Paths within a given space are constrained by the space's geometry. Any data generated by a path must then reflect the path's base space. A path's data constrained by the base space's geometry\u2014which we call the lifting property\u2014is the fundamental relation this thesis explores. Specifically, this thesis examines how mathematical functions\u2014and statistical algorithms\u2014that generate a path's data recover the base space's geometry\u2014termed predictive coding for cognitive mapping.
\r\n\r\nIt is shown that mathematical functions that generate a path's data recover the base space's geometry. We show that statistical estimators that generate a path's data can also recover the base space's geometry in a simulated natural environment. We also shown that mathematical functions recover a dynamical system if the function generates observations on a dynamical system's trajectory. We demonstrate applications of this principle to a bipedal robot's kinematics. Finally, we extend predictive coding for mapping\u2014predicting future observations from past observations to build an environmental map\u2014to plan and explore within the environment.
", "doi": "10.7907/rmb1-7w43", "publication_date": "2027", "thesis_type": "phd", "thesis_year": "2027" }, { "id": "thesis:18645", "collection": "thesis", "collection_id": "18645", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05272026-000328006", "primary_object_url": { "basename": "Andrew_Graven_PhD_Thesis_Final.pdf", "content": "final", "filesize": 5178414, "license": "other", "mime_type": "application/pdf", "url": "/18645/1/Andrew_Graven_PhD_Thesis_Final.pdf", "version": "v7.0.0" }, "type": "thesis", "title": "Weighted Quadrature Domains and the Faber Transform", "author": [ { "family_name": "Graven", "given_name": "Andrew J.", "orcid": "0000-0002-1998-3073", "clpid": "Graven-Andrew-J" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Isett", "given_name": "Philip", "orcid": "0000-0001-9038-5546", "clpid": "Isett-Phlip" }, { "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": "Sagman", "given_name": "Nathaniel", "orcid": "0000-0002-8485-7073", "clpid": "Sagman-Nathaniel-Levi" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "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:18726", "collection": "thesis", "collection_id": "18726", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05312026-224131162", "primary_object_url": { "basename": "thesis_yixinxu_final.pdf", "content": "final", "filesize": 7027464, "license": "other", "mime_type": "application/pdf", "url": "/18726/1/thesis_yixinxu_final.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Geometric Constraints on CFT-Like Data at Genus 0, 1, and 2.", "author": [ { "family_name": "Xu", "given_name": "Yixin", "orcid": "0009-0003-9849-1433", "clpid": "Xu-Yixin" } ], "thesis_advisor": [ { "family_name": "Simmons-Duffin", "given_name": "David", "orcid": "0000-0002-2937-9515", "clpid": "Simmons-Duffin-D" } ], "thesis_committee": [ { "family_name": "Ooguri", "given_name": "Hirosi", "orcid": "0000-0001-6021-3778", "clpid": "Ooguri-H" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Simmons-Duffin", "given_name": "David", "orcid": "0000-0002-2937-9515", "clpid": "Simmons-Duffin-D" }, { "family_name": "Wang", "given_name": "Yifan", "orcid": "0000-0001-9965-9777", "clpid": "Wang-Yifan" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis studies how self-consistency constraints on conformal field theory like observables can be used to extract information about CFT or geometric data. The observables considered are associated with genus-0, genus-1, and genus-2 geometries: the four-punctured sphere, the torus S1 x Sd-1, and the genus-2 manifold (S1 x Sd-1)\u266f(S1 x Sd-1). Although these observables differ in geometric and analytic complexity, they are united by a common principle: the same observable admits multiple equivalent decompositions, and the requirement that these descriptions agree imposes nontrivial constraints on the underlying spectrum and operator product expansion.
\r\n\r\nThe first part develops a hyperbolic bootstrap for compact hyperbolic spin surfaces. Using an analogy between harmonic analysis on hyperbolic manifolds and the conformal bootstrap, we derive rigorous bounds on Laplacian and Dirac spectral gaps by combining representation-theoretic methods with semidefinite programming. The second part studies thermal partition functions with angular twists. We show that, even in the absence of full modular invariance in dimensions greater than two, the high-temperature behavior of these observables is controlled by thermal effective field theory, leading to universal relations among twisted partition functions. The third part studies the genus-2 partition function in conformal field theory. By equating different decompositions of this observable, we derive a genus-2 crossing equation and use it to relate thermal one-point data to heavy operator product coefficients.
", "doi": "10.7907/36nw-bd70", "publication_date": "2026", "thesis_type": "phd", "thesis_year": "2026" }, { "id": "thesis:17383", "collection": "thesis", "collection_id": "17383", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06022025-234901151", "type": "thesis", "title": "The Topology of Cellular Ontogeny", "author": [ { "family_name": "Flores-Bautista", "given_name": "Emanuel", "orcid": "0000-0002-2810-1757", "clpid": "Flores-Bautista-Emanuel" } ], "thesis_advisor": [ { "family_name": "Thomson", "given_name": "Matthew", "orcid": "0000-0003-1021-1234", "clpid": "Thomson-M-W" } ], "thesis_committee": [ { "family_name": "Prober", "given_name": "David A.", "orcid": "0000-0002-7371-4675", "clpid": "Prober-D-A" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Pachter", "given_name": "Lior S.", "orcid": "0000-0002-9164-6231", "clpid": "Pachter-L" }, { "family_name": "Thomson", "given_name": "Matthew", "orcid": "0000-0003-1021-1234", "clpid": "Thomson-M-W" } ], "local_group": [ { "literal": "div_bbe" } ], "abstract": "A fundamental goal of modern biology is to build global, predictive models of gene regulation that encompass diverse physiological contexts. Single-cell transcriptomics has enabled the creation of developmental cell atlases--detailed catalogs of gene expression patterns and differentiation trajectories at an organismal scale. The widespread availability of cell atlases across metazoan model organisms presents an opportunity to construct global theories of cell-state control. In this thesis, we introduce a framework that uses persistent homology to decompose cell atlases into topological structures that provide signatures of gene regulation at the scale of an organism. Using this framework, we found that the topological structure of a broad set of developmental atlases contains only a discrete set of topological structures\u2014such as clusters, trees, and loops\u2014-revealing the recurrent use of global gene regulatory strategies. Our analysis revealed that the tree topology, while predominant, is not universal. Indeed, we identified non-trivial topologies containing loops in the development of human immune cells, seam-hypodermal cells in \\textit{C. elegans}, and the cnidocytes of multiple cnidarians. Analysis of cell-state manifolds with non-trivial topology demonstrated an important role of convergent structures in increasing cellular diversity along paths to a common cell fate, and of cyclic structures in self-renewal of progenitor-like states. Together, this work provides a global perspective on principles of cell-state regulation, and suggests that loops are important organizing structures for controlling cell differentiation.", "doi": "10.7907/t8hc-yq15", "publication_date": "2025", "thesis_type": "phd", "thesis_year": "2025" }, { "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:17444", "collection": "thesis", "collection_id": "17444", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06102025-233803574", "primary_object_url": { "basename": "Thesis.pdf", "content": "final", "filesize": 1146603, "license": "other", "mime_type": "application/pdf", "url": "/17444/2/Thesis.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Aspects of Topology and Measurement in Quantum Lattice Systems", "author": [ { "family_name": "Artymowicz", "given_name": "Adam Markus", "orcid": "0000-0002-5461-3991", "clpid": "Artymowicz-Adam-Markus" } ], "thesis_advisor": [ { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" } ], "thesis_committee": [ { "family_name": "Kitaev", "given_name": "Alexei", "orcid": "0000-0002-5777-642X", "clpid": "Kitaev-A" }, { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Ranard", "given_name": "Daniel", "orcid": "0000-0002-8199-7425", "clpid": "Ranard-D" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In the first part of this thesis, topological invariants of gapped phases on the lattice are studied. These include the Berry curvature, Thouless pump, the Hall conductance, and their higher-dimensional analogs. These invariants are proven to obstruct the promotion of a global symmetry to a gauge symmetry. Two of these invariants, the 1d higher Berry curvature and the 2d higher Thouless pump, are studied in detail. First, it is shown that they are related by a relation involving flux insertion, which can be interpreted physically as identifying the higher Thouless pump invariant with the excess Berry curvature of a fluxon. Second, it is proven that these two invariants take on quantized values in an invertible state.
