This thesis consists of three projects related to enumerative geometry and mirror symmetry, with particular emphasis on the mirror symmetry of flag varieties.
\r\n\r\nThe first project introduces and develops the theory of F-bundles, a framework for formulating mirror symmetry type results. We prove a spectral decomposition theorem for maximal F-bundles, and use it to obtain reconstruction and uniqueness results for certain decompositions of quantum cohomology related to birational geometry, complementing existing the existence theorem in the literature. In the third project, we further extend the F-bundle formalism to the equivariant setting and establish an unfolding theorem strengthening mirror symmetry statements from small to big quantum cohomology. As an application, we derive the equivariant mirror symmetry of general flag varieties in the third project, extending previous results which were previously known only at the level of small quantum cohomology.
\r\n\r\nThe second project focuses on the mirror symmetry of flag varieties. For complex partial flag varieties, we provide an explicit Pl\u00fccker coordinate superpotential formula that is sufficient to recover their small quantum cohomology on A-side, and we prove a folklore conjecture in mirror symmetry. Namely, we show that the eigenvalues of the action of the first Chern class on quantum cohomology are equal to the critical values of the superpotential.
", "doi": "10.7907/f6j9-0n16", "publication_date": "2026", "thesis_type": "phd", "thesis_year": "2026" }, { "id": "thesis:17279", "collection": "thesis", "collection_id": "17279", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05272025-235437474", "type": "thesis", "title": "Gromov-Witten Theory, Non-Archimedean Geometry, and Mirror Symmetry", "author": [ { "family_name": "Hinault", "given_name": "Thorgal Ga\u00ebtan", "orcid": "0000-0003-3420-3917", "clpid": "Hinault-Thorgal-Ga\u00ebtan" } ], "thesis_advisor": [ { "family_name": "Yu", "given_name": "Tony Yue", "orcid": "0000-0002-6019-8552", "clpid": "Yu-Tony-Yue" } ], "thesis_committee": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "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": "Xu", "given_name": "Weihong", "clpid": "Weihong-Xu" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis consists of three projects related to enumerative geometry and mirror symmetry, with an eye towards birational geometry.
\r\n\r\nThe first project studies how certain non-archimedean Gromov-Witten invariants of log Calabi-Yau surfaces, called infinitesimal cylinder counts, behave under blowup. We discuss the case of primitive cylinders, and establish a formula that expresses cylinder counts on a blow up of a toric surface in terms of counts in a simpler surface. The proof of the formula uses non-archimedean geometry techniques in an essential way to produce suitable degenerations of the geometric objects enumerated by the counts.
\r\n\r\nThe next two projects introduce and study the notion of F-bundle, a structure which can be used to formulate mirror symmetry type results using the language of differential geometry. Our spectral decomposition theorem provides a canonical decomposition for F-bundles satisfying a condition called maximality. We develop the theory of framing, and use it to obtain reconstruction theorems for isomorphisms between maximal F-bundles. As an application of this theory, we prove the uniqueness of certain decompositions of quantum cohomology related to birational geometry, complementing the existence results found in the literature. We also extend the framework of F-bundles to the setting of equivariant mirror symmetry, and prove an unfolding result which can be used to strengthen mirror symmetry statements from the small quantum cohomology to the big quantum cohomology. We apply this unfolding theorem to the equivariant mirror symmetry of general flag varieties, for which only the small quantum cohomology mirror symmetry was known until now.
", "doi": "10.7907/d7wf-ha89", "publication_date": "2025", "thesis_type": "phd", "thesis_year": "2025" }, { "id": "thesis:16515", "collection": "thesis", "collection_id": "16515", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06102024-225449252", "primary_object_url": { "basename": "Kulkarni_Neeraja_2024.pdf", "content": "final", "filesize": 521723, "license": "other", "mime_type": "application/pdf", "url": "/16515/1/Kulkarni_Neeraja_2024.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "A Kakeya Estimate for Sticky Sets Using a Planebrush", "author": [ { "family_name": "Kulkarni", "given_name": "Neeraja Raghavendra", "orcid": "0000-0001-7747-9177", "clpid": "Kulkarni-Neeraja-Raghavendra" } ], "thesis_advisor": [ { "family_name": "Conlon", "given_name": "David", "orcid": "0000-0001-5899-1829", "clpid": "Conlon-David" } ], "thesis_committee": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Conlon", "given_name": "David", "orcid": "0000-0001-5899-1829", "clpid": "Conlon-David" }, { "family_name": "Isett", "given_name": "Philip", "orcid": "0000-0001-9038-5546", "clpid": "Isett-Phlip" }, { "family_name": "Katz", "given_name": "Nets H.", "orcid": "0000-0002-6239-5429", "clpid": "Katz-N-H" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "A Besicovitch set is defined as a compact subset of \u211d\u207f which contains a line segment of length 1 in every direction. The Kakeya conjecture says that every Besicovitch set has Minkowski and Hausdorff dimensions equal to n. This thesis gives an improved Hausdorff dimension estimate, d \u2a7e 0.60376707287 n + O(1), for Besicovitch sets displaying a special structural property called \"stickiness.\" The improved estimate comes from using an incidence geometry argument called a \"k-planebrush,\" which is a higher dimensional analogue of Wolff's \"hairbrush\" argument from 1995.
\r\n \r\nIn addition, an x-ray transform estimate is obtained as a corollary of Zahl's k-linear estimate in 2019. The x-ray estimate, together with the estimate for sticky sets, implies that all Besicovitch sets in \u211d\u207f must have Minkowski dimension greater than (2 - \u221a2 + \u03b5)n. Though this Minkowski dimension estimate is not as good as one previously known from Katz-Tao(2000), it provides a new proof of the same result.
