## 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

## Departmental Ph.D. Thesis Exam – Keegan Dasilva Barbosa

Thursday, February 10, 2022 at 11:00 a.m.

PhD Candidate: Keegan Dasilva Barbosa
Supervisor: Stevo Todorcevic
Thesis title: Ramsey Degree Theory of Ordered and Directed Sets

***

We will study various Ramsey degree problems pertaining to categories of structures under various
embedding types. Using a technique originally penned by Laver, we show that the class of Aronszajn
lines ordered under the embedding relation is a better quasi-order when we assume PFA. As a corollary,
we deduce one dimensional Ramsey degrees for Aronszajn lines. We also devise a colouring algorithm to
colour Borel graphs coded by better quasi-orders on countable sets. We apply our algorithm to various
types of well established better quasi-orders and deduce that there is a whole class of better quasi-orders
that exists without a single known constructible example. We will conclude with some results pertaining
to the Kechris-Pestov-Todorcevic correspondence. This will include a categorical notion of precompact
expansion, which will prove to be more versatile in computing Ramsey degrees, as well as a weaker
notion of the Ramsey property which also corresponds to xed point properties of automorphism groups
of ultrahomogeneous structures. This includes an application to trees under various embedding types,
including the foundational strong embedding types studied by Milliken.

A copy of the thesis can be found here: Keegan Dasilva Barbosa Thesis

## Departmental PhD Thesis Exam – Jack Ding

Thursday, August 19, 2021
10:00 a.m.

PhD Candidate: Jack Ding
Supervisor: Lisa Jeffrey
Thesis title:  The Atiyah-Bott Lefschetz formula applied to the based loops on
SU(2)

***

We prove two generalizations of localization formulae for finite-dimensional spaces to the infinite-dimensional based loop group $\Omega G$.

The Atiyah-Bott-Lefschetz Formula is a well-known formula for computing the equivariant index of an elliptic operator on a compact smooth manifold. We provide an analogue of this formula for the based loop group $\Omega SU(2)$ with respect to the natural $(T \times S^1)$-action. This is accomplished by computing certain equivariant multiplicities in the K-theory of affine Schubert varieties. From this result we also derive an effective formula for computing characters of certain Demazure modules.

The based loop group for a compact Lie group $G$ has been studied intensively since the work of Atiyah and Pressley and the book of Pressley and Segal. It is an infinite-dimensional symplectic manifold equipped with a Hamiltonian torus action, where the torus is the product of a circle and the maximal torus of $G$. When $G = SU(2)$,
the fixed points for this action are in bijective correspondence with the integers. Our final result is a Duistermaat-Heckman type oscillatory integral over the based loop group, expanded around the fixed points of the torus action. To accomplish this we use Frenkel’s results (1984) on pinned Wiener measure for orbital integrals on the affine Lie algebra, as well as the results of Urakawa (1975) on the heat kernel for a compact Lie group and Fegan’s inversion formula (1978) for orbital integrals.

A copy of the thesis can be found here: thesis

## Departmental PhD Thesis Exam – Roger Bai

Monday, August 16, 2021
3:00 p.m. (sharp)

PhD Candidate: Roger Bai
Supervisor: Joel Kamnitzer
Thesis title:  Cluster Structure for Mirkovic-Vilonen Cycles and Polytopes

***

We look at the Mirkovic-Vilonen (MV) basis for semisimple Lie algebras and compare this to the associated cluster algebra to investigate the question of whether or not the cluster variables are in the MV basis.

We begin with finding analogues of the cluster structure among MV cycles and MV polytopes. In particular, we show the exchange relations correspond to an equation involving MV polytopes.

We extend a result of Baumann-Kamnitzer in relating valuations of an MV cycle and the dimension of homomorphism spaces of its associated preprojective algebra module. In doing so, we are able to give a partial result for an exchange relation involving MV cycles in low dimensions.

In joint work with Dranowski and Kamnitzer, we present a way to calculate the fusion product of MV cycles in type A through a generalization of the Mirkovic-Vybornov isomorphism.

Finally, we finish with examining the A_3 example and show directly that the cluster variables in this case are in the MV basis.

A copy of the thesis can be found here: Cluster_Structure_for_MV_Cycles_and_PolytopesFinal

## Departmental PhD Thesis Exam – Dylan Butson

Wednesday, August 18, 2021
10:00 a.m. (sharp)

PhD Candidate: Dylan Butson
Supervisor: Kevin Costello
Thesis title:  Equivariant Localization in Factorization Homology and Vertex
Algebras from Supersymmetric Gauge Theory

***

We develop a theory of equivariant factorization algebras on varieties with an action of a connected algebraic group $G$, extending the definitions of Francis-Gaitsgory [FG] and Beilinson-Drinfeld [BD1] to the equivariant setting. We define an equivariant analogue of factorization homology, valued in modules over $\textup{H}^\bullet_G(\text{pt})$, and in the case $G=(\mathbb{C}^\times)^n$ we prove an equivariant localization theorem for factorization homology, analogous to the classical localization theorem [AtB]. We establish a relationship between $\mathbb{C}^\times$ equivariant factorization algebras and filtered quantizations of their restrictions to the fixed point subvariety. These results provide a model for predictions from the physics literature about the $\Omega$-background construction introduced in [Nek1], interpreting factorization $\mathbb{E}_n$ algebras as observables in mixed holomorphic-topological quantum field theories.