\r\n\r\nIn the second part of this thesis, an algorithm is presented for learning Hamiltonian parameters from local expectation values of its Gibbs state via a local free-energy variational principle. The algorithm is benchmarked on the problem of black-box learning of a nearest-neighbour Hamiltonian in a 100-qubit spin chain, giving evidence of favourable scaling with system size. The theoretical analysis is then extended to incorporate measurement noise, as well as equipping the algorithm with certified a posteriori lower and upper error bounds on the inferred parameters. For commuting Hamiltonians, a priori convergence guarantees are also established.
", "doi": "10.7907/jacd-t049", "publication_date": "2025", "thesis_type": "phd", "thesis_year": "2025" }, { "id": "thesis:17348", "collection": "thesis", "collection_id": "17348", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06012025-234156081", "primary_object_url": { "basename": "PhD_Thesis (4).pdf", "content": "final", "filesize": 6885197, "license": "other", "mime_type": "application/pdf", "url": "/17348/1/PhD_Thesis (4).pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Mathematics of the Holographic Principle", "author": [ { "family_name": "Gesteau", "given_name": "Elliott M.", "orcid": "0000-0002-3189-0953", "clpid": "Gesteau-Elliott-M" } ], "thesis_advisor": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "thesis_committee": [ { "family_name": "Ooguri", "given_name": "Hirosi", "orcid": "0000-0001-6021-3778", "clpid": "Ooguri-H" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" }, { "family_name": "Liu", "given_name": "Hong", "clpid": "Liu-Hong" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The holographic principle, which states that quantum gravity in a given spacetime region admits an equivalent description in terms of a quantum system without gravity on its boundary, is a very promising candidate to lay the foundations of our understanding of quantum gravity. However, a precise general formulation of this principle, as well as its domain of applicability, are yet to be understood. This thesis explores the foundations of holography from a mathematical point of view. In particular, the theory of von Neumann algebras is exploited to understand features of the emergence of spacetime in holography, leveraging tools from quantum error correction in infinite dimensions as well as results on harmonic analysis. Some connections between hyperbolic geometry and recent developments in holography are also elucidated, an approach to the factorization of holographic theories based on Hopf algebras is developed, and a puzzle regarding the description of a closed universe within the context of the AdS/CFT correspondence is put forward.", "doi": "10.7907/2mkb-nz86", "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: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:16477", "collection": "thesis", "collection_id": "16477", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06032024-064737346", "primary_object_url": { "basename": "Ismail_Abouamal_thesis.pdf", "content": "final", "filesize": 602198, "license": "other", "mime_type": "application/pdf", "url": "/16477/1/Ismail_Abouamal_thesis.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Bost\u2013Connes\u2013Marcolli System for the Siegel Modular Variety", "author": [ { "family_name": "Abouamal", "given_name": "Ismail", "orcid": "0009-0004-0462-8177", "clpid": "Abouamal-Ismail" } ], "thesis_advisor": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "thesis_committee": [ { "family_name": "Mantovan", "given_name": "Elena", "orcid": "0000-0003-4521-2130", "clpid": "Mantovan-E" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Khalkhali", "given_name": "Masoud", "orcid": "0000-0003-2552-0208", "clpid": "Khalkhali-M" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Bost\u2013Connes\u2013Marcolli systems are an important type of quantum statistical mechanical systems which provide connections between the ergodic theory of dynamical systems, class field theory and von Neumann algebras. In this thesis, we generalize the GL\u2082-system to the case of the Siegel modular variety of degree two and study its various properties. We show that this dynamical system undergoes a spontaneous symmetry breaking phase transition and classify its equilibrium extremal states at different inverse temperatures. We next study its symmetry group and derive an intertwining equality between the action by symmetries and a subgroup of the Galois group of the Siegel modular field. Finally, we study the von Neumann algebras associated to the equilibrium states and classify their types.", "doi": "10.7907/hjp3-8a71", "publication_date": "2024", "thesis_type": "phd", "thesis_year": "2024" }, { "id": "thesis:16072", "collection": "thesis", "collection_id": "16072", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06032023-021309617", "primary_object_url": { "basename": "Sopenko_Nikita_2023.pdf", "content": "final", "filesize": 870905, "license": "other", "mime_type": "application/pdf", "url": "/16072/1/Sopenko_Nikita_2023.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Topological Invariants of Gapped Quantum Lattice Systems", "author": [ { "family_name": "Sopenko", "given_name": "Nikita A.", "orcid": "0000-0002-8479-1924", "clpid": "Sopenko-Nikita-A" } ], "thesis_advisor": [ { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" } ], "thesis_committee": [ { "family_name": "Chen", "given_name": "Xie", "orcid": "0000-0003-2215-2497", "clpid": "Chen-Xie" }, { "family_name": "Kitaev", "given_name": "Alexei", "orcid": "0000-0002-5777-642X", "clpid": "Kitaev-A" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In the first part of the thesis, a systematic way to construct topological invariants of gapped states of quantum lattices systems is proposed. It provides a generalization of the Berry phase and its equivariant analogue to systems with locality in arbitrary dimensions. For a smooth family of gapped ground states in d dimensions, it gives a closed (d + 2)-form on the parameter space which generalizes the curvature of the Berry connection. Its cohomology class is a topological invariant of the family. When the family is equivariant under the action of a compact Lie group G, topological invariants take values in the equivariant cohomology of the parameter space. These invariants unify and generalize the Hall conductance and the Thouless pump. We prove quantization properties of the invariants for low-dimensional invertible systems.
\r\n\r\nIn the second part, we discuss the properties of the invariant associated with the Hall conductance for 2d lattice systems with U(1)-symmetry. We define anyonic states associated with the flux insertions and relate their statistics to this invariant. We also provide the construction of states realizing chiral topological order with a non-trivial value of this invariant. The construction is based on the data of a unitary regular vertex operator algebra.
", "doi": "10.7907/715e-q184", "publication_date": "2023", "thesis_type": "phd", "thesis_year": "2023" }, { "id": "thesis:15183", "collection": "thesis", "collection_id": "15183", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05172023-003322589", "type": "thesis", "title": "Quantum States: With a View Toward Homological Algebra", "author": [ { "family_name": "Yang", "given_name": "Bowen", "orcid": "0000-0003-4778-831X", "clpid": "Yang-Bowen" } ], "thesis_advisor": [ { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" } ], "thesis_committee": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Chen", "given_name": "Xie", "orcid": "0000-0003-2215-2497", "clpid": "Chen-Xie" }, { "family_name": "Motrunich", "given_name": "Olexei I.", "orcid": "0000-0001-8031-0022", "clpid": "Motrunich-Olexei" }, { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The thesis comprises three papers covering different topics in quantum many-body physics. The first paper examines translationally invariant Pauli stabilizer codes, introducing invariants called charge modules and discussing their properties. The second paper explores invertible (G-invariant) states of 1D bosonic quantum lattice systems (or spin chains), demonstrating a full classification using group cohomology. The third paper analyzes the relation between ordinary correlators and Kubo's canonical correlators for thermal states of systems with short-range interactions. Overall, the thesis highlights the power of mathematics, especially homological methods, in understanding quantum states.
