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Graduate Thesis Defenses 2020

Andrew Ahn

Title: The Method of Moments in Convolved Random Matrix Models and Discrete Analogues
Date: Monday, April 13, 2020 | 2:00pm | Room: Remote (Via Zoom)
Committee: Vadim Gorin (thesis advisor), Alexei Borodin, and Alan Edelman

Abstract

We study the global and local asymptotics of Macdonald processes, its degenerations, and related models using the method of difference operators. We focus on three applications. First, we consider random plane partitions with interactions, arising from Macdonald processes with periodic weighting. For these models, the Macdonald parameters $q,t$ become interaction parameters for the underlying dimer model. We establish global limit shape and fluctuation theorems in the limit as the mesh size goes to $0$ and the interaction parameters tend to $1$. Second, we consider a particle system obtained by generalizing the notion of squared singular values of products of truncated orthogonal, unitary, and symplectic matrices to a one-parameter family of deformed models. This procedure is analogous to the extension of classical real, complex, and quaternion matrix ensembles to $\beta$-ensembles. A discrete time Markov chain is obtained by considering iterative multiplication of matrix factors and its appropriate generalization. We show global limit shapes and fluctuations when time is of constant order and the number of particles tend to infinity. We also establish local limit theorems at the right edge when time increases with the number of particles. Third, we discuss new developments in the method of difference operators to models beyond Macdonald processes. We apply this technique to obtain moments formulas for eigenvalues of sums of unitarily invariant random matrices and for measures derived from tensor products of representations of the unitary group.

Vishal Arul

Title: Explicit Division and Torsion Points on Superelliptic Curves and Jacobians
Date: Friday, April 3, 2020 | 10:30am | Room: Remote (Via Zoom)
Committee: Bjorn Poonen (thesis advisor and committee chair), Davesh Maulik, Wei Zhang

Abstract

We present some results on the arithmetic of superelliptic curves and jacobians.

As motivation, we first define and discuss elliptic curves and their group structure to understand torsion points and division by 2 in the elliptic curve case. The aim is to apply these ideas via 2-descent to understand the group of rational points of an elliptic curve.

Next, we discuss a formula for ``division by $1 - \zeta$'' on superelliptic curves and jacobians, which generalizes a result of Yuri Zarhin for division by 2 in the hyperelliptic case. As an application, we will give a new result on the Galois action on the $\ell$-torsion of the jacobian of a superelliptic $\ell$-to-1 covering of the projective line. If time permits, we will explain connections to Anderson-Ihara theory.

Finally, we classify torsion points on the curve $y^n = x^d + 1$ over the complex numbers, where $n$, $d$ are at least 2 and are coprime. The key technical idea is a new relationship between Jacobi sums and cyclotomic units. As an application, we show that there are no ``unexpected'' torsion points on the generic superelliptic curve, extending a result of Poonen and Stoll in the hyperelliptic setting. If time permits, we will explain connections with Vandiver's conjecture.

Pablo Boixeda Alvarez

Title: Affine Springer fibers and the representation theory of small quantum groups and related algebras
Date: Thursday, April 23, 2020 | 2:00pm | Room: Remote (via Zoom)
Committee: Prof Bezrukavnikov (advisor), Prof Etingof, Prof Yun

Abstract

This thesis deals with the connections of Geometry and Representation Theory. In particular we study the representation theory of small quantum groups and Frobenius kernels and the geometry of an equivalued affine Springer fiber $\mathcal{F}l_{ts}$ for $s$ a regular semisimple element.

We first relate the center of the small quantum group with the cohomology of the above affine Springer fiber.

The question of the center of the quantum group at a generic parameter was understood by Drinfeld in 1989, but the center of this algebra when you specialize at a root of unity remained an open question. By work of Lachowska, Qi and Bezrukavnikov we had some description of the center of the small quantum group. In joint work with Bezrukavnikov, Shen and Vasserot, we prove a more concrete description of the center of the small quantum group, in terms of the cohomology of this affine Springer fiber.

Secondly we study the geometry of the affine Springer fiber and in particular understand the fixed points of a torus action contained in each component a question that remains open for the general finite Springer fiber and which in this case could lead to a better understanding of the endomorphism of certain projectives in the category of small quntum group representations.

Lastly we prove a collection of algebraic results on the representation theory of Frobenius kernels. In particular we state some results pointing towards some construction of certain partial Verma functors and we compute this in the case of $SL_2$. We also compute the center of Frobenius kernels in the case of $SL_2$ and state a conjecture on a possible inductive construction of the general center.

Hood Chatham

Title: An Orientation Map for Height p-1 Real E theory
Date: Monday, April 6, 2020 | 4:30pm | Room: Remote (via Zoom)
Committee: Haynes Miller, Mike Hopkins, Zhouli Xu

Abstract

The real $\mathrm{K}$-theory spectrum $\mathrm{KO}$ is ``almost complex oriented''. Here are a collection of properties that demonstrate this:

(1) $\mathrm{KO}$ is the $C_2$ fixed points of a complex oriented cohomology theory $\mathrm{KU}$.

