This thesis develops the theory of weighted quadrature domains in parallel with the Faber transform as a tool for their analysis and explicit construction. Under this framework,e obtain a number of existence, uniqueness, and classification results for classical and weighted quadrature domains.
\r\n\r\nIn the classical setting, we derive explicit formulae relating the Riemann map of a simply connected quadrature domain to its quadrature function via the Faber transform. This reduces the inverse and direct problems to a problem of solving finitely many algebraic equations. Applying these results, along with tools from logarithmic potential theory, we obtain a complete classification of one-point quadrature domains with complex charge.
\r\n\r\nWe then introduce power-weighted quadrature domains (PQDs) - domains admitting a quadrature identity with respect to the weight \u03c1a(w) = \u2223w\u22232(a-1) for some a>0. A central structural result is that a simply connected domain is a PQD if and only if the ath power of the outer factor of its Riemann map extends to a rational function. This characterization yields Faber transform formulae analogous to the classical case, which we apply to partially classify one-point and monomial PQD families. Novel boundary phenomena - including the formation of boundary \"corners\" at the origin with angles that are integer multiples of \u03c0/a - are exhibited.
\r\n\r\nNext, we develop the theory of log-weighted quadrature domains (LQDs) - domains admitting a quadrature identity with respect to the weight \u03c10 = \u2223w\u2223-2, the limiting case a\u21920\u207a of \u03c1a |w|2(a-1). The non-integrable singularity at the origin introduces new phenomena: when the domain contains the origin, the quadrature function is no longer uniquely determined, but only up to the addition of a point charge q/w. Despite this loss of uniqueness, we show that a simply connected domain is an LQD if and only if the outer factor of its Riemann map extends to the exponential of a rational function. Classification results for null, monomial, and one-point LQD families are obtained, and the connection to classical quadrature domains via the exponential map is established.
\r\n\r\nFinally, we introduce algebraic quadrature domains (AQDs), defined with respect to weights of the form \u03c1R = |R'|\u00b2, where R is a non-constant rational function. This class subsumes both classical quadrature domains (R(w)=w) and integer power-weighted quadrature domains (R(w)=wn/n). We derive representation formulae in terms of the Faber transform and present several examples.
", "doi": "10.7907/tgqd-x974", "publication_date": "2026", "thesis_type": "phd", "thesis_year": "2026" }, { "id": "thesis:16730", "collection": "thesis", "collection_id": "16730", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:09162024-065933174", "type": "thesis", "title": "Spin Geometry and Quantum Diffusion", "author": [ { "family_name": "Gakkhar", "given_name": "Sitanshu", "orcid": "0000-0002-2895-9921", "clpid": "Gakkhar-Sitanshu" } ], "thesis_advisor": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "\u0106a\u0107i\u0107", "given_name": "Branimir Josip", "orcid": "0000-0002-4721-778X", "clpid": "\u0106a\u0107i\u0107-Branimir-Josip" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis studies diffusion processes on spinor endomorphism algebras. The spinor and connection laplacian generated heat semigroups are shown to quantum dynamical semigroups, and after spectral truncation the existence of Evans-Hudson flows is established. The vacuum state expectation of the process is related to the spectral action principle in noncommutative geometry. Examples where the flow is proven to exist for untruncated laplacians are given. Convergence of finite dimensional approximations, through discretization and truncation, to spectral triples encoding Riemannian geometry and their statespaces as quantum metric spaces is also considered.", "doi": "10.7907/ded0-hn95", "publication_date": "2025", "thesis_type": "phd", "thesis_year": "2025" }, { "id": "thesis:17252", "collection": "thesis", "collection_id": "17252", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05202025-052235420", "primary_object_url": { "basename": "zhang_jiaxin_2025_thesis.pdf", "content": "final", "filesize": 10784352, "license": "other", "mime_type": "application/pdf", "url": "/17252/1/zhang_jiaxin_2025_thesis.pdf", "version": "v6.0.0" }, "type": "thesis", "title": "On Multiple SLE Systems and their Deterministic Limits", "author": [ { "family_name": "Zhang", "given_name": "Jiaxin", "orcid": "0000-0002-5647-8949", "clpid": "Zhang-Jiaxin" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Hutchcroft", "given_name": "Thomas", "orcid": "0000-0003-0061-593X", "clpid": "Hutchcroft-Tom" }, { "family_name": "Yu", "given_name": "Tony Yue", "orcid": "0000-0002-6019-8552", "clpid": "Yu-Tony-Yue" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this thesis, we study multiple radial SLE(k) systems -- a family of random multi-curve systems in a simply-connected domain \u03a9, with marked boundary points z\u2081....z\u2099 \\in \u2202\u03a9 and a marked interior point q, where parameter k > 0 measures the randomness of the system. We also study the multiple radial SLE(0) systems as the deterministic limit of multiple radial SLE(k) systems.
\r\n\r\nAs a consequence of domain Markov property and conformal invariance, we derive that a multiple radial SLE(k) system is characterized by a conformally covariant partition function satisfying the null vector equations--a second-order PDE system. On the other hand, using the Coulomb gas method inspired by conformal field theory, we construct four types of solutions to the null vector equations, which can be classified according to topological link patterns.
\r\n\r\nWe construct the multiple radial SLE(0) systems from stationary relations by heuristically taking the classical limit of partition functions as k > 0. By constructing the field integrals of motion for the Loewner dynamics, we show that the traces of multiple radial SLE(0) systems are the horizontal trajectories of an equivalence class of quadratic differentials. These trajectories have limiting ends at the growth points and form a radial link pattern.
\r\n\r\nThe stationary relations connect the classification of multiple radial SLE(0) systems to the enumeration of critical points of the master function of trigonometric Knizhnik-Zamolodchikov (KZ) equations.
\r\n\r\nFor k > 0$, the partition functions of multiple radial SLE(k) systems correspond to eigenstates of the quantum Calogero-Sutherland (CS) Hamiltonian beyond the fermionic states. In the deterministic case of k=0, we show that the Loewner dynamics with a common parametrization of capacity form a special class of classical CS systems, restricted to a submanifold of phase space defined by the Lax matrix.
", "doi": "10.7907/spf7-9j65", "publication_date": "2025-06-13", "thesis_type": "phd", "thesis_year": "2025" }, { "id": "thesis:14306", "collection": "thesis", "collection_id": "14306", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:07162021-010608172", "primary_object_url": { "basename": "shubin_andrei_2021.pdf", "content": "final", "filesize": 808387, "license": "other", "mime_type": "application/pdf", "url": "/14306/1/shubin_andrei_2021.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Topics in Equidistribution and Exponential Sums", "author": [ { "family_name": "Shubin", "given_name": "Andrei", "orcid": "0000-0002-8702-3218", "clpid": "Shubin-Andrei" } ], "thesis_advisor": [ { "family_name": "Radziwi\u0142\u0142", "given_name": "Maksym", "orcid": "0009-0002-2756-5856", "clpid": "Radziwi\u0142\u0142-M" } ], "thesis_committee": [ { "family_name": "Ramakrishnan", "given_name": "Dinakar", "orcid": "0000-0002-0159-087X", "clpid": "Ramakrishnan-D" }, { "family_name": "Radziwi\u0142\u0142", "given_name": "Maksym", "orcid": "0009-0002-2756-5856", "clpid": "Radziwi\u0142\u0142-M" }, { "family_name": "Dunn", "given_name": "Alexander", "orcid": "0000-0003-1665-7114", "clpid": "Dunn-Alexander" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this thesis, we consider a few problems connected to the exponential sums which is one of the most important topics in analytic number theory.
\r\n\r\nIn the first part, we study the distribution of prime numbers in special subsets of integers and, in particular, the distribution of these primes in arithmetic progressions, small gaps between them, the behavior of the corresponding exponential sums over primes, and related questions. Big progress was made on these questions in recent years. The famous works of Zhang and Maynard gave the proof of existence of bounded gaps between consecutive primes. Applying the sieve of Selberg-Maynard-Tao and an analogue of the Bombieri-Vinogradov theorem, we obtain similar results for a large class of subsets of primes and improve some of the previous results. The proof of the analogue of the Bombieri-Vinogradov theorem is also connected to a breakthrough work of Bourgain, Demeter, and Guth on the proof of Vinogradov Mean Value Conjecture via l2-decoupling. Their result, in particular, has led to a significant improvement of the classical van der Corput estimates for a large class of exponential sums.