We give an account of the theory of factorization spaces, categories, functors, and algebras, following the approach of [Ras1]. We apply these results to give geometric constructions of factorization $\mathbb{E}_n$ algebras describing mixed holomorphic-topological twists of supersymmetric gauge theories in low dimensions. We formulate and prove several recent predictions from the physics literature in this language:

We recall the Coulomb branch construction of [BFN1] from this perspective. We prove a conjecture from [CosG] that the Coulomb branch factorization $\mathbb{E}_1$ algebra $\mathcal{A}(G,N)$ acts on the factorization algebra of chiral differential operators $\mathcal{D}^{\text{ch}}(Y)$ on the quotient stack $Y=N/G$. We identify the latter with the semi-infinite cohomology of $\mathcal{D}^{\text{ch}}(N)$ with respect to $\hat{ \mathfrak{g}}$, following the results of [Ras3]. Both these results require the hypothesis that $Y$ admits a Tate structure, or equivalently that $\mathcal{D}^{\text{ch}}(N)$ admits an action of $\hat{\mathfrak{g}}$ at level $\kappa=-\text{Tate}$.

We construct an analogous factorization $\mathbb{E}_2$ algebra $\mathcal{F}(Y)$ describing the local observables of the mixed holomorphic-B twist of four dimensional $\mathcal{N} =2$ gauge theory. We identify $S^1$ equivariant structures on $\mathcal{F}(Y)$ with Tate structures on $Y=N/G$, and prove that the corresponding filtered quantization of $\iota^!\mathcal{F}(Y)$ is given by the two-periodic Rees algebra of chiral differential operators on $Y$. This gives a mathematical account of the results of [Beem4]. Finally, we apply the equivariant cigar reduction principle to explain the relationship between these results and our account of the results of [CosG] described above.

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

## Departmental PhD Thesis Exam – Qin Deng

Wednesday, August 25 2021
2:00 p.m. (sharp)

PhD Candidate: Qin Deng
Supervisor: Vitali Kapovitch
Thesis title:  Hölder continuity of tangent cones and non-branching in RCD(K,N) spaces

***

This thesis is concerned with the study of the structure theory of metric measure spaces (X, d, m) satisfying the synthetic lower Ricci curvature bound condition RCD(K, N). We prove that such a space is non-branching and that tangent cones from the same sequence of rescalings are Holder continuous along the interior of every geodesic in X. More precisely, we show that the geometry of balls of small radius centred in the interior of any geodesic changes in at most a Holder continuous way along the geodesic in pointed Gromov-Hausdorff distance. This improves a result in the Ricci limit setting by Colding-Naber where the existence of at least one geodesic with such properties between any two points is shown. As in the Ricci limit case, this implies that the regular set of an RCD(K, N) space has m-a.e. constant dimension, a result recently established by Brue-Semola, and is m-a.e convex. It also implies that the top dimension regular set is weakly convex and, therefore, connected. In proving the main theorems, we develop in the RCD(K, N) setting the expected second
order interpolation formula for the distance function along the Regular Lagrangian flow of some vector field using its covariant derivative

A copy of the thesis can be found here:  Thesis Formatted

## Departmental PhD Thesis Exam – Andrew Colinet

Wednesday, August 18, 2021
11:00 a.m. (sharp)

PhD Candidate: Andrew Colinet
Supervisor: Robert Jerrard
Thesis title: Geometric Behaviour of Solutions to Equations of Ginzburg-Landau
Type on Riemannian Manifolds

***

In this thesis, we demonstrate the existence of complex-valued solutions to the Ginzburg-Landau equation
$-\Delta{}u+\frac{1}{\varepsilon^{2}}u(|u|^{2}-1)=0\hspace{20pt}\text{on }M,$

for $\varepsilon\ll1$, where $M$ is a three dimensional compact manifold without boundary, that have interesting geometric properties. Specifically, we argue the existence of solutions whose vorticity concentrates about an arbitrary closed nondegenerate geodesic on $M$.
In doing this, we extend the work of \cite{JSt} and \cite{Mes} who showed that there are solutions whose energy converges, after rescaling, to the arclength of a geodesic as above.

An important ingredient in the proof is a heat flow argument, which requires detailed information about limiting behaviour of solutions of the parabolic Ginzburg-Landau equation. Providing the necessary limiting behaviour is the other contribution of this thesis. In fact, more is achieved. Provided that $N\ge3$, we give a structural description of the limiting behaviour of solutions to the parabolic Ginzburg-Landau equation on an $N$-dimensional compact manifold without boundary $(M,g)$. More specifically, we are able to show that the limit of the renormalized energy measure orthogonally decomposes into a diffuse part, absolutely continuous with respect to the volume measure on $M$ induced by $g$, and a concentrated vortex part, supported on a codimension $2$ surface contained in $M$. Moreover, the diffuse part of the limiting energy has its time evolution governed by the heat equation while the concentrated part evolves in time according to a measure theoretic version of mean curvature flow. This extends the work of \cite{BOS2} who proved this for $N$-dimensional Euclidean space provided that $N\ge2$.

A copy of the thesis can be found here: Andrew_Colinet_Thesis