## Departmental PhD Thesis Exam – Jeremy Lane

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Friday, February 10, 2017
10:10 a.m.
BA1210

PhD Candidate:  Jeremy Lane
Supervisor:  Yael Karshon
Thesis title:  On the topology of collective integrable systems

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Abstract:

This thesis studies the topological properties of momentum maps of a large family of completely integrable systems called collective completely integrable systems.

The first result concerns the topological monodromy of a collective completely integrable system on a product of three two-spheres. This system is called a “Heisenberg spin-chain” for its connection with a quantum integrable system of the same name.

The remainder of the thesis is concerned with collective systems that are “constructed by Thimm’s trick” with the “action coordinates of Guillemin and Sternberg.”  We observe that the open dense subset of a symplectic manifold where such systems define a Hamiltonian torus action is connected. This observation was absent from the literature until this point. We also prove that if a system is constructed in this manner from a chain of Lie subalgebras then the image is given by an explicit set of inequalities called branching inequalities.

When the symplectic manifold in question is a multiplicity free Hamiltonian $U(n)$-manifold, the construction of Thimm’s trick with the action coordinates of Guillemin and Sternberg yields a completely integrable Hamiltonian torus action on a connected open dense subset. If the momentum map is proper, then we are able to prove lower bounds for the Gromov width of the symplectic manifold  from the classification of the connected open dense subset as a non-compact symplectic toric manifold. Convex multiplicity free manifolds of compact Lie groups have been classified by (Knop, 2010) and (Losev, 2009) and, accordingly, our lower bounds are given in terms of the combinatorics of the classifying data: the momentum set and a lattice. This result is the first estimate for the Gromov width of general multiplicity free manifolds of a nonabelian group.

This result  relies crucially on connectedness of the open dense subset and the explicit description of the momentum map image. The proof is a generalization of methods used to prove lower bounds for the Gromov width of $U(n+1)$ coadjoint orbits (Pabiniak, 2014)  which are an example of multiplicity free $U(n)$-manifolds.

A copy of the thesis can be found here: ut-thesis-revised-March-9

## Departmental PhD Thesis Exam – Asif Zaman

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, February 1, 2017
3:10 p.m.
BA6183

PhD Candidate:  Asif Zaman
Supervisor:  John Friedlander
Thesis title:  Analytic estimates for the Chebotarev Density Theorem and their applications

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Abstract:

In this thesis, we study the distribution of prime ideals within the Chebotarev Density Theorem. The theorem states that the Artin symbols attached to prime ideals are equidistributed within the Galois group of a given Galois extension.

We exhibit field-uniform unconditional bounds with explicit constants for the least prime ideal in the Chebotarev Density Theorem, that is, the prime ideal of least norm with a specified Artin symbol. Moreover, we provide a new upper bound for the number of prime ideals with a specified Artin symbol which is valid for a wide range and sharp, short of precluding a putative Siegel zero. To achieve these results, we establish explicit statistical information on the zeros of Hecke L-functions and the Dedekind zeta function. Our methods were inspired by works of Linnik, Heath-Brown, and Maynard in the classical case and the papers of Lagarias–Odlyzko, Lagarias–Montgomery–Odlyzko, and Weiss in the Chebotarev setting.

We include applications for primes represented by certain binary quadratic forms, congruences of coefficients for modular forms, and the group structure of elliptic curves reduced modulo a prime. In particular, we establish the best known unconditional upper bounds for the least prime represented by a positive definite primitive binary quadratic form and for the Lang–Trotter conjectures on elliptic curves.

A copy of the thesis can be found here: thesis_Zaman_v1

## Departmental PhD Thesis Exam – Jerrod Smith

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, February 1, 2017
2:10 p.m.
BA6183

PhD Candidate:  Jerrod Smith
Supervisor:  Fiona Murnaghan
Thesis title:  Construction of relative discrete series representations for $p$-adic $\mathbf{GL}_n$

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Abstract:

Let $F$ be a nonarchimedean local field of characteristic zero and odd residual characteristic.  Let $G$ be the $F$-points of a connected reductive group defined over $F$ and let $\theta$ be an $F$-rational involution of $G$.  Define $H$ to be the closed subgroup of $\theta$-fixed points in $G$.
The quotient variety $H \backslash G$ is a $p$-adic symmetric space.  A fundamental problem in the harmonic analysis on $H \backslash G$ is to understand the irreducible subrepresentations of the right-regular representation of $G$ acting on the space $L^2(H\backslash G)$ of complex-valued square integrable functions on $H\backslash G$.  The irreducible subrepresentations of $L^2(H\backslash G)$ are called relative discrete series representations.

