## Departmental PhD Thesis Exam – Lucas Ashbury-Bridgwood

Monday, August 22, 2022 at 10:00 a.m. (sharp)

PhD Candidate: Lucas Ashbury-Bridgwood
Supervisor: Balint Virag
Thesis title: Random Canonical Products and the Secular Function of the Stochastic Airy Operator

***

Secular functions of random matrices and their limits are of recent interest in random matrix theory. Such functions are entire with zeros the spectra of the corresponding operators. For example, the general beta ensembles, extending the joint eigenvalue law of classical random matrix ensembles, have a universal soft edge limit upon rescaling called the Airy beta point process. This process also arises as eigenvalues of a random operator called the stochastic Airy operator. It is proven here that secular functions of the general beta ensembles converge in distribution to that of the stochastic Airy operator. Furthermore, this convergence is realized in the context of regularized determinants of operators. This is done by proving new asymptotics of the Airy process and rigidity estimates of the general beta ensembles and establishing this convergence for more general random sequences. These results extend the currently known case for the Gaussian ensembles in Lambert and Paquette (2020). Growth asymptotics are proven for the secular function of the stochastic Airy operator, and as an application some open questions in Lambert and Paquette (2020) are answered. By applying and extending the work in Valkó and Virág (2020) in the bulk case, the secular function is proven to be a unique limiting solution of an ordinary differential equation. Additionally, new convergence laws for discrete matrix models limiting to the stochastic Airy operator are proven, including convergence of the derivatives of eigenfunctions.

***

A copy of the thesis can be found here: thesis lucas ashbury-bridgwood

## Departmental PhD Thesis Exam – Tristan Milne

Wednesday, September 7 at 10:00 a.m. (sharp)

PhD Candidate: Tristan Milne
Thesis title: Optimal Transport, Congested Transport, and Wasserstein Generative

***

Generative Adversarial Networks (GANs) are a method for producing a distribution $\mu$ that one can sample which approximates a distribution $\nu$ of real data. Wasserstein GANs with Gradient Penalty (WGAN-GP) [GAA+17] were designed to update $\mu$ by computing and then minimizing the Wasserstein 1 distance between $\mu$ and $\nu$. In the first part of this thesis we show that in fact, WGAN-GP do not compute this distance. Instead, they compute the minimum of a different optimal transport problem, the so-called congested transport [CJS08]. We then use this result to offer explanations of the observed performance of WGAN-GP. Our discovery also elucidates the role of the gradient penalty sampling strategy in WGAN-GP, and we show that by modifying this distribution one can ameliorate a transient form of mode collapse in the optimal mass flows.

The second part of this thesis presents new algorithms for generative modelling based on insights from optimal transport theory. The basic idea is to transform one distribution into another via iterated descent with an adaptive step size on learned Kantorovich potentials computed with WGAN-GP. We provide an initial convergence theory for this technique, as well as guarantees of convergence for an extension of this procedure when the target distribution is supported on a submanifold of Euclidean space with codimension at least two. As a proof of concept, we demonstrate via experiments that this provides a flexible and effective approach for several generative modelling problems, including image generation, translation, and denoising.

Further analysis of this algorithm reveals that it is connected to image restoration techniques via learned regularizers, which generalize the classical total variation denoising technique of Rudin-Osher-Fatemi (ROF) [ROF92]. We provide analogues of the results of [Mey01] on ROF to the learned regularizer setting. Leveraging this connection, we provide optimal transport versions of the iterated denoising [AXR+15] and multiscale image decompositions [TNV04] associated with ROF.

***

A copy of the thesis can be found here: thesis_July26th

## Departmental PhD Thesis Exam – Alexandru Gatea

Monday, September 12, 2022 at 3:00 p.m. (sharp)

PhD Candidate: Alexandru Gatea
Supervisor: Balint Virag
Thesis title:  Grid entropy in last passage percolation, a variational formula for Gibbs Free Energy, and applications to a “choose the best of D samples” model

***

Working in the setting of i.i.d. last-passage percolation on RD with no assumptions on the underlying edge-weight distribution, we develop the notion of grid entropy: a deterministic directed norm with negative sign that measures the proportion of empirical measures of edge weights (in a fixed direction or direction-free) which converge weakly to a given target
measure. We study various properties of grid entropy, including an upper bound on the sum of relative and grid entropies, upper semicontinuity in most cases, and the fact that grid entropy can be described as the negative convex conjugate of Gibbs Free Energy. We show that the direction-free case is nothing more than the direction-fixed case in the (1, 1, . . . , 1) direction. In addition, we derive a grid entropy variational formula for the point-to-point/point-to-hyperplane Gibbs Free Energies that answers a directed polymer version of a question of Hoffman. Shifting gears, we proceed to study the limiting behaviour of empirical measures in a model consisting of repeatedly taking D samples from a distribution and picking out one according to an omniscient “strategy.” We show that the set of limit points of empirical measures is almost surely the same whether or not we restrict ourselves to strategies which make the choices independently of all
past and future choices, and furthermore, that this set coincides with the set of measures with finite grid entropy. These sets are convex and weakly compact; we characterize their extreme points as those given by a natural “greedy” deterministic strategy and we compute the grid entropy of said extreme points to be 0. This yields a description of the set of limit points
of empirical measures as the closed convex hull of measures given by a density which is D ¨ Beta(1, D) distributed. We also derive a simplified version of a grid entropy-based variational formula for Gibbs Free Energy for this model, and we present the dual formula for grid entropy.

