Graduate Blog

Wednesday, December 13, 2023
3:00 p.m.

Zoom Web Conference

PhD Candidate: Min Seong Park
Supervisor: Adam Stinchcombe
Thesis title: Temporal Difference Learning for viscous incompressible flow

PhD Defense – Park, M

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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

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The draft of the thesis can be found here: KevinMinSeongParkThesisOct272023

Tuesday, December 12, 2023
1:00 p.m.

Zoom Web Conference

PhD Candidate: Adam Morgan
Supervisor: Fabio Pusateri
Thesis title: Scattering for a Generalized Benjamin-Bona-Mahony Equation

AMorgan_Defence_Poster

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The Benjamin-Bona-Mahony equation (BBM) is a nonlinear dispersive PDE that models surface gravity waves moving through a shallow layer of water. In the long-wave limit, BBM is expected to produce the same physical predictions as the more well-known Korteweg-de Vries equation (KdV). When describing intermediate or short wavelengths, however, KdV and BBM differ markedly. For example, the linearized BBM features finite speed of propagation and a dispersion relation with a nonzero inflection point while the linearized KdV does not. Now, global-in-time behaviour of small solutions of KdV-type equations with a nonlinear term of the form $u^{p}u_{x}$ has been well-understood since the turn of the century: provided $p>2$, small solutions tend to solutions of the linearized PDE as $t\rightarrow \infty$. In the jargon of nonlinear dispersive waves, this asymptotic “forgetting” of the nonlinearity is called \textbf{scattering}. For BBM-type equations with the same nonlinearity, however, scattering has only been established for $p>4$ and progress on the problem has almost ceased since the mid-1990s. In light of the similarities and differences between KdV and BBM, it is natural to ask if one can shrink the scattering exponent for BBM-type equations down to the KdV threshold $p>2$, and in this thesis I discuss some recent progress I have made towards resolving this question. Specifically, I establish linear dispersive decay and scattering for a BBM-type equation with $p=3$. In particular, the presence of a nonzero inflection point in the dispersion relation does not obstruct dispersive decay for this choice of nonlinear term. To deal with the difficulties that appear for $p=3<4$, I apply the method of space-time resonances, which reduces closing bootstrap estimates to bounding multidimensional oscillatory integrals.

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The draft of the thesis can be found here: amorgan_thesis_oct4_2023

Wednesday, September 13, 2023
10:00 a.m.

Zoom Web Conference

PhD Candidate: Hubert Dubé
Supervisor: Kumar Murty
Thesis title: On the Structure of Information Cohomology Exam

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The homological nature of Shannon entropy has been a subject of interest to some for well over fifty years, and yet new approaches are still being suggested studied. Most recently, P. Baudot and D. Bennequin [9], and later J. P. Vigneaux [10] have approached this using topoi whose cohomology of Ext functors contain the entropy function as a 1-cocycle. Moreover, both sets of authors have proven,
under some conditions on their respective constructions, that the entropy in fact generates all of H1. This leaves open questions regarding the general structure (algebraic and categorical) of information structures and the possible higher order cocycles.
This thesis is aimed at extending the theory behind the constructions of J. P. Vigneaux. In particular, we produce new results arising as analogues to results from other cohomology theories, namely the Mayer-Vietoris long exact sequence, to allow for decomposition of information structures into closed information structures, Shapiro’s lemma, to allow for some novel elementary computations, and lastly the Hochschild-Serre spectral sequence, to allow for cohomology computations by means of sub- and quotient structures.
We also provide structural results of algebraic and order-theoretic nature: we provide means to produce useful projective and injective sheaves over information structures, and furthermore prove a general structural result for projective sheaves. This enables easy computations of projective and injective resolutions that provide novel bounds on the cohomological dimension of information structures. Additionally, by taking into account the order-theoretic nature of information structures, we are able to produce an improved bound based on combinatorial invariants.
We finally utilize our results to study the second cohomology groups of some information structures and, in one particular case of interest, find new families of 2-cocycles.
Lastly, we provide with an alternative perspective in which we can view entropy function as a 1-cocycle by investigating it through the lense of operads rather than topoi. We prove that, in this new theory, the Shannon entropy also generates H1.

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The draft of the thesis can be found here: thesis

Monday, August 21, 2023
11:00 a.m.