\r\n\r\nIn the second part, we study the behavior of higher moments of Gauss sum twisted by a Mobius function. The moments of exponential sums are very important in number theory and harmonic analysis as they appear in many other problems. The sum with the Mobius function is of independent interest because of the famous Sarnak Conjecture which is on the edge of number theory, analysis, and dynamical systems. The bound we obtain for Lp-norm of the sum confirms that the Mobius function is uncorrelated with the quadratic phase \u03b1n2 for most \u03b1 \u03f5 [0; 1].
\r\n\r\nIn the third part, we study the distribution of lattice points on the surface of 3-dimensional sphere, which is known as Linnik problem. It turns out that the variance for such points is closely related to the behavior of certain GL(2) L-functions estimated at the central point 1/2. To evaluate the moments of these L-functions, we apply similar techniques used to evaluate the moments of Riemann zeta function on the critical line in the breakthrough works of Soundararajan and Harper. Their results have led to the sharp upper bounds for all positive moments of zeta function conditionally on Riemann Hypothesis and similar bounds for a broad class of L-functions in families conditionally on the corresponding Grand Riemann Hypothesis. We apply similar methods to get sharp upper bound for the variance of lattice points on the sphere. The connection of Weyl sums on the sphere to the sums of special values of GL(2) L-functions is a big output of the Langlands program, which has also gotten a lot of attention in recent years.
", "doi": "10.7907/153e-5r72", "publication_date": "2022", "thesis_type": "phd", "thesis_year": "2022" }, { "id": "thesis:14634", "collection": "thesis", "collection_id": "14634", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05272022-034901698", "primary_object_url": { "basename": "tao_zijian_2022.pdf", "content": "final", "filesize": 420897, "license": "other", "mime_type": "application/pdf", "url": "/14634/1/tao_zijian_2022.pdf", "version": "v6.0.0" }, "type": "thesis", "title": "Theory of Mathematical Optimization for Delegated Portfolio Management", "author": [ { "family_name": "Tao", "given_name": "Zijian", "clpid": "Tao-Zijian" } ], "thesis_advisor": [ { "family_name": "Cvitani\u0107", "given_name": "Jak\u0161a", "orcid": "0000-0001-6651-3552", "clpid": "Cvitani\u0107-J" } ], "thesis_committee": [ { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" }, { "family_name": "Cvitani\u0107", "given_name": "Jak\u0161a", "orcid": "0000-0001-6651-3552", "clpid": "Cvitani\u0107-J" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Sandomirskiy", "given_name": "Fedor", "orcid": "0000-0001-9886-3688", "clpid": "Sandomirskiy-Fedor" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We study the optimization problem of finding closed convex sets \u0393 ⊆ Rd containing the origin that minimize F(\u0393) = \u2211i=1k wi | \u03b8i/2 - p\u0393(\u03b8i) | 2, where w1, ..., wk > 0, \u03b81, ..., \u03b8k in Rd are given, and p\u0393(\u03b8i) are the closest points in \u0393 to \u03b8i, i = 1, ..., k. This problem is motivated by the topic of delegated portfolio management in finance. In Chapter 2, we will explore this connection. To approach the problem, we first prove existence of a solution for the general problem. To further study properties of the solution, we next introduce the semidefinite programming relaxation, for which we have a first-order characterization of optimality. We then explore the question of exactness of this relaxation, which turns out to be equivalent to the notion of localizability: the shape optimization problem embedded in higher dimensions must have solutions in the original dimension. Finally, we present special cases for which localizability holds.
", "doi": "10.7907/km2b-er60", "publication_date": "2022", "thesis_type": "phd", "thesis_year": "2022" }, { "id": "thesis:13868", "collection": "thesis", "collection_id": "13868", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:09062020-172055989", "primary_object_url": { "basename": "Thesis17_GeneRyanYoo.pdf", "content": "final", "filesize": 5463122, "license": "other", "mime_type": "application/pdf", "url": "/13868/1/Thesis17_GeneRyanYoo.pdf", "version": "v6.0.0" }, "type": "thesis", "title": "Learning Patterns with Kernels and Learning Kernels from Patterns", "author": [ { "family_name": "Yoo", "given_name": "Gene Ryan", "orcid": "0000-0002-5319-5599", "clpid": "Yoo-Gene-Ryan" } ], "thesis_advisor": [ { "family_name": "Owhadi", "given_name": "Houman", "orcid": "0000-0002-5677-1600", "clpid": "Owhadi-H" } ], "thesis_committee": [ { "family_name": "Stuart", "given_name": "Andrew M.", "orcid": "0000-0001-9091-7266", "clpid": "Stuart-A-M" }, { "family_name": "Owhadi", "given_name": "Houman", "orcid": "0000-0002-5677-1600", "clpid": "Owhadi-H" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Schroeder", "given_name": "Peter", "orcid": "0000-0002-0323-7674", "clpid": "Schr\u00f6der-P" }, { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "A major technique in learning involves the identification of patterns and their use to make predictions. In this work, we examine the symbiotic relationship between patterns and Gaussian process regression (GPR), which is mathematically equivalent to kernel interpolation. We introduce techniques where GPR can be used to learn patterns in denoising and mode (signal) decomposition. Additionally, we present the kernel flow (KF) algorithm which learns a kernels from patterns in the data with methodology inspired by cross validation. We further show how the KF algorithm can be applied to artificial neural networks (ANNs) to make improvements to learning patterns in images.
\r\n\r\nIn our denoising and mode decomposition examples, we show how kernels can be constructed to estimate patterns that may be hidden due to data corruption. In other words, we demonstrate how to learn patterns with kernels. Donoho and Johnstone proposed a near-minimax method for reconstructing an unknown smooth function u from noisy data u + \u03b6 by translating the empirical wavelet coefficients of u + \u03b6 towards zero. We consider the situation where the prior information on the unknown function u may not be the regularity of u, but that of \u2112u where \u2112 is a linear operator, such as a partial differential equation (PDE) or a graph Laplacian. We show that a near-minimax approximation of u can be obtained by truncating the \u2112-gamblet (operator-adapted wavelet) coefficients of u + \u03b6. The recovery of u can be seen to be precisely a Gaussian conditioning of u + \u03b6 on measurement functions with length scale dependent on the signal-to-noise ratio.
\r\n\r\nWe next introduce kernel mode decomposition (KMD), which has been designed to learn the modes vi = ai(t)yi(\u03b8i(t)) of a (possibly noisy) signal \u03a3ivi when the amplitudes ai, instantaneous phases \u03b8i, and periodic waveforms yi may all be unknown. GPR with Gabor wavelet-inspired kernels is used to estimate ai, \u03b8i, and yi. We show near machine precision recovery under regularity and separation assumptions on the instantaneous amplitudes ai and frequencies ˙\u03b8i.
\r\n\r\nGPR and kernel interpolation require the selection of an appropriate kernel modeling the data. We present the KF algorithm, which is a numerical-approximation approach to this selection. The main principle the method utilizes is that a \"good\" kernel is able to make accurate predictions with small subsets of a training set. In this way, we learn a kernel from patterns. In image classification, we show that the learned kernels are able to classify accurately using only one training image per class and show signs of unsupervised learning. Furthermore, we introduce the combination of the KF algorithm with conventional neural-network training. This combination is able to train the intermediate-layer outputs of the network simultaneously with the final-layer output. We test the proposed method on Convolutional Neural Networks (CNNs) and Wide Residual Networks (WRNs) without alteration of their structure or their output classifier. We report reduced test errors, decreased generalization gaps, and increased robustness to distribution shift without significant increase in computational complexity relative to standard CNN and WRN training (with Drop Out and Batch Normalization).
\r\n\r\nAs a whole, this work highlights the interplay between kernel techniques with pattern recognition and numerical approximation.
