098, and in part by the NSF grant PHY-94-07194 through ITP. H.R. and J.T. gratefully acknowledge the support of the A. Carl Helmholz Fellowship in the Department of Physics at the University of California, Berkeley.\n\n

Submitted - 9806171.pdf

", "abstract": "Spectra of glueball masses in non-supersymmetric Yang-Mills theory in three and four dimensions have recently been computed using the conjectured duality between superstring theory and large N gauge theory. The Kaluza-Klein states of supergravity do not correspond to any states in the Yang-Mills theory and therefore should decouple in the continuum limit. On the other hand, in the supergravity limit g_(YM)^2 N \u2192 \u221e, we find that the masses of the Kaluza-Klein states are comparable to those of the glueballs. We also show that the leading (g_(YM)^2N)^(-1) corrections do not make these states heavier than the glueballs. Therefore, the decoupling of the Kaluza-Klein states is not evident to this order.", "date": "1999-11-23", "date_type": "published", "id_number": "CaltechAUTHORS:20161219-110218427", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161219-110218427", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "PHY-95-14797" }, { "agency": "Department of Energy (DOE)", "grant_number": "DE-AC03-76SF00098" }, { "agency": "NSF", "grant_number": "PHY-94-07194" }, { "agency": "University of California, Berkeley" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.9911027v1", "primary_object": { "basename": "9806171.pdf", "url": "https://authors.library.caltech.edu/records/hgv8d-csh56/files/9806171.pdf" }, "pub_year": "1999", "author_list": "Ooguri, Hirosi; Robins, Harlan; et al." }, { "id": "https://authors.library.caltech.edu/records/vccg3-3as69", "eprint_id": 99180, "eprint_status": "archive", "datestamp": "2023-08-19 04:09:04", "lastmod": "2026-03-18 04:19:47", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Cohen-A-M", "name": { "family": "Cohen", "given": "Arjeh M." } }, { "id": "Steinbach-A", "name": { "family": "Steinbach", "given": "Anja" } }, { "id": "Ushirobira-R", "name": { "family": "Ushirobira", "given": "Rosane" } }, { "id": "Wales-D-B", "name": { "family": "Wales", "given": "David" } } ] }, "title": "Lie algebras generated by extremal elements", "ispublished": "unpub", "full_text_status": "public", "note": "

Submitted - 9903077.pdf

", "abstract": "We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals of L) over a field of characteristic distinct from 2. We prove that any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal number of extremal generators for the Lie algebras of type A_n (n\u22651), B_n (n\u22653), C_n (n\u22652), D_n (n\u22654), E_n (n = 6, 7, 8), F_4 and G_2 are shown to be n+1, n+1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.", "date": "1999-03-12", "date_type": "published", "id_number": "CaltechAUTHORS:20191009-092215716", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191009-092215716", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" }, { "id": "Physics-Department" } ] }, "doi": "10.48550/arXiv.9903077", "primary_object": { "basename": "9903077.pdf", "url": "https://authors.library.caltech.edu/records/vccg3-3as69/files/9903077.pdf" }, "pub_year": "1999", "author_list": "Cohen, Arjeh M.; Steinbach, Anja; et al." }, { "id": "https://authors.library.caltech.edu/records/023hn-a7782", "eprint_id": 79072, "eprint_status": "archive", "datestamp": "2023-08-19 02:37:43", "lastmod": "2026-03-09 21:40:20", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marcolli-M", "name": { "family": "Marcolli", "given": "Matilde" }, "orcid": "0000-0002-2045-2907" }, { "id": "Mathai-V", "name": { "family": "Mathai", "given": "Varghese" } } ] }, "title": "Twisted higher index theory on good orbifolds and fractional quantum numbers", "ispublished": "unpub", "full_text_status": "public", "keywords": "Fractional quantum numbers, Quantum Hall Effect, hyperbolic space, orbifolds, C\u2217-\nalgebras, K-theory, cyclic cohomology, Fuchsian groups, Harper operator, Baum-Connes conjecture.", "note": "(Submitted on 12 Mar 1998) \n\nThe second author thanks A. Carey and K. Hannabus for some clarifying comments concerning section 6.\n\n

Submitted - 9803051.pdf

", "abstract": "In this paper, we study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group, and we apply these results to obtain qualitative results, related\nto generalizations of the Bethe-Sommerfeld conjecture, on the spectrum of self adjoint elliptic operators which are invariant under a projective action of the orbifold fundamental group. We also compute the range of the higher traces on K-theory, which we then apply to compute the\nrange of values of the Hall conductance in the quantum Hall effect on the hyperbolic plane. The new phenomenon that we observe in this case is that the Hall conductance again has plateaus at all energy levels belonging to any gap in the spectrum of the Hamiltonian, where it is now shown to be equal to an integral multiple of a fractional valued invariant. Moreover the set of possible denominators is finite and has been explicitly determined. It is plausible that this might shed light on the mathematical mechanism responsible for fractional quantum numbers.", "date": "1998-03-12", "date_type": "published", "id_number": "CaltechAUTHORS:20170713-093723319", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170713-093723319", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.9803051", "primary_object": { "basename": "9803051.pdf", "url": "https://authors.library.caltech.edu/records/023hn-a7782/files/9803051.pdf" }, "pub_year": "1998", "author_list": "Marcolli, Matilde and Mathai, Varghese" }, { "id": "https://authors.library.caltech.edu/records/vq5xe-vp153", "eprint_id": 73068, "eprint_status": "archive", "datestamp": "2023-08-19 00:42:31", "lastmod": "2026-03-09 22:00:52", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Ooguri-H", "name": { "family": "Ooguri", "given": "Hirosi" }, "orcid": "0000-0001-6021-3778" }, { "id": "Yin-Zheng", "name": { "family": "Yin", "given": "Zheng" } } ] }, "title": "TASI Lectures on Perturbative String Theories", "ispublished": "unpub", "full_text_status": "public", "note": "We would like to thank the school organizer K.T. Mahanthappa and the program director B. Greene for their beautiful planning and organization of the school. We thank J. Harvey and J. Polchinski for their useful suggestions on the outline of the course and for J. Feng and M. Peskin for having a preliminary draft of their forthcoming book available to us. We are also grateful to C-S. Chu, Y. Oz and H. Steinacker for reading the draft. This work was supported in part by the National Science Foundation under grants PHY-951497 and in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098. Z.Y. is supported in part by a Graduate Research Fellowship of the U.S. Department of Education.\n\n

Submitted - 9612254.pdf

", "abstract": "These lecture notes are based on a course on string theories given by Hirosi Ooguri in the first week of TASI 96 Summer School at Boulder, Colorado. It is an introductory course designed to provide students with minimum knowledge before they attend more advanced courses on non-perturbative aspects of string theories in the School. The course consists of five lectures: 1. Bosonic String, 2. Toroidal Compactifications, 3. Superstrings, 4. Heterotic Strings, and 5. Orbifold Compactifications.", "date": "1996-12-31", "date_type": "published", "id_number": "CaltechAUTHORS:20161221-105050141", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161221-105050141", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "PHY-951497" }, { "agency": "Department of Energy (DOE)", "grant_number": "DE-AC03-76SF00098" }, { "agency": "Department of Education" } ] }, "local_group": { "items": [ { "id": "Mathematics-Department" } ] }, "doi": "10.48550/arXiv.9612254", "primary_object": { "basename": "9612254.pdf", "url": "https://authors.library.caltech.edu/records/vq5xe-vp153/files/9612254.pdf" }, "pub_year": "1996", "author_list": "Ooguri, Hirosi and Yin, Zheng" } ]