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Thesis Defenses

2026

  • Manan Bhatia

    Date: Friday, April 10, 2026 | 2:00pm | Room: 2-146

    Committee: Scott Sheffield, Alexei Borodin, Nike Sun

    Two flavours of random geometry: geodesics in the directed landscape and Liouville quantum gravity

    In recent years, random geometry has developed into a subject in its own right, with the directed landscape and Liouville quantum gravity serving as central examples. Despite their very different origins, these models share striking similarities, particularly in their coalescent geodesic geometry. In this thesis, we discuss a collection of works motivated by viewing these models through a common lens, highlighting the transfer of ideas and techniques between the two settings. We conclude by considering dynamically evolving random geometries and describing recent results on the exceptional behaviour arising in the directed landscape under such dynamics.

  • Byron Chin

    Date: Tuesday, April 14, 2026 | 10:00am | Room: 2-449 | Zoom Link

    Committee: Elchanan Mossel, Nike Sun, Ankur Moitra

    Finding thresholds and structure in discrete random models

    Thresholds capture the phenomenon of a drastic change in the behavior of a random model as a result of a marginal change in its parameters. Non-rigorous methods from physics have connected the location of thresholds with various structural properties of random models. Since then, the study of these connections has inspired a number of exciting mathematical developments.

    In this talk, I will discuss two works related to this story. The first is about finding communities in random graphs when the physics heuristics do not apply, and the second confirms a conjecture on the (non)existence of stable matchings from 1989.

  • David Cui

    Date: Friday, April 24, 2026 | 11:00am | Room: 2-136

    Committee: Peter Shor, Anand Natarajan, Jon Kelner

    Sum-of-Squares Methods for Nonlocal Games and Quantum Protocols

    Nonlocal games are a foundational tool for understanding entanglement and for constructing quantum protocols in settings involving multiple spatially separated quantum devices. This thesis applies combinatorial and sum-of-squares methods to the analysis of such games, with an eye towards structural guarantees and applications to verification. On the structural side, we study nonlocal games arising from graph-theoretic and multipartite settings. These include games motivated by graph-theoretic property testing, as well as games capturing monogamy phenomena in multipartite quantum systems. On the protocol side, we study compiled nonlocal games, in which the interaction is between a classical verifier and a single quantum device. In this setting, we give a complete quantum characterization of XOR games, an important subclass of nonlocal games, and develop a general computational framework for analyzing all compiled nonlocal games.

  • Travis Dillon

    Date: Thursday, April 9, 2026 | 1:00pm | Room: 2-449 | Zoom Link

    Committee: Henry Cohn, Jon Kelner, Yufei Zhao

    Adventures in Discrete Geometry

    After outlining some of my contributions to Serious Mathematical Research (in spherical designs and quantitative combinatorial geometry), I'll spend most of the time on two of my papers that I think are the most fun. The first is about chalk, straws, pretzel sticks, or anything cylindrical: How many can you arrange into a structure so that every cylinder is touching all the others? Martin Gardner popularized this question in 1957, and I'll discuss progress (joint with J. Koizumi and S. Luo) on a version of the problem with infinite cylinders. After that, I'll describe my solution to a problem posed in 2015 on empty polygons in the prime lattice. There will be jokes. You should come.

  • Ilya Dumanskiy

    Date: Thursday, April 30, 2026 | 2:30pm | Room: 2-449 | Zoom Link

    Committee: Roman Bezrukavnikov (chair), Ivan Losev, Zhiwei Yun

    Perverse coherent sheaves in representation theory

    A perverse coherent sheaf is a natural algebro-geometric concept arising in several core constructions related to geometric Langlands duality. However, some of the key tools developed for well-studied constructible perverse sheaves are not available in the coherent setting, thus the theory presents a new set of challenges. We develop the theory in three directions:
    1. We study the coherent Satake category by relating it to the category of modules over the quantum affine group, and construct short exact sequences which conjecturally are categorical cluster mutations.
    2. We define the non-commutative affine Schubert varieties and perversely-exotic t-structure on their resolutions in type A. We identify the corresponding basis of simples in equivariant K-theory with Lusztig's canonical basis for quantum affine groups. The proof goes by establishing properties of parabolic Bezrukavnikov equivalences.
    3. We propose the notion of perverse coherent sheaves on symplectic singularities and study its properties. We demonstrate the way in which this construction generalizes the two previously studied instances of perverse-coherent t-structure.