", "doi": "10.7907/fm81-v416", "publication_date": "2023", "thesis_type": "phd", "thesis_year": "2023" }, { "id": "thesis:15256", "collection": "thesis", "collection_id": "15256", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06012023-043437790", "type": "thesis", "title": "Regularities, Resurgence and R-Matrices in Chern Simons Theory", "author": [ { "family_name": "Gruen", "given_name": "Angus Fred Wilkinson", "orcid": "0000-0003-0284-009X", "clpid": "Gruen-Angus-Fred-Wilkinson" } ], "thesis_advisor": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" } ], "thesis_committee": [ { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" }, { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Ekholm", "given_name": "Tobias", "orcid": "0000-0003-2060-8462", "clpid": "Ekholm-Tobias" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis aims to address two related but distinct problems in Chern Simons theory:
\r\n\r\n1. In 2019, Gukov and Manolescu observed that for fixed a knot K, the family of coloured Jones polynomials Jk(K; q) display regularity in colour k and conjectured that this could be captured by a 2 variable series FK(x, q). Over the subsequent few years, Park proved that, for a large family of knots, FK(x, q) could be computed using the R-matrix for a particular Verma module.
\r\n\r\nWe will show that it is possible to extend the work of Park to compute the 2 variable series FNK(x, q) associated to other lie groups, slN, which capture a similar regularity in the quantum invariants PNk(K; q). Following on from this we will further show that in many cases these series FNK(x, q) themselves display a regularity in N, reminiscent of the HOMFLY-PT polynomial, allowing the construction of a 3 variable series FK(x, a, q) interpolating FNK(x, q) for all N.
\r\n\r\n2. Complex Chern Simons theory is a rare example of Quantum field theory with both interesting non-perturbative behaviour and whose perturbative expansion can be computed to high order. For a nice class of 3-manifolds, namely surgeries on knot complements, we will show how to predict aspects of the non-perturbative behaviour first semi-classically and then, using resurgence, through studying just the perturbative expansion around the trivial flat connection. Finally, we show that contrary to expectation, these families of 3-manifolds display regularity in the surgery coefficient.
", "doi": "10.7907/dr5q-3074", "publication_date": "2023", "thesis_type": "phd", "thesis_year": "2023" }, { "id": "thesis:14356", "collection": "thesis", "collection_id": "14356", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:09072021-163418163", "type": "thesis", "title": "Discrete Deligne Cohomology and Discretized Abelian Chern-Simons Theory", "author": [ { "family_name": "Norton", "given_name": "Thomas Clark", "orcid": "0000-0003-1438-8408", "clpid": "Norton-Thomas-Clark" } ], "thesis_advisor": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "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": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Pei", "given_name": "Du", "orcid": "0000-0001-5587-6905", "clpid": "Pei-Du" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The differential cohomology groups of a smooth manifold are discretized with respect to a triangulation. The realization of differential cohomology used is Deligne cohomology. A discretized version of the smooth Deligne double complex is constructed from cochain groups defined on simplices of the triangulation. The total cohomology of this double complex is studied and shown to satisfy exact sequences analogous to the standard structural sequences satisfied by differential cohomology. In the degree corresponding to line bundles with connection, our cohomology classes are shown to correspond to isomorphism classes of an existing notion of discrete line bundles with connection. Explicit examples of these discrete line bundles with connection are constructed. A ring structure is defined on the discrete Deligne cohomology groups; it is graded-commutative and non-associative (however, associativity is recovered in the continuum limit). The ring structure allows one to define a more general discrete Chern-Simons action than has previously appeared in the literature.
", "doi": "10.7907/aae8-re02", "publication_date": "2022", "thesis_type": "phd", "thesis_year": "2022" }, { "id": "thesis:14622", "collection": "thesis", "collection_id": "14622", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05252022-192257156", "primary_object_url": { "basename": "Jane_Panangaden_thesis_final.pdf", "content": "final", "filesize": 875985, "license": "other", "mime_type": "application/pdf", "url": "/14622/2/Jane_Panangaden_thesis_final.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Quantum Statistical Mechanics, Noncommutative Geometry, and the Boundary of Modular Curves", "author": [ { "family_name": "Panangaden", "given_name": "Jane Mariam", "orcid": "0000-0002-6214-2130", "clpid": "Panangaden-Jane-Mariam" } ], "thesis_advisor": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "thesis_committee": [ { "family_name": "Mantovan", "given_name": "Elena", "orcid": "0000-0003-4521-2130", "clpid": "Mantovan-E" }, { "family_name": "Rains", "given_name": "Eric M.", "clpid": "Rains-E-M" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "orcid": "0000-0002-0159-087X", "clpid": "Ramakrishnan-D" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The Bost-Connes system is a C*-dynamical system whose partition function, KMS states, and symmetries are related to the explicit class field theory of the field of rational numbers. In particular, its zero-temperature KMS states, when evaluated on certain points in an arithmetic sub-algebra, yield the generators of the maximal abelian extension of the rationals. The Bost-Connes system can be viewed in terms of a geometric picture of 1-dimensional Q-lattices. The GL\u2082 system is an extension of this idea to the setting of 2-dimensional Q-lattices. A specialization of the GL\u2082-system introduced in by Connes, Marcolli, and Ramachandran, is related in a similar way to the explicit class field theory of imaginary quadratic extensions.
\r\n\r\nInspired by the philosophy of Manin's real multiplication program, we define a boundary version of the GL\u20822-system. In this viewpoint we see the projective line under a certain PGL(2,Z) action (which is related to the shift of the continued fraction expansion) as a moduli space characterizing degenerate elliptic curves. These degenerate elliptic curves can be realized as noncommutative 2-tori. This moduli space of the non-commutative tori is interpreted as an invisible boundary of the moduli space of elliptic curves. In fact, we define a family of such boundary GL\u2082 systems indexed by a choice of continued fraction algorithm. We analyze their partition functions, KMS states, and ground states. We also define an arithmetic algebra of unbounded multipliers in analogy with the GL\u2082 case. We show that the ground states when evaluated on points in the arithmetic algebra give pairings of the limiting modular symbols introduced by Manin and Marcolli with weight-2 cusp forms.
\r\n\r\nWe also begin the project of extending this picture to the higher weight setting by defining a higher-weight limiting modular symbol. We use as a starting point the Shokurov modular symbols, which are constructed using Kuga modular varieties, which are non-singular projective varieties over the modular curves. We subject these modular symbols to a limiting procedure. We then show, using the coding space setting of Kessenbohmer and Stratmann, that these limiting modular symbols can be written as a Birkhoff ergodic average everywhere.
", "doi": "10.7907/xrba-8471", "publication_date": "2022", "thesis_type": "phd", "thesis_year": "2022" }, { "id": "thesis:13859", "collection": "thesis", "collection_id": "13859", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:08272020-235955490", "primary_object_url": { "basename": "Caltech_Thesis_Minyoung_You.pdf", "content": "final", "filesize": 730006, "license": "other", "mime_type": "application/pdf", "url": "/13859/1/Caltech_Thesis_Minyoung_You.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Topological Phases of Matter: Classification, Stacking Law, and Relation to Topological Quantum Field Theory", "author": [ { "family_name": "You", "given_name": "Minyoung", "orcid": "0000-0002-8251-953X", "clpid": "You-Minyoung" } ], "thesis_advisor": [ { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" } ], "thesis_committee": [ { "family_name": "Chen", "given_name": "Xie", "orcid": "0000-0003-2215-2497", "clpid": "Chen-Xie" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Motrunich", "given_name": "Olexei I.", "orcid": "0000-0001-8031-0022", "clpid": "Motrunich-Olexei" }, { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We study aspects of gapped phases of matter, focusing on their classification, including the group law under stacking, and their relation to topological quantum field theories (TQFT). In one spatial dimension, it is well-known that Matrix Product States (MPS) efficiently approximate ground states of gapped systems; by showing that these states arise naturally in 1 + 1-dimensional lattice TQFT, which in turn are closely related to continuum TQFT, we provide a concrete connection between ground states of lattice systems and TQFT in 1 + 1 dimensions. We generalize this to systems with symmetries and fermions, and obtain a classification and group law for the stacking of 1 + 1-dimensional symmetry-protected topological phases. Further, we study the effect of turning on/off interactions for the classification: the phase classification of a given symmetry class of Hamiltonians can be different depending on whether we allow interactions or not, and in low dimensions we provide some concrete formulas relating the phases under the non-interacting classification and those under the interacting classification. Lastly, we study the phases of the 2 + 1-dimensional topological superconductor, and show that for all 16 phases braiding statistics of vortices, which determine the underlying TQFT, can be obtained by stacking layers of the basic p + ip superconductor.