", "doi": "10.7907/japt-b214", "publication_date": "2024", "thesis_type": "phd", "thesis_year": "2024" }, { "id": "thesis:16471", "collection": "thesis", "collection_id": "16471", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06022024-205427263", "primary_object_url": { "basename": "Trung-PhDthesis.pdf", "content": "final", "filesize": 479513, "license": "other", "mime_type": "application/pdf", "url": "/16471/1/Trung-PhDthesis.pdf", "version": "v6.0.0" }, "type": "thesis", "title": "On Arithmetic Invariants of Special Families of K3-Type Surfaces", "author": [ { "family_name": "Can", "given_name": "Tran Thanh Trung", "orcid": "0000-0002-2043-3335", "clpid": "Can-Tran-Thanh-Trung" } ], "thesis_advisor": [ { "family_name": "Mantovan", "given_name": "Elena", "orcid": "0000-0003-4521-2130", "clpid": "Mantovan-E" } ], "thesis_committee": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" }, { "family_name": "Zhao", "given_name": "Roy", "clpid": "Zhao-Roy" }, { "family_name": "Mantovan", "given_name": "Elena", "orcid": "0000-0003-4521-2130", "clpid": "Mantovan-E" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis studies applications of Shimura varieties in positive characteristic to questions on arithmetic invariants of special families of K3-type surfaces.
\r\n\r\nThe first main result determines the Newton polygons and Artin invariants of 144 special families of K3-type surfaces. The second is a refinement of a conjecture of Serre for K3 surfaces over number field.
", "doi": "10.7907/z5mc-g704", "publication_date": "2024", "thesis_type": "phd", "thesis_year": "2024" }, { "id": "thesis:14644", "collection": "thesis", "collection_id": "14644", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05272022-215641742", "primary_object_url": { "basename": "zhang_victor_2022_thesis.pdf", "content": "final", "filesize": 1220288, "license": "other", "mime_type": "application/pdf", "url": "/14644/1/zhang_victor_2022_thesis.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Twisted Heisenberg Central Extensions and the Affine ADE Basic Representation", "author": [ { "family_name": "Zhang", "given_name": "Victor", "clpid": "Zhang-Victor" } ], "thesis_advisor": [ { "family_name": "Zhu", "given_name": "Xinwen", "clpid": "Zhu-Xinwen" } ], "thesis_committee": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" }, { "family_name": "Rains", "given_name": "Eric M.", "clpid": "Rains-E-M" }, { "family_name": "Zhu", "given_name": "Xinwen", "clpid": "Zhu-Xinwen" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We study various aspects of the representation theory of loop groups, all with the aim of giving geometric constructions, parameterized by conjugacy classes of the Weyl group, of the basic representation of the affine Lie algebras associated to a simply laced simple Lie algebra as a restriction isomorphism on dual sections of the level 1 line bundle on the affine Grassmannian. Along the way, we obtain various results on the structure of loop tori, the definition of a notion of a Heisenberg Central extension as an alternative for twisted modules over the lattice vertex algebra and the determination of their representation theory, some computations on central extensions of a torus over a field by K2, and a new proof of the classification of the conjugacy classes of the Weyl group by parabolic induction.
", "doi": "10.7907/42v3-ws41", "publication_date": "2022", "thesis_type": "phd", "thesis_year": "2022" }, { "id": "thesis:14205", "collection": "thesis", "collection_id": "14205", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05302021-184629266", "primary_object_url": { "basename": "Thesis- Lingfei Yi.pdf", "content": "final", "filesize": 731445, "license": "other", "mime_type": "application/pdf", "url": "/14205/1/Thesis- Lingfei Yi.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Geometric Langlands for Hypergeometric Sheaves", "author": [ { "family_name": "Yi", "given_name": "Lingfei", "clpid": "Yi-Lingfei" } ], "thesis_advisor": [ { "family_name": "Zhu", "given_name": "Xinwen", "clpid": "Zhu-Xinwen" } ], "thesis_committee": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Campbell", "given_name": "Christopher J.", "clpid": "Campbell-Christopher-J" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "orcid": "0000-0002-0159-087X", "clpid": "Ramakrishnan-D" }, { "family_name": "Zhu", "given_name": "Xinwen", "clpid": "Zhu-Xinwen" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Generalized hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. In this paper, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems in the setting of geometric Langlands program. We\r\nwork in the framework of rigid automorphic data that is mainly due to Zhiwei Yun. The key point is to choose a proper collection of level structures and compute the space of automorphic forms that are equivariant with respect to these level structures. For those hypergeometric sheaves with wild ramification, we also generalize the construction of Hecke eigensheaves to other classical groups. We study part of their eigenvalues in the de Rham setting by giving an alternative construction of their Hecke eigen D-module using quantization of Hitchin system.
", "doi": "10.7907/257d-6p75", "publication_date": "2021", "thesis_type": "phd", "thesis_year": "2021" }, { "id": "thesis:14269", "collection": "thesis", "collection_id": "14269", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06092021-045504824", "primary_object_url": { "basename": "Lieber_Joshua_2021.pdf", "content": "final", "filesize": 882840, "license": "other", "mime_type": "application/pdf", "url": "/14269/1/Lieber_Joshua_2021.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "\u00c9tudes in Homotopical Thinking: F\u2081-geometry, Concurrent Computing, and Motivic Measures", "author": [ { "family_name": "Lieber", "given_name": "Joshua Franklin", "orcid": "0000-0002-6936-5054", "clpid": "Lieber-Joshua-Franklin" } ], "thesis_advisor": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "thesis_committee": [ { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Mantovan", "given_name": "Elena", "clpid": "Mantovan-E" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis weaves together three papers, each of which provides a use of homotopical intuition in a different field of mathematics. The first applies it to the study of various models of F\u2081-geometry, focusing mainly on the Bost-Connes algebra. The second endeavors to compare two homotopical models for concurrent computing before introducing a new one as well. Finally, the last paper provides a construction for obtaining derived motivic measures from an abstract six functors formalism and, in particular, applies this idea to obtain a lift of the Gillet-Soul\u00e9 motivic measure.", "doi": "10.7907/a4zm-1f28", "publication_date": "2021", "thesis_type": "phd", "thesis_year": "2021" }, { "id": "thesis:13784", "collection": "thesis", "collection_id": "13784", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06072020-124705473", "primary_object_url": { "basename": "Panda_Corina_2020.pdf", "content": "final", "filesize": 804687, "license": "other", "mime_type": "application/pdf", "url": "/13784/1/Panda_Corina_2020.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Generalizations of a Theorem of Hecke", "author": [ { "family_name": "Panda", "given_name": "Corina Bianca", "orcid": "0000-0002-6637-211X", "clpid": "Panda-Corina-Bianca" } ], "thesis_advisor": [ { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" } ], "thesis_committee": [ { "family_name": "Mantovan", "given_name": "Elena", "clpid": "Mantovan-E" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Yom Din", "given_name": "Alexander", "clpid": "Yom-Din-Alexander" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Let p > 3 be an odd prime, p \u2261 3 mod 4 and let \u03c0\u207a, \u03c0\u207b be the pair of cuspidal representations of SL\u2082(𝔽p). It is well known by Hecke that the difference m\u03c0\u207a - m\u03c0\u207b in the multiplicities of these two irreducible representations occurring in the space of weight 2 cusps forms with respect to the principal congruence subgroup \u0393(p), equals the class number h(-p) of the imaginary quadratic field ℚ(\u221a(-p)).
