## Departmental PhD Thesis Exams

### Dinushi Munasinghe

Tuesday, July 16, 2024

11:00 a.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate: Dinushi Munasinghe

Supervisor: Dror Bar-Natan, Ben Webster

Thesis title: Schur Algebras in Type B

Complete copy of Munashinghe's Thesis

We compare two type B generalizations of the $q$-Schur algebra: the cyclotomic $q$-Schur algebra of Dipper, James, and Mathas, and an algebra constructed to maintain the type B Schur duality of Bao, Wang, and Watanabe, introduced by Lai and Luo. By writing the latter algebra as an idempotent truncation of the former, we leverage its properties to establish cellularity and study the crystal graph structure of the simples of the endomorphism algebra, investigating parameter values at which these algebras are Morita equivalent and quasi-hereditary. We then investigate its blocks, also by comparison with those of the cyclotomic $q$-Schur algebra and type B Hecke algebra.

## Past PhD Defenses

### Keirn Munro

Wednesday, June 26, 2024

10:00 a.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate: Keirn Munro

Supervisor: Marco Gualtieri

Thesis title: Simplicial Approximation of the Hodge Laplacian Using Cauchy Sequences of Hilbert Complexes

Complete copy of Keirn's thesis

Discrete differential geometry arises from the use of discrete spaces such as graphs, simplicial, cubical, or polyhedral complexes for modeling geometric structures on manifolds. A common practice in this work is to transport structures on smooth manifolds to discrete counterparts in a process referred to as discretization. Discretizations often appear as elements of a sequence that approximates the smooth structure on the manifold through some measure of convergence. Algorithms which produce such sequences are highly sought after for computational applications but frequently ignore deeper structural relationships between successive discrete models.

This thesis makes contributions to the discretization of Hodge theory through the construction of a framework that serves to axiomatize a foundational set of results in the field. The salient feature of this framework is the ability to directly measure the difference in approximation accuracy between discretizations without reference to the overarching smooth structure. This provides a Cauchy-type characterization of sequences of discretizations while opening the scope of inquiry to a much larger class of problems involving the analysis of Hodge Theory through Cauchy sequences.

### Belal Abuelnasr

Wednesday, June 26, 2024

2:00 p.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate: Belal Abuelnasr

Supervisor: Adam Stinchcombe

Thesis title: Modelling and Simulating Retinal Dynamics and Physiology

Complete copy of Belal's thesis

We present a detailed physiological model of the (human) retina that includes the biochemistry and electrophysiology of phototransduction, neuronal electrical coupling, and the spherical geometry of the eye. The model is a parabolic-elliptic system of partial differential equations based on the mathematical framework of the bi-domain equations, which we have generalized to account for multiple cell-types. We discretize in space with non-uniform finite differences and step through time with a custom adaptive time-stepper that employs a backward differentiation formula and an inexact Newton method. A refinement study confirms the accuracy and efficiency of our numerical method. We generalize our time-stepping scheme to higher order and derive estimates for the corresponding local truncation errors. Numerical simulations using the model compare favourably with experimental findings, such as desensitization to light stimuli and calcium buffering in photoreceptors. Other numerical simulations suggest an interplay between photoreceptor gap junctions and inner segment, but not outer segment, calcium concentration. Applications of this model and simulation include analysis of retinal calcium imaging experiments, the design of electroretinograms, the design of visual prosthetics, and studies of ephaptic coupling within the retina.

### Julian Ransford

Thursday, June 27, 2024

11:00 a.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate: Julian Ransford

Supervisor: Balint Virag

Thesis title: Directed polymers in the intermediate disorder regime and the Seppäläinen–Johansson model

Complete copy of Julian's thesis

See poster for abstract.