", "doi": "10.7907/c5fn-ac81", "publication_date": "2021", "thesis_type": "phd", "thesis_year": "2021" }, { "id": "thesis:10207", "collection": "thesis", "collection_id": "10207", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05252017-092503939", "primary_object_url": { "basename": "main.pdf", "content": "final", "filesize": 1104452, "license": "other", "mime_type": "application/pdf", "url": "/10207/1/main.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Mathematical Results on Quantum Many-Body Physics", "author": [ { "family_name": "Lemm", "given_name": "Marius Christopher", "orcid": "0000-0001-6459-8046", "clpid": "Lemm-Marius-Christopher" } ], "thesis_advisor": [ { "family_name": "Frank", "given_name": "Rupert L", "orcid": "0000-0001-7973-4688", "clpid": "Frank-R-L" } ], "thesis_committee": [ { "family_name": "Frank", "given_name": "Rupert L.", "orcid": "0000-0001-7973-4688", "clpid": "Frank-R-L" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Motrunich", "given_name": "Olexei I.", "orcid": "0000-0001-8031-0022", "clpid": "Motrunich-Olexei" }, { "family_name": "Simon", "given_name": "Barry M.", "orcid": "0000-0003-2561-8539", "clpid": "Simon-B" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The collective behavior exhibited by a large number of microscopic quantum particles is at the heart of some of the most striking phenomena in condensed matter physics such as Bose-Einstein condensation and superconductivity. Physicists and mathematicians have made great progress in understanding when and how these collective phenomena emerge through the interplay of particle statistics, particle interaction and the value of thermodynamic parameters like the temperature or the chemical potential. Due to the extreme complexity of realistic many-body systems, it is natural to introduce appropriate simplifications to render their analysis feasible. Three examples of such simplifications which have proven themselves as viable starting points for a fruitful and mathematically rigorous analysis of many-body systems are the following: (a) the study of integrable models; (b) the derivation of effective theories, valid on a macroscopic scale, from more fundamental microscopic theories under appropriate coarse-graining; and (c) the use of quantum information theory to understand general connections between correlation, entanglement and particle statistics.
\r\n\r\nIn this thesis, we present mathematically rigorous results that were obtained in these three directions. (1) We prove anomalous quantum many-body transport in XY quantum spin chains for certain choices of the external magnetic field. The anomalous transport is described via new kinds of anomalous Lieb-Robinson bounds, including one of power-law type. We note that the XY spin chain is integrable as it can be mapped to free fermions via the non-local Jordan-Wigner transformation. (2) We derive effective macroscopic theories of Ginzburg-Landau type from the microscopic BCS theory of superconductivity in certain circumstances. We study the case of a multi-component order parameter for translation-invariant systems and the condensation of fermion pairs at zero temperature in a domain with a hard boundary. (3) We use techniques from quantum information-theory to derive bounds on the entropy of fermionic reduced density matrices, a measure of the entanglement inherent to a fermionic quantum state.
", "doi": "10.7907/Z9D21VNV", "publication_date": "2017", "thesis_type": "phd", "thesis_year": "2017" }, { "id": "thesis:9960", "collection": "thesis", "collection_id": "9960", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:10262016-165619613", "primary_object_url": { "basename": "melissayeung_thesis.pdf", "content": "final", "filesize": 6669444, "license": "other", "mime_type": "application/pdf", "url": "/9960/1/melissayeung_thesis.pdf", "version": "v7.0.0" }, "type": "thesis", "title": "Applied Computational Topology for Point Clouds and Sparse Timeseries Data", "author": [ { "family_name": "Yeung", "given_name": "Melissa L.", "clpid": "Yeung-Melissa-L" } ], "thesis_advisor": [ { "family_name": "Desbrun", "given_name": "Mathieu", "orcid": "0000-0003-3424-6079", "clpid": "Desbrun-M" } ], "thesis_committee": [ { "family_name": "Desbrun", "given_name": "Mathieu", "orcid": "0000-0003-3424-6079", "clpid": "Desbrun-M" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The proliferation of sensors and advancement of technology has led to the production and collection of unprecedented amounts of data in recent years. The data are often noisy, non-linear, and high-dimensional, and the effectiveness of traditional tools may be limited. Thus, the technological advances that enable the ubiquitous collection of data from the cosmological scale to the subatomic scale also necessitate the development of complementary tools that address the new nature of the data.
\r\n\r\nRecently, there has been much interest in and success with developing topologically-motivated techniques for data analysis. These approaches are especially useful when a topological method is sensitive to large- and small-scale features that might not be detected by methods that require a level of geometric detail that is not provided by the data or by methods that may obscure geometric features, such as principal component analysis (PCA), multi\u2013dimensional scaling (MDS), and cluster analysis.
\r\n\r\nOur work explores topological data analysis through two frameworks.
\r\n\r\nIn the first part, we provide a tool for detecting material coherence from a set of spatially sparse particle trajectories via the study of a map induced on homology by the braid corresponding to the motion of particles. While the theory of coherent structures has received a great deal of attention and benefited from many advances in recent years, many of these techniques are limited when the data are sparse. We demonstrate through various examples that our work provides a practical and scalable tool for identifying coherent sets from a sparse set of particle trajectories using eigenanalysis.
\r\n\r\nIn the second part, we formalize the local-to-global structure captured by topology in the setting of point clouds. We extend existing tools in topological data analysis and provide a theoretical framework for studying topological features of a point cloud over a range of resolutions, enabling the analysis of topological features using statistical methods. We apply our tools to the analysis of high-dimensional geospatial sensor data and provide a statistic for quantifying climate anomalies.
", "doi": "10.7907/Z9D798DF", "publication_date": "2017", "thesis_type": "phd", "thesis_year": "2017" }, { "id": "thesis:8915", "collection": "thesis", "collection_id": "8915", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05292015-072640484", "primary_object_url": { "basename": "BSkinnerFinalThesis.pdf", "content": "final", "filesize": 455400, "license": "other", "mime_type": "application/pdf", "url": "/8915/1/BSkinnerFinalThesis.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Logarithmic Potential Theory on Riemann Surfaces", "author": [ { "family_name": "Skinner", "given_name": "Brian Paul", "clpid": "Skinner-Brian-Paul" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Alberts", "given_name": "Tom", "clpid": "Alberts-T" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We develop a logarithmic potential theory on Riemann surfaces which generalizes logarithmic potential theory on the complex plane. We show the existence of an equilibrium measure and examine its structure. This leads to a formula for the structure of the equilibrium measure which is new even in the plane. We then use our results to study quadrature domains, Laplacian growth, and Coulomb gas ensembles on Riemann surfaces. We prove that the complement of the support of the equilibrium measure satisfies a quadrature identity. Furthermore, our setup allows us to naturally realize weak solutions of Laplacian growth (for a general time-dependent source) as an evolution of the support of equilibrium measures. When applied to the Riemann sphere this approach unifies the known methods for generating interior and exterior Laplacian growth. We later narrow our focus to a special class of quadrature domains which we call Algebraic Quadrature Domains. We show that many of the properties of quadrature domains generalize to this setting. In particular, the boundary of an Algebraic Quadrature Domain is the inverse image of a planar algebraic curve under a meromorphic function. This makes the study of the topology of Algebraic Quadrature Domains an interesting problem. We briefly investigate this problem and then narrow our focus to the study of the topology of classical quadrature domains. We extend the results of Lee and Makarov and prove (for n ≥ 3) c ≤ 5n-5, where c and n denote the connectivity and degree of a (classical) quadrature domain. At the same time we obtain a new upper bound on the number of isolated points of the algebraic curve corresponding to the boundary and thus a new upper bound on the number of special points. In the final chapter we study Coulomb gas ensembles on Riemann surfaces.", "doi": "10.7907/Z9Q52MK8", "publication_date": "2015", "thesis_type": "phd", "thesis_year": "2015" }, { "id": "thesis:8975", "collection": "thesis", "collection_id": "8975", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06032015-011726054", "primary_object_url": { "basename": "linghu_daiqi_2015_thesis.pdf", "content": "final", "filesize": 1214103, "license": "other", "mime_type": "application/pdf", "url": "/8975/1/linghu_daiqi_2015_thesis.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Chains of Non-Regular de Branges Spaces", "author": [ { "family_name": "Linghu", "given_name": "Daiqi", "clpid": "Linghu-Daiqi" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Frank", "given_name": "Rupert L.", "clpid": "Frank-R-L" }, { "family_name": "Katz", "given_name": "Nets H.", "orcid": "0000-0002-6239-5429", "clpid": "Katz-N-H" }, { "family_name": "Silva", "given_name": "Elwadura Prabath S.", "clpid": "Silva-E-P-S" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We consider canonical systems with singular left endpoints, and discuss the concept of a scalar\r\nspectral measure and the corresponding generalized Fourier transform associated with a canonical\r\nsystem with a singular left endpoint. We use the framework of de Branges\u2019 theory of Hilbert spaces\r\nof entire functions to study the correspondence between chains of non-regular de Branges\r\nspaces, canonical systems with singular left endpoints, and spectral measures.