  • Jose Guzman

    Date: Thursday, April 23, 2026 | 3:00pm | Room: 2-136

    Committee: Davesh Maulik (committee chair), Paul Seidel, and Miguel Moreira

    Logarithmic cobordism and Donaldson-Thomas invariants

    In this thesis we construct a ‘logarithmic’ cobordism ring of varieties equipped with an snc (simple normal crossings) divisor. The relations in this ring come from simple normal crossings degenerations, and since invariants of snc pairs satisfy a degeneration formalism, this allows us to use our cobordism ring to study invariants of snc pairs. In particular, we apply our log cobordism ring to prove a conjecture of Maulik-Ranganathan on zero dimensional log Donaldson-Thomas invariants. Furthermore, motivated by the fact that invariants in log geometry are invariant under logarithmic blow ups, we prove that the log cobordism ring with this further log blow up relation collapses to the algebraic cobordism ring of Levine-Pandharipande. This latter theorem serves as an illustration that although log enumerative geometry creates many more cases by allowing for varieties to have an snc boundary, the invariants are highly constrained due to the invariance under degeneration and log blow ups.

  • Serina Hu

    Date: Wednesday, April 29, 2026 | 1:00pm | Room: 4-153 | Zoom Link

    Committee:

    The higher Verlinde category Ver_4^+ and superLie theory in characteristic 2

    We develop representation theory of the simplest higher Verlinde category in characteristic 2, Ver_4^+ . We consider two perspectives when studying this category. First, it is a nontrivial reduction of the category of supervector spaces to characteristic 2, hence studying Lie theory in this category produces a notion of superLie theory in characteristic 2. From this perspective, we analyze supergroups, Lie superalgebras, and superpolynomial functors in characteristic 2, and relate these to existing notions of Lie theory in characteristic 2. Second, the higher Verlinde categories are the conjectured analogs of supervector spaces in characteristic p in Deligne’s theorem, which classifies all moderate growth symmetric tensor categories in characteristic 0 as representation categories of affine supergroup schemes. Benson, Etingof, and Ostrik conjectured that in characteristic p, one replaces supervector spaces with the union of the higher Verlinde categories in characteristic p. Hence, it is conjecturally sufficient to understand representation theory in the higher Verlinde categories in order to characterize symmetric tensor categories of moderate growth, and we develop the theory to do so in our category as a starting point for more general cases. We first provide some background on symmetric tensor categories and define Ver_4^+ and its objects of interest. We then define and study Lie algebras and Lie superalgebras in this category, including classifying the low-dimensional Lie (super)algebras and relating them to Lie superalgebras in characteristic 2. Finally, we discuss representation theory of general linear groups in Ver_4^+ , classifying their irreducible representations and their irreducible polynomial representations (hence also the simple polynomial functors in Ver_4^+) via a highest weight theory.

  • Elena Kim

    Date: Tuesday, April 14, 2026 | 1:30pm | Room: 13-4101

    Committee: Semyon Dyatlov, Richard Melrose, Larry Guth

    The Support of Semiclassical Measures in Higher Dimensions

    Semiclassical measures characterize the mass of Laplacian eigenfunctions in the high frequency limit. The quantum unique ergodicity conjecture of Rudnick and Sarnak states that on compact negatively curved manifolds, the only semiclassical measure is the Liouville measure; in other words, all eigenfunctions equidistribute. Motivated by this conjecture, we prove the first result on the support of semiclassical measures for real hyperbolic n-manifolds. The proof adapts the lower dimensional argument of Dyatlov and Jin for semiclassical measures on hyperbolic surfaces using the higher-dimensional fractal uncertainty principle of Cohen. We use similar methods to characterize semiclassical measures for quantum cat maps, a discrete toy model. We prove that semiclassical measures for quantum cat maps generically have full support. Additionally, we construct an explicit counterexample to quantum unique ergodicity for cat maps: a semiclassical measure supported on the union of two transversal symplectic subtori.

