Thesis Defenses

2023

• Yan Sheng Ang

  Date: Tuesday, April 25, 2023 | 10:00am | Room: 2-449 | Zoom Link

  Committee: Laura DeMarco (advisor), Bjorn Poonen (committee chair), Yufei Zhao, Giulio Tiozzo (University of Toronto)

  Dynamical statistics for power series and polynomials with restricted coefficients

  In this thesis, we study statistical properties and related results in two different dynamical settings. In the first part, we consider a family of fractals arising as limit sets of pairs of similitudes; these fractals are closely related to power series with all coefficients equal to ±1. Motivated by the Julia-Mandelbrot correspondence, we construct a natural measure in the parameter space satisfying analogous properties for this family. Viewing the natural measure as an average root-counting measure, we establish its asymptotics and angular equidistribution. We also prove an anti-concentration inequality for the limit sets, and use this to bound the variation of the number of roots of the typical random power series from its expected value.

  In the second part, in joint work with Jit Wu Yap, we consider pairs of polynomials with rational coefficients of bounded height. In the generic case, we control the structure of the Julia sets and some notions of arithmetic complexity at most places. Using this, we prove that the average number of common preperiodic points of the two polynomials goes to 0 as height increases. We also obtain lower and upper bounds for the essential minimum of the sum of canonical heights of the two polynomials.

• Aaron Berger

  Date: Thursday, April 20, 2023 | 2:00pm | Room: 2-449 | Zoom Link

  Committee: Yufei Zhao (committee chair), Dor Minzer, Lisa Sauermann

  Three applications of the regularity method in combinatorics

  This thesis covers three applications of the regularity method in combinatorics. The first application concerns a classical question in extremal combinatorics: how many triangles can there be in a graph if no two triangles may share an edge? We give a new proof of the best-known bounds for this problem. The next two applications are "popular differences" results. Roth's theorem states that any set of integers with positive upper density contains a three-term arithmetic progression. However, a typical (or random) set has many three-term arithmetic progressions. Although it is not true in general that any particular set of integers must have as many progressions as a random set, one can show that for any set there is a nonzero popular difference such that the set has as many progressions with this common difference as a random set would. We give two multidimensional generalizations of this result.

• Sinho Chewi

  Date: Monday, April 24, 2023 | 11:00am | Room: 2-361 | Zoom Link

  Committee: Philippe Rigollet (supervisor) , Jonathan Kelner, Ankur Moitra

  An optimization perspective on log-concave sampling and beyond

  The primary contribution of this thesis is to advance the theory of complexity for sampling from a continuous probability density over R^d. Some highlights include: a new analysis of the proximal sampler, taking inspiration from the proximal point algorithm in optimization; an improved and sharp analysis of the Metropolis-adjusted Langevin algorithm, yielding new state-of-the-art guarantees for high-accuracy log-concave sampling; the first lower bounds for the complexity of log-concave sampling; an analysis of mirror Langevin Monte Carlo for constrained sampling; and the development of a theory of approximate first-order stationarity in non-log-concave sampling. We further illustrate the main tools in this work---diffusions and Wasserstein gradient flows---through applications to functional inequalities, the entropic barrier, Wasserstein barycenters, variational inference, and diffusion models.

• Yuqiu Fu

  Date: Friday, April 14, 2023 | 3:00pm | Room: 2-151

  Committee: Larry Guth (advisor), Gigliola Staffilani, and Andrew Lawrie

  Fourier decoupling for convex sequences

  We study the decoupling theory for functions on $\mathbb{R}$ with Fourier transform supported in a neighborhood of a convex sequence $\{a_n\}_{n=1}^{N} \subset \mathbb{R}$, where $a_n = g(\frac{n}{N}),$ and $g: [0,1] \rightarrow \mathbb{R}$ is a $C^2$ function satisfying $g'(x) > 0,$ $g''(x) > 0$ for every $x \in [0,1].$ We utilize the wave packet structure of functions with frequency support in a neighborhood of an arithmetic progression. This is joint work with Larry Guth and Dominique Maldague.

• Shengwen Gan

  Date: Wednesday, April 19, 2023 | 3:00pm | Room: 2-255

  Committee: Larry Guth, David Jerison, Yufei Zhao

  The restricted projection problems

  Given a non-degenerate curve in R^3, its tangent directions form a one-dimensional family of lines and its normal directions form a one-dimensional family of planes. I study the orthogonal projections of any set A in R^3 onto these restricted family of lines and planes.