\r\n\r\nWe find sufficient integrability conditions on a Hamiltonian H which ensure the existence of a\r\nchain of de Branges functions in the first generalized P\u00f3lya class with Hamiltonian H. This result\r\ngeneralizes de Branges\u2019 Theorem 41, which showed the sufficiency of stronger integrability\r\nconditions on H for the existence of a chain in the P\u00f3lya class. We show the conditions that de\r\nBranges came up with are also necessary. In the case of Krein\u2019s strings, namely when the Hamiltonian\r\nis diagonal, we show our proposed conditions are also necessary.
\r\n\r\nWe also investigate the asymptotic conditions on chains of de Branges functions as t approaches\r\nits left endpoint. We show there is a one-to-one correspondence between chains of de Branges\r\nfunctions satisfying certain asymptotic conditions and chains in the P\u00f3lya class. In the case of\r\nKrein\u2019s strings, we also establish the one-to-one correspondence between chains satisfying certain\r\nasymptotic conditions and chains in the generalized P\u00f3lya class.
", "doi": "10.7907/Z9C24TD4", "publication_date": "2015", "thesis_type": "phd", "thesis_year": "2015" }, { "id": "thesis:7716", "collection": "thesis", "collection_id": "7716", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05162013-102038765", "primary_object_url": { "basename": "Robin Tucker-Drob, Edited Thesis.pdf", "content": "final", "filesize": 1273374, "license": "other", "mime_type": "application/pdf", "url": "/7716/1/Robin Tucker-Drob, Edited Thesis.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Descriptive Set Theory and the Ergodic Theory of Countable Groups", "author": [ { "family_name": "Tucker-Drob", "given_name": "Robin Daniel", "clpid": "Tucker-Drob-Robin-Daniel" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Sokic", "given_name": "Miodrag", "clpid": "Sokic-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The primary focus of this thesis is on the interplay of descriptive set theory and the ergodic theory of group actions. This incorporates the study of turbulence and Borel reducibility on the one hand, and the theory of orbit equivalence and weak equivalence on the other. Chapter 2 is joint work with Clinton Conley and Alexander Kechris; we study measurable graph combinatorial invariants of group actions and employ the ultraproduct construction as a way of constructing various measure preserving actions with desirable properties. Chapter 3 is joint work with Lewis Bowen; we study the property MD of residually finite groups, and we prove a conjecture of Kechris by showing that under general hypotheses property MD is inherited by a group from one of its co-amenable subgroups. Chapter 4 is a study of weak equivalence. One of the main results answers a question of Ab\u00e9rt and Elek by showing that within any free weak equivalence class the isomorphism relation does not admit classification by countable structures. The proof relies on affirming a conjecture of Ioana by showing that the product of a free action with a Bernoulli shift is weakly equivalent to the original action. Chapter 5 studies the relationship between mixing and freeness properties of measure preserving actions. Chapter 6 studies how approximation properties of ergodic actions and unitary representations are reflected group theoretically and also operator algebraically via a group's reduced C*-algebra. Chapter 7 is an appendix which includes various results on mixing via filters and on Gaussian actions.", "doi": "10.7907/ER96-5D06", "publication_date": "2013", "thesis_type": "phd", "thesis_year": "2013" }, { "id": "thesis:7252", "collection": "thesis", "collection_id": "7252", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:11032012-213910304", "primary_object_url": { "basename": "Chang-Sha-2013.pdf", "content": "final", "filesize": 666640, "license": "other", "mime_type": "application/pdf", "url": "/7252/1/Chang-Sha-2013.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Hele-Shaw Flow Near Cusp Singularities", "author": [ { "family_name": "Chang", "given_name": "Sha", "clpid": "Chang-Sha" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Simon", "given_name": "Barry M.", "orcid": "0000-0003-2561-8539", "clpid": "Simon-B" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Lee", "given_name": "Seung-Yeop", "clpid": "Lee-S-Y" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis discusses the radial version of the Hele-Shaw problem. Different from the channel version, traveling-wave solutions do not exist in this version. Under algebraic potentials, in the case that the droplets expand, in finite time, cusps will appear on the boundary and classical solutions may not exist afterwards. Physicists have suggested that for (2p+1,2)-cusps, that near cusp singularities of Hele-Shaw flow, after scaling X, Y by some powers of time t respectively, the main part of Y(X, t) is a one-parameter family and does not depend on time t. They have also suggested that the solutions of the Hele-Shaw problem are connected with dispersionless KdV (dKdV) hierarchy. In this study, we rigorously proved that this is the case for (3,2)-cusps when the droplets are simply connected and the external potentials are algebraic. We gave exact solutions and showed that the main parts of the exact solutions are some special solutions of the dispersionless string equation. More over, borrowed from the physical paper$\\cite{Teo}$ with a little more details, we showed the arguments of how these special solutions are related to dKdV hierarchy.", "doi": "10.7907/YJEK-W376", "publication_date": "2013", "thesis_type": "phd", "thesis_year": "2013" }, { "id": "thesis:7749", "collection": "thesis", "collection_id": "7749", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05242013-033054707", "type": "thesis", "title": "On Reconstruction Theorems in Noncommutative Riemannian Geometry", "author": [ { "family_name": "\u0106a\u0107i\u0107", "given_name": "Branimir Josip", "orcid": "0000-0002-4721-778X", "clpid": "\u0106a\u0107i\u0107-Branimir-Josip" } ], "thesis_advisor": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "thesis_committee": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Markovic", "given_name": "Vladimir", "clpid": "Markovic-V" }, { "family_name": "Venselaar", "given_name": "Jan Jitse", "clpid": "Venselaar-J-J" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We present a novel account of the theory of commutative spectral triples and their two closest noncommutative generalisations, almost-commutative spectral triples and toric noncommutative manifolds, with a focus on reconstruction theorems, viz, abstract, functional-analytic characterisations of global-analytically defined classes of spectral triples. We begin by reinterpreting Connes's reconstruction theorem for commutative spectral triples as a complete noncommutative-geometric characterisation of Dirac-type operators on compact oriented Riemannian manifolds, and in the process clarify folklore concerning stability of properties of spectral triples under suitable perturbation of the Dirac operator. Next, we apply this reinterpretation of the commutative reconstruction theorem to obtain a reconstruction theorem for almost-commutative spectral triples. In particular, we propose a revised, manifestly global-analytic definition of almost-commutative spectral triple, and, as an application of this global-analytic perspective, obtain a general result relating the spectral action on the total space of a finite normal compact oriented Riemannian cover to that on the base space. Throughout, we discuss the relevant refinements of these definitions and results to the case of real commutative and almost-commutative spectral triples. Finally, we outline progess towards a reconstruction theorem for toric noncommutative manifolds.", "doi": "10.7907/WRT2-7630", "publication_date": "2013", "thesis_type": "phd", "thesis_year": "2013" }, { "id": "thesis:7677", "collection": "thesis", "collection_id": "7677", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05082013-134706988", "primary_object_url": { "basename": "style_PhD.pdf", "content": "final", "filesize": 901327, "license": "other", "mime_type": "application/pdf", "url": "/7677/1/style_PhD.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Dirac Spectra, Summation Formulae, and the Spectral Action", "author": [ { "family_name": "Teh", "given_name": "Kevin Kai-Wen", "clpid": "Teh-Kevin-Kai-Wen" } ], "thesis_advisor": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "thesis_committee": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Markovic", "given_name": "Vladimir", "clpid": "Markovic-V" }, { "family_name": "Venselaar", "given_name": "Joannes Jitse", "clpid": "Venselaar-J-J" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Noncommutative geometry is a source of particle physics models with matter Lagrangians coupled to gravity. One may associate to any noncommutative space (A, H, D) its spectral action, which is defined in terms of the Dirac spectrum of its Dirac operator D. When viewing a spin manifold as a noncommutative space, D is the usual Dirac operator. In this paper, we give nonperturbative computations of the spectral action for quotients of SU(2), Bieberbach manifolds, and SU(3) equipped with a variety of geometries. Along the way we will compute several Dirac spectra and refer to applications of this computation.", "doi": "10.7907/Z545-KK47", "publication_date": "2013", "thesis_type": "phd", "thesis_year": "2013" }, { "id": "thesis:6242", "collection": "thesis", "collection_id": "6242", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:02132011-055211033", "primary_object_url": { "basename": "Nahid_Walji-thesis.pdf", "content": "final", "filesize": 509547, "license": "other", "mime_type": "application/pdf", "url": "/6242/1/Nahid_Walji-thesis.pdf", "version": "v6.0.0" }, "type": "thesis", "title": "Supersingular Distribution, Congruence Class Bias, and A Refinement of Strong Multiplicity One", "author": [ { "family_name": "Walji", "given_name": "Nahid", "clpid": "Walji-Nahid" } ], "thesis_advisor": [ { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" } ], "thesis_committee": [ { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Jorza", "given_name": "Andrei", "clpid": "Jorza-A" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis consists of four chapters, including an introduction.