(2) Complex oriented cohomology theories have trivial Hurewicz image, whereas $\mathrm{KO}$ has a small Hurewicz image -- it detects $\eta$ and $\eta^2$.

(3) Complex oriented cohomology theories receive a ring map from $\mathrm{MU}$. $\mathrm{KO}$ receives no ring map from $\mathrm{MU}$ but it receives one from $\mathrm{MSU}$.

(4) If $E$ is a complex orientable cohomology theory, every complex vector bundle $V$ is $E$-orientable. Not every complex vector bundle $V$ is $\mathrm{KO}$-orientable, but $V\oplus V$ and $V^{\otimes 2}$ are.

Higher real $E$ theory $\mathrm{EO}$ is an odd primary analogue of $\mathrm{KO}$. At $p=3$, $\mathrm{EO}$ is closely related to $\mathrm{TMF}$. $\mathrm{EO}$ is defined as the $C_p$ fixed points of a complex oriented cohomology theory, and it has a small but nontrivial Hurewicz image, so it satisfies analogues of properties (1) and (2). I prove that it also satisfies analogues of properties (3) and (4). In particular, I produce a unital orientation map from a Thom spectrum to $\mathrm{EO}$ and prove that for any complex vector bundle $V$ the bundles $pV$ and $V^{\otimes p}$ are complex oriented.

Because this is an electronic talk, I will focus on spectral sequence demonstrations using my in-progress spectral sequence software.

Atticus Christensen

Title: A Topology on Points on Stacks
Date: Tuesday, April 28, 2020 | 2:00pm | Room: Remote (via Zoom)
Committee: Bjorn Poonen (advisor), Wei Zhang, Davesh Maulik

Abstract

For a variety over certain topological rings R, like Z_p or C, there is a well-studied way to topologize the R-points on the variety. In this paper, we generalize this definition to algebraic stacks. For an algebraic stack Xf over many topological rings R, we define a topology on the isomorphism classes of R-points of X. We prove expected properties of the resulting topological spaces including functoriality. Then, we extend the definition to the case when $R$ is the ring of adeles of some global field. Finally, we use this last definition to strengthen the local-global compatibility for stacky curves of Bhargava--Poonen to a strong approximation result.

Miles Couchman

Title: The stability of bound states in pilot-wave hydrodynamics
Date: Thursday, April 16, 2020 | 2:30pm | Room: Remote (via Zoom)
Committee: John Bush (thesis advisor), Rodolfo Rosales, Anand Oza

Abstract

Millimetric droplets bouncing on the surface of a vertically vibrating fluid bath may self-propel through a resonant interaction with their own wavefield, displaying behaviors previously thought to be exclusive to the microscopic quantum realm. We investigate the stability of quantized bound states comprised of multiple droplets interacting through their shared wavefield, using an integrated experimental and theoretical approach. We consider the behavior of droplet pairs, rings, and chains as the bath's vibrational acceleration is increased progressively, and uncover a rich variety of dynamical states including periodic oscillations and traveling waves. The instability observed is dependent on the droplet number and size, and whether the drops are bouncing in- or out-of-phase relative to their neighbors. We develop a new theoretical model that accounts for the coupling between a drop's horizontal and vertical motion, enabling us to rationalize the majority of our experimental findings. We thus demonstrate that variations in a drop's impact phase with the bath have a critical influence on the stability of bouncing-droplet bound states. Our work provides insight into the complex interactions and collective motions that arise in bouncing-droplet aggregates, and forges new mathematical links with extant models of microscopic physics.

Jacob Mitchell Gold

Title: Self-organized fine-tuned response in a driven spin glass
Date: Thursday, September 10, 2020 | 3:00pm | Room: Remote (via Zoom)
Zoom Link: https://mit.zoom.us/j/94490538691
Committee: Jeremy Englhttps://mit.zoom.us/j/94490538691and, Jörn Dunkel, John Bush

Abstract

In this thesis, I investigate the principles that that can be used to predict the behavior of a many-bodied system when an external drive is applied. I consider a spin glass as a prototypical model of such a system, and investigate these principles through simulation. I find that spins differentiate into slow spins which decouple from the drive and fast spins which couple more strongly to the drive, resulting in macroscopic quantities like work absorption rate and internal energy decreasing as compared to the near-equilibrium distribution. Which spins fall into which categories is specific to a particular realization of the external drive; changing to another drive changes which spins are fast and which are slow, revealing a drive-specific adaptation. I investigate limits on the memory of the system, and demonstrate the system's capability to identify changes in real-world images.