• Feng Gui

  Date: Friday, April 14, 2023 | 2:00pm | Room: 2-449

  Committee: William Minicozzi, Tobias Colding, Tristan Collins

  Liouville Properties and Dimensionality Bounds for Harmonic and Caloric Functions

  Classical Liouville type theorems claim that solutions to certain elliptic or parabolic PDE are trivial provided some generic constraints about the function and the underlying space. When the solution space is not trivial, one can ask whether it is a linear space with finite dimension. In this thesis, we study several Liouville properties in geometric analysis. First, we prove a Hamilton type and a Souplet-Zhang type gradient estimates which imply a strong Liouville theorem for ancient $f$-caloric functions with certain growth assumption on smooth metric measure spaces. Second, we generalize Colding-Minicozzi's result to estimate the dimension of polynomial growth $f$-caloric functions. We apply some of these results to Ricci solitons. Lastly, we prove a dimensionality bound for exponential growth solutions to a parabolic type equation on an infinite strip.

• Jackson Hance

  Date: Monday, May 22, 2023 | 10:00am | Room: 2-361

  Committee: Tobias H. Colding (chair), William P. Minicozzi, Tristan Collins

  Regularity of the Level Set Equation for Mean Curvature Flow with an Axis of Symmetry

  In this thesis we study the regularity of viscosity solutions to the level set equation for mean curvature flow. We describe a set of hypotheses under which we can prove that the level sets of these solutions are C^(1,1) submanifolds of spacetime with well understood behavior near singular times. We then relate the derivatives of the solution of the level set flow to the solutions of certain evolution equations along fixed level sets. Finally we carry out this program to show that certain solutions with an axis of symmetry are in fact classical solutions of the level set problem.

• Sergei Korotkikh

  Date: Monday, April 24, 2023 | 3:00pm | Room: 2-255

  Committee: Alexei Borodin, Alex Postnikov, Pavel Etingof

  New degrees of freedom in integrable models with q-Hahn weights and their applications to symmetric functions and probability

  We present three groups of results about integrable lattice models constructed from orthogonality weights of q-Hahn polynomials. First, we establish that the q-Hahn orthogonality weights appear as matrix coefficients in certain isomorphisms between tensor products of representations of quantum affine sl_2 algebra. This allows us to find new integrable degrees of freedom in q-Hahn models by constructing an integrable vertex model on a square lattice with weights coming not from an R-matrix, as usually the case, but from our isomorphisms.

  Second, we use the partition function of our new vertex model to construct a generalization of t=0 Macdonald symmetric functions, which we call inhomogeneous spin q-Whittaker polynomials. Using integrability we are able to extend several classical properties of symmetric functions to our generalization, in particular, we prove analogues of the Cauchy and dual Cauchy identities. Moreover, we are able to characterize spin q-Whittaker polynomials by vanishing at certain points, which leads to a discovery of interpolation analogues of q-Whittaker and elementary symmetric polynomials.

  Finally, we introduce a (colored) stochastic version of our vertex model and prove explicit integral expressions for q-deformed moments of the (colored) height functions of it. Following known techniques our stochastic model can be interpreted as a q-discretization of the Beta polymer model with three families of integrable parameters, and we are able to extend the known results about Tracy-Widom large-scale fluctuations to our generalization of this polymer model.

• Luis Kumanduri

  Date: Wednesday, April 26, 2023 | 11:00am | Room: 2-449

  Committee: Larry Guth (chair), Tomasz Mrowka, Daniel Alvarez-Gavela

  Homotopically nontrivial area-contracting maps

  In this thesis, we investigate the existence of area contracting maps in a given homotopy class. In particular, we give various generalizations of Guth's h-principle for k-dilation of maps between spheres to other manifolds. This includes all maps from highly connected domains, maps to highly connected targets with a homology vanishing condition, and necessary and sufficient conditions for a homotopy class of maps X^m -> Y^n between closed and oriented manifolds to have representatives with small k-dilation for k = n, n-1 when k > (m+1)/2.