In this thesis, we give an explicit construction of relative discrete series representations for three $p$-adic symmetric spaces, all of which are quotients of the general linear group.  We consider $\mathbf{GL}_n(F) \times \mathbf{GL}_n(F) \backslash \mathbf{GL}_{2n}(F)$, $\mathbf{GL}_n(F) \backslash \mathbf{GL}_n(E)$, and $\mathbf {U}_{E/F} \backslash \mathbf{GL}_{2n}(E)$, where $E$ is a quadratic Galois extension of $F$ and $\mathbf {U}_{E/F}$ is a quasi-split unitary group.  All of the representations that we construct are parabolically induced from $\theta$-stable parabolic subgroups admitting a certain type of Levi subgroup.  In particular, we give a sufficient condition for the relative discrete series representations that we construct to be non-relatively supercuspidal.  Finally, in an appendix, we describe all of the relative discrete series of $\mathbf{GL}_{n-1}(F)\times \mathbf{GL}_{1}(F) \backslash \mathbf{GL}_{n}(F)$.

A copy of the thesis can be found here: thesis_jmsmith-27-01-2017

## Departmental PhD Thesis Exam – Louis-Philippe Thibault

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Thursday, September 8, 2016
11:10 a.m.
BA6183

PhD Candidate:  Louis-Philippe Thibault
Supervisor:  Ragnar Buchweitz
Thesis title: Tensor product of preprojective algebras and preprojective structure on skew-group algebras

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Abstract:

We investigate properties of finite subgroups $G<SL(n, k)$ for which the skew-group algebra $k[x_1,\ldots, x_n]\#G$ does not have a grading structure of (higher) preprojective algebra. Namely, we prove that if a finite subgroup $G<SL(n, k)$ is conjugate to a finite subgroup of $SL(n_1, k)\times SL(n_2, k)$, for some $n_1, n_2\geq 1$, $n_1+n_2 =n$, then the skew-group algebra $R\#G$ is not Morita equivalent to a (higher) preprojective algebra. Motivated by this question, we study preprojective algebras over Koszul algebras. We give a quiver construction for the preprojective algebra over a basic Koszul $n$-representation-infinite algebra. Moreover, we show that such algebras are derivation-quotient algebras whose relations are given by a superpotential. The main problem is also related to the preprojective algebra structure on the tensor product $\Pi:=\Pi_1\otimes_k \Pi_2$ of two Koszul preprojective algebras. We prove that a superpotential in $\Pi$ is given by the shuffle product of  superpotentials in $\Pi_1$ and $\Pi_2$. Finally, we prove that if $\Pi$ has a grading structure such that it is $n$-Calabi-Yau of Gorentstein parameter $1$, then its degree $0$ component is the tensor product of a Calabi-Yau algebra and a higher representation-infinite algebra. This implies that it is infinite-dimensional, which means in particular that $\Pi$ is not a preprojective algebra.

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

## Departmental PhD Thesis Exam – Ivan Livinskyi

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, August 3, 2016
2:10 p.m.
BA6183

PhD Candidate:  Ivan Livinskyi
Supervisor:  Steve Kudla
Thesis title:  On the integrals of the Kudla-Millson theta series

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Abstract:

The Kudla-Millson theta series $\theta_{km}$ of a pseudoeuclidean space $V$ of signature $(p,q)$ and lattice $L$ is a differential form on the symmetric space $\mathcal D$ attached to the pseudoorthogonal group $\mathrm{O}(p,q)$ that transforms like a genus $n$ Siegel modular form of weight $(p+q)/2$. Any integral of $\theta_{km}$ inherits the modular transformation law and becomes a nonholomorphic Siegel modular form. A special case of such integral is the well-known Zagier Eisenstein series $\mathcal{F}(\tau)$ of weight $3/2$ as showed by Funke.

We show that for $n=1$ and $p=1$ the integral of $\theta_{km}$ along a geodesic path coincides with the Zwegers theta function $\widehat{\Theta}_{a,b}$. We construct a higher-dimensional generalization of Zwegers theta functions as integrals of $\theta_{km}$ over geodesic simplices for $n\geq 2$.