Ricardo Grande Izquierdo

Title: The role of smoothing effect in some dispersive equations
Date: Monday, April 27, 2020 | 11:00am | Room: Remote (via Zoom)
Committee: Gigliola Staffilani (advisor), David Jerison, Andrew Lawrie

Abstract

In this thesis, we will study the role of smoothing effect in the local well-posedness theory of dispersive partial differential equations in three different contexts. First, we use it to overcome a loss of derivatives in a family of nonlocal dispersive equations. Second, we exploit a discrete version of the smoothing effect to study a discrete system of particles and how to approximate it by a continuous dispersive equation. Third, we use an anisotropic version of the smoothing effect to establish the local well-posedness theory of the two-dimensional Dysthe equation, which is used to model oceanic rogue waves.

Campbell Hewett

Title: Computability of Rational Points on Curves over Function Fields in Characteristic $p$
Date: Thursday, April 30, 2020 | 9:30am | Room: Remote (via Zoom)
Committee: Bjorn Poonen, Andrew Sutherland, Wei Zhang

Abstract

The motivating problem of this thesis is that of explicitly computing the $K$-rational points of a regular nonsmooth curve $X$ over a finitely generated field $K$ of characteristic $p$. We start with an in-depth study of such curves in general and the tools exclusive to characteristic $p$ geometry needed to compute their $K$-points. We describe a combined going-down and going-up approach to compute $X(K)$ that generalizes and makes effective the proof of finiteness of $X(K)$ given by Voloch. We break the problem up into three separate cases according to the absolute genus of $X$. In the absolute genus 0 case, we give an algorithm to compute $X(K)$ that is an effective version of a method given by Jeong. We also implement a special case of this algorithm in Sage and apply it to example curves. In the absolute genus 1 case, we give an algorithm to compute $X(K)$ that works when we make extra assumptions about $X$, and we make some remarks in the case where those assumptions are removed. In the absolute genus at least 2 case, we give an unconditional algorithm to compute $X(K)$.

Some tools and algorithms we provide along the way do not directly involve regular nonsmooth curves and are interesting in their own right. We describe ways to effectively descend curves with respect to transcendental or purely inseparable field extensions. We explore the methods of $p$-descent on elliptic curves in characteristic $p$ and provide explicit equations defining $\mathbb{Z}/p\mathbb{Z}$- and $\mu_p$-torsors over them. We prove an effective de Franchis-Severi theorem for characteristic $p$ that generalizes the one given by Baker, et al. over number fields. Lastly, we use a height bound proved by Szpiro to give an algorithm to compute $Y(K)$ for any smooth nonisotrivial curve $Y$ over $K$ followed by an algorithm to compute $Y(K^{1/p^{\infty}})$, which was proved to be finite by Kim.

James Hirst

Title: Coupling sparse models and dense extremal problems
Date: Friday, July 24, 2020 | 4:00pm | Room: Remote (via Zoom)
Committee: Henry Cohn, Jon Kelner, Zilin Jiang

Abstract

We study the problem of coupling a stochastic block model with a planted bisection to a uniform random graph having the same average degree. Focusing on the regime where the average degree is a constant relative to the number of vertices $n$, we show that the distance to which the models can be coupled undergoes a phase transition from $O(\sqrt{n})$ to $\Omega(n)$ as the planted bisection in the block model varies. This settles half of a conjecture of Bollob\'{a}s and Riordan and has some implications for sparse graph limit theory. In particular, for certain ranges of parameters, a block model and the corresponding uniform model produce samples which must converge to the same limit point. This implies that any notion of convergence for sequences of graphs with $\Theta(n)$ edges which allows for samples from a limit object to converge back to the limit itself must identify these models.

On the other hand, we demonstrate that the existing theory of dense graph limits is a powerful tool for dealing with extremal problems on graphs with $\Theta(n^2)$ edges. The language of graphons along with the flag algebra method allow us to obtain many results which would otherwise be out of reach or at least difficult to manage. We study graph profiles which captures correlations between different graphs in a larger network. Further, we give proofs in the flag algebra of some inducibility-like problems which have gained some particular interest recently.

Ethan Jaffe

Title: Asymptotic description of the formation of black holes from short-pulse data
Date: Wednesday, March 11, 2020 | 1:00pm | Room: 2-449
Committee: Richard Melrose, Peter Hintz, David Jerison

Abstract

In this thesis we present partial progress towards the dynamic formation of black holes in the four-dimensional Einstein vacuum equations from Christodoulou's short-pulse ansatz. We identify natural scaling in a putative solution metric and use the technique of real blowup to propose a desingularized manifold and an associated rescaled tangent bundle (which we call the "short-pulse tangent bundle") on which the putative solution remains regular. We prove the existence of a solution solving the vacuum Einstein equations formally at each boundary face of the blown-up manifold and show that for an open set of restricted short-pulse data, the formal solution exhibits curvature blowup at a hypersurface in one of the boundary hypersurfaces of the desingularized manifold.