\r\n\r\nThis thesis consists of two main parts. In the first part, we extend Hecke's result to all fundamental discriminants of imaginary quadratic fields, including the even case. The proof is geometric in nature and uses the holomorphic Lefschetz number.
\r\n\r\nIn the second part, we consider generalizations to groups with higher ℚ-rank. In particular, we focus on the rank 2 special unitary group SU(2, 2). On the representation theory side, we prove the regular unipotent classes have positive contribution to an alternating sum of multiplicities of certain irreducible cuspidal representations of SU(2, 2) over the finite field of p elements. We also show that the semisimple classes have zero contribution, which is again a direct generalization of the SL\u2082 case. To obtain these two results, we make use of the Deligne-Lusztig theory and the connection of the traces to the Gelfand-Graev representations.
", "doi": "10.7907/bxgs-4825", "publication_date": "2020", "thesis_type": "phd", "thesis_year": "2020" }, { "id": "thesis:11273", "collection": "thesis", "collection_id": "11273", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:11142018-032432585", "type": "thesis", "title": "Special Values of Zeta-Functions for Proper Regular Arithmetic Surfaces", "author": [ { "family_name": "Siebel", "given_name": "Daniel A.", "clpid": "Siebel-Daniel-A" } ], "thesis_advisor": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" } ], "thesis_committee": [ { "family_name": "Mantovan", "given_name": "Elena", "clpid": "Mantovan-E" }, { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Amir-Khosravi", "given_name": "Zavosh", "clpid": "Amir-Khosravi-Zavosh" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We explicate Flach's and Morin's special value conjectures in [8] for proper regular arithmetic surfaces \u03c0 : X \u2192 Spec Z and provide explicit formulas for the conjectural vanishing orders and leading Taylor coefficients of the associated arithmetic zeta-functions. In particular, we prove compatibility with the Birch and Swinnerton-Dyer conjecture, which has so far only been known for projective smooth X. Further, we derive a direct sum decomposition of R\u03c0*Z(n) into motivic degree components.", "doi": "10.7907/YMHN-2T74", "publication_date": "2019", "thesis_type": "phd", "thesis_year": "2019" }, { "id": "thesis:10920", "collection": "thesis", "collection_id": "10920", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05212018-182542598", "primary_object_url": { "basename": "chen_ruiyuan_2018.pdf", "content": "final", "filesize": 1188145, "license": "other", "mime_type": "application/pdf", "url": "/10920/1/chen_ruiyuan_2018.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Definability and Classification of Equivalence Relations and Logical Theories", "author": [ { "family_name": "Chen", "given_name": "Ruiyuan", "orcid": "0000-0002-5891-8717", "clpid": "Chen-Ruiyuan" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Lupini", "given_name": "Martino", "orcid": "0000-0003-1588-7057", "clpid": "Lupini-M" }, { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis consists of four independent papers.
\r\n\r\nIn the first paper, joint with Kechris, we study the global aspects of structurability in the theory of countable Borel equivalence relations. For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class. We show that K-structurability interacts well with various preorders commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of K-structurable equivalence relations for various K, under inclusion, and show that it is a distributive lattice. Finally, we consider the effect on K-structurability of various model-theoretic properties of K; in particular, we characterize the K such that every K-structurable equivalence relation is smooth.
\r\n\r\nIn the second paper, we consider the classes of Kn-structurable equivalence relations, where Kn is the class of n-dimensional contractible simplicial complexes. We show that every Kn-structurable equivalence relation Borel embeds into one structurable by complexes in Kn with the further property that each vertex belongs to at most Mn := 2n-1(n2+3n+2)-2 edges; this generalizes a result of Jackson-Kechris-Louveau in the case n=1.
\r\n\r\nIn the third paper, we consider the amalgamation property from model theory in an abstract categorical context. A category is said to have the amalgamation property if every pushout diagram has a cocone. We characterize the finitely generated categories I such that in every category with the amalgamation property, every I-shaped diagram has a cocone.
\r\n\r\nIn the fourth paper, we prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic Lω1ω: every countable Lω1ω-theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories (L,T) and (L',T'), every Borel functor Mod(L',T') → Mod(L,T) between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some L'ω1ω-interpretation of T in T', which generalizes a recent result of Harrison-Trainor, Miller, and Montalban in the case where T, T' are ℵ0-categorical.
