## Departmental PhD Thesis Exam – Ali Cheraghi

PhD Candidate: Ali Cheraghi
Supervisor: Jacob Tsimerman
Thesis title: Special Correspondences of Abelian Varieties and Eisenstein Series

****

****

A draft of the thesis can be found here: 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

****

****

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: :

****

****

A draft of the thesis can be found here:

## 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

## Departmental PhD Thesis Exam – Assaf Bar-Natan

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

****

****

A draft of the thesis can be found here:

## 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

****

****

A draft of the thesis can be found here:

## 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 – 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