## 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

## Departmental PhD Thesis Exam – Stefan Dawydiak

Thursday, June 23, 2022 at 10:00 a.m. (sharp)

PhD Candidate: Stefan Dawydiak
Supervisor: Alexander Braverman
Thesis title: Three pictures of Lusztig’s asymptotic Hecke algebra

****

Let  W ̃   be  an  extended  affine  Weyl  group,  H be  the  its  Hecke  algebra  over  the  ring  Z[q, q−1]  with standard basis {T_w}w∈W ̃ , and J  be Lusztig’s asymptotic Hecke algebra, viewed as a based ring with basis.  This thesis studies the algebra J  from several perspectives, proves theorems about various  incarnations  of  J ,  and  provides  tools  to  be  applied  for  future work.  We  prove  three  types of  results.   In  the  second  and  third  chapters,  we  investigate  J  as  a  subalgebra  of  the  (q−1)-adic completion  of  H  via  Lusztig’s  map  φ.  In  the  second  chapter,  we  use  Harish-Chandra’s  Plancherel formula for p-adic groups to show that the coefficient of T_x in t_w is a rational function of q, depending only  on  the  two-sided  cell  containing  w,  with  no  poles  outside  of  a  finite set  of  roots  of  unity that  depends  only  on W ̃.  In  type  A ̃_n  and  type (C_2 ) ̃,  we  show that  the  denominators  all  divide a  power  of  the  Poincaré  polynomial  of  the  finite Weyl  group.   As  an  application,  we  conjecture that these denominators encode more detailed information about the failure of the Kazhdan-Lusztig classification of H-modules at roots of the Poincaré polynomial than is currently known.  In the third chapter, we reprove the results of the second chapter without using any tools from harmonic analysis in the special case G = SL_2.  In this case we also prove a positivity property for the coefficients of  T_x in t_w, that we conjecture holds in general.  We also produce explicit formulas for the action of J on the Iwahori invariants S^I of the Schwartz space of the basic affine space. In the fourth chapter, we  give  a  triangulated  monoidal  category  of  coherent  sheaves  whose Grothendieck  group  surjects onto  J_0  ⊂ J ,  the  based  ring  of  the  lowest  two  sided  cell of W ̃,  equipped  with  a  monoidal  functor from  the  category  of  coherent  sheaves  on  the derived  Steinberg  variety.  We  show  that  this  partial categorification  acts  on  natural  coherent  categorifications  of  S^I .   In  low  rank  cases,  we  construct complexes lifting the basis elements t_w of  J_0 and their structure constants.

****

A draft of the thesis can be found here: Stefan-Dawydiak-Thesis-v4.2

## Departmental PhD Thesis Exam – Kenneth Chiu

Wednesday, June 1, 2022 at 11:00 a.m. (sharp)

PhD Candidate: Kenneth Chiu
Supervisor: Jacob Tsimerman
Thesis title: Functional transcendence in mixed Hodge theory

*****

Ax-Schanuel theorem is a function field analogue of the Schanuel’s conjecture in transcendental number theory. Building on the works of Bakker, Gao, Klingler, Mok, Pila, Tsimerman, Ullmo and Yafaev, we extend the Ax-Schanuel theorem to mixed period mappings. Using this together with the Ax-Schanuel theorem for foliated principal bundles by Blázquez-Sanz, Casale, Freitag, and Nagloo, we further extend the Ax-Schanuel theorem to the derivatives of mixed period mappings. The linear subspaces in the Ax-Schanuel theorem are replaced by weak Mumford-Tate domains, which are certain group orbits of mixed Hodge structures. In particular, we prove that these domains have complex structures, and that their real-split retractions can be decomposed into semisimple and unipotent parts. We prove that the image of a mixed period mapping is contained in the weak Mumford-Tate domain that arises from the monodromy group of the variation. O-minimal geometry, namely the definable Chow theorem and the Pila-Wilkie counting theorem, are used in the proof of our extension of the Ax-Schanuel theorem.

****

The draft of the thesis can be found here:Thesis

## Departmental PhD Thesis Exam – Joshua Lackman

Tuesday, April 12, 2022 at 11:00 a.m. (sharp)

PhD Candidate: Joshua Lackman
Supervisor: Marco Gualtieri
Thesis title: The van Est Map on Geometric Stacks

***

We generalize the van Est map and isomorphism theorem in three ways. First, we generalize the van Est map from a comparison map between Lie groupoid cohomology and Lie algebroid cohomology to a (more conceptual) comparison map between the cohomology of a stack $\mathcal{G}$ and the foliated cohomology of a stack $\mathcal{H}\to\mathcal{G}$ mapping into it. At the level of Lie grouoids, this amounts to describing the van Est map as a map from Lie groupoid cohomology to the cohomology of a particular LA-groupoid. We do this by, essentially, associating to any
(nice enough) homomorphism of Lie groupoids $f:H\to G$ a natural foliation
of the stack $[H^0/H]\,.$ In the case of a wide subgroupoid $H\xhookrightarrow{}G\,,$ this foliation can be thought of as equipping
the normal bundle of $H$ with the structure of an LA-groupoid. This generalization allows us to derive results that couldn’t be obtained with the usual van Est map for Lie groupoids. In particular, we recover classical results, including van Est’s isomorphism theorem about the maximal compact subgroup, which we generalize to proper subgroupoids, as well as the Poincar\'{e} lemma. Secondly, we generalize the functions that we can take cohomology of in the context of the van Est map; instead of using functions valued in representations, we can use functions valued in modules — for example, we can use $S^1$-valued functions and $\mathbb{Z}$-valued functions. This allows us to obtain classical results about linearizing group actions, as well as results about lifting group actions to gerbes. Finally, everything we do works in the holomorphic category in addition to the smooth category.