", "doi": "10.7907/7BP3-VZ93", "publication_date": "2018", "thesis_type": "phd", "thesis_year": "2018" }, { "id": "thesis:11006", "collection": "thesis", "collection_id": "11006", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06012018-160828760", "primary_object_url": { "basename": "meehan_connor_2018.pdf", "content": "final", "filesize": 791323, "license": "other", "mime_type": "application/pdf", "url": "/11006/1/meehan_connor_2018.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Definable Combinatorics of Graphs and Equivalence Relations", "author": [ { "family_name": "Meehan", "given_name": "Connor George Walmsley", "orcid": "0000-0002-7596-2437", "clpid": "Meehan-Connor-George-Walmsley" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Lupini", "given_name": "Martino", "orcid": "0000-0003-1588-7057", "clpid": "Lupini-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Let D = (X, D) be a Borel directed graph on a standard Borel space X and let \u03c7B(D) be its Borel chromatic number. If F0, \u2026, Fn-1: X \u2192 X are Borel functions, let DF0, \u2026, Fn-1 be the directed graph that they generate. It is an open problem if \u03c7B(DF0, \u2026, Fn-1) \u2208 {1, \u2026, 2n + 1, \u21350}. Palamourdas verified the foregoing for commuting functions with no fixed points. We show here that for commuting functions with the property that there is a path from each x \u2208 X to a fixed point of some Fj, there exists an increasing filtration X = \u22c3m < \u03c9 Xm such that \u03c7B(DF0, \u2026, Fn-1\u21be Xm) \u2264 2n for each m. We also prove that if n = 2 in the previous case, then \u03c7B(DF0, F1) \u2264 4. It follows that the approximate measure chromatic number \u03c7apM(D) \u2264 2n + 1 when the functions commute.
\r\n\r\nIf X is a set, E is an equivalence relation on X, and n \u2208 \u03c9, then define [X]nE = {(x0, ..., xn - 1) \u2208 nX: (\u2200i,j)(i \u2260 j \u2192 \u00ac(xi E xj))}. For n \u2208 \u03c9, a set X has the n-J\u00f3nsson property if and only if for every function f: [X]n= \u2192 X, there exists some Y \u2286 X with X and Y in bijection so that f[[Y]n=] \u2260 X. A set X has the J\u00f3nsson property if and only for every function f : (\u22c3n \u2208 \u03c9 [X]n=) \u2192 X, there exists some Y \u2286 X with X and Y in bijection so that f[\u22c3n \u2208 \u03c9 [Y]n=] \u2260 X. Let n \u2208 \u03c9, X be a Polish space, and E be an equivalence relation on X. E has the n-Mycielski property if and only if for all comeager C \u2286 nX, there is some Borel A \u2286 X so that E \u2264B E \u21be A and [A]nE \u2286 C. The following equivalence relations will be considered: E0 is defined on \u03c92 by x E0 y if and only if (\u2203n)(\u2200k > n)(x(k) = y(k)). E1 is defined on \u03c9(\u03c92) by x E1 y if and only if (\u2203n)(\u2200k > n)(x(k) = y(k)). E2 is defined on \u03c92 by x E2 y if and only if \u2211{1\u2044(n + 1): x(n) \u2260 y(n)} < \u221e. E3 is defined on \u03c9(\u03c92) by x E3 y if and only if (\u2200n)(x(n) E0 y(n)). Holshouser and Jackson have shown that \u211d is J\u00f3nsson under AD. The present research will show that E0 does not have the 3-Mycielski property and that E1, E2, and E3 do not have the 2-Mycielski property. Under ZF + AD, \u03c92/E0 does not have the 3-J\u00f3nsson property.
\r\n\r\nLet G = (X, G) be a graph and define for b \u2265 1 its b-fold chromatic number \u03c7(b)(G) as the minimum size of Y such that there is a function c from X into b-sets of Y with c(x) \u2229 c(y) = \u2205 if x G y. Then its fractional chromatic number is \u03c7f(G) = infb \u03c7(b)(G)\u2044b if the quotients are finite. If X is Polish and G is a Borel graph, we can also define its fractional Borel chromatic number \u03c7fB(G) by restricting to only Borel functions. We similarly define this for Baire measurable and \u03bc-measurable functions for a Borel measure \u03bc. We show that for each countable graph G, one may construct an acyclic Borel graph G' on a Polish space such that \u03c7fBM(G') = \u03c7f(G) and \u03c7BM(G') = \u03c7(G), and similarly for \u03c7f\u03bc and \u03c7\u03bc. We also prove that the implication \u03c7f(G) = 2 \u21d2 \u03c7(G) = 2 is false in the Borel setting.
", "doi": "10.7907/45E4-MC27", "publication_date": "2018", "thesis_type": "phd", "thesis_year": "2018" }, { "id": "thesis:10934", "collection": "thesis", "collection_id": "10934", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05252018-080955604", "primary_object_url": { "basename": "final_thesis_05252018.pdf", "content": "final", "filesize": 414444, "license": "other", "mime_type": "application/pdf", "url": "/10934/1/final_thesis_05252018.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Self-Gluing Formula of the Monopole Invariant and its Application on Symplectic Structures", "author": [ { "family_name": "Jeong", "given_name": "Gahye", "orcid": "0000-0003-3273-7691", "clpid": "Jeong-Gahye" } ], "thesis_advisor": [ { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" } ], "thesis_committee": [ { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" }, { "family_name": "Markovic", "given_name": "Vladimir", "clpid": "Markovic-V" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Vafaee", "given_name": "Faramarz", "orcid": "0000-0003-2086-9558", "clpid": "Vafaee-F" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Seiberg-Witten theory has been an important tool in studying a class of 4-manifolds. Moreover, the Seiberg-Witten invariants have been used to compute for simple structures of symplectic manifolds. The normal connected sum operation on 4- manifolds has been used to construct 4-manifolds. In this thesis, we demonstrate how to compute the Seiberg-Witten invariant of 4-manifolds obtained from the normal connected sum operation. In addition, we introduce the application of the formula on the existence of symplectic structures of manifolds given by the normal connected sum.