  • Daniel Lazarev

    Date: Friday, March 20, 2026 | 12:30pm | Room: 2-147 | Zoom Link

    Committee: Bonnie Berger (chair), Benjamin M. Neale, Henry Cohn

    Entropy-Extremizing Representations

    Exploring alternative representations of mathematical and physical objects is central to progress in mathematics and science. From changes of variables to integral transformations, the discovery of effective representations often reveals hidden structure and enables the solution of otherwise difficult, or even intractable problems.

    In the era of large-scale and high-dimensional data, this challenge has taken on renewed importance. Modern machine learning methods seek representations that extract meaningful structure from increasingly complex datasets, yet the discovery of such representations typically remains heuristic and problem-specific. Both in mathematics and in data science, a general theory explaining why certain representations are optimal, and how to construct them systematically, remains largely absent.

    This thesis presents a unifying framework based on entropy-extremizing representations, in which optimal representations arise as solutions to variational principles defined by entropy under structural constraints. Within this framework, entropy serves as a measure of information content relative to a specified mathematical structure, and extremizing it identifies canonical representations satisfying the desired constraints.

    Here, I develop both the theoretical foundations and practical applications of this idea. First, I introduce a theory of universal entropy in structure spaces, showing that entropies associated with different mathematical contexts arise as instances of a common universal construction. Building on this perspective, I develop entropy-extremizing formulations of several mathematical problems, including representations of uniformly distributed subspaces and duality relations based on the generalized Stokes theorem.

    The framework is then applied to computational biology and machine learning. I develop GUIDE (Generic Unmixing by Independent Decomposition), a statistical method based on entropy minimization that uncovers latent, biologically meaningful structure in complex genetic architectures. I further introduce w-values, a statistically grounded measure derived from hyperspherical geometry for evaluating neural network weights with applications to model compression and pruning. Finally, I present DiffEvol, a diffusion-based model of evolutionary dynamics that recasts evolution as a mutation-driven diffusion within a constrained subspace of genotype space.

    Together, these results demonstrate how entropy-extremizing principles can provide both a mathematical foundation and a practical methodology for discovering informative representations across mathematics, machine learning, and biology.

  • Zhenhao Li

    Date: Friday, April 10, 2026 | 1:00pm | Room: 4-270

    Committee: Semyon Dyatlov, Richard Melrose, Larry Guth

    Microlocal structures of internal waves in a 2D aquarium

    We study a model of forced internal waves in an effectively two-dimensional aquarium. In bounded domains, we provide a precise microlocal description of the singular profile that forms during the long-time propagation of internal waves when the underlying classical dynamics feature attractors. By allowing for domains with corners, our model aligns with the experimental setups of Maas et al. We discover additional mild singularities that propagate from these corners. On the other hand, we show that forced internal waves remain bounded in energy when the underlying classical dynamics are sufficiently chaotic.

    Beyond bounded aquaria, we also consider internal waves in a channel with subcritical bottom topography. We construct the scattering matrix for the internal waves problem in a channel with straight ends, mapping incoming data to outgoing data. Furthermore, we study the long-time evolution of internal waves in these channels and show that the leading profiles of the solutions are the outgoing solutions of the stationary equation.

  • Kyle Mckee

    Date: Wednesday, April 22, 2026 | 10:00am | Room: 4-370 | Zoom Link

    Committee: John Bush, Keaton Burns, and Ruben Rosales

    Geometry, Transport, and Control in Laminar Fluid Flows

    In this thesis, we investigate a collection of problems involving fluid flows and transport phenomena. For example, we investigate how the shape of bodies can be engineered so that unexpected forces arise when the body is immersed in a background flow. We then investigate the role of domain symmetries in heat transfer problems, and thereby resolve a recent conjecture in the field. We elucidate how boundary geometry and material properties govern wave instabilities beneath levitating drops. We demonstrate that electromagnetic forcing enables a new class of flow topologies in Hele-Shaw flows, both theoretically and experimentally. We also study the problem of advection-diffusion in multiply-connected (many-body) potential flows. We lastly show that particle control in generic Stokes flows is governed by a Riemannian metric defined over the fluid domain, the geodesics of which correspond to energy-optimal transport paths. This formulation is developed explicitly for magnetohydrodynamic Hele-Shaw flows, and its generalization to more general Stokes flows in 2D and 3D is described.