", "doi": "10.7907/4xq7-gh61", "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:11505", "collection": "thesis", "collection_id": "11505", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05092019-193947900", "primary_object_url": { "basename": "Tao_Jim_2019_thesis.pdf", "content": "final", "filesize": 434533, "license": "other", "mime_type": "application/pdf", "url": "/11505/25/Tao_Jim_2019_thesis.pdf", "version": "v14.0.0" }, "type": "thesis", "title": "Analysis on Vector Bundles over Noncommutative Tori", "author": [ { "family_name": "Tao", "given_name": "Jim", "orcid": "0000-0002-0751-9273", "clpid": "Tao-Jim" } ], "thesis_advisor": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "thesis_committee": [ { "family_name": "Rains", "given_name": "Eric M.", "clpid": "Rains-E-M" }, { "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": "Fathizadeh", "given_name": "Farzad", "orcid": "0000-0002-7863-4009", "clpid": "Fathizadeh-F" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Noncommutative geometry is the study of noncommutative algebras, especially C*-algebras, and their geometric interpretation as topological spaces. One C*-algebra particularly important in physics is the noncommutative n-torus, the irrational rotation C*-algebra A\u0398 with n unitary generators U1, . . . , Un which satisfy UkUj = e2\u03c0i\u03b8j,kUjUk and Uj* = Uj-1, where \u0398 \u2208 Mn(\u211d) is skew-symmetric with upper triangular entries that are irrational and linearly independent over \u211a. We focus on two projects: an analytically detailed derivation of the pseudodifferential calculus on noncommutative tori, and a proof of an index theorem for vector bundles over the noncommutative two torus. We use Raymond's definition of an oscillatory integral with Connes' construction of pseudodifferential operators to rederive the calculus in more detail, following the strategy of the derivations in Wong's book on pseudodifferential operators. We then define the corresponding analog of Sobolev spaces on noncommutative tori, for which we prove analogs of the Sobolev and Rellich lemmas, and extend all of these results to vector bundles over noncommutative tori. We extend Connes and Tretkoff's analog of the Gauss-Bonnet theorem for the noncommutative two torus to an analog of the McKean-Singer index theorem for vector bundles over the noncommutative two torus, proving a rearrangement lemma where a self-adjoint idempotent e appears in the denominator but does not commute with the k2 already there from the rearrangement lemma proven by Connes and Tretkoff.
", "doi": "10.7907/C4QF-GF45", "publication_date": "2019", "thesis_type": "phd", "thesis_year": "2019" }, { "id": "thesis:11515", "collection": "thesis", "collection_id": "11515", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05132019-171340673", "primary_object_url": { "basename": "GekhtmanDmitriThesis2019Final.pdf", "content": "final", "filesize": 450965, "license": "other", "mime_type": "application/pdf", "url": "/11515/1/GekhtmanDmitriThesis2019Final.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Two Holomorphic Extremal Problems in Teichm\u00fcller Theory", "author": [ { "family_name": "Gekhtman", "given_name": "Dmitri", "orcid": "0000-0002-6473-1115", "clpid": "Gekhtman-Dmitri" } ], "thesis_advisor": [ { "family_name": "Markovic", "given_name": "Vladimir", "clpid": "Markovic-V" } ], "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": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Ivrii", "given_name": "Oleg", "orcid": "0000-0002-8939-3108", "clpid": "Ivrii-O" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this thesis, we study the complex geometry of the Teichm\u00fcller space of conformal structures on a finite-type Riemann surface. We give partial answers to two structural questions: (1) Which holomorphic disks in Teichm\u00fcller space are holomorphic retracts of Teichm\u00fcller space? (2) What are the holomorphic and Kobayashi-isometric submersions between Teichm\u00fcller spaces? In both cases, the answers have to do with the geometry of the underlying surfaces, while the methods require developing and applying novel analytic tools.
\r\n\r\nQuestion (1) is equivalent to asking the following: on which pairs of points in Teichm\u00fcller space do the Carath\u00e9odory and Teichm\u00fcller metrics coincide? Markovic showed that the Carath\u00e9odory and Teichm\u00fcller metrics on Teichm\u00fcller space are not the same. On the other hand, Kra earlier showed that the metrics coincide when restricted to a Teichm\u00fcller disk generated by a differential with no odd-order zeros. We conjecture the converse: the Carath\u00e9odory and Teichm\u00fcller metrics agree on a Teichm\u00fcller disk if and only if the Teichm\u00fcller disk is generated by a differential with no odd-order zeros. We prove this conjecture for the Teichm\u00fcller spaces of the five-times punctured sphere and the twice-punctured torus. As a key analytic step in the proof, we study the family of holomorphic retractions from the polydisk onto its diagonal. In particular, we analyze the asymptotics of the orbit of such a retraction under the conjugation action of a unipotent subgroup of PSL2(\u211d).
\r\n\r\nQuestion (2) concerns holomorphic and isometric submersions between Teichm\u00fcller spaces of finite-type surfaces. We prove that, with potential exceptions coming from low-genusphenomena, any such map is a forgetful map \u03c4g,n \u2192 \u03c4g,m obtained by filling in punctures. This generalizes a classical result of Royden and Earle-Kra asserting that biholomorphisms between finite-type Teichm\u00fcller spaces arise from mapping classes. As a key step in the argument, we prove that any \u2102-linear embedding Q(X) \u21aa Q(Y) between spaces of holomorphic integrable quadratic differentials is, up to scale, pull-back by a holomorphic map. We accomplish this step by adapting methods developed by Markovic to study isometries of infinite-type Teichm\u00fcller spaces. The main analytic tool used is a theorem of Rudin on isometries of Lp spaces.
", "doi": "10.7907/XKMM-8591", "publication_date": "2019", "thesis_type": "phd", "thesis_year": "2019" }, { "id": "thesis:11735", "collection": "thesis", "collection_id": "11735", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06102019-125514401", "type": "thesis", "title": "Supersymmetric Scattering Amplitudes and Algebraic Aspects of Holography from the Projective Line", "author": [ { "family_name": "Heydeman", "given_name": "Matthew Thomas Edwin", "orcid": "000-0001-7033-9075", "clpid": "Heydeman-Matthew-Thomas-Edwin" } ], "thesis_advisor": [ { "family_name": "Schwarz", "given_name": "John H.", "orcid": "0000-0001-9861-7559", "clpid": "Schwarz-J-H" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "thesis_committee": [ { "family_name": "Cheung", "given_name": "Clifford W.", "orcid": "0000-0002-9983-9425", "clpid": "Cheung-Clifford" }, { "family_name": "Schwarz", "given_name": "John H.", "orcid": "0000-0001-9861-7559", "clpid": "Schwarz-J-H" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Carroll", "given_name": "Sean M.", "orcid": "0000-0002-4226-5758", "clpid": "Carroll-S-M" } ], "local_group": [ { "literal": "Walter Burke Institute for Theoretical Physics" }, { "literal": "div_pma" } ], "abstract": "In this thesis, we consider two topics in string theory and quantum field theory which are related by the common appearance of one-dimensional projective geometry. In the first half of the thesis, we study six-dimensional (6D) supersymmetric quantum field theories and supergravity at the leading (tree) approximation and compute the complete S-matrix for these theories as world-sheet integrals over the punctured Riemann sphere. This exploits the analytic structure of tree amplitudes which are rational and holomorphic in the kinematics and naturally related to the geometry of points on the complex projective line. The 6D n-particle S-matrix makes many symmetries and hidden properties manifest and generalizes the well-studied formulas for four-dimensional amplitudes in the form of twistor string theory and the rational curves formalism. While the systems we study are all field theories, they are in essence low-energy effective field theory limits of string theory and M-theory backgrounds. This includes theories such as those with 6D (2,0) supersymmetry which contain U(1) self-dual tensor fields which are difficult to treat from a Lagrangian point of view. Our formulas circumvent this difficulty and allow a generalization and unification of a large class of 6D scattering amplitudes which permit a sensible classical limit, including the abelian world-volume of the M-theory Five-brane. Dimensional reduction to four dimensions is also possible, leading to new formulas for 4D physics from 6D.