Vishesh Jain

Title: Quantitative invertibility of random matrices: a combinatorial perspective
Date: Tuesday, March 31, 2020 | 3:00pm | Room: Remote (via Zoom)
Committee: Elchanan Mossel (chair), Ankur Moitra, Nike Sun

Abstract

In this thesis, we develop a novel framework for investigating the lower tail behavior of the least singular value of random matrices -- a subject which has been intensely studied in the past two decades. Our focus is on obtaining high probability bounds, rather than on estimating the least singular value of a `typical' realisation of the random matrix.

In our main application, we consider random matrices of the form $M_n := M + N_n$, where $M$ is a fixed complex matrix with operator norm at most $\exp(n^{c})$, and $N_n$ is a random matrix, each of whose entries is an independent copy of a complex random variable with mean $0$ and variance $1$. This setting, with some additional restrictions, has been previously considered in a series of influential works by Tao and Vu, most notably in connection with the strong circular law, and the smoothed analysis of the condition number, and our results extend and improve upon theirs in a couple of ways.

As opposed to all previous works obtaining such bounds with error rate better than $n^{-1}$, our proof makes no use either of the inverse Littlewood--Offord theorems, or of any sophisticated net constructions. Instead, we show how to reduce the optimization problem characterizing the smallest singular value from the (complex) sphere to (Gaussian) integer vectors, where it is solved using direct combinatorial arguments.

Borys Kadets

Title: Arboreal representations, sectional monodromy groups, and abelian varieties over finite fields
Date: Tuesday, March 31, 2020 | 1:00pm | Room: Remote (via Zoom)
Committee: Bjorn Poonen (chair and thesis advisor), Davesh Maulik, Andrew Sutherland

Abstract

To an embedded algebraic curve $X \subset \mathbb{P}^n$ one can associate a certain combinatorial invariant, the sectional monodromy group $G_X$; it is a subgroup of the symmetric group $S_{\deg X}$. Informally, the group $G_X$ parametrizes the monodromy of the hyperplane section $H \cap X$ as the hyperplane $H$ varies.

When the ground field is the field of complex numbers, a theorem of Castelnuovo says that $G_X = S_d$ for any curve $X$. This result has numerous applications in algebraic geometry, for example to the degree-genus problem. However, when the ground field has positive characteristic, many curves have unusual section monodromy groups.

I will describe a method for computing $G_X$ for various classes of curves $X$. This method gives a classification of nonstrange nondegenerate curves $X \subset \mathbb{P}^r$ with $r>2$ and $G_X \neq A_d, S_d$. In a different direction, I will completely classify sectional monodromy groups of plane monomial curves, the problem studied previously by Abhyankar, Cohen, Smith, and Uchida. The answers include finite groups of Lie type and some sporadic simple groups.

Dmitry Kubrak

Title: Cohomologically proper stacks over $\mathbb{Z_p}$: algebra, geometry and representation theory
Date: Monday, April 20, 2020 | 2:00pm | Room: Remote (via Zoom)
Committee: Roman Bezrukavnikov (chair and advisor), Zhiwei Yun, Wei Zhang

Abstract

In this thesis, we study a class of so-called cohomologically proper stacks from various perspectives, mainly concentrating on the $p$-adic context. Cohomological properness is a relaxed properness condition on a stack which roughly asks the cohomology of any coherent sheaf to be finitely generated over the base. We extend some of the techniques available for smooth proper schemes to smooth cohomologically proper stacks, featuring in particular recently developed theory of prismatic cohomology and the classical Deligne-Illusie method for the Hodge-to-de Rham degeneration. As main applications we prove the Totaro's conjectural inequality between the dimensions of the de Rham and the singular $\mathbb{F_p}$-cohomology of the classifying stack of a reductive group, compute the ring of prismatic characteristic classes at non-torsion primes and give some new examples of the Hodge-to-de Rham degeneration for stacks in characteristic 0. We also study some descent properties of certain Brauer group classes on conical resolutions, a question having some applications to the theory of Fedosov quantizations in characteristic $p$. Some surprising results about the $\mathbb{G_m}$-weights of differential 1-forms that are obtained along the way, originally motivated the attempt to generalize the integral $p$-adic Hodge theory to the setting of cohomologically proper stacks.

Zhenkun Li

Title: Contributions to sutured monopole and sutured instanton Floer homology theories
Date: Tuesday, March 31, 2020 | 3:00pm | Room: Remote (via Zoom)
Committee: Tomasz S. Mrowka (advisor, committee chair), Paul Seidel, Matthew Stoffregen

Abstract

In this thesis, we present development of some aspects of sutured monopole and sutured instanton Floer homology theories. Sutured monopole and instanton Floer homologies were introduced by Kronheimer and Mrowka. They are the adaption of monopole and instanton Floer theories to the case of balanced sutured manifolds, which are compact oriented $3$-manifolds together with some special data on the boundary called the suture. We constructed the gluing and cobordism maps in these theories, construct gradings associated to a properly embedded surface inside the balanced sutured manifolds, and use these tools to further construct minus versions of knot Floer homologies in monopole and instanton theories. These constructions contribute to lying down a solid basis in sutured monopole and sutured instanton Floer homology theories, upon which we could develop further applications.