    This thesis involves a combination of theoretical, experimental, and numerical work. Theoretical analysis allows us to rationalize the emergence of waves beneath a levitating drop via the asymptotic solution to an eigenvalue problem. Other classical applied mathematical techniques, including conformal mapping and complex analysis, are shown to apply beyond their usual domains of application. For example, we demonstrate their application to advection-diffusion problems in geometrically complex domains and also to formulating electromagnetic control strategies in Hele-Shaw cells.

    This collection of work showcases the manner in which geometry may be exploited to design and control systems involving fluid flow and transport.

  • Mikayel Mkrtchyan

    Date: Thursday, January 8, 2026 | 9:00am | Room: 2-449 | Zoom Link

    Committee: Zhiwei Yun, Wei Zhang, Benjamin Howard

    Higher Siegel-Weil formulae over function fields

    In their seminal work, Feng-Yun-Zhang introduced function field analogues of Kudla-Rapoport cycles for moduli spaces of shtukas, and initiated the study of their intersection theory. They proved a higher Siegel-Weil formula in the case of unitary groups and non-degenerate Fourier coefficients, relating the degrees of these cycles to higher derivatives of Siegel-Eisenstein series. In this talk, we will discuss two generalizations of their result: 1) we prove a higher Siegel-Weil formula for unitary groups for corank-1 degenerate coefficients, and 2) we introduce analogous special cycles on moduli spaces of symplectic shtukas, and prove a higher Siegel-Weil formula for such cycles in the non-degenerate case.

  • Hao Peng

    Date: Friday, April 24, 2026 | 1:00pm | Room: 2-449

    Committee: Wei Zhang (chair), Andrew Sutherland, and Bjorn Poonen

    On Beilinson–Bloch–Kato conjecture for polarized motives

    We study the Beilinson--Bloch--Kato conjecture for polarized motives. In the conjugate self-dual case, we show that if the central $L$-value does not vanish, then the associated Bloch--Kato Selmer group with coefficients in a suitable local field vanishes. In the self-dual analytic rank-zero case, we reduce the conjecture to a conjecture in the endoscopic Rankin--Selberg case related to the orthogonal Gross--Prasad periods.

  • Vijay Srinivasan

    Date: Friday, April 24, 2026 | 2:30pm | Room: 2-449 | Zoom Link

    Committee: Wei Zhang, Bjorn Poonen, Andrew Sutherland

    Genus 2 curves and modular forms on Shimura curves

    This thesis studies geometric aspects of modular forms on Shimura curves over the rational numbers. The motives associated to modular forms of weight 2 have been long understood as isogeny factors in the Jacobians of Shimura curves. For modular forms of weight at least 3, we provide a new incarnation of the motive associated to such a modular form, built from the universal genus 2 curve over a Shimura curve. To illustrate the utility of such a description, we construct generalized diagonal cycles relevant to the central vanishing properties of triple-product L-series (via the conjectures of Beilinson).

  • Songchen Tan

    Date: Thursday, April 23, 2026 | 2:00pm | Room: 2-449 | Zoom Link

    Committee: Alan Edelman (chair), John Urschel, Steven Johnson

    Compiler-Enhanced Numerical Algorithms beyond Automatic Differentiation

    Automatic differentiation occupies a central role in numerical algorithms for nonlinear equations, differential equations, and optimization. This thesis reinterprets automatic differentiation as a paradigm of program transformation enabled by modern compiler techniques, and explores how such techniques can systematically generate not only derivatives but also a broader class of algorithmic components from user-defined programs. By introducing principles of symbolic-numeric computing that leverage symbolic intermediate representations, the thesis establishes a unified framework for arbitrary-order differentiation and the automatic synthesis of high-level algorithmic components. These methods are then applied to develop advanced algorithms for solving nonlinear equations, ordinary and stochastic differential equations, and physics-informed neural networks. Through both theoretical insights and practical tool development, the thesis demonstrates that compiler techniques can significantly enhance the efficiency, flexibility, and capabilities of numerical computation.