If $\Gamma$ is a discrete group of isometries of $V$ that preserve the lattice $L$ and act trivially on the cosets $L^\ast/L$, then the fundamental region $\Gamma\backslash \mathcal D$ is an arithmetic locally symmetric space. We prove that the integral of $\theta_{km}$ over $\Gamma\backslash \mathcal D$ converges and compute it in some cases. In particular, we extend the results of Kudla to the cases $p=1$, and $q$ odd.

A copy of the thesis can be found here: Livinsky_Thesis

## Departmental PhD Thesis Exam – Jeremy Voltz

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, July 20, 2016
11:10 a.m.
BA6183

PhD Candidate:  Jeremy Voltz
Supervisor:  Kostya Khanin
Thesis title:  Two results on Asymptotic Behaviour of Random Walks in Random Environment

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Abstract:

In the first chapter of this thesis, we consider a model of directed polymer in $1+1$ dimensions in a product-type random environment $\omega(t,x) = b(t) F(x)$,  where the  fields $F$ and $b$ are i.i.d., with $F(x)$ continuous, symmetric and bounded, and $b(t) = \pm 1$ with probability $1/2$.  Thus $\omega$ can be viewed as the field $F$ oscillating in time.  We consider directed last-passage percolation through this random field; namely, we investigate the behavior of the length $n$ polymer path with maximal action, where the action of a path is simply the sum of the environment variables it moves through.

We prove a law of large numbers for the maximal action of the path from the origin to a fixed endpoint $(n, \lfloor \alpha n \rfloor)$, and investigate the limiting shape function $a(\alpha)$.  We prove that this shape function is non-linear, and has a corner at $\alpha = 0$, thus indicating that this model does not belong to the KPZ universality class.  We conjecture that this shape function has a linear piece near $\alpha = 0$.

With probability tending to $1$, the maximizing path with free endpoint will localize on an edge with $F$ values far from each other.  Under an assumption on the arrival time to this localization site, we prove that the path endpoint and the centered action of the path, both rescaled by $n^{-2/3}$, converge jointly to a universal law, given by the maximizer and value of a functional on a Poisson point process.

In the second chapter, we consider a class of multidimensional random walks in random environment, where the environment is of the type $p_0 + \gamma \xi$, with $p_0$ a deterministic, homogeneous environment with underlying drift, and $\xi$ an i.i.d. random perturbation.   Such environments were considered by Sabot in \cite{Sabot2004}, who finds a third-order expansion in the perturbation for the non-null velocity (which is guaranteed to exist by Sznitman and Zerner’s LLN \cite{Sznitman1999}).  We prove that this velocity is an analytic function of the perturbation, by applying perturbation theory techniques to the Markov operator for a certain Markov chain in the space of environments.

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

## Departmental PhD Thesis Exam – Andrew Stewart

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Tuesday, June 28, 2016
3:10 p.m.
BA6183

PhD Candidate:  Andrew Stewart
Co-Supervisors:  Balint Virag
Thesis title:  On the scaling limit of the range of a random walk bridge on regular trees

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Abstract:

Consider a nearest neighbour random walk $X_n$ on the $d$-regular tree ${\Bbb T}_d$, where $d\geq 2$, conditioned on $X_n = X_0$. This is known as the random walk bridge. We derive Gaussian-like tail bounds for the return probabilities of the random walk bridge on the scale of $n^{1/2}$. This contrasts with the case of the unconditioned random walk, where Gaussian-like tail bounds exists on the scale of $n$.

We introduce the notion of the infinite bridge, which is known to arise as the distributional limit of the random walk bridge. We also establish some preliminary facts about the infinite bridge.

By showing that the Brownian Continuum Random Tree (BCRT) is characterized by its random self-similarity property, we prove that the range of the random walk bridge converges in distribution to the BCRT when rescaled by $Cn^{-1/2}$ for an appropriate constant $C$

A draft of the thesis can be found here: andrew-stewart-thesis

## Departmental PhD Thesis Exam – Michal Kotowski

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Tuesday, June 28, 2016
1:10 p.m.
BA6183

PhD Candidate:  Michal Kotowski
Co-Supervisors:  Balint Virag
Thesis title:  Return probabilities on groups and large deviations for permuton processes

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Abstract:

The topic of this thesis are random processes on finite and infinite groups. More specifically, we are concerned with random walks on finitely generated amenable groups and stochastic processes which arise as limits of trajectories of the interchange process on a line.