Svetlana Makarova

Title: Strange Duality on Elliptic and K3 Surfaces
Date: Monday, April 20, 2020 | 2:00pm | Room: Remote (via Zoom)
Committee: Davesh Maulik, Chenyang Xu, Alina Marian

Abstract

In this talk, I will summarize the result of my work on the Strange Duality conjecture. First, I will explain the statement of the conjecture: how we can get a morphism between global sections of two theta line bundles on moduli spaces of sheaves. Then, I will explain the mechanism that allows to generalize a series of results from Hilbert schemes: we were able to generalize a nice trick first used by Marian—Oprea that relies on Bridgeland's construction of birationalities of moduli over elliptic surfaces. Finally, I will explain how our construction, which extensively uses Alper's theory of good moduli spaces for stacks, allows to extend the formulation of the Strange Duality conjecture on the quasipolarized locus in the moduli of K3 surfaces, and bring all the ingredients together to obtain generic Strange Duality for degree two K3 surfaces.

Lucas Mason-Brown

Title: Unipotent Representations of Real Reductive Groups
Date: Wednesday, April 29, 2020 | 4:30pm | Room: Remote (via Skype)
Committee: David Vogan (advisor), Roman Bezrukavnikov, George Lusztig

Abstract

Let $G$ be a real reductive group and let $\widehat{G}$ be the set of irreducible unitary representations of $G$. The determination of $\widehat{G}$ (for arbitrary $G$) is one of the fundamental unsolved problems in representation theory. In the early 1980s, Arthur introduced a finite set $\mathrm{Unip}(G)$ of (conjecturally unitary) irreducible representations of $G$ called \emph{unipotent representations}. In a certain sense, these representations form the building blocks of $\widehat{G}$. Hence, the determination of $\widehat{G}$ requires as a crucial ingredient the determination of $\mathrm{Unip}(G)$. In this thesis, we prove three results on unipotent representations. First, we study unipotent representations by restriction to $K \subset G$, a maximal compact subgroup. We deduce a formula for this restriction in a wide range of cases, proving (in these cases) a long-standing conjecture of Vogan. Next, we study the unipotent representations attached to induced nilpotent orbits. We find that $\mathrm{Unip}(G)$ is `generated' by an even smaller set $\mathrm{Unip}'(G)$ consisting of representations attached to \emph{rigid} nilpotent orbits. Finally, we study the unipotent representations attached to the principal nilpotent orbit. We provide a complete classification of such representations, including a formula for their $K$-types.

Gweneth McKinley

Title: Probabilistic and extremal behavior in graphs and matrices
Date: Wednesday, February 5, 2020 | 4:15pm | Room: 2-449
Committee: Henry Cohn, Nike Sun, Laszlo Lovasz

Abstract

This thesis deals with several related questions in probabilistic and extremal graph theory and discrete random matrix theory.

First, for any bipartite graph H containing a cycle, we prove an upper bound of 2^O(ex(n,H)) on the number of labeled H-free graphs on n vertices, given only a fairly natural assumption on the growth rate of ex(n, H). Bounds of the form 2^O(ex(n,H have been proven only for relatively few special graphs H, often with considerable difficulty, and our result unifies all previously known special cases.

Next, we give a variety of bounds on the clique numbers of random graphs arising from the theory of graphons. A graphon is a symmetric measurable function W from [0,1]^2 to [0,1], and each graphon gives rise naturally to a random graph distribution, denoted G(n,W), that can be viewed as a generalization of the Erdös-Rényi random graph. Recently, Doležal, Hladký, and Máthé gave an asymptotic formula of order log(n) for the size of the largest clique in G(n,W) when W is bounded away from 0 and 1. We show that if W is allowed to approach 1 at a finite number of points, and displays a moderate rate of growth near these points, then the clique number of G(n,W) will be of order √n almost surely. We also give a family of examples with clique number of order n^c for any c in (0,1), and some conditions under which the clique number of G(n,W) will be o(√n) or omega(√n).

Finally, for an n×m matrix M of independent Rademacher (±1) random variables, it is well known that if nm, then M is of full rank with high probability; we show that this property is resilient to adversarial changes to M. More precisely, if mn + n^(1−ε/6) , then even after changing the sign of (1 − ε)m/2 entries, M is still of full rank with high probability. This is asymptotically best possible, as one can easily make any two rows proportional with at most m/2 changes. Moreover, this theorem gives an asymptotic solution to a slightly weakened version of a conjecture made by Van Vu in 2008.

Tudor Padurariu

Title: K-theoretic Hall algebras for quivers with potential
Date: Monday, April 13, 2020 | 10:00am | Room: Remote (via Zoom)
Committee: Davesh Maulik (advisor and committee chair), Roman Bezrukavnikov, Andrei Negut

Abstract

Given a quiver with potential, Kontsevich-Soibelman constructed a Hall algebra on the cohomology of the stack of representations of (Q,W). In particular cases, one recovers positive parts of Yangians as defined by Maulik-Okounkov. For general (Q,W), the Hall algebra has nice structure properties, for example Davison-Meinhardt proved a PBW theorem for it using the decomposition theorem.