\r\n\r\nIn Chapter 1, we study the Seiberg-Witten theory for various types of 3- and 4- manifolds. We review the Seiberg-Witten equation and invariants for 4-manifolds with cylindrical ends as well as closed and smooth 4-manifolds . Furthermore, we explain how to compute the Seiberg-Witten invariants for two types of 4-manifolds: the products of a circle and a 3-manifold and sympectic manifolds.
\r\n\r\nIn Chapter 2, we prove that the Seiberg-Witten invariant of a new manifold obtained from the normal connected sum can be represented by the Seiberg-Witten invariant of the original manifolds. In [Tau01], the author has proved the case of the operation along tori. In [MST96], the authors have proved the case of the operation along surfaces with genus at least 2 when the product of the circle and the surface is separating in the ambient 4-manifold. In this thesis, we show the proof of the remaining case.
\r\n\r\nIn Chapter 3, we prove the existence of certain symplectic structures on manifolds obtained from the normal connected sum of two 4-manifolds using the multiple gluing formula stated in Chapter 2. We explain how to construct covering spaces of the manifold and compute the Seiberg-Witten invariant of the covering spaces by the gluing formula. From the relation between the Seiberg-Witten invariants and symplectic structures, we prove the main application.
", "doi": "10.7907/BH06-KS91", "publication_date": "2018", "thesis_type": "phd", "thesis_year": "2018" }, { "id": "thesis:10170", "collection": "thesis", "collection_id": "10170", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05122017-151540545", "primary_object_url": { "basename": "seunghee_ye_2017_thesis.pdf", "content": "final", "filesize": 501743, "license": "other", "mime_type": "application/pdf", "url": "/10170/1/seunghee_ye_2017_thesis.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "The Geometry of Moduli Spaces of Maps from Curves", "author": [ { "family_name": "Ye", "given_name": "Seunghee", "orcid": "0000-0002-1250-2935", "clpid": "Ye-Seunghee" } ], "thesis_advisor": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" } ], "thesis_committee": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Solis", "given_name": "Pablo R.", "clpid": "Solis-P-R" }, { "family_name": "Zhu", "given_name": "Xinwen", "clpid": "Zhu-Xinwen" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "A class of moduli spaces that has long been the interest of many algebraic geometers is the class of moduli spaces parametrizing maps from curves to target spaces. Different such moduli spaces have distinct geometry and also invariants associated to them. In this thesis, we will study the geometry of three such moduli spaces. By understanding the global geometry of each moduli space, we will produce a stratification, which plays a central role in proving a result about invariants associated to the space.", "doi": "10.7907/Z9MP5191", "publication_date": "2017", "thesis_type": "phd", "thesis_year": "2017" }, { "id": "thesis:10156", "collection": "thesis", "collection_id": "10156", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05052017-144643636", "primary_object_url": { "basename": "Nasrollahpoursamami_Emad_2017.pdf", "content": "final", "filesize": 406991, "license": "other", "mime_type": "application/pdf", "url": "/10156/1/Nasrollahpoursamami_Emad_2017.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Periods of Feynman Diagrams", "author": [ { "family_name": "Nasrollahpoursamami", "given_name": "Emad", "orcid": "0000-0002-9658-1529", "clpid": "Nasrollahpoursamami-Emad" } ], "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": "Rains", "given_name": "Eric M.", "clpid": "Rains-E-M" }, { "family_name": "Zhu", "given_name": "Xinwen", "clpid": "Zhu-Xinwen" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We study differential equations for Feynman amplitudes and show that the corresponding D-module is isomorphic to a GKZ D-modules. We show that the sheaf of solutions to the D-module is isomorphic to a certain relative homology and that the amplitudes are periods of a relative motive. Using these ideas, we develop a method of regularization which specializes to dimensional regularization and analytic regularization.
", "doi": "10.7907/Z9GX48MR", "publication_date": "2017", "thesis_type": "phd", "thesis_year": "2017" }, { "id": "thesis:10236", "collection": "thesis", "collection_id": "10236", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05312017-155530848", "primary_object_url": { "basename": "chan_william_2017.pdf", "content": "final", "filesize": 735746, "license": "other", "mime_type": "application/pdf", "url": "/10236/1/chan_william_2017.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Aspects of Definability for Equivalence Relations", "author": [ { "family_name": "Chan", "given_name": "William", "orcid": "0000-0002-0661-1764", "clpid": "Chan-William" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" }, { "family_name": "Lupini", "given_name": "Martino", "orcid": "0000-0003-1588-7057", "clpid": "Lupini-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis will show that in the constructible universe L and set forcing extensions of L, there are no almost Borel reductions of the well-ordering equivalence relation into the admissibility equivalence relation and no Borel reductions of the isomorphism relation of any counterexample to Vaught's conjecture into the admissibility equivalence relation.
\r\n\r\nLet E be an analytic equivalence relation on a Polish space X with all classes Borel. Let I be a sigma-ideal on X such that its associated forcing of I-positive Borel subsets is a proper forcing. Assuming sharps of appropriate sets exist, it will be shown that there is an I-positive Borel subset of X on which the restriction of E is a Borel equivalence relation.
\r\n\r\nAssuming there are infinitely many Woodin cardinals below a measurable cardinal, then for any equivalence relation E in L(R) with all Borel classes and sigma-ideal I whose associated forcing is proper, there is an I-positive Borel set on which the restriction of E is Borel.