***

A copy of the thesis can be found here: thesis

## Departmental PhD Thesis Exam – Kathlyn Dykes

Thursday, August 18, 2022 at 10:00 a.m. (sharp)

PhD Candidate: Kathlyn Dykes
Supervisor: Joel Kamnitzer
Thesis title: MV polytopes and reduced double Bruhat cells

***

When $G$ is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of $G$. By fixing $w$ from the Weyl group, we can define MV polytopes whose highest vertex is labelled by $w$. We show that these polytopes are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by $w^{-1}$. To do this, we define a collection of generalized minor functions $\Delta_\gamma^\text{new}$ which tropicalize on the reduced Bruhat cell to the BZ data of an MV polytope of highest vertex $w$.

We also describe the combinatorial structure of MV polytopes of highest vertex $w$. We explicitly describe the map from the Weyl group to the subset of elements bounded by $w$ in the Bruhat order which sends $u \mapsto v$ if the vertex labelled by $u$ coincides with the vertex labelled by $v$ for every MV polytope of highest vertex $w$. As a consequence of this map, we prove that these polytopes have vertices labelled by Weyl group elements less than $w$ in the Bruhat order.

A motivation for studying MV polytopes of highest vertex $w$ is that they are the finite-type equivalent of lower affine MV polytopes for $\widehat{SL_2}$. We show that for $\ell(w) \leq 3$, lower affine MV polytopes with highest vertex $w$ are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by $w^{-1}$ for $\widehat{SL_2}$.

Finally, MV polytopes in the finite case are defined by the tropical Pl\”{u} relations while rank 2 affine MV polytopes are defined by “diagonal relations”. We prove that for $B_2$ polytopes, these diagonal relations hold and are equivalent to the tropical Pl\”{u}cker relations.

***

A copy of the thesis can be found here: Dykes_Kathlyn_thesis

## Departmental PhD Thesis Exam – Lennart Doppenschmitt

Wednesday, August 31, 2022 at 11:00 a.m. (sharp)

PhD Candidate: Lennart Doppenschmitt
Supervisor: Marco Gualtieri
Thesis title: Hamiltonian Geometry of Generalized Kähler Metrics

***

Generalized Kähler structures are a natural generalization of Kähler metrics. In this thesis, we pose and investigate the question of finding a generalized Kähler metric with a prescribed volume form in a given generalized Kähler class. This is a natural generalization of the famous Calabi conjecture. We define a generalized Kähler class as a homotopy class of bisections in a holomorphic symplectic Morita equivalence between holomorphic Poisson manifolds. To answer this question we introduce holomorphic families of branes, a novice concept to study variations of generalized complex branes with a complex parameter. We then apply this to families of Lagrangian brane bisections in a symplectic Morita equivalence to analyze variations in generalized Kähler metrics. We construct an almost Kähler metric on the infinite-dimensional space of prequantized generalized Kähler metrics and set up a Hamiltonian group action by gauge transformations. This setup leads to a downward gradient flow of a functional on the space of generalized Kähler metrics towards the metric with prescribed volume form.

***

A copy of the thesis can be found here: thesis

## Departmental PhD Thesis Exam – David Urbanik

Thursday, August 11, 2022 at 10:00 a.m. (sharp)

PhD Candidate: David Urbanik
Supervisor: Jacob Tsimerman
Thesis title: Algebraic Cycle Loci at the Integral Level

****

Suppose f is a smooth projective family from X to S defined over the ring of integers I of a number field K. For each prime of I with residue field k, we consider the algebraic loci in S_k above which cohomological cycle conjectures predict the existence of non-trivial families of algebraic cycles, generalizing the Hodge loci of the generic fibre S_K. We develop a technique for studying all such loci, together, at the integral level. As a consequence we give a non-Zariski density criterion for the union of non-trivial ordinary algebraic cycle loci in S. The criterion is quite general, depending only on the level of the Hodge flag in a fixed cohomological degree w and the Zariski density of the associated geometric monodromy representation.

****

A draft of the thesis can be found here: David Urbanik – Math PhD Thesis

## Departmental PhD Thesis Exam – Georgios Papas

Tuesday, July 19, 2022 at 12:00 p.m.