One can define a K-theoretic version of this algebra using certain categories of singularities that depend on the stack of representations of (Q,W). In particular cases, these Hall algebras are positive parts of quantum affine algebras. We show that some of the structure properties in cohomology, such as the PBW theorem, can be lifted to K-theory, replacing the use of the decomposition theorem with semi-orthogonal decompositions.

Jiewon Park

Title: Convergence of complete Ricci-flat manifolds
Date: Monday, April 13, 2020 | 10:00am | Room: Remote (via Zoom)
Committee: Tobias Colding (advisor and chair), Bill Minicozzi, and Tristan Collins

Abstract

This thesis is focused on the convergence at infinity of complete Ricci flat manifolds. In the first part of this thesis, we will give a natural way to identify between two scales, potentially arbitrarily far apart, in the case when a tangent cone at infinity has smooth cross section. The identification map is given as the gradient flow of a solution to an elliptic equation. We use an estimate of Colding-Minicozzi of a functional that measures the distance to the tangent cone. In the second part of this thesis, we prove a matrix Harnack inequality for the Laplace equation on manifolds with suitable curvature and volume growth assumptions, which is a pointwise estimate for the integrand of the aforementioned functional. This result provides an elliptic analogue of matrix Harnack inequalities for the heat equation or geometric flows.

Joshua Pfeffer

Title: Liouville quantum gravity and the interplay between quantum and Euclidean scales
Date: Friday, May 1st | 3:15pm | Room: Remote (via Zoom)
Committee: Scott Sheffield (chair), Yilin Wang, Alexei Borodin

Abstract

My talk is an introduction to Liouville quantum gravity (LQG), a universal model of random geometry in two dimensions, through the lens of several works from my doctoral thesis. My talk focuses on the interplay between quantum (i.e. LQG) and Euclidean scales in the study of these random fractal surfaces.

  • I describe how the Euclidean Hausdorff dimension of a random fractal on a surface relates to its quantum Hausdorff dimension. I highlight the particular example of the Hausdorff dimension of LQG, which relates the scaling of LQG areas and distances.
  • I demonstrate the power of regularizing LQG on a quantum scale rather than a Euclidean scale to (1) relate the rigorous construction of LQG to its original heuristic definition in the physics literature, and (2) define a nondegenerate model of LQG for values of central charge in the range $(1,25)$.

This talk is based on joint works with Morris Ang, Julien Dubedat, Hugo Falconet, Ewain Gwynne, Nina Holden, Minjae Park, Guillaume Remy, Scott Sheffield, and Xin Sun.

Christopher Ryba

Title: Stable Characters for Symmetric Groups and Wreath Products
Date: Wednesday, April 15, 2020 | 1:00pm | Room: Remote (via Zoom)
Committee: Pavel Etingof (advisor and committee chair), Andrei Negut, Thomas Lam

Abstract

Representations of the symmetric group $\mathcal{S}_n$ satisfy certain stability properties as $n \to \infty$. One of these, Murnaghan's stability of Kronecker coefficients, asserts that certain tensor product multiplicities have finite limits as $n$ becomes large. This allows us to define a ``limiting Grothendieck ring'', which is roughly the $n \to \infty$ limit of the Grothendieck ring of $\mathcal{S}_n$-mod. The construction comes with a basis corresponding to irreducible objects. We will discuss three facets of this theory.

Firstly, let $\mathcal{R}$ be a Hopf algebra. Representations of the wreath product $\mathcal{R} \wr \mathcal{S}_n$ satisfy analogous stability properties, and we may similarly define a limiting Grothendieck ring, $\mathcal{G}_\infty$. This ring is not necessarily commutative (depending on $\mathcal{R}$); we show that over $\mathbb{Q}$, $\mathcal{G}_\infty$ is isomorphic to a particular tensor product of universal enveloping algebras.

Secondly, we consider the algebraic structure of $\mathcal{G}_\infty$. We show that it is a Hopf algebra, and that it is given by distributions on a certain formal group scheme.

Thirdly, in the case where $\mathcal{R} = \mathbb{C}$ (so we are dealing with representations of symmetric groups), it turns out $\mathcal{G}_\infty$ is isomorphic to the ring of symmetric functions. However, the basis yielded by our construction is not any of the standard symmetric function bases. We categorify this basis by producing a resolution of irreducible representations of $\mathcal{S}_n$ by modules restricted from $GL_n(\mathbb{C})$.