", "doi": "10.7907/Z90P0X3M", "publication_date": "2017", "thesis_type": "phd", "thesis_year": "2017" }, { "id": "thesis:9788", "collection": "thesis", "collection_id": "9788", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05272016-115010946", "primary_object_url": { "basename": "hwang_brian_2016_thesis.pdf", "content": "final", "filesize": 308442, "license": "other", "mime_type": "application/pdf", "url": "/9788/11/hwang_brian_2016_thesis.pdf", "version": "v9.0.0" }, "type": "thesis", "title": "Constructing Self-Dual Automorphic Representations on General Linear Groups", "author": [ { "family_name": "Hwang", "given_name": "Brian W.", "clpid": "Hwang-Brian-W" } ], "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": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Mantovan", "given_name": "Elena", "clpid": "Mantovan-E" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We prove a globalization theorem for self-dual representations of GLN over a totally real number field F, which gives a positive existence criterion for self-dual cuspidal automorphic representations of GLN(AF) with prescribed local components at a finite set of finite places. A byproduct of our argument is that the automorphic representations that we construct are cohomological (equivalently, regular algebraic) and so fall into the class of automorphic representations on GLN for which there is a well-established theory for how to attach Galois representations, using the etale cohomology of certain Shimura varieties. The primary motivation is to give a sort of \"bare-handed\" or \"low tech\" proof of a result that is implied by the philosophy of twisted endoscopy in the Langlands program. While we are guided by this overarching picture, in the argument itself, we obtain all our results by working directly on GLN and the group obtained by twisting it under the \"inverse-transpose\" involution. In particular, we do not appeal to any general results on twisted endoscopic transfer or assume any big \"black box\" results like the (conjectured) stabilization of the twisted trace formula. Hence, such results are unconditional as stated, and we remark throughout on why the particular assumptions that we impose turn out to be necessary, indicating the (often substantial amount of) additional work required to generalize the stated results.
\r\n\r\nIn an appendix, in stark contrast to our approach above, we give an abridged argument for proving a globalization theorem on GLN in great generality, assuming a couple of major technical hypotheses (albeit, ones that are widely believed to be true) and yielding to Arthur's endoscopic classiffication of representations of symplectic and special orthogonal groups. Our hope is for such an argument to provide an outline for how we might ultimately prove results like generalizations of the globalization criterion above in the future.
", "doi": "10.7907/Z9WD3XKB", "publication_date": "2016", "thesis_type": "phd", "thesis_year": "2016" }, { "id": "thesis:8863", "collection": "thesis", "collection_id": "8863", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05182015-134458833", "type": "thesis", "title": "On the Construction of Higher \u00e9tale Regulators", "author": [ { "family_name": "Fan", "given_name": "Sin Tsun Edward", "clpid": "Fan-Sin-Tsun-Edward" } ], "thesis_advisor": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" } ], "thesis_committee": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Mantovan", "given_name": "Elena", "clpid": "Mantovan-E" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We present three approaches to define the higher \u00e9tale regulator maps \u03a6r,net : Hret(X,Z(n)) → HrD(X,Z(n)) for regular arithmetic schemes. The first two approaches construct the maps on the cohomology level, while the third construction provides a morphism of complexes of sheaves on the \u00e9tale site, along with a technical twist that one needs to replace the Deligne-Beilinson cohomology by the analytic Deligne cohomology inspired by the work of Kerr, Lewis, and M\u00fcller-Stach. A vanishing statement of infinite divisible torsions under \u03a6r,net is established for r > 2n + 1.", "doi": "10.7907/Z9BZ63Z1", "publication_date": "2015", "thesis_type": "phd", "thesis_year": "2015" }, { "id": "thesis:8920", "collection": "thesis", "collection_id": "8920", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05292015-095300444", "primary_object_url": { "basename": "Thesis.pdf", "content": "", "filesize": 1360403, "license": "other", "mime_type": "application/pdf", "url": "/8920/1/Thesis.pdf", "version": "v1.0.0" }, "type": "thesis", "title": "Topological Strings, Double Affine Hecke Algebras, and Exceptional Knot Homology", "author": [ { "family_name": "Elliot", "given_name": "Ross Filip", "clpid": "Elliot-Ross-Filip" } ], "thesis_advisor": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" } ], "thesis_committee": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "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": "In this thesis, we consider two main subjects: refined, composite invariants and exceptional knot homologies of torus knots. The main technical tools are double affine Hecke algebras (\"DAHA\") and various insights from topological string theory.
\r\n\r\nIn particular, we define and study the composite DAHA-superpolynomials of torus knots, which depend on pairs of Young diagrams and generalize the composite HOMFLY-PT polynomials from the full HOMFLY-PT skein of the annulus. We also describe a rich structure of differentials that act on homological knot invariants for exceptional groups. These follow from the physics of BPS states and the adjacencies/spectra of singularities associated with Landau-Ginzburg potentials. At the end, we construct two DAHA-hyperpolynomials which are closely related to the Deligne-Gross exceptional series of root systems.
\r\n\r\nIn addition to these main themes, we also provide new results connecting DAHA-Jones polynomials to quantum torus knot invariants for Cartan types A and D, as well as the first appearance of quantum E6 knot invariants in the literature.
