Departmental PhD Thesis Exam – Artane Siad

Monday, July 12, 2021
1:00 p.m. (sharp)

PhD Candidate:  Artane Siad
Supervisor:   Arul Shankar
Thesis title: Monogenic Fields with Odd Class Number

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We prove an upper bound on the average number of 2-torsion elements in the class group monogenised fields of any degree $n \ge 3$, and, conditional on a widely expected tail estimate, compute this average exactly. As an application, we show that there are infinitely many number fields with odd class number in any even degree and signature. This completes a line of results on class number parity going back to Gauss.

A copy of the thesis can be found here: thesis v3

Departmental PhD Thesis Exam – Jamal Kawach

Thursday, June 24, 2021
2:00 p.m. (sharp)

PhD Candidate:  Jamal Kawach
Supervisor:   Stevo Todorcevic
Thesis title: Approximate Ramsey Methods in Functional Analysis

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We study various aspects of approximate Ramsey theory and its interactions with functional analysis. In particular, we consider approximate versions of the structural Ramsey property and the amalgamation property within the context of multi-seminormed spaces, Fréchet spaces and other related structures from functional analysis. Along the way, we develop the theory of Fraïssé limits of classes of finitedimensional Fréchet spaces, and we prove a version of the Kechris-Pestov-Todorčević correspondence relating the approximate Ramsey property to the topological dynamics of the isometry groups of certain
infinite-dimensional Fréchet spaces. Motivated by problems regarding the structural Ramsey theory of Banach spaces, we study various generalizations of the Dual Ramsey Theorem of Carlson and Simpson.
Specifically, using techniques from the theory of topological Ramsey spaces we obtain versions of the Dual Ramsey Theorem where ω is replaced by an arbitrary countable ordinal. Moving toward block Ramsey theory, we prove an infinite-dimensional version of Gowers’ approximate Ramsey theorem concerning the oscillation stability of S(c0), the unit sphere of the Banach space c0. We then show that results of this form can be parametrized by products of infinitely many perfect sets of reals, and we use this result to
obtain a parametrized version of Gowers’ c0 theorem.

A copy of the thesis can be found here: Kawach ut-thesis

Departmental PhD Thesis Exam – Seong Hyun Park

Monday, April 12, 2021
1:00 p.m. (sharp)

PhD Candidate:  Seong Hyun Park
Supervisor:   Jérémie Lefebvre
Thesis title: Adaptive myelination and its synchronous dynamics in the Kuramoto network model with state-
dependent delays

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White matter pathways form a complex network of myelinated axons that play a critical role in brain function by facilitating the timely transmission of neural signals. Recent evidence reveals that white matter networks are adaptive and that myelin undergoes continuous reformation through behaviour and learning during
both developmental stages and adulthood in the mammalian lifecycle. Consequently, this allows axonal conduction delays to adjust in order to regulate the timing of neuron signals propagating between different brain regions. Despite its newly founded relevance, the network distribution of conduction delays have yet
to be widely incorporated in computational models, as the delays are typically assumed to be either constant or ignored altogether. From its clear influence towards temporal dynamics, we are interested in how adaptive myelination affects oscillatory synchrony in the brain. We introduce a plasticity rule into the delays
of a weakly coupled oscillator network, whose positions relative to its natural limit cycle oscillations is described through a coupled phase model. From this, the addition of slowly adaptive network delays can potentially lead coupled oscillators to a more phase synchronous limit cycle. To see how adaptive white matter remodelling can shape synchronous dynamics, we modify the canonical Kuramoto model by enabling all connections with phase-dependent delays that change over time. We directly compare the synchronous behaviours of the Kuramoto
model equipped with static delays and adaptive delays by analyzing the synchronized equilibria and stability of the system’s phases. Our mathematical analysis of the model with dirac and exponentially distributed connection delays, supported by numerical simulations, demonstrates that larger, more widely varying distributions of delays generally impede synchronization in the Kuramoto network. Adaptive delays act as a stabilizing mechanism for the synchrony of the network by adjusting towards a more optimal distribution of delays. Adaptive delays also make global synchronization more resilient to perturbations and injury
towards network architecture. Our results provide insights about the potential significance of activity-dependent myelination. For future works, we hope that these results lay out the groundwork to computationally study the influence of adaptative myelination towards large-scale brain synchrony.

A copy of the thesis can be found here: SHP_Dissertation

Departmental PhD Thesis Exam – Larissa Richards

Wednesday, March 24, 2021
11:00 a.m. (sharp)

PhD Candidate:  Larissa Richards
Supervisor:   Ilia Binder
Thesis title: Convergence rates of random discrete model curves approaching SLE curves in the scaling limit