\r\n\r\nIn Chapter 2, we take an averaging approach to the question of the distribution of supersingular primes of degree one, for elliptic curves over a number field. We begin by modifying the Lang-Trotter heuristic to address the case of an abelian extension, then we show that it holds on average (up to a constant) for a family of elliptic curves by using ideas of David-Pappalardi.
\r\n\r\nIn Chapter 3, we prove constructively that there exists an infinite number of (arbitrarily) thin families of rational elliptic curves for which the Lang-Trotter conjecture holds on average, in part by using techniques of Fouvry-Murty.
\r\n\r\nIn Chapter 4, we obtain a result related to the strong multiplicity one theorem for non-dihedral cuspidal automorphic representations for GL(2), with trivial central character and non-twist-equivalent symmetric squares. Given a real algebraic number, we also find a lower bound for the lower density of the set of finite places for which the associated Hecke eigenvalue is not equal to that algebraic number.
", "doi": "10.7907/WF31-3096", "publication_date": "2011", "thesis_type": "phd", "thesis_year": "2011" }, { "id": "thesis:5919", "collection": "thesis", "collection_id": "5919", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06072010-004607725", "primary_object_url": { "basename": "kozhan_thesis.pdf", "content": "final", "filesize": 880362, "license": "other", "mime_type": "application/pdf", "url": "/5919/1/kozhan_thesis.pdf", "version": "v6.0.0" }, "type": "thesis", "title": "Asymptotics for Orthogonal Polynomials, Exponentially Small Perturbations and Meromorphic Continuations of Herglotz Functions", "author": [ { "family_name": "Kozhan", "given_name": "Rostyslav", "clpid": "Kozhan-Rostyslav" } ], "thesis_advisor": [ { "family_name": "Simon", "given_name": "Barry M.", "orcid": "0000-0003-2561-8539", "clpid": "Simon-B" } ], "thesis_committee": [ { "family_name": "Simon", "given_name": "Barry M.", "orcid": "0000-0003-2561-8539", "clpid": "Simon-B" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Duits", "given_name": "Maurice", "clpid": "Duits-M" }, { "family_name": "Ryckman", "given_name": "Eric", "clpid": "Ryckman-E" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The thesis consists of a series of results on the theory of orthogonal polynomials on the real line.
\r\n\r\n1. We establish Szego asymptotics for matrix-valued measures under the assumption that the absolutely continuous part satises Szego's condition and the mass points satisfy a Blaschke-type condition. This generalizes the scalar analogue of Peherstorfer-Yuditskii [PY01] and the matrix-valued result of AptekarevNikishin [AN83], which handles only a finite number of mass points.
\r\n\r\n2. We obtain matrix-valued Jost asymptotics for a block Jacobi matrix under an L1-type condition on parameters, and give a necessary and sufficient condition for an analytic matrix-valued function to be the Jost function of a block Jacobi matrix with exponentially converging parameters. This establishes the matrix-valued analogue of Damanik-Simon [DS06b].
\r\n\r\n3. The latter results allow us to fully characterize the matrix-valued Weyl-Titchmarsh m-functions of block Jacobi matrices with exponentially converging parameters.
\r\n\r\n4. We find a necessary and sufficient condition for a finite gap Herglotz function m to be the m-function of a Jacobi matrix with the prescribed \"distance\" from the isospectral torus of periodic Jacobi matrices associated with a given finite gap set (with all gaps open). The condition is in terms of meromorphic continuations of the function m to a natural Riemann surface, and the structure of poles and zeros of m.
\r\n\r\n5. The results from parts 3 and 4 give certain corollaries on the point perturbations of measures. Namely, we find conditions on when adding or removing a pure point preserves the exponential rate of convergence of Jacobi parameters. The method applies in the matrix-valued case of exponential convergence to the free block Jacobi matrix, and in the scalar case of exponential convergence to a periodic Jacobi matrix. This extends Geronimo's results from [Ger94].
\r\n\r\n6. We obtain two results on the equivalence classes of block Jacobi matrices: first, that the Jacobi matrix of type 2 in the Nevai class has An coefficients converging to 1, and second, that under an L1-type condition on the Jacobi coefficients, equivalent Jacobi matrices of type 1, 2, and 3 are pairwise asymptotic.