\r\n\r\nIn the second half of the thesis, we discuss the projective algebraic and geometric structure of the AdS3/CFT2 correspondence. In the usual statement of this correspondence, two-dimensional conformal field theory (CFT) on the Riemann sphere or a higher-genus surface is holographically dual to features of topological gravity in three dimensions with negative curvature. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. We generalize the AdS (anti-de Sitter space)/CFT correspondence according to this principle using projective geometry over the p-adic numbers, Qp. The result is a formulation of holography in which the bulk geometry is discrete---the Bruhat--Tits tree for PGL(2,Qp)---but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. Parallel to the usual holographic correspondence, semi-classical dynamics of fields in the bulk compute the correlation functions of local operators on the boundary. Beyond correlators on the p-adic line, we propose a tensor network model in which the patterns of entanglement on the boundary are computed by discrete geometries in the bulk. We suggest that this forms the natural geometric setting for tensor networks that have been proposed as models of bulk reconstruction via quantum error correcting codes. The model is built from tensors based on projective geometry over finite fields, Fp, and correctly computes the Ryu-Takayanagi formula, holographic entanglement and black hole entropy, and multiple interval entanglement inequalities.
\r\n\r\nIn Chapter 2, we present tree-level n-particle on-shell scattering amplitudes of various brane theories with 16 conserved supercharges which are generalizations of Dirac--Born--Infeld theory. These include the world-volume theory of a probe D3-brane or D5-brane in 10D Minkowski spacetime as well as a probe M5-brane in 11D Minkowski spacetime, which describes self interactions of an abelian tensor supermultiplet with 6D (2,0) supersymmetry. We propose twistor-string-like formulas for tree-level scattering amplitudes of all multiplicities for each of these theories, and the amplitudes are written as integrals over the moduli space of certain rational maps localized on the (n-3)! solutions of the scattering equations. The R symmetry of the D3-brane theory is shown to be SU(4) x U(1), and the U(1) factor implies that its amplitudes are helicity conserving. Each of 6D theories (D5-brane and M5-brane) reduces to the D3-brane theory by dimensional reduction. As special cases of the general M5-brane amplitudes, we present compact formulas for examples involving only the self-dual B field with n=4,6,8.
\r\n\r\nIn Chapter 3, we extend this formalism to n-particle tree-level scattering amplitudes of six-dimensional N=(1,1) super Yang--Mills (SYM) and N=(2,2) supergravity (SUGRA). The SYM theory arises on the world volume of coincident D5-branes, and the supergravity is the result of toroidal compactification of string theory. These theories have non-abelian interactions which allow for both even and odd-point amplitudes, unlike the branes of Chapter 2. Due to the properties of spinor-helicity variables in six dimensions, the even-n and odd-n formulas are quite different and have to be treated separately. We first propose a manifestly supersymmetric expression for the even-n amplitudes of N=(1,1) SYM theory and perform various consistency checks. By considering soft-gluon limits of the even-n amplitudes, we deduce the form of the rational maps and the integrand for n odd. The odd-n formulas obtained in this way have a new redundancy that is intertwined with the usual SL(2,C) invariance on the Riemann sphere. We also propose an alternative form of the formulas, analogous to the Witten--RSV (Roiban, Spradlin, and Volovich) formulation, and explore its relationship with the symplectic (or Lagrangian) Grassmannian. Since the amplitudes are formulated in a way that manifests double-copy properties, formulas for the six-dimensional N=(2,2) SUGRA amplitudes follow. These six-dimensional results allow us to deduce new formulas for five-dimensional SYM and SUGRA amplitudes, as well as massive amplitudes of four-dimensional N=4 SYM on the Coulomb branch.
\r\n\r\nIn Chapter 4, we consider half-maximal supergravity and present a twistor-like formula for the complete tree-level S matrix of chiral 6D (2,0) supergravity coupled to 21 abelian tensor multiplets. This is the low-energy effective theory that corresponds to Type IIB superstring theory compactified on a K3 surface. As in previous chapters, the formula is expressed as an integral over the moduli space of certain rational maps of the punctured Riemann sphere; the new ingredient is an integrand which successfully incorporates both gravitons and multiple flavors of tensors. By studying soft limits of the formula, we are able to explore the local moduli space of this theory, SO(5,21)/(SO(5) x SO(21)). Finally, by dimensional reduction, we also obtain a new formula for the tree-level S-matrix of 4D N=4 Einstein--Maxwell theory.
\r\n\r\nIn Chapter 5, we introduce p-adic AdS/CFT and discuss several physical and mathematical features of the holographic correspondence between conformal field theories on P1(Qp) and lattice models on the Bruhat--Tits tree of PGL(2,Qp), an infinite tree of p+1 valence which has the p-adic projective line as its boundary. We review the p-adic numbers, the Bruhat--Tits tree, and some of their applications to physics including p-adic CFT. A key feature of these constructions is the discrete and hierarchical nature of the tree and the corresponding field theories, which serve as a toy model of holography in which there are no gravitons and no conformal descendants. Standard holographic results for massive free scalar fields in a fixed background carry over to the tree; semi-classical dynamics in the bulk compute correlation functions in the dual field theory and we obtain a precise relationship between the bulk mass and the scaling dimensions of local operators. It is also possible to interpret the vertical direction in the tree a renormalization-group scale for modes in the boundary CFT. Higher-genus bulk geometries (the BTZ black hole and its generalizations) can be understood straightforwardly in our setting and their construction parallels the story in AdS_3 topological gravity.
\r\n\r\nIn Chapter 6, we consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat--Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a p-adic version of entropy which obeys a Ryu--Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one p-adic backgrounds, along with a Bekenstein--Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated). In addition, we construct infinite classes of perfect tensors directly from semi-classical states in phase spaces over finite fields, generalizing the CRSS algorithm. These codes and the resulting networks provide a natural bulk geometric interpretation of non-Archimedean notions of entanglement in holographic boundary states.