", "doi": "10.7907/Z9ST7MRK", "publication_date": "2015", "thesis_type": "phd", "thesis_year": "2015" }, { "id": "thesis:8879", "collection": "thesis", "collection_id": "8879", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05222015-133207861", "primary_object_url": { "basename": "Dawra_Nakul_2015_thesis.pdf", "content": "final", "filesize": 539248, "license": "other", "mime_type": "application/pdf", "url": "/8879/1/Dawra_Nakul_2015_thesis.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "On the Link Floer Homology of L-space Link", "author": [ { "family_name": "Dawra", "given_name": "Nakul", "clpid": "Dawra-Nakul" } ], "thesis_advisor": [ { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" } ], "thesis_committee": [ { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Vafaee", "given_name": "Faramarz", "orcid": "0000-0003-2086-9558", "clpid": "Vafaee-F" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We will prove that, for a 2 or 3 component L-space link, HFL- is completely determined by the multi-variable Alexander polynomial of all the sub-links of L, as well as the pairwise linking numbers of all the components of L. We will also give some restrictions on the multi-variable Alexander polynomial of an L-space link. Finally, we use the methods in this paper to prove a conjecture of Yajing Liu classifying all 2-bridge L-space links.", "doi": "10.7907/Z9CZ353T", "publication_date": "2015", "thesis_type": "phd", "thesis_year": "2015" }, { "id": "thesis:8460", "collection": "thesis", "collection_id": "8460", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06012014-162932302", "primary_object_url": { "basename": "Thesis_Liling_Gu.pdf", "content": "final", "filesize": 519404, "license": "other", "mime_type": "application/pdf", "url": "/8460/1/Thesis_Liling_Gu.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Integral Finite Surgeries on Knots in S\u00b3", "author": [ { "family_name": "Gu", "given_name": "Liling", "clpid": "Gu-Liling" } ], "thesis_advisor": [ { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" } ], "thesis_committee": [ { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Liu", "given_name": "Yi", "clpid": "Liu-Y" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Using the correction terms in Heegaard Floer homology, we prove that if a knot in S3 admits a positive integral T-, O-, or I-type surgery, it must have the same knot Floer homology as one of the knots given in our complete list, and the resulting manifold is orientation-preservingly homeomorphic to the p-surgery on the corresponding knot.", "doi": "10.7907/E7VQ-SV60", "publication_date": "2014", "thesis_type": "phd", "thesis_year": "2014" }, { "id": "thesis:7807", "collection": "thesis", "collection_id": "7807", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05312013-164406051", "primary_object_url": { "basename": "thesis.pdf", "content": "final", "filesize": 416503, "license": "other", "mime_type": "application/pdf", "url": "/7807/1/thesis.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Relative Mirror Symmetry and Ramifications of a Formula for Gromov-Witten Invariants", "author": [ { "family_name": "van Garrel", "given_name": "Michel", "clpid": "van-Garrel-Michel" } ], "thesis_advisor": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" } ], "thesis_committee": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" }, { "family_name": "Tian", "given_name": "Zhiyu", "clpid": "Tian-Z" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "For a toric Del Pezzo surface S, a new instance of mirror symmetry, said relative, is introduced and developed. On the A-model, this relative mirror symmetry conjecture concerns genus 0 relative Gromov-Witten of maximal tangency of S. These correspond, on the B-model, to relative periods of the mirror to S. Furthermore, for S not necessarily toric, two conjectures for BPS state counts are related. It is proven that the integrality of BPS state counts of the total space of the canonical bundle on S implies the integrality for the relative BPS state counts of S. Finally, a prediction of homological mirror symmetry for the open complement is explored. The B-model prediction is calculated in all cases and matches the known A-model computation for the projective plane.", "doi": "10.7907/9EQP-PD83", "publication_date": "2013", "thesis_type": "phd", "thesis_year": "2013" }, { "id": "thesis:7656", "collection": "thesis", "collection_id": "7656", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05022013-153358975", "primary_object_url": { "basename": "52.pdf", "content": "final", "filesize": 298220, "license": "other", "mime_type": "application/pdf", "url": "/7656/1/52.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Determinantal Hypersurface from a Geometric Perspective", "author": [ { "family_name": "Huang", "given_name": "Jingjing", "clpid": "Huang-Jingjing" } ], "thesis_advisor": [ { "family_name": "Rains", "given_name": "Eric M.", "clpid": "Rains-E-M" } ], "thesis_committee": [ { "family_name": "Rains", "given_name": "Eric M.", "clpid": "Rains-E-M" }, { "family_name": "Hassibi", "given_name": "Babak", "clpid": "Hassibi-B" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Wilson", "given_name": "Richard M.", "clpid": "Wilson-R-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this paper, we give a geometric interpretation of determinantal forms, both in the case of general matrices and symmetric matrices. We will prove irreducibility of the determinantal singular loci and state its dimension. We also provide detailed description of the singular locus for small dimensions.", "doi": "10.7907/KMK9-6493", "publication_date": "2013", "thesis_type": "phd", "thesis_year": "2013" }, { "id": "thesis:6756", "collection": "thesis", "collection_id": "6756", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:12212011-095330433", "type": "thesis", "title": "On the Weil-\u00e9tale Cohomology of S-Integers", "author": [ { "family_name": "Chiu", "given_name": "Yi-Chih", "clpid": "Chiu-Yi-Chih" } ], "thesis_advisor": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" } ], "thesis_committee": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Jorza", "given_name": "Andrei", "clpid": "Jorza-A" }, { "family_name": "Mantovan", "given_name": "Elena", "clpid": "Mantovan-E" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We generalize the Lichtenbaum's prototype of Weil-\u00e9tale cohomology to S-integers and study its relation to the Tate sequences. In the final part, we present a more natural way to define Weil-\u00e9tale cohomology for one-dimensional arithmetic schemes motivated by a dual quasi-isomorphism between Weil-\u00e9tale cohomology and \u00e9tale cohomology. ", "doi": "10.7907/W4CA-6489", "publication_date": "2012", "thesis_type": "phd", "thesis_year": "2012" }, { "id": "thesis:7057", "collection": "thesis", "collection_id": "7057", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05212012-102942842", "primary_object_url": { "basename": "awalker_thesis_2012.pdf", "content": "final", "filesize": 1413763, "license": "other", "mime_type": "application/pdf", "url": "/7057/1/awalker_thesis_2012.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Surface Maps into Free Groups", "author": [ { "family_name": "Walker", "given_name": "Alden Kent", "clpid": "Walker-Alden-Kent" } ], "thesis_advisor": [ { "family_name": "Calegari", "given_name": "Danny C.", "orcid": "0009-0007-9304-2822", "clpid": "Calegari-D" } ], "thesis_committee": [ { "family_name": "Markovic", "given_name": "Vladimir", "clpid": "Markovic-V" }, { "family_name": "Calegari", "given_name": "Danny C.", "orcid": "0009-0007-9304-2822", "clpid": "Calegari-D" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We exploit the combinatorial properties of surface maps into free groups to prove several new results in the field of stable commutator length and bounded cohomology. We show that random homomorphisms between free groups are isometries of scl; we prove interesting properties of the scl unit ball; we describe a transfer construction for quasimorphisms and give an infinite family of chains whose scl it certifies; we linearize the dynamics of endomorphisms on free groups and use this to prove that random endomorphisms can be realized by surface immersions, which provides many examples of surface subgroups of HNN extensions of free groups; and finally, we give an algorithm to compute scl in free products of finite or infinite cyclic groups that generalizes and improves previous work.", "doi": "10.7907/FDRS-9S44", "publication_date": "2012", "thesis_type": "phd", "thesis_year": "2012" }, { "id": "thesis:5861", "collection": "thesis", "collection_id": "5861", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05272010-144845323", "type": "thesis", "title": "The Arithmetic and Geometry of a Class of Algebraic Surfaces of General Type and Geometric Genus One", "author": [ { "family_name": "Lyons", "given_name": "Christopher Michael", "clpid": "Lyons-Christopher-Michael" } ], "thesis_advisor": [ { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" } ], "thesis_committee": [ { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Mantovan", "given_name": "Elena", "clpid": "Mantovan-E" }, { "family_name": "Gholampour", "given_name": "Amin", "clpid": "Gholampour-A" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We study of a class of algebraic surfaces of general type and geometric genus one, with a view toward arithmetic results. These surfaces, called CC surfaces here, have been classified over the complex numbers by Catanese and Ciliberto. At the heart of our work is a large monodromy result for a family containing all members of a large subclass of CC surfaces, called the admissible CC surfaces. This result is obtained by an analysis of degenerations of admissible CC surfaces.