", "doi": "10.7907/KK1A-Z663", "publication_date": "2010", "thesis_type": "phd", "thesis_year": "2010" }, { "id": "thesis:1963", "collection": "thesis", "collection_id": "1963", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05222009-125630", "primary_object_url": { "basename": "Thesis.pdf", "content": "final", "filesize": 432084, "license": "other", "mime_type": "application/pdf", "url": "/1963/1/Thesis.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Structure-Preserving Discretization of Incompressible Fluids", "author": [ { "family_name": "Pavlov", "given_name": "Dmitry", "clpid": "Pavlov-Dmitry" } ], "thesis_advisor": [ { "family_name": "Marsden", "given_name": "Jerrold E.", "clpid": "Marsden-J-E" }, { "family_name": "Desbrun", "given_name": "Mathieu", "clpid": "Desbrun-M" } ], "thesis_committee": [ { "family_name": "Marsden", "given_name": "Jerrold E.", "clpid": "Marsden-J-E" }, { "family_name": "Breuer", "given_name": "Jonathan", "clpid": "Breuer-J" }, { "family_name": "Desbrun", "given_name": "Mathieu", "clpid": "Desbrun-M" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The geometric nature of Euler fluids has been clearly identified and extensively studied in mathematics. However computational approaches to fluid mechanics, mostly derived from a numerical-analytic point of view, are rarely designed with structure preservation in mind, and often suffer from spurious numerical artifacts. In contrast, we geometrically derive discrete equations of motion for fluid dynamics from first principles. Our approach uses a finite-dimensional Lie group to discretize the group of volume-preserving diffeomorphisms, and the discrete Euler equations are derived from a variational principle with non-holonomic constraints. The resulting discrete equations of motion induce a structure-preserving time integrator with good long-term energy behavior, for which an exact discrete Kelvin circulation theorem holds. Possible extensions of our method to magnetohydrodynamics, viscous flows, optimal transport and a link to Brenier's generalized flows are also discussed. ", "doi": "10.7907/KCQK-QF78", "publication_date": "2009", "thesis_type": "phd", "thesis_year": "2009" }, { "id": "thesis:2253", "collection": "thesis", "collection_id": "2253", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05292008-085545", "primary_object_url": { "basename": "TheThesis.pdf", "content": "final", "filesize": 1128742, "license": "other", "mime_type": "application/pdf", "url": "/2253/2/TheThesis.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Braid Forcing, Hyperbolic Geometry, and Pseudo-Anosov Sequences of Low Entropy", "author": [ { "family_name": "Venzke", "given_name": "Rupert William", "clpid": "Venzke-Rupert-William" } ], "thesis_advisor": [ { "family_name": "Calegari", "given_name": "Danny C.", "orcid": "0009-0007-9304-2822", "clpid": "Calegari-D" } ], "thesis_committee": [ { "family_name": "Calegari", "given_name": "Danny C.", "orcid": "0009-0007-9304-2822", "clpid": "Calegari-D" }, { "family_name": "Aschbacher", "given_name": "Michael", "orcid": "0000-0002-8380-0921", "clpid": "Aschbacher-M" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Ghiggini", "given_name": "Paolo", "clpid": "Ghiggini-P" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We view braids as automorphisms of punctured disks and define a partial order on pseudo-Anosov braids called the \"forcing order\". The order measures whether one automorphism induces another given automorphism on the surface. Pseudo-Anosov growth rate decreases relative to the order and appears to give a good measure of braid complexity. Unfortunately it appears difficult computationally to determine explicitly the partial order structure by hand. We use several computer algorithms to study the bottom part of the partial order when the number of braid strands is fixed. From the algorithms, we build sequences of low entropy pseudo-Anosov n-strand braids that are minimal in the sense that they do not force any other pseudo-Anosov braids on the same number of strands. The sequences are an extension of work done by Hironaka and Kin, and we conjecture the sequences to achieve minimal entropy among certain non-trivial classes of braids. In general, the lowest entropy pseudo-Anosov braids appear to have mapping tori that come from Dehn surgery on very low volume hyperbolic 3-manifolds, and we begin to analyze the relation between entropy and hyperbolic volume. Moreover, the low-growth families contain non-trivial low-growth families of horseshoe braids and we proceed to study dynamics of the horseshoe map as well.\r\n", "doi": "10.7907/290Y-BY53", "publication_date": "2008", "thesis_type": "phd", "thesis_year": "2008" }, { "id": "thesis:1584", "collection": "thesis", "collection_id": "1584", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05022008-113702", "primary_object_url": { "basename": "thesis-final.pdf", "content": "final", "filesize": 727712, "license": "other", "mime_type": "application/pdf", "url": "/1584/1/thesis-final.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Amenability, Countable Equivalence Relations, and Their Full Groups", "author": [ { "family_name": "Tsankov", "given_name": "Todor Dimitrov", "clpid": "Tsankov-Todor-Dimitrov" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Caicedo", "given_name": "Andr\u00e9s", "clpid": "Caicedo-A" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis consists of an introduction and four independent chapters.
\r\n\r\nIn Chapter 1, we study homeomorphism groups of metrizable compactifications of the natural numbers. Those groups can be represented as almost zero-dimensional Polishable subgroups of the group S\u221e. We show that all Polish groups are continuous homomorphic images of almost zero-dimensional Polishable subgroups of S\u221e. We also find a sufficient condition for these groups to be one dimensional.
\r\n\r\nIn Chapter 2, we study the connections between properties of the action of a countable group \u0393 on a countable set X and the ergodic theoretic properties of the corresponding shift action of \u0393 on MX, where M is a measure space. In particular, we show that the action of \u0393 on X is amenable iff the corresponding shift has almost invariant sets. This is joint work with Alexander Kechris.
\r\n\r\nIn Chapter 3, we prove that if the Koopman representation associated to a measure-preserving action of a countable group on a standard non-atomic probability space is non-amenable, then there does not exist a countable-to-one Borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action (i.e., an action on the boundary of a countably splitting tree), generalizing previous results of Hjorth and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. This is joint work with Inessa Epstein.
\r\n\r\nIn Chapter 4, we study full groups of countable, measure-preserving equivalence relations. Our main results include that they are all homeomorphic to the separable Hilbert space and that every homomorphism from an ergodic full group to a separable group is continuous. We also find bounds for the minimal number of generators of a dense subgroup of full groups allowing us to distinguish full groups of equivalence relations generated by free, ergodic actions of the free groups Fn and Fm if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group iff its full group has a finitely generated dense subgroup. This is joint work with John Kittrell.\r\n
", "doi": "10.7907/N91C-HV48", "publication_date": "2008", "thesis_type": "phd", "thesis_year": "2008" }, { "id": "thesis:1896", "collection": "thesis", "collection_id": "1896", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05202005-160741", "primary_object_url": { "basename": "thesis.pdf", "content": "final", "filesize": 491373, "license": "other", "mime_type": "application/pdf", "url": "/1896/1/thesis.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Descriptive Properties of Measure Preserving Actions and the Associated Unitary Representations", "author": [ { "family_name": "Wei", "given_name": "Tzer-jen", "clpid": "Wei-Tzer-jen" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Wales", "given_name": "David B.", "clpid": "Wales-D-B" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis consists of two independent parts.
\r\n\r\nIn the first part, we study the descriptive complexity of full groups [E]. The main result is
\r\n\r\n i) If E is not smooth, then [E] is Sigma\u2070\u20833 complete;
\r\n\r\n ii) If E is smooth, then [E] is closed.
In the second part, we study descriptive properties of the Koopman unitary repreesentation associated with the measure preserving action. We characterize the smoothness and compressibility of the equivalence induced by the unitary representaion. We also study many connections between the equivalence relation on L^2(X) and the equivalence relation on X.