", "doi": "10.7907/HFPD-JX10", "publication_date": "2019", "thesis_type": "phd", "thesis_year": "2019" }, { "id": "thesis:11588", "collection": "thesis", "collection_id": "11588", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05312019-162611521", "primary_object_url": { "basename": "Turzillo-THESIS-final.pdf", "content": "final", "filesize": 1553691, "license": "other", "mime_type": "application/pdf", "url": "/11588/1/Turzillo-THESIS-final.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Short-Range Entangled Phases of Fermions", "author": [ { "family_name": "Turzillo", "given_name": "Alexander Michael", "orcid": "0000-0003-4293-4293", "clpid": "Turzillo-Alexander-Michael" } ], "thesis_advisor": [ { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" } ], "thesis_committee": [ { "family_name": "Motrunich", "given_name": "Olexei I.", "orcid": "0000-0001-8031-0022", "clpid": "Motrunich-Olexei" }, { "family_name": "Chen", "given_name": "Xie", "orcid": "0000-0003-2215-2497", "clpid": "Chen-Xie" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis is an attempt to understand the physics of short-range entangled phases of fermions through several related approaches. The first angle is topological quantum field theory. We discuss the classification of interacting fermionic short-range entangled phases by spin cobordism and give an algebraic characterization of unoriented equivariant bosonic topological quantum field theories in one spatial dimension. A second tool is tensor network representation. We develop the formalism of fermionic matrix product states and use it to derive the stacking group law for one dimensional symmetry-enriched fermionic short-range entangled phases. We also study its relationship with state sum constructions of topological quantum field theories and develop a state sum construction for pin-minus theories in one spatial dimension. The third approach is topological band theory. We classify free fermionic phases enriched by a unitary symmetry in any dimension and determine the map into the interacting classification.", "doi": "10.7907/3JR4-8G12", "publication_date": "2019", "thesis_type": "phd", "thesis_year": "2019" }, { "id": "thesis:9896", "collection": "thesis", "collection_id": "9896", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:07282016-144651775", "primary_object_url": { "basename": "Saberi_Ingmar_2016.pdf", "content": "final", "filesize": 1992686, "license": "other", "mime_type": "application/pdf", "url": "/9896/1/Saberi_Ingmar_2016.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Knots, Trees, and Fields: Common Ground Between Physics and Mathematics", "author": [ { "family_name": "Saberi", "given_name": "Ingmar Akira", "clpid": "Saberi-Ingmar-Akira" } ], "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": "Schwarz", "given_name": "John H.", "orcid": "0000-0001-9861-7559", "clpid": "Schwarz-J-H" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Spiropulu", "given_name": "Maria", "orcid": "0000-0001-8172-7081", "clpid": "Spiropulu-M" } ], "local_group": [ { "literal": "Caltech Theory" }, { "literal": "div_pma" } ], "abstract": "One main theme of this thesis is a connection between mathematical physics (in particular, the three-dimensional topological quantum field theory known as Chern-Simons theory) and three-dimensional topology. This connection arises because the partition function of Chern-Simons theory provides an invariant of three-manifolds, and the Wilson-loop observables in the theory define invariants of knots. In the first chapter, we review this connection, as well as more recent work that studies the classical limit of quantum Chern-Simons theory, leading to relations to another knot invariant known as the -polynomial. (Roughly speaking, this invariant can be thought of as the moduli space of flat connections on the knot complement.) In fact, the connection can be deepened: through an embedding into string theory, categorifications of polynomial knot invariants can be understood as spaces of BPS states.
\r\n\r\nWe go on to study these homological knot invariants, and interpret spectral sequences that relate them to one another in terms of perturbations of supersymmetric theories. Our point is more general than the application to knots; in general, when one perturbs any modulus of a supersymmetric theory and breaks a symmetry, one should expect a spectral sequence to relate the BPS states of the unperturbed and perturbed theories. We consider several diverse instances of this general lesson. In another chapter, we consider connections between supersymmetric quantum mechanics and the de Rham version of homotopy theory developed by Sullivan; this leads to a new interpretation of Sullivan's minimal models, and of Massey products as vacuum states which are entangled between different degrees of freedom in these models.
\r\n\r\nWe then turn to consider a discrete model of holography: a Gaussian lattice model defined on an infinite tree of uniform valence. Despite being discrete, the matching of bulk isometries and boundary conformal symmetries takes place as usual; the relevant group is , and all of the formulas developed for holography in the context of scalar fields on fixed backgrounds have natural analogues in this setting. The key observation underlying this generalization is that the geometry underlying AdS3/CFT2 can be understood algebraically, and the base field can therefore be changed while maintaining much of the structure. Finally, we give some analysis of -polynomials under change of base (to finite fields), bringing things full circle.
", "doi": "10.7907/Z9VX0DHZ", "publication_date": "2017", "thesis_type": "phd", "thesis_year": "2017" }, { "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:10298", "collection": "thesis", "collection_id": "10298", "cite_using_url": "https://resolver.caltech.edu/CaltechThesis:06062017-094601489", "primary_object_url": { "basename": "Fu_Lei-main.pdf", "content": "final", "filesize": 454305, "license": "other", "mime_type": "application/pdf", "url": "/10298/1/Fu_Lei-main.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Immersing Essential Surfaces in Odd Dimensional Closed Hyperbolic Manifolds", "author": [ { "family_name": "Fu", "given_name": "Lei", "orcid": "0000-0003-0076-138X", "clpid": "Fu-Lei" } ], "thesis_advisor": [ { "family_name": "Markovic", "given_name": "Vladimir", "clpid": "Markovic-V" } ], "thesis_committee": [ { "family_name": "Markovic", "given_name": "Vladimir", "clpid": "Markovic-V" }, { "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" }, { "family_name": "Zhang", "given_name": "Tengren", "clpid": "Zhang-Tengren" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The surface subgroup theorem, proved by Kahn and Markovic, states that the fundamental group of every closed hyperbolic 3-manifold contains a closed hyperbolic surface subgroup. The criterion of incompressibility, a criterion to ensure that an immersing surface to be essential, has played an important role in their proof.
\r\n\r\nIn this thesis, we generalize the criterion of incompressibility from dimension three to all higher dimensions. Then we use the mixing property of the geodesic flow to construct a closed immersed surface which satisfies the assumption of our criterion when the hyperbolic manifold is in an odd dimension. Together, we prove the surface subgroup theorem in all odd dimensions.
", "doi": "10.7907/Z9XK8CKN", "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: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:9852", "collection": "thesis", "collection_id": "9852", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06072016-152814803", "primary_object_url": { "basename": "StoicaBogdan2016thesis.pdf", "content": "final", "filesize": 439230, "license": "other", "mime_type": "application/pdf", "url": "/9852/1/StoicaBogdan2016thesis.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Boundary Relative Entropy as Quasilocal Energy: Positive Energy Theorems and Tomography", "author": [ { "family_name": "Stoica", "given_name": "Bogdan", "clpid": "Stoica-Bogdan" } ], "thesis_advisor": [ { "family_name": "Ooguri", "given_name": "Hirosi", "orcid": "0000-0001-6021-3778", "clpid": "Ooguri-H" } ], "thesis_committee": [ { "family_name": "Ooguri", "given_name": "Hirosi", "orcid": "0000-0001-6021-3778", "clpid": "Ooguri-H" }, { "family_name": "Schwarz", "given_name": "John H.", "orcid": "0000-0001-9861-7559", "clpid": "Schwarz-J-H" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Chen", "given_name": "Yanbei", "orcid": "0000-0002-9730-9463", "clpid": "Chen-Yanbei" } ], "local_group": [ { "literal": "Caltech Theory" }, { "literal": "div_pma" } ], "abstract": "We argue that for a spherical region R on the boundary, relative entropy between the vacuum and an arbitrary holographic excited state can be computed in the bulk as a quasilocal energy associated to the volume between R and the minimal surface B\u0303 ending on the boundary \u2202R. Since relative entropy is monotonic and positive in any well-defined quantum theory, the associated quasilocal energy must also be positive and monotonic. This gives rise to an infinite number of constraints on the gravitational bulk, which must be satisfied in any theory of quantum gravity with a well-defined UV completion. For small regions $R$, these constraints translate into integrated positivity conditions of the bulk stress-energy tensor. When the bulk is Einstein gravity coupled to scalar fields, the boundary relative entropy can be related to an integral of the bulk action on the minimal surface B\u0303. Near the boundary, this expression can be inverted via the inverse Radon transform, to obtain the bulk stress energy tensor at a point in terms of the boundary relative entropy.", "doi": "10.7907/Z9ZW1HW3", "publication_date": "2016", "thesis_type": "phd", "thesis_year": "2016" }, { "id": "thesis:9799", "collection": "thesis", "collection_id": "9799", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05272016-153537481", "primary_object_url": { "basename": "NiThesis-2016.pdf", "content": "final", "filesize": 749282, "license": "other", "mime_type": "application/pdf", "url": "/9799/1/NiThesis-2016.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Rota-Baxter Algebras, Renormalization on Kausz Compactifications and Replicating of Binary Operads", "author": [ { "family_name": "Ni", "given_name": "Xiang", "clpid": "Ni-Xiang" } ], "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": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" }, { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" }, { "family_name": "Fathizadeh", "given_name": "Farzad", "orcid": "0000-0002-7863-4009", "clpid": "Fathizadeh-F" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis is divided into two parts:
\r\n\r\nIn the first part, we consider Rota-Baxter algebras of meromorphic forms with poles along a (singular) hypersurface in a smooth projective variety and the associated Birkhoff factorization for algebra homomorphisms from a commutative Hopf algebra. In the case of a normal crossings divisor, the Rota-Baxter structure simplifies considerably and the factorization becomes a simple pole subtraction. We apply this formalism to the unrenormalized momentum space Feynman amplitudes, viewed as (divergent) integrals in the complement of the determinant hypersurface. We lift the integral to the Kausz compactification of the general linear group, whose boundary divisor is normal crossings. We show that the Kausz compactification is a Tate motive and the boundary divisor is a mixed Tate configuration. The regularization of the integrals that we obtain differs from the usual renormalization of physical Feynman amplitudes, and in particular it gives mixed Tate periods in cases that have non-mixed Tate contributions in the usual form. This part is based on joint work with Matilde Marcolli (see (80)).