Zoom Web Conference

PhD Candidate: Pouya Honaryar
Supervisor: Kasra Rafi
Thesis title: Lattice point counts in Teichmüller space and negative curvature

PhD Defense – Honaryar
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In this thesis we present two different, but related, results; one in the setting of
Teichmüller theory, and the other in the setting of negative curvature. For the first
result, let be a pseudo-Anosov homeomorphism of a compact orientable surface
Sg, and let L denote the axis of the action of on the Teichmüller space of Sg,
denoted by Tg. In Chapter 3 we obtain asymptotics for the number of translates
of L that intersect a Teichmüller ball of radius R centered at a fixed X 2 Tg, as
R ! 1. For the second result, let M be a compact closed manifold of variable
negative curvature. We fix two points x; y in the universal cover fM of M, fix an
element id 6= in the fundamental group 􀀀 of M, and denote the set of elements
in 􀀀 that are conjugate to by Conj . In Chapter 4 we obtain asymptotics for the
number of Conj –orbits of y that lie in a ball of radius R centered at x, as R ! 1.
If M is two-dimensional, or of dimension n 3 and curvature bounded above by
􀀀1 and below by 􀀀(n􀀀1
n􀀀2 )2, we find a power saving error term for this count.
Since the two results are written in different settings, their similarities might
be hidden at first glance. This is why we included Chapter 2, in which we present
a unified approach to both results in the setting of constant negative curvature.
Writing the arguments in this simple setting helps us emphasize the underlying
ideas shared by both results.

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The draft of the thesis can be found here: honaryar_thesis_v2

Wednesday, September 6, 2023
10:00 a.m. (sharp)

Zoom Web Conference

PhD Candidate: Yichao Chen
Co-Supervisors: Luis Seco, Sebastian Jaimungal
Thesis title: Principal Agent Mean-Field Problems and Multi-Period Compliance Problems With Applications in REC Markets

PhD Defense – Chen, Y

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Mean field type control problems (MFC) and mean field games (MFGs) are common models to
characterize the asymptotic behavior of a spectrum of interacting agents. One of the successful
application of MFG lies in the study of Renewable Energy Certificate (REC) markets, where there are
interacting firms regulated by a regulator through some payment structures imposed at the end of the
(possibly multiple) compliance periods. The structure of the REC markets motivates us to model it
with principal agent games and multi-period games in a mean-field approach. This thesis investigates a
family of principal agent mean field problems and a family form of multi-period mean filed games with
applications in REC market modelling.

The thesis contains three parts. First we formulate a family of MF-PA problems as an principal’s
optimization problem linked to the terminal conditions of a collection of MV-FBSDE systems. This
fomulation describes the scenario where the principal affect the agents’ equilibria through the agents’
terminal cost. Under suitable assumptions, we proved the well-posedness of the proposed family of
MF-PA problems and we showed the approximation consistency with respect to the principal’s objective
where the MV-FBSDE systems are replaced by its discretized versions. We provide some examples of
PA-MF problems and verify the well-posedness and the approximation consistency. Second, we use
principal agent MFG to model the regulating problem of the REC markets, where the agents form a
Nash equilibria according to the principal’s penalty function, and the principal evaluates the resulting
equilibria. We propose and implement an alternating optimization scheme, based on deep-BSDE
method, to numerically solve the PA-MFG for the REC markets. Our numerical results demonstrate
the efficacy of the algorithm and provide intriguing insights of the REC market regulating modelling
in the mean-field limit. Last but not the least, we introduce a mean field game framework for the
multi-period compliance problem in REC markets. We study a broad family of terminal penalties
generalizing the simple penalty that is proportional to the amount of lacking in the inventory. We argue
for the convexity of the cumulative terminal penalties with respect to the cumulative inventory. Under
suitable regularity condition of the objective function and in both indefinite banking and finite banking
scenarios, we apply variational analysis to the agents’ objective and derive the optimal controls of the
agents together with an explicit equilibrium price that clears the market. We then derive a MV-FBSDE
system that characterizes the equilirbium of the multi-period MFG.

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The draft of the thesis can be found here: Yichao_Chen_Thesis

Wednesday, August 9, 2023
12:00 p.m. (sharp)

Zoom Web Conference

PhD Candidate: Clovis Hamel Ascanio
Supervisor: Frank Tall
Thesis title: New Results in Model Theory and Set Theory

PhD Defense – Hamel
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Traditionally, the role of general topology in model theory has been mainly limited to the study of compacta that arise in first-order logic. In this context, the topology tends to be so trivial that it turns into combinatorics, motivating a widespread approach that focuses on the combinatorial component while usually hiding the topological one. This popular combinatorial approach to
model theory has proved to be so useful that it had become rare to see more advanced topology in model-theoretic articles. Prof. Franklin D. Tall has led the re-introduction of general topology as a valuable tool to push the boundaries of model theory. Most of this thesis is directly influenced by and builds on this idea.

The first part will answer a problem of T. Gowers on the undefinability of pathological Banach spaces such as Tsirelson space. The topological content of this chapter is centred around Grothendieck spaces. In a similar spirit, the second part will show a new connection between the
notion of metastability introduced by T. Tao and the topological concept of pseudocompactness. We shall make use of this connection to show a result of X. Caicedo, E. Due˜nez, J. Iovino in a much simplified manner.