### Daniel Calderon Wilches

Wednesday, July 3, 2024

2:00 p.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate: Daniel Calderon Wilches

Supervisor: Stevo Todorcevic & Ilijas Farah

Thesis title: Forcing in Analysis and Combinatorics

Complete copy of Daniel's thesis

In this thesis, we explore some new applications of the forcing technique in the context of Analysis and Combinatorics by:

- Constructing a model of Set Theory in which strong measure zero subsets of the real line are meager-additive while Borel’s conjecture fails, answering a long-standing question due to Bartoszy´nski and Judah.
- Constructing a model of Set Theory in which Jensen’s ♢ℵ1 fails, there is a counterexample to Naimark’s Problem, and there is a separably represented C∗-algebra with exactly two inequivalent irreducible representations. Such a C∗-algebra cannot satisfy the conclusion of Glimm’s Dichotomy Theorem.
- Studying families of infinite block sequences of elements of the space FINk, Ramsey properties of such families, and Ramsey properties localized on selective or semiselective coideals and ultrafilters.

### Amirmasoud Geevechi

Tuesday, July 9, 2024

11:00 a.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate: Amirmasoud Geevechi

Supervisor: Robert Jerrard

Thesis title: A Gluing Problem for a Gauged Hyperbolic PDE

Complete copy of Amirmasoud's thesis

In this thesis, we study the dynamic Abelian Higgs model in dimension 3 at the critical coupling. This is a system of partial differential equations which enjoys local symmetries known as gauge transformations. The stationary finite energy solutions to these equations in dimension 2 have been classified by Jaffe and Taubes in 1980, the so called vortex configurations. In 1992, Stuart has proved that one can construct solutions near the critical coupling regime in dimension 1+2 whose dynamics are approximated by a finite dimension Hamiltonian system on the moduli space which reduces to the geodesic flow at the critical coupling.

In this project, we study how one can glue the vortex configurations to find dynamic solutions in dimension 3. More precisely, we prove that one can construct solutions which are approximated by wave maps to the moduli space of vortex configurations. The proof involves an ansatz to construct approximate solutions and then add perturbations. In the ansatz, we go through an iterative mechanism to reduce the error of the approximate solution so that it is prepared to be perturbed to find an honest solution.

In both steps of the project, the ansatz and perturbation, the choice of gauge is crucial. It is noteworthy that the choices of gauge are different in these two steps. We proceed by a choice for gauge, simplify the equations and then we have to decompose the quantities into two components, the zero modes (the tangent vectors to the moduli space) and the orthogonal complement to zero modes. According to the Higgs mechanism, stability is available for the components orthogonal to zero modes. In this regard, in the ansatz, the dynamics of zero modes is designed in such a way that the orthogonality condition to zero modes is satisfied. In the perturbation part, the dynamics of zero modes is forced by the evolution of orthogonal components. Obtaining desired estimates for the tangential

part requires taking advantage of explicit structure of equations, rather than the usual estimates. Also, the number of iterations in the ansatz should be high enough so that the desired estimates hold for the tangential part.

### Ilia Kirillov

Monday, June 24, 2024

11:00 a.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate:Ilia Kirillov

Supervisor: Boris Khesin

Thesis title: Coadjoint orbits of symplectomorphism groups of surfaces

Complete copy of Ilia's thesis

In this thesis, we classify generic coadjoint orbits for the action of symplectic (equivalently, areapreserving) diffeomorphisms of compact symplectic surfaces with or without boundary. This completes V. Arnold’s program of studying Casimir invariants of incompressible fluids in 2D. To obtain this classification, we first solve an auxiliary problem, which is of interest by itself: classify generic Morse functions on surfaces with respect to the action of area-preserving diffeomorphisms. As a technical tool, we prove an analog of Morse-Darboux lemma in the case of a singular point on the boundary. We also generalize all the results above to the case of non-orientable surfaces without boundary.

### David Pechersky

Wednesday, June 19, 2024

11:00 a.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate: David Pechersky

Supervisor: Ilia Binder

Thesis title: Discrete Complex Analysis and Convergence of Observables on Orthodiagonal Maps

Complete copy of David's thesis

Discrete complex analysis is the study of discrete holomorphic functions. These are functions defined on graphs embedded in the plane that satisfy some discrete analogue of the Cauchy-Riemann equations. While the subject is classical, it has seen a resurgence in the past 20-30 years with the work of Kenyon, Mercat, Smirnov, and many others demonstrating the power of discrete complex analysis as a tool for understanding 2D statistical physics at criticality.

In this talk, we’ll discuss how discrete complex analysis can be applied to solve a purely deterministic problem for a very general class of discretizations of 2D space accommodating a notion of discrete

complex analysis.