\r\n\r\nWe apply this monodromy theorem to prove the Tate and semisimplicity conjectures for all admissible CC surfaces over finitely-generated fields of characteristic zero, which are statements about the Galois representations on their cohomology. We also apply the theorem to produce an example of an algebraic cycle on a Shimura variety of orthogonal type that is not contained in any proper special subvariety; this we do by using the period map of the aforementioned family. Finally, we deduce the existence of complex CC surfaces with the minimum possible Picard number.
", "doi": "10.7907/67VN-D890", "publication_date": "2010", "thesis_type": "phd", "thesis_year": "2010" }, { "id": "thesis:5820", "collection": "thesis", "collection_id": "5820", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05212010-060144793", "primary_object_url": { "basename": "DissCrepFlop.pdf", "content": "final", "filesize": 873224, "license": "other", "mime_type": "application/pdf", "url": "/5820/1/DissCrepFlop.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Gromov-Witten Invariants: Crepant Resolutions and Simple Flops", "author": [ { "family_name": "Cheong", "given_name": "Wan Keng", "clpid": "Cheong-Wan-Keng" } ], "thesis_advisor": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" } ], "thesis_committee": [ { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Aschbacher", "given_name": "Michael", "clpid": "Aschbacher-M" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Wales", "given_name": "David B.", "clpid": "Wales-D-B" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Let S be any smooth toric surface. We establish a ring isomorphism between the equivariant extended Chen-Ruan cohomology of the n-fold symmetric product stack [Symn(S)] of S and the equivariant extremal quantum cohomology of the Hilbert scheme Hilbn(S) of n points in S. This proves a generalization of Ruan's Cohomological Crepant Resolution Conjecture for the case of Symn(S).
\r\n\r\nMoreover, we determine the operators of small quantum multiplication by divisor classes on the orbifold quantum cohomology of [Symn(Ar)], where Ar is the minimal resolution of the cyclic quotient singularity C2/Zr+1. Under the assumption of the nonderogatory conjecture, these operators completely determine the quantum ring structure, which gives an affirmative answer to Bryan-Graber's Crepant Resolution Conjecture on [Symn(Ar)] and Hilbn(Ar). More strikingly, this allows us to complete a tetrahedron of equivalences relating the Gromov-Witten theories of [Symn(Ar)]/Hilbn(Ar) and the relative Gromov-Witten/Donaldson-Thomas theories of Ar x P1.
\r\n\r\nFinally, we prove a closed formula for an excess integral over the moduli space of degree d stable maps from unmarked curves of genus one to the projective space Pr for positive integers r and d. The result generalizes the multiple cover formula for Pr and reveals that any simple Pr flop of smooth projective varieties preserves the theory of extremal Gromov-Witten invariants of arbitrary genus. It also provides examples for which Ruan's Minimal Model Conjecture holds.
\r\n", "doi": "10.7907/6KZZ-MT72", "publication_date": "2010", "thesis_type": "phd", "thesis_year": "2010" }, { "id": "thesis:1827", "collection": "thesis", "collection_id": "1827", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05152009-115934", "primary_object_url": { "basename": "thesis.pdf", "content": "final", "filesize": 587586, "license": "other", "mime_type": "application/pdf", "url": "/1827/1/thesis.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "A Geometric Study of Commutator Subgroups", "author": [ { "family_name": "Zhuang", "given_name": "Dongping", "clpid": "Zhuang-Dongping" } ], "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": "Day", "given_name": "Matthew B.", "clpid": "Day-M-B" }, { "family_name": "Aschbacher", "given_name": "Michael", "orcid": "0000-0002-8380-0921", "clpid": "Aschbacher-M" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Let G be a group and G' its commutator subgroup. Commutator length (cl) and stable commutator length (scl) are naturally defined concepts for elements of G'. We study cl and scl for two classes of groups. First, we compute scl in generalized Thompson's groups and their central extensions. As a consequence, we find examples of finitely presented groups in which scl takes irrational (in fact, transcendental) values. Second, we study large scale geometry of the Cayley graph of a commutator subgroup with respect to the canonical generating set of all commutators. When G is a non-elementary hyperbolic group, we prove that, for any n, there exists a quasi-isometrically embedded, dimension n integral lattice in this graph. Thus this graph is not hyperbolic, has infinite asymptotic dimension, and has only one end. For a general finitely presented group, we show that this graph is large scale simply connected.\r\n", "doi": "10.7907/J566-2537", "publication_date": "2009", "thesis_type": "phd", "thesis_year": "2009" } ]