\r\n\r\nIn the second part, we consider the notions of the replicators, including the duplicator and triplicator, of a binary operad. We show that taking replicators is in Koszul dual to taking successors in (9) for binary quadratic operads and is equivalent to taking the white product with certain operads such as Perm. We also relate the replicators to the actions of average operators. After the completion of this work (in 2012; see (85)), we realized that the closely related notions di-Var-algebra and tri-Var-algebra have been introduced independently in (48) (in 2011; see also (63; 64)) by Kolesnikov and his coauthors. In fact their notions also apply to not necessarily binary operads (64). In this regard, the second part of this thesis provides an alternative and more detailed treatment of these notations for binary operads. This part is based on joint work with Chengming Bai, Li Guo, and Jun Pei (see (85)).
", "doi": "10.7907/Z9SB43QZ", "publication_date": "2016", "thesis_type": "phd", "thesis_year": "2016" }, { "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: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: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:8875", "collection": "thesis", "collection_id": "8875", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05212015-124038753", "primary_object_url": { "basename": "Kasatkin_Victor_2015_thesis.pdf", "content": "final", "filesize": 1050740, "license": "cc_by", "mime_type": "application/pdf", "url": "/8875/1/Kasatkin_Victor_2015_thesis.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Some Constructions, Related to Noncommutative Tori; Fredholm Modules and the Beilinson\u2013Bloch Regulator", "author": [ { "family_name": "Kasatkin", "given_name": "Victor", "clpid": "Kasatkin-Victor" } ], "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": "Fathizadeh", "given_name": "Farzad", "orcid": "0000-0002-7863-4009", "clpid": "Fathizadeh-F" }, { "family_name": "Mantovan", "given_name": "Elena", "clpid": "Mantovan-E" }, { "family_name": "Rains", "given_name": "Eric M.", "clpid": "Rains-E-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "A noncommutative 2-torus is one of the main toy models of noncommutative geometry, and a noncommutative n-torus is a straightforward generalization of it. In 1980, Pimsner and Voiculescu in [17] described a 6-term exact sequence, which allows for the computation of the K-theory of noncommutative tori. It follows that both even and odd K-groups of n-dimensional noncommutative tori are free abelian groups on 2n-1 generators. In 1981, the Powers-Rieffel projector was described [19], which, together with the class of identity, generates the even K-theory of noncommutative 2-tori. In 1984, Elliott [10] computed trace and Chern character on these K-groups. According to Rieffel [20], the odd K-theory of a noncommutative n-torus coincides with the group of connected components of the elements of the algebra. In particular, generators of K-theory can be chosen to be invertible elements of the algebra. In Chapter 1, we derive an explicit formula for the First nontrivial generator of the odd K-theory of noncommutative tori. This gives the full set of generators for the odd K-theory of noncommutative 3-tori and 4-tori.
\r\n\r\nIn Chapter 2, we apply the graded-commutative framework of differential geometry to the polynomial subalgebra of the noncommutative torus algebra. We use the framework of differential geometry described in [27], [14], [25], [26]. In order to apply this framework to noncommutative torus, the notion of the graded-commutative algebra has to be generalized: the \"signs\" should be allowed to take values in U(1), rather than just {-1,1}. Such generalization is well-known (see, e.g., [8] in the context of linear algebra). We reformulate relevant results of [27], [14], [25], [26] using this extended notion of sign. We show how this framework can be used to construct differential operators, differential forms, and jet spaces on noncommutative tori. Then, we compare the constructed differential forms to the ones, obtained from the spectral triple of the noncommutative torus. Sections 2.1-2.3 recall the basic notions from [27], [14], [25], [26], with the required change of the notion of \"sign\". In Section 2.4, we apply these notions to the polynomial subalgebra of the noncommutative torus algebra. This polynomial subalgebra is similar to a free graded-commutative algebra. We show that, when restricted to the polynomial subalgebra, Connes construction of differential forms gives the same answer as the one obtained from the graded-commutative differential geometry. One may try to extend these notions to the smooth noncommutative torus algebra, but this was not done in this work.
\r\n\r\nA reconstruction of the Beilinson-Bloch regulator (for curves) via Fredholm modules was given by Eugene Ha in [12]. However, the proof in [12] contains a critical gap; in Chapter 3, we close this gap. More specifically, we do this by obtaining some technical results, and by proving Property 4 of Section 3.7 (see Theorem 3.9.4), which implies that such reformulation is, indeed, possible. The main motivation for this reformulation is the longer-term goal of finding possible analogs of the second K-group (in the context of algebraic geometry and K-theory of rings) and of the regulators for noncommutative spaces. This work should be seen as a necessary preliminary step for that purpose.
\r\n\r\nFor the convenience of the reader, we also give a short description of the results from [12], as well as some background material on central extensions and Connes-Karoubi character.
", "doi": "10.7907/Z91R6NGH", "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: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: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:7898", "collection": "thesis", "collection_id": "7898", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06172013-011507797", "primary_object_url": { "basename": "paul_skerritt_2013_thesis.pdf", "content": "final", "filesize": 1159181, "license": "other", "mime_type": "application/pdf", "url": "/7898/1/paul_skerritt_2013_thesis.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Geometric Quantization and Foliation Reduction", "author": [ { "family_name": "Skerritt", "given_name": "Paul Michael", "clpid": "Skerritt-Paul-Michael" } ], "thesis_advisor": [ { "family_name": "Marsden", "given_name": "Jerrold E.", "clpid": "Marsden-J-E" }, { "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": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Porter", "given_name": "Frank C.", "orcid": "0000-0003-1948-8889", "clpid": "Porter-F-C" }, { "family_name": "Preskill", "given_name": "John P.", "orcid": "0000-0002-2421-4762", "clpid": "Preskill-J" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "A standard question in the study of geometric quantization is whether symplectic reduction interacts nicely with the quantized theory, and in particular whether “quantization commutes with reduction.” Guillemin and Sternberg first proposed this question, and answered it in the affirmative for the case of a free action of a compact Lie group on a compact K\u00e4hler manifold. Subsequent work has focused mainly on extending their proof to non-free actions and non-K\u00e4hler manifolds. For realistic physical examples, however, it is desirable to have a proof which also applies to non-compact symplectic manifolds.