### Jim Shaw

Thursday, June 13, 2024

11:00 a.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate: Jim Shaw

Supervisor: Yun William Yu

Thesis title: Practical and theoretical problems in biological sequence comparison.pdf

DNA sequencing technologies have revolutionized the study of biology, allowing unprecedented access to the blueprint of life – our genomes. As these technologies have matured, massive amounts of DNA sequences are now becoming available for computers to analyze. The increasing throughput of these technologies has rendered old algorithms unusable. In this thesis, we first build a principled mathematical foundation for approximate DNA string matching, also called sequence alignment. We then show that using theoretically-backed approaches can result in faster and better software implementations, resulting in useful new tools for biologists.

In Chapter 3, we rigorously analyze how to subsample DNA strings to speed up alignment while retaining sensitivity. We show how to determine a sampling algorithm’s conservation, which is a measure of how sensitively it can match strings. Surprisingly, different sampling methods have vastly different conservation even when retaining the same amount of “information”. We show that modifying existing software to use better subsampling algorithms gives more sensitive results.

In Chapter 4, we provide the first non-trivial runtime and accuracy bounds on a widely-used DNA alignment algorithm called seed-chain-extend. We break the worst-case quadratic runtime barrier of sequence alignment by performing an average-case analysis under a probabilistic evolutionary model of DNA sequence. Our results are concordant with algorithmic results on real data and provide new insights into the rigorous analysis of sequence alignment.

In Chapter 5, we utilize the subsampling and the seed-chain-extend approaches analyzed in Chapters 3 and 4 to build a new genome-genome comparison method and software called skani. skani can estimate the evolutionary divergence between two genomes > 25 times faster than previous algorithms. We show that skani can compare hundreds of thousands of genomes in seconds on a standard desktop computer, enabling large-scale comparisons not possible before.

### Adrian She

Wednesday, June 12, 2024

11:00 a.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate : Adrian She

Supervisors: Toniann Pitassi, Henry Yuen

Thesis title: Algebraic Methods in Query and Proof Complexity

Complete copy of Adrian's thesis

Algebraic methods have become a powerful tool for analyzing the complexity of various computational models, including low-depth circuits, algebraic proofs, and quantum query algorithms. In particular, the complexity of computing a function in these models is related to whether or not the function admits a lowdegree polynomial approximation. In this thesis, we present two novel applications of algebraic methods in computational complexity theory.

In the first part of the thesis, we study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether or not it satisfies some property. In addition to containing the classical query complexity model as a special case, this model also contains “inherently quantum” problems that have no classical analogue. Our main contribution is a generalized polynomial method for analyzing the complexity of unitary property testing problems. By leveraging connections with

invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a subspace, and approximating the entanglement entropy of a state. We also present a candidate problem towards an oracle separation of QMA and QMA(2), a long standing open question in quantum complexity theory.

In the second part of the thesis, we study the tensor isomorphism problem (TI), which has recently emerged as having connections to multiple areas of research, including quantum information theory, postquantum cryptography, and computational algebra. However, the current best upper bound is essentially the brute force algorithm. Being an algebraic problem, the study of tensor isomorphism naturally lends itself to algebraic and semi-algebraic proof systems such as the polynomial calculus (PC) and sum-of squares

(SoS). We show a Ω(n) lower bound on PC degree or SoS degree for tensor isomorphism and a non-trivial upper bound for testing isomorphism of tensors of bounded rank. Along the way, we also show that PC cannot perform basic linear algebra in sublinear degree, such as comparing the rank of two matrices. We introduce a strictly stronger proof system, called PC + Inv, which enables linear algebra to be done in low degree. We conjecture that even PC + Inv cannot solve TI in polynomial time either, and highlight many other open questions about proof complexity approaches to TI.

### Joaquin Sanchez Garcia

Wednesday, June 12, 2024

2:00 p.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate: Joaquin Sanchez Garcia

Supervisor: Robert McCann

Thesis title: 4 Problems in optimal transportation

Complete copy of Joaquin's thesis

This thesis is devoted to the study of 4 different problems for which we use the theory of optimal transportation:

1. The generalization of the Schrödinger Problem to synthetic Lorentzian geometries. 2. The small time existence for solutions for the aggregation equation on compact Riemannian manifolds for non-regular interaction potentials via the minimizing movement scheme. 3. The technique of measure pre-conditioning general Machine-Learning tasks and Domain Adaptation transfer learning. 4.The generalization of an economic model of Roy for partition of labor including occupational choice as a constraint.