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Recently, A. Kempannien and S. Smirnov provided a framework for showing convergence of discrete
model interfaces to the corresponding SLE curves. They show that given a uniform bound on specific
crossing probabilities one can deduce that the interface has subsequential scaling limits that can be
described almost surely by Löwner evolutions. This leads to the natural question to investigate the
rate of convergence to the corresponding SLE curves. F. Johansson Viklund has developed a framework
for obtaining a power-law convergence rate to an SLE curve from a power-law convergence rate for the
driving function provided some additional geometric information along with an estimate on the growth
of the derivative of the SLE map. This framework is applied to the case of the loop-erased random
walk. In this thesis, we show that if your interface satisfies the uniform annulus condition proposed by
Kempannien and Smirnov then one can deduce the geometric information required to apply Viklund’s
framework. As an application, we apply the framework to the critical percolation interface. The first
step in this direction for critical percolation was done by I. Binder, L. Chayes and H.K. Lei where they
proved that the convergence rate of the Cardy-Smirnov observable is polynomial in the size of the lattice.
It relies on a careful analysis of the boundary behaviour of conformal maps and their discrete analytic
approximations as well as a Percolation construction of the Harris systems. Further, we exploit the
toolbox developed by D. Chelkak for discrete complex analysis on isoradial graphs to show polynomial
rate of convergence for the discrete martingale observables for harmonic explorer and the FK Ising
model to the corresponding continuum objects. Then, we apply the framework developed above to gain
a polynomial convergence rate for the corresponding curves.

A copy of the thesis can be found here:  LarissaRichardsThesis

Departmental PhD Thesis Exam – Mihai Alboiu

Wednesday, March 24, 2021
10:00 a.m. (sharp)

PhD Candidate:  Mihai Alboiu
Supervisor:   George Elliott
Thesis title: The Stable Rank of Diagonal Ash Algebras

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Building on the work of Lutley, we study a certain subclass of recursive subhomogeneous algebras, called DSH algebras, in which the pullback maps are all diagonal in a suitable sense. We examine inductive limits of DSH algebras, where each bonding map is itself diagonal in an appropriate way, and show that every simple algebra thus obtained has stable rank one. We are therefore able to show that every simple dynamical crossed product has stable rank one and that the Toms-Winter Conjecture holds for such algebras.

We also introduce the class of non-unital DSH algebras and make partial progress towards showing that inductive limits of such algebras with diagonal maps have stable rank one. Moreover, we investigate more intrinsic notions of a diagonal map and matrix unit compatibility and show that in the full matrix algebra setting they agree with their usual (global) counterparts.

A copy of the thesis can be found here:  ALBOIU_Thesis_Draft_Updated

Departmental PhD Thesis Exam – Selim Tawfik

Monday, January 11, 2021
1:00 p.m.

PhD Candidate:  Selim Tawfik
Supervisor:   Eckhard Meinrenken
Thesis title:  Fusion Product of D/G-Valued Moment Maps

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A fusion product is defined for Hamiltonian spaces with moment maps valued in a Lie group $D$ generalizing those of Alekseev-Malkin-Meinrenken. An analogous theory for these general Hamiltonian spaces is developed and, among other results, versions of symplectic reduction, duality and the shifting trick are derived. The Hamiltonian spaces with moment maps valued in a homogeneous space $D/G$ of Alekseev-Kosmann-Schwarzbach are shown to be equivalent to certain Hamiltonian spaces with group-valued moment maps. The aforementioned theory is consequently brought to bear on that of $D/G$-valued moment maps, thereby defining a fusion product for these. This fusion product affords many new examples of $D/G$-valued moment maps, of which there was hitherto a paucity. Among said examples are moduli spaces of flat principal bundles over certain surfaces with boundary.

A copy of the thesis can be found here: main

Departmental PhD Thesis Exam – Arthur Mehta

Friday, January 22, 2021
4:00 p.m.

PhD Candidate:  Arthur Mehta
Supervisor:   Henry Yuen
Thesis title:  Entanglement and non-locality in games and graphs

***

This thesis is primarily based on two collaborative works written by the author and several coauthors. These works are presented in Chapters 4 and 5 and are on the topics of quantum graphs, and self-testing via non-local games, respectively.

Quantum graph theory, also known as non-commutative graph theory, is an operator space generalization of graph theory. The independence number, and Lova’sz theta function were generalized to this setting by Duan, Severini, and Winter and two different version of the chromatic number were introduced by Stahlke and Paulsen. In Chapter 4, we introduce two new generalizations of the chromatic number to non-commutative graphs and provide an upper bound on the parameter of Stahlke. We provide a generalization of the graph complement and show the chromatic number of the orthogonal complement of a non-commutative graph is bounded below by its theta number. We also provide a generalization of both Sabidussi’s Theorem and Hedetniemi’s conjecture to non-commutative graphs.

The study of non-local games considers scenarios in which separated players collaborate to provide satisfying responses to questions given by a referee. The condition of separating players makes non-local games an excellent setting to gain insight into quantum phenomena such as entanglement and non-locality. Non-local games can also provide protocols known as self-tests. Self-testing allows an experimenter to interact classically with a black box quantum system and certify that a specific entangled state was present, and a specific set of measurements were performed. The most studied self-test is the CHSH game which certifies the presence of a single EPR entangled state and the use of anti-commuting Pauli measurements. In Chapter 5, we introduce an algebraic generalization of CHSH and obtain a self-test for non-Pauli operators resolving an open question posed by Coladangelo and Stark (QIP 2017). Our games also provide a self-test for states other than the maximally entangled state, and hence resolves the open question posed by Cleve and Mittal (ICALP 2012).