• Chen Lu

  Date: Tuesday, August 15, 2023 | 10:00am | Room: 2-361

  Committee: Philippe Rigollet, Jonathan Kelner, Subhabrata Sen

  Upper and Lower Bounds for Sampling

  This thesis studies the problem of drawing samples from a probability distribution. Despite the prevalence of sampling problems in applications, the quantitative behavior of sampling algorithms remains poorly understood. This thesis aims to contribute to the theoretical understanding of sampling by giving upper bounds and more importantly lower bounds for various sampling algorithms and problem classes. On the upper bound side, we propose new sampling algorithms, motivated by the perspective of sampling as optimization, and give convergence guarantees for them. We also obtain state-of-the-art convergence results for the popular Metopolis-Adjusted Langevin Algorithm. On the lower bound side, we establish the query complexity for strongly log-concave sampling in all constant dimensions. Our lower bounds rely on simple geometric constructions, which can hopefully be of aid to similar results in high dimensions.

• Oron Propp

  Date: Wednesday, April 26, 2023 | 11:00am | Room: 2-255

  Committee: Roman Bezrukavnikov, Zhiwei Yun, Pavel Etingof

  A coherent categorification of the asymptotic affine Hecke algebra

  We give a new realization of Lusztig's asymptotic affine Hecke algebra $J$ in terms of coherent sheaves on a moduli stack of Deligne–Langlands parameters. More precisely, we show that $J$ arises from a certain restriction of the "coherent Springer sheaf," which is associated to the (usual) affine Hecke algebra by means of the categorical local Langlands correspondence. We then prove a conjecture of Ben-Zvi–Chen–Helm–Nadler and Zhu stating that the coherent Springer sheaf lies in cohomological degree $0$, i.e., is a sheaf rather than a complex. Finally, we upgrade this sheaf-theoretic realization of $J$ to a categorification via certain coherent sheaves on Springer fibers, confirming (a linearized version of) a conjecture of Qiu–Xi.

• Andrew Salmon

  Date: Wednesday, April 19, 2023 | 4:00pm | Room: 2-142

  Committee: Zhiwei Yun (advisor / committee chair), Roman Bezrukavnikov, Wei Zhang

  Nearby cycles over general bases and the tamely ramified Langlands correspondence for function fields

  Vincent Lafforgue gave a construction of a semisimple Langlands parameter from an automorphic form for any reductive group over a function field by using excursion operators. Our aim is to give a general approach for proving certain local-global compatibilities satisfied by these Langlands parameters. The main consequence for the Langlands correspondence is to show that Lafforgue's construction is compatible with Lusztig's theory of character sheaves at a given point of a smooth curve over a finite field. Namely, using the theory of character sheaves, one attaches a torus character and a two-sided cell to an irreducible representation of a reductive group over a finite field. If our automorphic form lives in an isotypic component determined by this irreducible representation, we show that the torus character and two-sided cell determine the semisimple and unipotent parts of the image of the tame generator under the Langlands correspondence, respectively.

  The main theorem in the general approach to local-global compatibility is that nearby cycles commute with pushforward of certain perverse sheaves from the stack of global shtukas to a power of a curve. To prove this result, the main technical ingredient is the notion of what we call $\Psi$-factorizability, where nearby cycles over a general base are independent of the composition of specializations chosen, and the $\Psi$-factorizability statements we make give some answers to a question raised by Genestier-Lafforgue. Using this theorem, to compute the action of framed excursion operators, one may instead compute the monodromy of certain nearby cycles sheaves on certain restricted shtukas. In our case of interest, perverse sheaves over restricted shtukas are related to the monodromic affine Hecke category by a horocycle correspondence. That is, one may reduce certain questions in the global function field Langlands program to questions in local geometric Langlands.

• Felipe Suarez

  Date: Thursday, August 3, 2023 | 10:00am | Room: 2-361 | Zoom Link

  Committee: Philippe Rigollet (Advisor), Michel Goemans, Suvrit Sra.

  Perspectives on Geometry and Optimization: from Measures to Neural Networks

  This thesis explores geometrical aspects of matrix completion, interior point methods, unbalanced optimal transport, and neural network training. We use these examples to illustrate four ways in which geometry plays key yet fundamentally different roles in optimization.