1 Part 1: The Schrödinger problem in synthetic Lorentzian geometries

The Schrödinger problem refers to the minimization of relative entropy with respect to a reference measure. The Schrödinger problem is usually analyzed in two related formulations: the static and the dynamic Schrödinger problems. We study both approaches. One of the main questions of the Schrödinger problem is whether or not the solutions to the entropically regularized optimal transport problem converge to solutions of the optimal transport problem. In the dynamical setting, this property amounts to study the Large Deviation Principles of the reference measure. We study a Levy-like construction which emulates the behaviour of Brownian bridges which allows us to recover a partial version of the entropic convergence in the non-smooth Lorentzian case.

2 Part 2: The aggregation equation via the minimizing movement scheme in compact Riemannian manifolds

We study the small-time existence of solutions for non-smooth potentials via the minimizing movement scheme. The theory of gradient flows in metric spaces does not consider a potential non-regularity of potentials in the cut-locus. The presence of the cut-locus presents a difficulty for the JKO scheme to choose a direction, nevertheless we show explicitly a time bound for which we can flow the minimizing movement scheme.

3 Part 3: Measure Pre-conditioning in Machine-Learning

We study a new technique to improve convergence of algorithms for specific ML-tasks. We show that if the modifications of the problem at level n (sample size) are done in a specific way (full learner recovery

systems) we can show analytical subsequential convergence to the original model. This technique seems to be specifically important for Domain Adaptation in transfer learning.

4 Part 4: Generalizing an economic model of Roy for labor partition using occupational choice as a constraint

We study the analytical properties of a generalization of the economic model for labor force partition studied by Dr. Roy. The new model proposed by Dr. Siow, includes occupational choice as a constraint rather than a consequence. This difference allows us to rewrite the problem in an analytically useful way.

### Saeyon Mylvaganam

Monday, June 10, 2024

2:00 p.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate : Saeyon Mylvaganam

Supervisors: Stefanos Aretakis

Thesis title: The Null Gluing Problem and Conservation Laws for Maxwell's Equations

Complete copy of Saeyon's thesis

We study the null gluing problem for Maxwell’s equations along null hypersurfaces. By studying a weaker formulation of the gluing problem, which we call the kth-order gluing problem, we classify all possible conservation laws by proving that they are the only obstructions to gluing. We derive sets of conserved charges for the zeroth-order gluing problem along general null hypersurfaces and the first-order gluing problem along extremal horizons. We derive an elliptic structure related to a foliation with 2-spheres of a null hypersurface, using a similar method introduced in [9] by Aretakis. We also show the non-existence of zeroth-order conservation laws along extremal horizons and the non-existence of kth-order conservation laws for spherically symmetric extremal horizons by using a hierarchy of v-weighted integrals of the Maxwell equations. Finally, we determine how the space of these conserved charges changes under a change of foliation by understanding the gauge covariance of the elliptic structure.

### Tomas Dominguez Chiozza

Monday, February 12th, 2024

10:00 a.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate: Tomas Dominguez Chiozza

Supervisor: Dmitry Panchenko

Thesis title: A Hamilton-Jacobi approach to the stochastic block model

Complete copy of Tomas' thesis

This thesis addresses the problem of recovering the community structure in the stochastic block model with two communities. The stochastic block model is a random graph model with planted clusters widely employed as the canonical model to study clustering and community detection. The focus is on the fundamental limits of community detection, quantified by the asymptotic mutual information between the observed network and the actual community structure. This mutual information is studied using the Hamilton-Jacobi approach, pioneered by Jean-Christophe Mourrat.

The first contribution of this thesis is a detailed description of the Hamilton-Jacobi approach, and its application to computing the limit of the mutual information in the dense stochastic block model, where the average degree of a node diverges with the total number of nodes. The main novelty is a wellposedness theory for Hamilton-Jacobi equations on positive half-space that leverages the monotonicity of the non-linearity to circumvent the imposition of an artificial boundary condition as previously done in the literature.