  • Zixuan Xu

    Date: Friday, April 17, 2026 | 10:00am | Room: 2-361 | Zoom Link

    Committee: Lisa Sauermann (Supervisor), Dor Minzer (Chair), John Urschel

    Extremal Hyperplane Problems on the Hypercube

    The hypercube [-1,1]^n in R^n is a fundamental object that is widely studied in combinatorics and theoretical computer science. In this thesis, we consider two types of geometric problems on the hypercube: covering every vertex of the hypercube using affine hyperplanes and slicing every edge of the hypercube using affine hyperplanes. These problems are well studied in extremal combinatorics and have connections to proof complexity, circuit complexity, and perceptron models. We prove new lower bounds for the minimum size of nondegenerate hyperplane collections covering every vertex of the hypercube and for the minimum size of hyperplane collections slicing every edge. In this talk, I will focus on the connections between these problems and highlight some main proof ideas.

  • Yuan Yao

    Date: Wednesday, April 29, 2026 | 4:15pm | Room: 2-132

    Committee: Henry Cohn, Alexander Postnikov, Lauren Williams (Harvard University)

    The Combinatorics of Triangulations of Products of Two Simplices

    We study the combinatorial structure of triangulations of the Cartesian product of two standard simplices, a seemingly simple object that is deeply connected to many objects spanning across polyhedral geometry, tropical geometry, topology, algebraic geometry, and matroid theory. In the first part, we extend our current understanding in the case where one of the two simplices has dimension two or three. In the second part, we establish several new cryptomorphisms between triangulations and related objects, addressing some gaps in the current literature. Finally, we study a generalization where the triangulated polytope is a sub-polytope of the product of two simplices, and generalize the related combinatorial characterizations accordingly.

  • Kai Zhe Zheng

    Date: Wednesday, April 29, 2026 | 10:00am | Room: 2-449 | Zoom Link

    Committee: Dor Minzer (chair), Henry Cohn, and Michael Sipser

    Probabilistically Checkable Proofs and Applications

    Probabilistically Checkable Proofs (PCPs) are proof systems that allow a verifier to check the correctness of a proof by reading only a few locations — sometimes as few as two. Since their introduction in the 1990s, PCPs have become a central tool in theoretical computer science, with applications to areas such as hardness of approximation, coding theory, and cryptography.

    In this talk, I will present several new developments in PCPs, including the first PCPs achieving nearly optimal alphabet–soundness tradeoff, yielding new hardness-of-approximation results, as well as the first 3-query low-soundness PCPs of proximity, which resolve a conjecture about local decoding from 2005.

  • Aleksandr Zimin

    Date: Tuesday, April 14, 2026 | 3:00pm | Room: 2-361

    Committee: Yury Polyanskiy, Philippe Rigollet (chair), Michel Goemans

    Advances in Probability and Statistics Using AI Methods and Agents

    This thesis studies two directions of interaction between mathematics and computation: using mathematics to understand transformer models, and using computation to obtain rigorous results in probability and combinatorics. On the transformer side, it develops a mathematical perspective in which transformers are viewed as maps on probability measures and as optimization-based dynamics on token configurations. One part introduces a transformer method for Gaussian mixture recovery, while another uses an optimization interpretation of transformer blocks to derive YuriiFormer, a momentum-based transformer architecture. On the mathematical side, the thesis gives a counterexample to the bunkbed conjecture and develops new non-simulability results and inequalities for bond, site, and hypergraph percolation. It also surveys recent progress in Lean formalization and AI-assisted proof search, and discusses how these tools can support mathematical research.