  The first part explores the benign properties of exploiting the intrinsic symmetries in matrix completion. In the second problem we study the emergence of Fisher-Rao flows in entropic linear programs and explore its relationship to interior point methods. The third problem concerns unbalanced optimal transport. Inspired by a Lagrangian formulation of curvature for curves of measures, we present an algorithm for interpolation in Wasserstein-Fisher-Rao space. Lastly, we study the non-convex dynamics of neural network training for large step sizes and show that a simplified model of a two-layer neural network exhibits a phase transition and a self-stabilizing property known as the "edge of stability".

• Ethan Sussman

  Date: Monday, April 24, 2023 | 3:00pm | Room: 2-361

  Committee: Peter Hintz (Advisor), Gigliola Staffilani, Semyon Dyatlov

  Scattering at threshold

  We discuss the production of full asymptotic expansions for solutions to (1) the Klein--Gordon equation at null infinity and (2) the time-independent Schrodinger equation at low energy in the presence of an attractive Coulomb potential.

• Sarah Tammen

  Date: Tuesday, July 11, 2023 | 2:00pm | Room: 2-361

  Committee: Larry Guth, Gigliola Staffilani, and Yufei Zhao

  Incidence Problems for Slabs

  In this thesis, I prove incidence estimates for slabs which are formed by intersecting small neighborhoods of well-spaced hyperplanes in $\mathbb{R}^d$? with the unit cube $[0,1]^d$. My work is an analogue of a theorem of Guth, Solomon, and Wang, who proved a version of the Szemerédi-Trotter theorem for thin tubes that satisfy a certain strong spacing condition. My proof uses induction on scales and the high-low method of Vinh, along with new geometric insights.

• Roger Van Peski

  Date: Thursday, June 8, 2023 | 12:30pm | Room: 2-449

  Committee: Alexei Borodin (advisor, chair), Ju-Lee Kim, Melanie Matchett Wood

  Asymptotics, exact results, and analogies in p-adic random matrix theory

  After giving some background on classical and p-adic random matrix theory and their appearances in physics and number theory, I will present some new structural and probabilistic results on p-adic random matrices. On the probabilistic side, we prove local limit theorems for the singular numbers of p-adic matrix products with explicit densities given by surprisingly intricate q-series/exponential sums, giving what may be viewed as a p-adic analogue of the sine process. These, and other results in the thesis which I won't mention in the talk, are made possible by connecting probabilistic questions to spherical functions on p-adic groups (Hall-Littlewood polynomials), giving structural results analogous to known ones for complex random matrices.

• Adela Zhang

  Date: Tuesday, July 25, 2023 | 2:00pm | Room: 2-361

  Committee: Haynes Miller (chair), Jeremy Hahn, Mike Hopkins

  Koszul duality and the bar spectral sequence

  The bar spectral sequence for algebras over a spectral operad relates Koszul duality phenomena in several contexts. In this thesis, we apply this classical tool to the Koszul dual pair given by the (non-unital) E∞ operad and the spectral Lie operad over Fp. The bar spectral sequence for E∞ algebras yields the structure of operations on mod p TAQ cohomology and spectral partition Lie algebras, building on the work of Brantner-Mathew. In the colimit, the unary operations are Koszul dual to the Dyer-Lashof algebra. On the other hand, the bar construction against certain spectral Lie algebras models labeled configuration spaces by a theorem of Knudsen. The associated bar spectral sequence yields new results on their mod p homology at low weights, as well as interesting patterns of universal differentials. We also record preliminary work with Andrew Senger on detecting these differentials via deformation of comonads.

• Rose Zhang

  Date: Monday, April 24, 2023 | 2:00pm | Room: 56-114

  Committee: Larry Guth, David Jerison, Yufei Zhao

  Application of high-low method to distance problems

  In this thesis, I used the high-low method to study incidence problems arising form variants of the distance set conjectures.

  In the complex space, my collaborator Sarah Tammen and I proved analogues of the incidence estimates of Guth, Solomon and Wang \cite{WellSpacedTubes} for tubes obeying strong spacing conditions, and we used one of our new estimates to resolve a discretized variant of Falconer's distance set problem in $\mathbb{C}^2$.

  In the real space, I proved an incidence estimate for special boxes in $\mathbb{R}^6$, and used the estimate to derive a bound for a discretized variant of Falconer's problem in $\mathbb{R}^3$