", "doi": "10.7907/HECM-A179", "publication_date": "2005", "thesis_type": "phd", "thesis_year": "2005" }, { "id": "thesis:1772", "collection": "thesis", "collection_id": "1772", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05132004-233348", "primary_object_url": { "basename": "thesisfinal7.pdf", "content": "final", "filesize": 392765, "license": "other", "mime_type": "application/pdf", "url": "/1772/1/thesisfinal7.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Conformal Laminations", "author": [ { "family_name": "Gupta", "given_name": "Vineet", "clpid": "Gupta-Vineet" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Calegari", "given_name": "Danny C.", "orcid": "0009-0007-9304-2822", "clpid": "Calegari-D" }, { "family_name": "Hersonsky", "given_name": "Saar", "clpid": "Hersonsky-S" }, { "family_name": "Schlag", "given_name": "Wilhelm", "clpid": "Schlag-W" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "A lamination on a circle is an equivalence relation on the points of the circle. Laminations can be induced on a circle by a map that is continuous on the closed disc and injective in the interior. Such laminations are characterized topologically, as being flat and closed. In this paper we investigate the conditions under which a closed, flat lamination is induced by a conformal mapping. We show that if the set of multiple points of the lamination form a Cantor set, whose end points are identified under the lamination, then the lamination is conformal. More generally, the union of such laminations is also conformal. We also show conjecture that any closed, flat lamination, such that the set of multiple points is of logarithmic capacity zero, is conformal.", "doi": "10.7907/MQAY-KR87", "publication_date": "2004", "thesis_type": "phd", "thesis_year": "2004" }, { "id": "thesis:3079", "collection": "thesis", "collection_id": "3079", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-08102004-142550", "primary_object_url": { "basename": "D_Zhan_thesis.pdf", "content": "final", "filesize": 648843, "license": "other", "mime_type": "application/pdf", "url": "/3079/1/D_Zhan_thesis.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Random Loewner Chains in Riemann Surfaces", "author": [ { "family_name": "Zhan", "given_name": "Dapeng", "orcid": "0000-0001-5528-8142", "clpid": "Zhan-Dapeng" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Borodin", "given_name": "Alexei", "clpid": "Borodin-A" }, { "family_name": "Simon", "given_name": "Barry M.", "orcid": "0000-0003-2561-8539", "clpid": "Simon-B" }, { "family_name": "Berger", "given_name": "Noam", "clpid": "Berger-N" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The thesis describes an extension of O. Schramm's SLE processes to complicated plane domains and Riemann surfaces. First, three kinds of new SLEs are defined for simple conformal types. They have properties similar to traditional SLEs. Then harmonic random Loewner chains (HRLC) are defined in finite Riemann surfaces. They are measures on the space of Loewner chains, which are increasing families of closed subsets satisfying certain properties. An HRLC is first defined on local charts using Loewner's equation. Since the definitions in different charts agree with each other, these local HRLCs can be put together to construct a global HRLC. An HRLC in a plane domain can be described by differential equations involving canonical plane domains. Those old and new SLEs are special cases of HRLCs. An HRLC is determined by a parameter K \u2265 0, a starting point and a target set. When K = 6, the HRLC satisfies the locality property. When K = 2, the HRLC preserves some observable that resembles the observable for the corresponding loop-erased random walk (LERW). So HRLC\u2082 should be the scaling limit of LERW. With reasonable assumptions, HRLC8/3 differs from a restriction measure by a conformally invariant density; for K \u2208 (0,8/3), HRLCK differs from a pre-restriction measure by a conformally invariant density. A restriction measure could be constructed from a pre-restriction measure by adding Brownian bubbles.", "doi": "10.7907/TK0N-3F57", "publication_date": "2004", "thesis_type": "phd", "thesis_year": "2004" }, { "id": "thesis:1936", "collection": "thesis", "collection_id": "1936", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05222003-114151", "primary_object_url": { "basename": "thesis.pdf", "content": "final", "filesize": 502565, "license": "other", "mime_type": "application/pdf", "url": "/1936/1/thesis.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Sum Rules and the Szeg\u00f6 Condition for Jacobi Matrices", "author": [ { "family_name": "Zlato\u0161", "given_name": "Andrej", "orcid": "0000-0003-0660-7404", "clpid": "Zlato\u0161-Andrej" } ], "thesis_advisor": [ { "family_name": "Simon", "given_name": "Barry M.", "orcid": "0000-0003-2561-8539", "clpid": "Simon-B" } ], "thesis_committee": [ { "family_name": "Simon", "given_name": "Barry M.", "orcid": "0000-0003-2561-8539", "clpid": "Simon-B" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Killip", "given_name": "Rowan", "clpid": "Killip-R" }, { "family_name": "Schlag", "given_name": "Wilhelm", "clpid": "Schlag-W" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We consider Jacobi matrices J with real b_n on the diagonal, positive a_n on the next two diagonals, and with u'(x) the density of the absolutely continuous part of the spectral measure. In particular, we are interested in compact perturbations of the free matrix J_0, that is, such that the a_n go to 1 and b_n go to 0. We study the Case sum rules for such matrices. These are trace formulae relating sums involving the a_n's and b_n's on one side and certain quantities in terms of the spectral measure on the other. We establish situations where the sum rules are valid, extending results of Case and Killip-Simon.
\r\n\r\nThe matrix J is said to satisfy the Szego condition whenever the integral
\r\n\r\nint_{0}^{pi} log [u'(2 cos t)] dt,
\r\n\r\nwhich appears in the sum rules, is finite. Applications of our results include an extension of Shohat's classification of certain Jacobi matrices satisfying the Szego condition to cases with an infinite point spectrum, and a proof that if n(a_n - 1) go to a, nb_n go to b, and 2a < |b|, then the Szego condition fails. Related to this, we resolve a conjecture by Askey on the Szego condition for Jacobi matrices which are Coulomb perturbations of J_0. More generally, we prove that if
\r\n\r\na_n = 1 + a/n^c + O(n^{-1-eps}) and b_n = b/n^c + O(n^{-1-eps})
\r\n\r\nwith 0 < \u03b3 \u2264 1 and eps > 0, then the Szego condition is satisfied if and only if 2a \u2265|b|
", "doi": "10.7907/DBVE-VF23", "publication_date": "2003", "thesis_type": "phd", "thesis_year": "2003" }, { "id": "thesis:4960", "collection": "thesis", "collection_id": "4960", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-12122003-115521", "primary_object_url": { "basename": "Garduno_ah_2002.pdf", "content": "final", "filesize": 2501082, "license": "other", "mime_type": "application/pdf", "url": "/4960/1/Garduno_ah_2002.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Regularization of the Amended Potential around a Symmetric Configuration", "author": [ { "family_name": "Gardu\u00f1o", "given_name": "Antonio Hern\u00e1ndez", "clpid": "Gardu\u00f1o-Antonio-Hern\u00e1ndez" } ], "thesis_advisor": [ { "family_name": "Gabai", "given_name": "David", "clpid": "Gabai-David" }, { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" } ], "thesis_committee": [ { "family_name": "Marsden", "given_name": "Jerrold E.", "clpid": "Marsden-J-E" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "orcid": "0000-0002-0159-087X", "clpid": "Ramakrishnan-D" }, { "family_name": "Gabai", "given_name": "David", "clpid": "Gabai-David" }, { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Relative equilibria are periodic trajectories that, in a dynamical system with continuous symmetry, correspond to fixed points in the projected dynamics to the quotient space. In Hamiltonian systems with symmetry, it is of interest to understand the structure of relative equilibria near symmetric states. In this context, we give a method that in some cases of simple mechanical systems with compact symmetry group gives information about the relative equilibria bifurcating from a set of relative equilibria with isotropy subgroup isomorphic to S^1. This method is based on the blowing-up of the amended potential.", "doi": "10.7907/NBWA-EQ57", "publication_date": "2002", "thesis_type": "phd", "thesis_year": "2002" }, { "id": "thesis:6773", "collection": "thesis", "collection_id": "6773", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:01242012-162804546", "primary_object_url": { "basename": "Erdogan_mb_2002.pdf", "content": "final", "filesize": 12337548, "license": "other", "mime_type": "application/pdf", "url": "/6773/1/Erdogan_mb_2002.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Mapping Properties of Certain Averaging Operators", "author": [ { "family_name": "Erdo\u011fan", "given_name": "Mehmet Burak", "clpid": "Erdo\u011fan-Mehmet-Burak" } ], "thesis_advisor": [ { "family_name": "Wolff", "given_name": "Thomas H.", "clpid": "Wolff-T-H" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "orcid": "0000-0002-0159-087X", "clpid": "Ramakrishnan-D" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Hundertmark", "given_name": "Dirk", "orcid": "0000-0002-0643-0138", "clpid": "Hundertmark-Dirk" }, { "family_name": "Simon", "given_name": "Barry M.", "orcid": "0000-0003-2561-8539", "clpid": "Simon-B" }, { "family_name": "Wales", "given_name": "David B.", "clpid": "Wales-D-B" }, { "family_name": "Wolff", "given_name": "Thomas H.", "clpid": "Wolff-T-H" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "orcid": "0000-0002-0159-087X", "clpid": "Ramakrishnan-D" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this thesis, we investigate the mapping properties of two averaging operators.
\r\n\r\nIn the first part, we consider a model rigid well-curved line complex G_d in R^d. The X-ray transform, X, restricted to G_d is defined as an operator from functions on R^d to functions on G_d in the following way:\r\n\r\nXf(l) = \u222b_lf, l \u03f5 G_d.\r\n\r\nWe obtain sharp mixed norm estimates for X in R^4 and R^5.
\r\n\r\nIn the second part, we consider the elliptic maximal function M. Let \u03b5 be the set of all ellipses in R^2 centered at the origin with axial lengths in [1/2,2].\r\nLet f : R^2 -> R, then M f : R^2 -> R is defined in the following way:\r\n\r\nMf(x) = ^(sup)_(E\u03f5\u03b5) ^1/_(|E|) \u222b_E f(x+s)d\u03c3(s),\r\n\r\nwhere d\u03c3 is the arclength measure on E and |E| is the length of E.
\r\n\r\nIn this part of the thesis, we investigate the L^P mapping properties of M.