Ao Sun

Title: Singular Behaviour and Long Time Behaviour of Mean Curvature Flow
Date: Thursday, April 16, 2020 | 3:00pm | Room: Remote (Via Zoom)
Committee: Larry Guth, David Jerison​, William Minicozzi (Thesis advisor)

Abstract

In this thesis, we investigate two asymptotic behaviours of the mean curvature flow. The first one is the asymptotic behaviour of singularities of the mean curvature flow, and the asymptotic limit is modelled by the tangent flows. The second one is the asymptotic behaviour of the mean curvature flow as time goes to infinity. We will study several problems related to the asymptotic behaviours.

The first problem is the partial regularity of the limit. The partial regularity of mean curvature flow without any curvature assumptions was first studied by Ilmanen. We will follow the idea of Ilmanen to study the partial regularity of other asymptotic limit. In particular, we introduce a generalization of Colding-Minicozzi's entropy in a closed manifold, which plays a significant role.

The second problem is the genericity of the tangent flows of mean curvature flow. The generic mean curvature flow was introduced by Colding-Minicozzi. Furthermore, they introduced mean curvature flow entropy and use it to study the generic tangent flows of mean curvature flow. We study the multiplicity of the generic tangent flow. In particular, we prove that the generic compact tangent flow of mean curvature flow of surfaces has multiplicity $1$. This result partially addresses the famous multiplicity $1$ conjecture of Ilmanen. One key idea is defining a local version of Colding-Minicozzi's entropy.

We also discuss some related results. These results include a joint work with Zhichao Wang and a joint work with Julius Baldauf.​

Piotr Suwara

Title: Semi-infinite Homology of Floer Spaces
Date: Thursday, July 16, 2020 | 3:00pm | Room: Remote (via Zoom)
Committee: Tomasz Mrowka (chair), Paul Seidel, Matthew Stoffregen.

Abstract

This dissertation presents a framework for defining Floer homology of infinite-dimensional spaces with a functional. This approach is meant to generalize the traditional constructions of Floer homologies which mimic the construction of the Morse-Smale-Witten complex. To define Floer homology we use cycles modelled on infinite-dimensional manifolds with corners, as described by Maksim Lipyanskiy, where the key is to impose appropriate compactness and polarization axioms on the cycles. We separate and carefully inspect these two types of axioms, paying special attention to correspondences, generalizing the definition of a polarization as well as axiomatizing the notion of a family of perturbations. The latter is used to define an intersection pairing and maps induced on Floer homology by correspondences. Moreover, we prove suspension isomorphisms and prove that this Floer homology agrees with Morse homology for finite-dimensional manifolds with a Palais-Smale functional. Finally, we explain how to apply this framework to Seiberg-Witten-Floer theory, defining Floer homology groups associated to rational homology spheres and their spinc-structures. Importantly, we prove moduli spaces of solutions to Seiberg-Witten equations induce maps on Floer homology in a functorial fashion and that a trivial cobordism induces the identity map.

Brandon Tran

Title: Building and Using Robust Representations in Image Classification
Date: Thursday, April 16, 2020 | 1:30pm | Room: Remote (via Zoom)
Committee: Jon Kelner, Ankur Moitra & Aleksander Madry (thesis advisor)

Abstract

One of the major appeals of the deep learning paradigm is the ability to learn high-level feature representations of complex data. These learned representations obviate manual data pre-processing, and are versatile enough to generalize across tasks. However, they are not yet capable of fully capturing abstract, meaningful features of the data. For instance, the pervasiveness of adversarial examples---small perturbations of correctly classified inputs causing model misclassification---is a prominent indication of such shortcomings.

The goal of this thesis is to work towards building learned representations that are more robust and human-aligned. To achieve this, we turn to adversarial (or robust) training, an optimization technique for training networks less prone to adversarial inputs. Typically, robust training is studied purely in the context of machine learning security (as a safeguard against adversarial examples)---in contrast, we will cast it as a means of enforcing an additional prior onto the model. Specifically, it has been noticed that, in a similar manner to the well-known convolutional or recurrent priors, the robust prior serves as a "bias" that restricts the features models can use in classification---it does not allow for any features that change upon small perturbations.

We find that the addition of this simple prior enables a number of downstream applications, from feature visualization and manipulation to input interpolation and image synthesis. Most importantly, robust training provides a simple way of interpreting and understanding model decisions. Besides diagnosing incorrect classification, this also has consequences in the so-called "data poisoning" setting, where an adversary corrupts training samples with the hope of causing misbehaviour in the resulting model. We find that in many cases, the prior arising from robust training significantly helps in detecting data poisoning.

