## Departmental Ph.D. Thesis Exam – Vincent Girard

Monday, November 28, 2022 at 2:00 (sharp)

65 St George St, Room 101
Zoom Web Conference

PhD Candidate: Vincent Girard
Supervisor: Fiona Murnaghan
Thesis title: Relatively Supercuspidal Representations of the Symplectic p-adic Groups

***

In this thesis, we construct in a concrete manner a family of non-supercuspidal, relatively supercuspidal
representations of symplectic p-adic groups, based on the work of Murnaghan [Mur]. We cover
the symmetric pairs (Sp(4n, F), Sp(2n, E)), (Sp(2n, E), Sp(2n, F)) and (Sp(2n, F), Sp(2k, F) x Sp(2(n-k), F))
where F is a p-adic field of odd residual characteristic, and E is a quadratic field extension of F.

We also look at the symmetric pairs (Sp(2n, F), GL(n, F)) and (Sp(2n, F), U(n, E/F, ε)) for ε an invertible
Hermitian matrix over E/F. For these additional pairs, the above construction doesn’t result
in any non-supercuspidal, relatively supercuspidal representations (despite these pairs admitting
distinguished supercuspidals).

We end with an in-depth look at the case of Sp(2, F) = SL(2, F). We show that in this low-rank
example, for all of the above pairs, all irreducible relatively supercuspidal representations of SL2(F)
are either supercuspidal or obtained from our construction. In particular, all irreducible H-relatively
supercuspidal representations of SL(2, F), for H either GL(1, F) or U(1, E/F, ε), are supercuspidal.

A copy of the thesis can be found here: Girard_Vincent_202212_PhD_thesis

## MSc Thesis Presentation – Turner Silverthorne

Thursday, September 22, 2022 at 12:00 p.m.
BA6183

MSc Candidate: Turner Silverthorne
Thesis title: A mathematical model of promoter methylation in the circadian clock

***

Living systems use genetic networks to produce complex dynamics necessary for survival. Examples include limit-cycle oscillations, bistable switching, and the attenuation of molecular noise. The mammalian circadian clock is a particularly interesting example, consisting of two interacting genetic feedback loops that achieve a delicate balance between robustness and plasticity.

The clock’s period (it’s most important output) is vital for normal biological function, and regulated by a variety of factors. Recent experiments have provided strong evidence that DNA methylation plays a role in controlling the period of the circadian clock. The connections between epigenetic factors (such as DNA methylation) and the circadian clock are multifaceted and poorly understood. In this thesis, I investigate epigenetic regulation of the circadian clock from a mathematical perspective.

Building on an earlier model of the primary feedback loop in the circadian clock, we add a layer of epigenetic regulation. From this extended model, we derive a perturbative estimate of the clock’s period that quantifies the influence of DNA methylation. We then use timescale separation arguments to derive an approximate model with the structure of a monotone cyclic feedback system. Such systems obey a generalization of the Poincaré-Bendixson theorem and hence have a relatively-tame bifurcation theory. Using our reduced model, we show that methylation can induce Hopf bifurcations, alter the period, and remove bistability. Together, our analysis of the reduced and full model add a new perspective to epigenetic regulation of one of the most important biological oscillators: the circadian clock.

***

A copy of the thesis can be found here: masters_thesis_silverthorne

## 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 MSc Thesis Presentation – Mahmud Azam

Tuesday, August 23 2022 at 3:00 p.m. (sharp)

MSc Candidate: Mahmud Azam
Supervisor: Alexander Kupers
Thesis title: Semidirect Products of ∞–Operads

***

We provide a construction of an $\infty$–operad from a functor $BG \to \Op_\infty$ encoding the action of a group $G$ on a given unital $\infty$–operad whose underlying $\infty$–category is a Kan complex.  This construction, restricted to classical operads in $\Set$ viewed as $\infty$–operads, coincides with the semidirect product construction. Taking this as the definition of semidirect product of $\infty$–operads, we show that the action of $G$ on the given $\infty$–operad is equivalent to the trivial action if and only if the corresponding semidirect products
are equivalent. We then outline how one might generalize this result to operads in $\Top$
and use this to show that the semidirect product of the real version of the little $n$–disks operad with $SO(n – 1)$ or $SO(n – 2)$ for $n$ even or odd respectively corresponding to the usual action is equivalent to the semidirect product corresponding to the trivial action.

***

A copy of the thesis can be found here:Operads

## 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

## Mathematics Graduate Career: Alumni Panel Discussion – June 20, 2022

The Department of Mathematics, in collaboration with the MGSA and UofT Career Center, is hosting a panel discussion with UofT mathematics alumni who are working in the exciting fields of data science, consulting, finance etc.

Please come to this panel to discover:

career options available to advanced degree holders in mathematics,
what skills you can cultivate for a specific career,
what kinds of mathematics are used industry-specific careers, and much more.

Please see the event poster for further details: Graduate Career Poster – 2022.