In the first part of the thesis we construct a new class of finitely generated groups, called bubble groups. Analysis of the random walk on such groups shows that they are non-Liouville, but have return probability exponents close to $1/2$. Such behavior was previously unknown for random walks on groups. Our construction is based on permutational wreath products over tree-like Schreier graphs and the analysis of large deviations of inverted orbits on such graphs.

In the second part of we analyze large deviations of the interchange process on a line, which can be thought of as a random walk in the group of all permutations, with adjacent transpositions as generators. This is done in the setting of random permuton processes, which provide a notion of a limit for a permutation-valued stochastic processes. More specifically, we provide bounds on the probability that the trajectory of the interchange process (as a permuton process) is close in distribution to a deterministic permuton process. As an application, we show that short paths joining the identity and the reverse permutation in the Cayley graph of $\mathcal{S}_{n}$ are typically close to the so-called sine curve process, which is the conjectured limit of random sorting networks. The analysis is done in the framework of interacting particle systems.

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

## Departmental PhD Thesis Exam – Marcin Kotowski

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Tuesday, June 28, 2016
11:10 a.m.
BA6183

PhD Candidate:  Marcin Kotowski
Co-Supervisors:  Balint Virag
Thesis title:  Random Schroedinger operators with connections to spectral properties of groups and directed polymers

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Abstract:

This thesis studies random Schroedinger operators with connections to group theory and models from statistical physics. First, we study 1D operators obtained as perturbations of the standard adjacency operator on $\Bbb Z$ by putting random i.i.d. noise with finite logarithmic variance on the edges. We study their expected spectral measures $\mu_H$ near zero. We prove that the measure exhibits a spike of the form $\mu_H(-\varepsilon,\varepsilon) \sim \frac{C}{\sim{\log\varepsilon}^2}$, which was first observed by Dyson for a specific choice of the edge weight distribution. We prove the result in generality, without assuming any regularity of edge weights.

We also identify the limiting local eigenvalue distribution, obtained by counting crossings of the Brownian motion derived from the operator. The limiting distribution is different from Poisson and the usual random matrix statistics. The results also hold in the setting where the edge weights are not independent, but are sufficiently ergodic, e.g. exhibit mixing. In conjunction with group theoretic tools, we then use the result to compute Novikov-Shubin invariants, which are group invariants related to the spectral measure, for various groups, including lamplighter groups and lattices in the Lie group Sol.

Second, we study similar operators in the two dimensional setting. We construct a random Schroedinger operator on a subset of the hexagonal lattice and study its smallest eigenvalues. Using a combinatorial mapping, we relate these eigenvalues to the partition function of the directed polymer model on the square lattice. For a specific choice of the edge weight distribution, we obtain a model known as the log-Gamma polymer, which is integrable. Recent results about the fluctuations of free energy for the log-Gamma polymer allow us to prove Tracy-Widom type fluctuations for the smallest eigenvalue of the original random Schroedinger operator.

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

## Departmental PhD Thesis Exam – Trefor Bazett

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, May 11, 2016
11:10 a.m.
BA6183

PhD Candidate:  Trefor Bazett
Co-Supervisors:  Lisa Jeffrey/Paul Selick
Thesis title: The equivariant K-theory of commuting 2-tuples in SU(2)

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Abstract:

In this thesis, we study the space of commuting n-tuples in SU(2), $Hom(\mathbb{Z}^n, SU(2))$. We describe this space geometrically via providing an explicit G-CW complex structure, an equivariant analog of familiar CW- complexes. For the n=2 case, this geometric description allows us to compute various cohomology theories of this space, in particular the G-equivariant K-Theory $K_G^*(Hom(\mathbb{Z}^2, SU(2)))$, both as an $R(SU(2))$-module and as an $R(SU(2))$-algebra. This space is of particular interest as $\phi^{-1}(e)$ in a quasi-Hamiltonian system $M\xrightarrow{\phi} G$ consisting of the G-space $SU(2)\times SU(2)$, together with a moment map $\phi$ given by the commutator map. Finite dimensional quasi-Hamiltonian spaces have a bijective correspondence with certain infinite dimensional Hamiltonian spaces, and we additionally compute relevant components of this larger picture in addition to $\phi^{-1}(e)=Hom(\mathbb{Z}^2, SU(2))$ for this example.

A copy of the thesis can be found here: TreforBazettThesis