Sam Turton

Title: Theoretical modeling of pilot-wave hydrodynamics
Date: Friday, April 24, 2020 | 11:00am | Room: Remote (via Zoom)
Committee: John Bush (thesis advisor), Rodolfo Rosales, Pedro Sáenz

Abstract

In this thesis, we develop and apply a number of theoretical models describing the dynamics of a walking droplet. We first review the hierarchy of theoretical models introduced to study the dynamics of a droplet walking on a vibrating bath. We discuss the stroboscopic model introduced by Oza et al. (2013), and elucidate the role of spatial wave-damping and the effect of the droplet's vertical dynamics. We then revisit the weak-acceleration limit of the stroboscopic equations developed by Bush et al. (2014), demonstrating its connection to the Rayleigh oscillator model proposed by Labousse & Perrard (2014). We extend this framework in order to consider droplet interactions with weakly-variable topography, and compare our model predictions with the results of an accompanying experimental study. Particular attention is given to outlining the physical limits in which the topographical effects may be captured by an effective force. We also investigate theoretically the dynamics of one- and two-dimensional droplet spin lattices, and demonstrate that their collective behavior is readily captured by a generalized Kuramoto model, which we explicitly derive from the boost framework. Finally we reconsider the stability of the steady walking state, motivated by the statistical signature reported in the trajectories of droplets interacting with wells (Sáenz et al. 2020). By extending the hydrodynamic setting to a generalized pilot-wave framework, we discover states in which the walker's speed oscillates over a scale close to the Faraday wavelength, in addition to a regime in which walker motion is unstable and undergoes random-walk-like motion. We demonstrate how either of these two mechanisms may play a role in the emergence of quantum-like statistics from pilot-wave dynamics. We conclude with a discussion of the implications of this work and suggest fruitful directions of future research.

Jake Wellens

Title: Assorted results in boolean function complexity, uniform sampling and clique partitions of graphs
Date: Tuesday, April 21, 2020 | 2:00pm | Room: Remote (via Zoom)
Committee: Henry Cohn (advisor), Elchanan Mossel and Jon Kelner

Abstract

In the first part of the talk, I'll discuss some new bounds on the number of relevant variables for an arbitrary boolean function in terms of its degree, sensitivity, certificate and decision tree complexities, as well some constant-factor improvements on the best-known polynomial relationships between these measures. In the second part, I'll briefly discuss some progress on a conjecture of De Caen, Erdos, Pullman and Wormald on clique partitions of a graph and its complement, building on ideas of Keevash and Sudakov.

Allen Yuan

Title: On the higher Frobenius
Date: Monday, March 9, 2020 | 4:30pm | Room: 2-131
Committee: Haynes Miller (chair), Jacob Lurie (advisor), Jeremy Hahn

Abstract

Algebraic topology is the study of spaces via algebraic invariants. Given such an invariant of a space $X$, one can ask: how much of $X$ is captured by that invariant? For instance, can one recover $X$ itself (up to homotopy)?

This question was first addressed in work of Quillen and Sullivan on rational homotopy theory in the 1960's and in work of Dwyer-Hopkins and Mandell on $p$-adic homotopy theory in the 1990's. They showed that various algebraic enhancements of the notion of cohomology allow one to recover various approximations to a space $X$, such as its rationalization or $p$-completion.

In this thesis, we describe how to unify these ideas and recover a space in its entirety, rather than up to an approximation, using deeper invariants. The approach is centered around an insight of Nikolaus and Scholze, who demonstrate that the classical Frobenius endomorphism for rings in characteristic $p$ naturally generalizes to a phenomenon in higher algebra (more precisely, for $E_{\infty}$-ring spectra), which we call the higher Frobenius. Our main result is that there is an action of the circle group on (a certain subcategory of) $p$-complete $E_{\infty}$-rings whose monodromy is the higher Frobenius. Using this higher Frobenius action, we give a fully faithful model for a simply connected finite complex $X$ in terms of Frobenius-fixed $E_{\infty}$-rings.

Guangi Yue

Title: Combinatorics of Affine Springer Fibers and Combinatorial Wall-Crossing
Date: Tuesday, April 7, 2020 | 2:00pm | Room: Remote (via Zoom)
Committee: Roman Bezrukavnikov, Richard Stanley, Zhiwei Yun

Abstract

This thesis deals with several combinatorial problems in representation theory.

The first part of the thesis studies the combinatorics of affine Springer fibers of type A. In particular, we give an explicit description of irreducible components of $\mathcal{F} l_{tS}$ and calculate the relative positions between two components. We also study the two-sided Kazhdan-Lusztig cell containing column-type affine permutations and establish a connection with the affine Springer fibers, which is compatible with the affine matrix ball construction algorithm. The results also proved a special case of Lusztig's conjecture. The work in this part include joint work with Pablo Boixeda.

In the second part, we define the combinatorial wall-crossing transformation and the generalized column regularization on partitions and prove that a certain composition of these two transformations has the same effect on the one-row partition. This result gives a special situation where column regularization, can be used to understand the complicated Mullineux map, and also proves a special case of Bezrukavnikov's conjecture. Furthermore, we prove a condition under which the two maps are exactly the same, generalizing the work of Bessenrodt, Olsson and Xu. The combinatorial constructions is related to the Iwahori-Hecke algebra and the global crystal basis of the basic $U_q(\mathfrak{sl}_b)$-module and we provide several conjectures regarding the $q$-decomposition numbers and generalizations of results due to Fayers. This part is a joint work with Panagiotis Dimakis and Allen Wang.