\r\n\r\nIn this thesis we give a proof of the quantization-reduction problem for general symplectic manifolds. This is accomplished by working in a particular wavefunction representation, associated with a polarization that is in some sense compatible with reduction. While the polarized sections described by Guillemin and Sternberg are nonzero on a dense subset of the K\u00e4hler manifold, the ones considered here are distributional, having support only on regions of the phase space associated with certain quantized, or \"admissible\", values of momentum.
\r\n\r\nWe first propose a reduction procedure for the prequantum geometric structures that “covers” symplectic reduction, and demonstrate how both symplectic and prequantum reduction can be viewed as examples of foliation reduction. Consistency of prequantum reduction imposes the above-mentioned admissibility conditions on the quantized momenta, which can be seen as analogues of the Bohr-Wilson-Sommerfeld conditions for completely integrable systems.
\r\n\r\nWe then describe our reduction-compatible polarization, and demonstrate a one-to-one correspondence between polarized sections on the unreduced and reduced spaces.
\r\n\r\nFinally, we describe a factorization of the reduced prequantum bundle, suggested by the structure of the underlying reduced symplectic manifold. This in turn induces a factorization of the space of polarized sections that agrees with its usual decomposition by irreducible representations, and so proves that quantization and reduction do indeed commute in this context.
\r\n\r\nA significant omission from the proof is the construction of an inner product on the space of polarized sections, and a discussion of its behavior under reduction. In the concluding chapter of the thesis, we suggest some ideas for future work in this direction.
", "doi": "10.7907/Q72T-Y013", "publication_date": "2013", "thesis_type": "phd", "thesis_year": "2013" }, { "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:6466", "collection": "thesis", "collection_id": "6466", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05272011-153933399", "primary_object_url": { "basename": "thesis_Dapeng_Zhang.pdf", "content": "final", "filesize": 442558, "license": "other", "mime_type": "application/pdf", "url": "/6466/1/thesis_Dapeng_Zhang.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Projective Dirac Operators, Twisted K-Theory, and Local Index Formula", "author": [ { "family_name": "Zhang", "given_name": "Dapeng", "clpid": "Zhang-Dapeng" } ], "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": "Agarwala", "given_name": "Susama", "clpid": "Agarwala-S" }, { "family_name": "Calegari", "given_name": "Danny C.", "orcid": "0009-0007-9304-2822", "clpid": "Calegari-D" }, { "family_name": "Kitaev", "given_name": "Alexei", "orcid": "0000-0002-5777-642X", "clpid": "Kitaev-A" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We construct a canonical noncommutative spectral triple for every oriented closed Riemannian manifold, which represents the fundamental class in the twisted K-homology of the manifold. This so-called \"projective spectral triple\" is Morita equivalent to the well-known commutative spin spectral triple provided that the manifold is spin-c. We give an explicit local formula for the twisted Chern character for K-theories twisted with torsion classes, and with this formula we show that the twisted Chern character of the projective spectral triple is identical to the Poincare dual of the A-hat genus of the manifold.", "doi": "10.7907/B2Z4-P206", "publication_date": "2011", "thesis_type": "phd", "thesis_year": "2011" }, { "id": "thesis:5807", "collection": "thesis", "collection_id": "5807", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05132010-155930760", "primary_object_url": { "basename": "Louwsma_Thesis.pdf", "content": "final", "filesize": 498211, "license": "other", "mime_type": "application/pdf", "url": "/5807/1/Louwsma_Thesis.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Extremality of the Rotation Quasimorphism on the Modular Group", "author": [ { "family_name": "Louwsma", "given_name": "Joel Ryan", "clpid": "Louwsma-Joel-Ryan" } ], "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": "Day", "given_name": "Matthew B.", "clpid": "Day-M-B" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "For any element A of the modular group PSL(2,Z), it follows from work of Bavard that scl(A) is greater than or equal to rot(A)/2, where scl denotes stable commutator length and rot denotes the rotation quasimorphism. Sometimes this bound is sharp, and sometimes it is not. We study for which elements A in PSL(2,Z) the rotation quasimorphism is extremal in the sense that scl(A)=rot(A)/2. First, we explain how to compute stable commutator length in the modular group, which allows us to experimentally determine whether the rotation quasimorphism is extremal for a given A. Then we describe some experimental results based on data from these computations.
\r\n\r\nOur main theorem is the following: for any element of the modular group, the product of this element with a sufficiently large power of a parabolic element is an element for which the rotation quasimorphism is extremal. We prove this theorem using a geometric approach. It follows from work of Calegari that the rotation quasimorphism is extremal for a hyperbolic element of the modular group if and only if the corresponding geodesic on the modular surface virtually bounds an immersed surface. We explicitly construct immersed orbifolds in the modular surface, thereby verifying this geometric condition for appropriate geodesics. Our results generalize to the 3-strand braid group and to arbitrary Hecke triangle groups.
", "doi": "10.7907/0Y6J-VP31", "publication_date": "2011", "thesis_type": "phd", "thesis_year": "2011" }, { "id": "thesis:5941", "collection": "thesis", "collection_id": "5941", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06082010-161504738", "primary_object_url": { "basename": "thesis_template.pdf", "content": "final", "filesize": 577447, "license": "other", "mime_type": "application/pdf", "url": "/5941/1/thesis_template.pdf", "version": "v12.0.0" }, "type": "thesis", "title": "Geography and Botany of Irreducible Symplectic 4-Manifolds with Abelian Fundamental Group", "author": [ { "family_name": "Torres Ruiz", "given_name": "Rafael", "clpid": "Torres-Ruiz-Rafael" } ], "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": "Calegari", "given_name": "Danny C.", "orcid": "0009-0007-9304-2822", "clpid": "Calegari-D" }, { "family_name": "Mantovan", "given_name": "Elena", "orcid": "0000-0003-4521-2130", "clpid": "Mantovan-E" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "orcid": "0000-0002-0159-087X", "clpid": "Ramakrishnan-D" }, { "family_name": "Morin", "given_name": "Baptiste J.", "clpid": "Morin-B-J" }, { "family_name": "Duits", "given_name": "Maurice", "clpid": "Duits-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this thesis the geography and botany of irreducible symplectic 4-manifolds with abelian fundamental group of small rank are studied. It resembles an anthology of the contribution obtained by the author during his infatuation with 4-dimensional topology by studying its recent developments. As such, each chapter is independent from each other and the reader is welcomed to start reading whichever one seems more appealing. We now give an outline for the sake of convenience.
\r\n\r\nThe first chapter of the thesis deals with the existence and (lack of) uniqueness of smooth irreducible symplectic non-spin 4-manifolds with cyclic fundamental group (both finite and infinite). Chapter 2 does the same for 4-manifolds with abelian, yet non-cyclic \u03c01; the use of the homeomorphism criteria on these manifolds due to I. Hambleton and M. Kreck is of interest. In Chapter 3, the Spin geography for abelian fundamental groups of small rank is studied. A couple of subtle relations between simply connected and non-simply connected exotic 4-manifolds are explored through out the fourth chapter.
\r\n\r\nChapter 5 gives closure to a question raised in Chapter 4, and describes current research projects pursued by the author. These projects came naturally through the results presented in previous chapters. The thesis ends by describing two research progress that are being pursued. Chapter 6 contains the current situation regarding the geography and botany of spin manifolds with zero signature.
\r\n\r\nThe current state of the joint work of the author with Jonathan Yazinski (at McMaster University at the time of writing) is described in the seventh and final chapter.
", "doi": "10.7907/A9WD-BN29", "publication_date": "2010", "thesis_type": "phd", "thesis_year": "2010" } ]