The results of Chapter 5 make use of sums of squares techniques in the settings of group rings and *-algebras. In Chapter 3, we review these techniques and discuss how they relate to the study of non-local games. We also provide a weak sum of squares property for the ring of integers. We show that if a Hermitian element is positive under all unitary representations then it must be expressible as a sum of Hermitian squares.

A copy of the thesis can be found here:  Thesis_Version_3 (1)-1

Departmental PhD Thesis Exam – Adam Gardner

Monday, November 30, 2020
5:00 p.m.

PhD Candidate:  Adam Gardner
Supervisor:   Michael Sigal
Thesis title: Instability of electroweak homogeneous vacua in strong magnetic fields

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We consider the classical (local) vacua of the Weinberg-Salam (WS) model of electroweak forces. These are defined as no-particle, static solutions to the WS equations minimizing the WS energy locally. In the absence of particles, the Weinberg-Salam model reduces to the Yang-Mills-Higgs (YMH) equations for the gauge group U(2).

We consider the WS system in a constant external magnetic field, b, and prove that (i) there is a magnetic field threshold b* such that for b<b*, the vacua are translationally invariant, while, for b>b*, they are not, (ii) for b>b*, there are non-translationally invariant solutions with lower energy per unit volume and with the discrete translational symmetry of a 2D lattice in the plane transversal to the external magnetic field, and (iii) the lattice minimizing the energy per unit volume approaches the hexagonal one as the magnetic field strength approaches the threshold b*.

A copy of the thesis can be found here: Adam-Gardner-Thesis

Departmental PhD Thesis Exam – Nathan Carruth

Friday, November 13, 2020
11:00 a.m.

PhD Candidate:  Nathan Carruth
Supervisor:   Spyros Alexakis
Thesis title: Focussed Solutions to the Einstein Vacuum Equations

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We construct solutions to the Einstein vacuum equations in polarised translational symmetry in $3 + 1$ dimensions which have $H^1$ energy concentrated in an arbitrarily small region around a two-dimensional null plane and large $H^2$ initial data. Specifically, there is a parameter $k$ and coordinates $s$, $x$, $v$, $y$ such that the null plane is given by $x = k^{-1/2}/2$, $v = T\sqrt{2} – k^{-1}/2$ for some $T$ independent of $k$, the $H^1$ energy of the solution is concentrated on the region $[0, T’] \times [0, k^{-1/2}] \times [T\sqrt{2} – k^{-1}, T\sqrt{2}] \times \R^1$, and the $H^2$ norm of the initial data is bounded below by a multiple of $k^{3/4}$. The time $T’$ has a lower bound independent of $k$.

This result relies heavily on a new existence theorem for the Einstein vacuum equations with characteristic initial data which is large in $H^2$. This result is proved using parabolically scaled coordinates in a null geodesic gauge.

A copy of the thesis can be found here: thesis_comm

Departmental PhD Thesis Exam – Justin Ko

Wednesday, August 25, 2020
11:00 a.m.

PhD Candidate:  Justin Ko
Supervisor:   Dmitry Panchenko
Thesis title: The Free Energy of Spherical Vector Spin Glasses

We study a class of vector spin models with configurations restricted to subsets of the sphere. We will prove a constrained free energy formula for these models. This formula defines a large deviations principle for the limiting distribution of the overlaps under the asymptotic Gibbs measure. The thesis builds on the mathematical results used to prove free energy formulas for the classical Sherrington–Kirkpatrick spin glass, spherical spin models, and vector spin glass models. The free energy formula proved in this thesis are true generalizations of the classical results, in the sense that these vector spin formulas restricted to one dimension coincide with the known results for classical models.

The first contribution of this thesis is a variational formula for contrained copies of classical spherical spin glasses sampled at different temperatures. The free energy for multiple systems of spherical spin glasses with constrained overlaps was first studied by Panchenko and Talagrand. They proved an upper bound of the constrained free energy using Guerra’s interpolation. In this thesis, we prove this upper bound is sharp. Our approach combines the ideas of the Aizenman–Sims–Starr scheme and the synchronization mechanism used in the vector spin models. We derive a vector version of the Aizenman–Sims–Starr scheme for spherical spin glass and use the synchronization property of arrays obeying the overlap-matrix form of the Ghirlanda–Guerra identities to prove the matching lower bound.

The second contribution of this thesis is the simplification of this variational formula to the form originally discovered for the classical spherical spin glass model by Crisanti and Sommers. In particular, we prove the analogue of the Crisanti–Sommers variational formula for spherical spin glasses with vector spins. This formula is derived from the discrete Parisi variational formula for the limit of the free energy of constrained copies of spherical spin glasses. In vector spin models, the variations of the functional order parameters must preserve the monotonicity of matrix paths which introduces a new challenge in contrast to the derivation of the classical Crisanti–Sommers formula.

A copy of the thesis can be found here: ut-thesis-Ko-updated