***

The draft of the thesis can be found here: Thesis Draft March 21

## Departmental Ph.D. Thesis Exam – Saied Sorkhou

Tuesday, April 5, 2022 at 2:00 p.m. (sharp)

PhD Candidate: Saied Sorkhou
Supervisor: Joe Repka
Thesis title: Levi Decomposable Subalgebras of Classical Lie Algebras with Regular
Simple Levi Factor

***

This thesis describes and characterizes a significant class of subalgebras of the classical Lie algebras, namely those which are Levi decomposable with regular and simple Levi factor, with select exceptions. Such subalgebras are entirely determined by their Levi factors and radicals. The possible Levi factors are well-established in the literature and so the contribution of this thesis is a characterization of the radicals. The radicals naturally decompose into nontrivial and trivial components. The nontrivial component is found to be fully classified by subsets of the parent root system and Weyl group. However, a classification of the trivial component requires solving the open problem of classifying solvable subalgebras of classical Lie algebras. Nonetheless, this thesis establishes a criterion on the trivial components for determining when two such subalgebras are conjugate. This thesis also briefly explores the ramifications of relaxing
simplicity of the Levi factor to allow for semisimplicity.

***

A draft of the thesis is available here: thesis_draft_Feb_23_2022

## Departmental Ph.D. Thesis Exam – Malors Espinosa Lara

Everyone is welcome to attend the presentation. Refreshments will be served in the lounge at 1:30 p.m.!

Wednesday, April 6, 2022 at 2:00 p.m.

PhD Candidate: Malors Espinosa Lara
Supervisor: Jim Arthur
Thesis title: Explorations on Beyond Endoscopy

***

In this thesis we provide a description of the first paper on Beyond Endoscopy by Altu˘g and explain how to generalize to totally real fields, based on a joint work of the author with Melissa Emory, Debanjana Kundu and Tian An Wong, and is a work in preparation. This part is mostly expository, and we refer the reader to the relevant paper [7] Furthermore, we prove a conjecture of Arthur. In his original paper on Beyond
Endoscopy, Langlands provides a formula for certain product of orbital integrals in GL(2, Q), subsequently used by Altu˘g to manipulate the regular elliptic part of the trace formula with the goal of isolating the contribution of the trivial representation. Arthur predicts this formula should coincide with a product of polynomials associated to zeta functions of orders constructed by Zhiwei Yun. We prove this is the case by finding the explicit polynomials and recovering the original formula from them.
We also explain how some aspects of the strategy used can be interpreted as problems of independent interest and importance of their own.

***

A draft of the thesis is available here: Malors_Espinosa_PhD_Thesis_8FEB2022

## Departmental Ph.D. Thesis Exam – Wenbo Li

Everyone is welcome to attend the presentation. Refreshments will be served in the lounge at 3:30 p.m.!

Tuesday, April 5, 2022 at 4:00 p.m.

PhD Candidate: Wenbo Li
Supervisor: Ilia Binder
Thesis title: Quasiconformal Geometry of Metric Measure Spaces
and its Application to Stochastic Processes

***

We study three topics of quasiconformal geometry in this dissertation; the quasisymmetric embeddability of metric Sierpi\’nski carpets, the quasisymmetric embeddability of weak tangents and the conformal dimension of stochastic spaces. The common tools we use to attack these three topics are different versions of Moduli and the convergence of spaces.

For the first problem, the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpi\’nski carpet into the plane, we use transboundary modulus to study it. This allows us to give a complete characterization in the case of dyadic slit carpets. Every such slit carpet $X$ can be embedded into a “pillowcase sphere” $\widehat{X}$ which is a metric space homeomorphic to the sphere $\mathbb{S}^2$. We show that $X$ can be quasisymmetrically embedded into the plane if and only if $\widehat{X}$ is quasisymmetric to $\mathbb{S}^2$ if and only if $\widehat{X}$ is Ahlfors $2$-regular.

For the second problem, the problem of quasisymmetric embeddability of weak tangents of metric spaces, we first show that quasisymmetric embeddability is hereditary, i.e., if $X$ can be quasisymmetrically embedded into $Y$, then every weak tangent of $X$ can be quasisymmetrically embedded into some weak tangent of $Y$, given that $X$ is proper and doubling. However, the converse is not true in general; we will illustrate this with several counterexamples. In special situations, we are able to show that the embeddability of weak tangents implies global or local embeddability of the ambient space. Finally, we apply our results to Gromov hyperbolic groups and visual spheres of expanding Thurston maps.

For the third problem, the conformal dimension of stochastic spaces, we develop tools related to the Fuglede modulus to study it. In order to achieve this goal, we study the conformal dimension of deterministic and random Cantor sets and investigate the situation of conformal dimension $1$. We apply our techniques to construct minimal(in terms of conformal dimension) planar graph. We further develop this line of inquiry by proving that a “natural” object, the graph of one dimensional Brownian motion, is almost surely minimal.

***

A draft of the thesis is available here: Wenbo Li Ph.D. Dissertation UofT Mathematics