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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

    Geodesics in the directed landscape and Liouville quantum gravity: a comparative study

    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.

  • 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.

  • 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: 4-144

    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.

  • 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).

  • 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.

  • 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.