PhD Candidate: Georgios Papas
Supervisor: Jacob Tsimerman
Thesis title: Some topics in the arithmetic of Hodge structures and an Ax-Scanuel theorem for GLn

****

In the first part of this thesis we consider smooth projective morphisms $f:X\rightarrow S$ of $K$-varieties with $S$ an open curve and $K$ a number field. We establish upper bounds of the Weil height $h(s)$ by $[K(s):K]$ at certain points $s\in S(\bar{K})$ that are “exceptional” with respect to the variation of Hodge structures $R^n(f^{an})_{*}(\Q_{X^{an}_{\C}})$, where $n=\dim X-1$. We work under the assumption that the generic special Mumford-Tate group of this variation is $Sp(\mu,\Q)$, the variation degenerates in a strong fashion over some fixed point $s_0$ of a proper curve that contains $S$, the Hodge conjecture holds, and that what we define as a “good arithmetic model” exists for the morphism $f$ over the ring $\mathcal{O}_K$.

Our motivation comes from the field of unlikely intersections, where analogous bounds were used to settle unconditionally certain cases of the Zilber-Pink conjecture.

In the second part of this thesis, we prove an Ax-Schanuel type result for the exponential functions of general linear groups over $\mathbb{C}$. We prove the result first for the group of upper triangular matrices and then for the group $GL_n$ of all $n\times n$ invertible matrices over $\mathbb{C}$. We also obtain Ax-Lindemann type results for these maps as a corollary, characterizing the bi-algebraic subsets of these maps.

Our motivation comes from the fact that Ax-Schanuel and Ax-Lindemann type results are an important tool in the theory of unlikely intersections, in the context of the Pila-Zannier method.

****

A draft of the thesis can be found here: G.Papas, Thesis

## Departmental Ph.D. Thesis Exam – Carrie Clark

Wednesday, June 29, 2022 at 10:00 a.m.

PhD Candidate: Carrie Clark
Supervisor: Almut Burchard
Thesis title: Droplet formation in simple nonlocal aggregation models

****

We interaction energies given by various kernels, and investigate how  these kernels drive the formation of multiple flocks within a larger population. We show that for a class of kernels having a “well-barrier” shape that the energy is minimized by a sequence of indicators of finitely many balls whose supports become infinitely far apart from one another. The dichotomy case of the concentration compactness principle is a key ingredient in our proof. We also consider a toy model which forbids points in the support of an admissible density from being within a certain range of distances from one another. We show in one dimensions, that no matter the width of this range the energy is minimized by the indicator of a union of well separated intervals of length 1 and one smaller interval. Finally, we also consider weakly repulsive kernels and show that Wasserstein $d_{\infty}$ local minimizers must saturate the density constraint.

****

A draft of the thesis can be found here: thesis copy

## Departmental PhD Thesis Exam – Jeffrey Im

Friday, June 17, 2022 at 3:00 p.m. (sharp)

PhD Candidate: Jeffrey Im
Supervisor: George Elliott
Thesis title: : Coloured Isomorphism of Classifiable C*-algebras

****

It is shown that the coloured isomorphism class of a unital, simple, Z-stable, separable amenable C-algebra satisfying the Universal Coefficient Theorem (UCT) is determined by its tracial simplex. This is a joint work with George A. Elliott.

****

A draft of the thesis can be found here: thesis-1

## Departmental PhD Thesis Exam – Assaf Bar-Natan

Tuesday, August 16, 2022 at 12:00 p.m. (sharp)

PhD Candidate: Assaf Bar-Natan
Supervisor: Kasra Rafi
Thesis title: Geodesic Envelopes in Teichmuller Space Equipped with the Thurston Metric

****

The Thurston metric on Teichm\”{u}ller space, first introduced by W. P. Thurston is an asymmetric metric on Teichm\”{u}ller space defined by $d_{Th}(X,Y) = \frac12 \log\sup_{\alpha} \frac{l_{\alpha}(Y)}{l_{\alpha}(X)}$. This metric is geodesic, but geodesics are far from unique. In this thesis, we show that in the once-punctured torus, and in the four-times punctured sphere, geodesics stay a uniformly-bounded distance from each other. In other words, we show that the \textit{width} of the \textit{geodesic envelope}, $E(X,Y)$ between any pair of points $X,Y \in \mc{T}(S)$ (where $S = S_{1,1}$ or $S = S_{0,4}$) is bounded uniformly. To do this, we first identify extremal geodesics in $Env(X,Y)$, and show that these correspond to \textit{stretch vectors}. We then compute Fenchel-Nielsen twisting along these paths, and use these computations, along with estimates on earthquake path lengths, to prove the main theorem.

****

A draft of the thesis can be found here: thesis