The second contribution of this thesis is a novel well-posedness theory for an infinite-dimensional Hamilton-Jacobi equation posed on the set of non-negative measures and with a monotonic non-linearity.

Such an infinite-dimensional Hamilton-Jacobi equation appears naturally when applying the Hamilton- Jacobi approach to the sparse stochastic block model, where the total number of nodes diverges while the average degree of a node remains bounded. The solution to the infinite-dimensional Hamilton-Jacobi equation is defined as the limit of the solutions to an approximating family of finite-dimensional Hamilton- Jacobi equations on positive half-space. In the special setting of a convex non-linearity, a Hopf-Lax variational representation of the solution is also established.

The third contribution of this thesis is a conjecture for the limit of the mutual information in the sparse stochastic block model, and a proof that this conjectured limit provides a lower bound for the asymptotic mutual information. In the case when links across communities are more likely than links within communities, the asymptotic mutual information is known to be given by a variational formula. It is also shown that the conjectured limit coincides with this formula in this case.

### Kevin (Min Seong) Park

Wednesday, December 13, 2023

3:00 p.m.

BA6183/ Zoom Web Conference

PhD Candidate: Kevin Min Seong Park

Supervisor: Adam Stinchcombe

Thesis title: Temporal Difference Learning for viscous incompressible flow

This thesis presents a stochastic numerical method for computing viscous incompressible flow. By Itô’s lemma, the solution to a linear parabolic PDE is a martingale over an appropriate probability measure induced by Brownian motion. Given an initial boundary value problem, a functional corresponding to the martingale condition is minimized numerically through deep reinforcement learning. This methodology is well-suited for high dimensional PDEs over irregular domains, as it is mesh-free and sampling techniques can avoid the curse of dimensionality.

The extension to computing viscous incompressible flow is done by first formulating a martingale condition for the viscous Burgers’ equation. Its solution is obtained by a fixed point iteration for which a proof of convergence in L2 is provided. The constrained minimization problem subject to divergence-free vector fields is designed for the incompressible Navier-Stokes equations. The velocity is determined without the pressure gradient. The stochastic numerical method avoids difficulties arising from coupling of velocity and pressure terms by globally maintaining incompressibility. Furthermore, pressure can be recovered from the computed velocity in a post-processing step.

The numerical implementation details are provided, including errors from statistical sampling. Simulations of various flow scenarios are showcased, including those with analytical solutions such as Stokes’ flow in a revolving ball, Poiseuille flow, and the Taylor-Green vortex. Additional validation is acquired from comparing against numerical solutions for cavity flow, and flow past a disk. Analysis is undertaken to determine bounds on the statistical and numerical error. A number of improvements in deep learning and generalizations to broader classes of PDES are proposed as possible avenues of future research. Software is available at github.com/mskpark/DRLPDE

### Tomas Kojar

Thursday, May 23, 2024

11:00 a.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate: Tomas Kojar

Supervisor: Ilia Binder

Thesis title: Inverse of the Gaussian multiplicative chaos.pdf

Complete copy of Tomas' thesis

In this thesis we do a fundamental study of the Gaussian multiplicative chaos (GMC) on the real line enroute to an interesting problem in the field of Schramm Loewner evolution (SLE) curve. This is a singular measure that is the limit of the integral of an exponentiated Gaussian field with logarithmic covariance. It is strictly monotonic and so it has an inverse that we study its single-point and multi-point moments and correlation structure.

The interesting problem is a coupling between the Gaussian free field and the SLE curve. In the 2010 - work "Conformal weldings of random surfaces: SLE and the quantum gravity zipper" S.Sheffield initiated an approach of coupling those two random objects by constructing the quantum zipper joint process. Here we follow an alternative perspective initiated around the same time in the 2009-work "Random Conformal Weldings" K. Astala, P. Jones, A. Kupiainen, E. Saksman using the Beltrami equation and the Lehto

estimates. In particular, they proved a conformal welding result for the GMC measure on the unit circle. In the thesis, we prove the analogous result for the inverse of the GMC measure on the unit circle.