", "doi": "10.7907/JRFS-4S52", "publication_date": "2002", "thesis_type": "phd", "thesis_year": "2002" }, { "id": "thesis:13594", "collection": "thesis", "collection_id": "13594", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:11212019-172159302", "primary_object_url": { "basename": "kovrijkine-oe-2000.pdf", "content": "final", "filesize": 2582332, "license": "other", "mime_type": "application/pdf", "url": "/13594/1/kovrijkine-oe-2000.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Some Estimates of Fourier Transforms", "author": [ { "family_name": "Kovrijkine", "given_name": "Oleg E.", "clpid": "Kovrijkine-Oleg-E" } ], "thesis_advisor": [ { "family_name": "Wolff", "given_name": "Thomas H.", "clpid": "Wolff-T-H" } ], "thesis_committee": [ { "family_name": "Wolff", "given_name": "Thomas H.", "clpid": "Wolff-T-H" }, { "family_name": "Keel", "given_name": "Markus", "clpid": "Keel-Markus" }, { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This work consists of two independent parts. In the first part we prove several results related to the theorem of Logvinenko and Sereda on determining sets for functions with Fourier transforms supported in a parallelepiped. We obtain a polynomial instead of exponential bound in this theorem, and we extend it to the case of functions with Fourier transforms supported in the union of a bounded number of parallelepipeds. When dimension d = 1 we also consider the case of infinitely many lacunary intervals. We generalize the Zygmund theorem for lacunary series whose Fourier coefficients are replaced with polynomials of uniformly bounded degree. We give also a necessary condition for the support of Fourier transforms for which the Logvinenko-Sereda theorem still holds.
\r\n\r\nIn the second part we prove that the L\u00b2([0,1]d x SO(d)) norm of periodizations of a function from L\u00b9(\u211dd) is equivalent to the L\u00b2(\u211dd) norm of the function itself in higher dimensions. We generalize the statement for functions from Lp(\u211dd) where 1 \u2264 p < (2d)/(d + 2) spirit of the Stein-Tomas theorem. We also show that the following theorem due to M. Kolountzakis and T. Wolff does not hold if dimension d = 2: if periodizations of a function from L\u00b9(\u211dd) are constants, then the function is continuous and bounded provided that the dimension d is at least three.
", "doi": "10.7907/0p2k-ah86", "publication_date": "2000", "thesis_type": "phd", "thesis_year": "2000" }, { "id": "thesis:13588", "collection": "thesis", "collection_id": "13588", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:11212019-103601328", "primary_object_url": { "basename": "jackson-fy-1998.pdf", "content": "final", "filesize": 2203082, "license": "other", "mime_type": "application/pdf", "url": "/13588/1/jackson-fy-1998.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Sun-Dual Characterizations of the Translation Group of \u211d", "author": [ { "family_name": "Jackson", "given_name": "Frances Yvonne", "clpid": "Jackson-Frances-Yvonne" } ], "thesis_advisor": [ { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" } ], "thesis_committee": [ { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Wolff", "given_name": "Thomas H.", "clpid": "Wolff-T-H" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Let E be a Banach space. The mapping t \u2192 T (t) of \u211d (real numbers) into Lb(E), the Banach algebra of all bounded linear operators on E, is called a strongly continuous group or a C\u2080-group, if G = {T(t) : t \u2208 \u211d} defines a group representation of (\u211d, +) into the multiplicative group of Lb(E), and if \u2200f \u2208 E,
\r\n\r\n[equation; see abstract in scanned thesis for details].
\r\n\r\nFor example, if E = C\u2080(\u211d), the function space which consists of all continuous, complex functions that vanish at infinity, then (\u2200t \u2208 \u211d) (\u2200f \u2208 C\u2080(\u211d)), the function T(t)f(x) = f(x + t), x \u2208 \u211d, defines a strongly continuous group, since each f \u2208 E is uniformly continuous; this group is called the translation group. If we now consider E = B(\u211d), the space of bounded, continuous complex functions on \u211d, then although the translation group on E is not strongly continuous, it is strongly continuous on the subspace BUC(\u211d) of E, which consists of bounded, uniformly continuous functions. BUC(\u211d) is the largest subspace of E on which the translation group is strongly continuous.
\r\n\r\nThe adjoint family of a C\u2080-group defined on a Banach space E, need not be strongly continuous on the Banach dual E* of E. Let E\u2299 (pronounced E-sun) be the largest linear subspace of E* relative to which the adjoint family is a C\u2080-group:
\r\n\r\n[equation; see abstract in scanned thesis for details].
\r\n\r\nE\u2299 is called the sun-dual or sun-space of E. If E = C\u2080(\u211d), then it follows from a well-known result of A. Plessner that E\u2299 = L\u00b9(\u211d) ([Ple]). This research paper contains a characterization of the sun-dual of BUC(\u211d) and of the subspace W AP(\u211d) of BUC(\u211d), which consists of weakly almost periodic functions on \u211d.
", "doi": "10.7907/3y5x-kg66", "publication_date": "1998", "thesis_type": "phd", "thesis_year": "1998" }, { "id": "thesis:3554", "collection": "thesis", "collection_id": "3554", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-09152006-144938", "primary_object_url": { "basename": "Smirnov_sk_1996.pdf", "content": "final", "filesize": 9203792, "license": "other", "mime_type": "application/pdf", "url": "/3554/1/Smirnov_sk_1996.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Spectral Analysis of Julia Sets", "author": [ { "family_name": "Smirnov", "given_name": "Stanislav K.", "clpid": "Smirnov-Stanislav-K" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Kahn", "given_name": "Jeremy", "clpid": "Kahn-Jeremy" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" } ], "local_group": [ { "literal": "Caltech Distinguished Alumni Award" }, { "literal": "div_pma" } ], "abstract": "We investigate different measures defined geometrically or dynamically on polynomial Julia sets and their scaling properties. Our main concern is the relationship between harmonic and Hausdorff measures.
\r\n\r\nWe prove that the fine structure of harmonic measure at the more exposed points of an arbitrary polynomial Julia set is regular, and dimension spectra or pressure for the corresponding (negative) values of parameter are real-analytic. However, there is a precisely described class of polynomials, where a set of preperiodic critical points can generate a unique very exposed tip, which manifests in the phase transition for some kinds of spectra.
\r\n\r\nFor parabolic and subhyperbolic polynomials, and also semihyperbolic quadratics we analyze the spectra for the positive values of parameter, establishing the extent of their regularity.
\r\n\r\nResults are proved through spectral analysis of the transfer (Perron-Frobenius-Ruelle) operator.
", "doi": "10.7907/X37M-D376", "publication_date": "1996", "thesis_type": "phd", "thesis_year": "1996" }, { "id": "thesis:17522", "collection": "thesis", "collection_id": "17522", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:07142025-185158120", "primary_object_url": { "basename": "Schlag_W_1996.pdf", "content": "final", "filesize": 18223525, "license": "other", "mime_type": "application/pdf", "url": "/17522/1/Schlag_W_1996.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "L\u1d56 to L\u146b Estimates for the Circular Maximal Function", "author": [ { "family_name": "Schlag", "given_name": "Wilhelm", "orcid": "0000-0002-6418-1715", "clpid": "Schlag-Wilhelm" } ], "thesis_advisor": [ { "family_name": "Wolff", "given_name": "Thomas H.", "clpid": "Wolff-T-H" } ], "thesis_committee": [ { "family_name": "Wolff", "given_name": "Thomas H.", "clpid": "Wolff-T-H" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Wilson", "given_name": "Richard M.", "clpid": "Wilson-R-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this thesis we establish sharp Lp \u2192 Lq bounds for the circular maximal function\r\nin the plane. This is accomplished by interpolating a L5/2 \u2192 L5 endpoint estimate\r\nwith Bourgain's well-known Lp \u2192 Lp bounds. The endpoint estimate is proved by\r\ncombining the geometric/combinatorial method of Kolasa- Wolff with a L2 inequality\r\non a small ball. The Lp \u2192 Lq estimates for the circular maximal function established\r\nin this thesis would be a consequence of C. Sogge's sharp local smoothing conjecture\r\nfor the wave equation.", "doi": "10.7907/4tq4-p076", "publication_date": "1996", "thesis_type": "phd", "thesis_year": "1996" } ]