The third part of the thesis will carry a higher set-theoretic content as we shall use forcing and descriptive set theory to show that the well-known theorem of M. Morley on the trichotomy concerning the number of models of a first-order countable theory is undecidable if one considers second-order countable theories instead. The only part that did not originate from model-theoretic questions will be the fourth one. We show that ZF+DC+“all Turing invariant sets of reals have the perfect set property” implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations. This result provides evidence in favour of a long-standing conjecture asking whether Turing Determinacy implies the
Axiom of Determinacy.

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The draft of the thesis can be found here: Thesis – Clovis E. Hamel Ascanio

Zoom Web Conference

PhD Candidate: Kaidi Ye
Supervisor: Lisa Jeffrey
Thesis title: The SO(4) Verlinde Formula Using Real Polarization

PhD Defense – Ye
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We adapt the construction of Jeffrey and Weitsman [5] to interpret the SO(4) Verlinde formula through a real polarization.
To accomplish this objective, our study focuses on the moduli space of flat G connections on a compact, oriented two-manifold of genus g denoted as Σg . We explore this space for various Lie groups G and establish their relationship through a double covering map. By utilizing the Goldman flow, we identify the Hamiltonian flows with a period of 1. These period 1 Hamiltonian flows then allow us to determine the Bohr-Sommerfeld fibres. Through this process, we establish a one-to-one correspondence between the number of Bohr-Sommerfeld fibres and integer labellings of a trivalent graph derived from the pants decomposition of Σg under certain conditions. The count of such labellings provides the precise dimension of the Hilbert space H, which emerges from the geometric quantization of the moduli space.

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The draft of the thesis can be found here: PHD_thesis_draft-2

Thursday, August 10, 2023
1:00 p.m. (sharp)

Zoom Web Conference

PhD Candidate: Vasiliki Liontou
Co-Supervisors: Matilde Marcolli/Boris Khesin
Thesis title: Gabor Frames and Contact Geometry: From models of the primary visual cortex to higher dimensional signal analysis on manifolds

PhD Defense – Liontou
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This thesis has two objectives: first, to provide a model of the functional architecture of the primary visual cortex (V1) in terms of both geometry and signal analysis and second to provide a mathematical framework for signal analysis on certain classes of contact manifolds. It is organized in three main parts.
Firstly, we introduce a model of the primary visual cortex (V1), which allows the compression and decomposition of a signal by a discrete family of orientation and position dependent receptive profiles. We show in particular that a specific framed sampling set and an associated Gabor system is determined by the Legendrian circle bundle structure of the 3-manifold of contact elements on a surface (which models the V1−cortex), together with the presence of an almost complex structure on the tangent bundle of the surface (which models the retinal surface). We identify a maximal area of the signal planes, deter-
mined by the retinal surface, that provides a finite number of receptive profiles, sufficient for good encoding and decoding. We consider the extension of this model for receptive fields dependent on position, orientation, frequency and phase.

Moreover, we provide a construction of Gabor Frames that encode local linearizations of a signal detected on a curved smooth manifold of arbitrary dimension. In particular we use Gabor Filters that can detect higher-dimensional boundaries on the manifolds. We describe an application in configuration spaces in robotics with sharp constrains. The construction is a generalization of the geometric framework, developed for the study of the visual cortex.

Finally, we present a general construction of Gabor analysis on manifolds with coori-entable contact distribution, equipped with a Legendrian fibration and an almost CR-Structure. This construction is suitable for studying the stability of Gabor frames under contact transformations of the manifold. We prove that Gabor frames with a specific class of window functions are stable under a certain class of contact transformations.

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The draft of the thesis can be found here: Thesis_Liontou-2

Thursday, July 27, 2023
1:00 p.m.

Zoom Web Conference

PhD Candidate: Mehmet Durlanik
Supervisor: Arul Shankar
Thesis title: Non-vanishing and 1-level density for Artin L-functions of D4 fields

PhD Defense – Durlanik

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We study families of Artin $L$-functions associated to the 2-dimensional irreducible representations of the Galois group of $D_4$ number fields, ordered by their conductors. We compute the first moment of the central values of these $L$-functions, and as a consequence prove that infinitely many $L$-functions in each family are non-vanishing at $\frac{1}{2}$.

In order to obtain these results, we prove asymptotic formulas with power saving error-terms for these families. As a consequence, we are also able to study the 1-level density of the low-lying zeros of the $L$-functions in these families and verify the Katz-Sarnak conjecture.

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The draft of the thesis can be found here: Thesis – Durlanik

Tuesday, August 15, 2023
2:00 p.m. (sharp)

Zoom Web Conference

PhD Candidate: Mateusz Olechnowicz
Co-Supervisors: Jacov Tsimerman/Patrick Ingram
Thesis title: Preperiodicity in arithmetic dynamics

PhD Defense – Olechnowicz

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Motivated by the Uniform Boundedness Conjecture of Morton and Silverman, we prove various
results related to preperiodic points of algebraic dynamical systems.

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The draft of the thesis can be found here: Olechnowicz_Mateusz_G_202311_PhD_thesis_DRAFT2