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:

A copy of the thesis can be found here:

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_18-01-2017

Funeral service for Yuri Cher

Monday, October 24, 2016

York Cemetery and Funeral Centre
160 Beecroft Rd, North York
ON M2N 5Z5

Visitation: 11:00 a.m.-12:00 p.m.

Service: 12:00 – 1:00 p.m.

Burial: 1:00 p.m.

MSc supervised project presentation – Yuguang Bai

Everyone is welcome to attend.

Thursday, September 22, 2016
4:10 p.m.
BA6180

MSc Candidate:  Yuguang Bai
Supervisor:  Pierre Milman
Project title: Illustrative Proof of Hirzebruch-Riemann-Roch Theorem for Algebraic Curves

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Abstract:

I will be going over something I did for my Master’s project this past summer. Namely, the idea for the proof of the Hirzebruch-Riemann-Roch Theorem for Algebraic Curves, which shows how the topological genus is equal to an algebraic invariant, called the arithmetic genus.

The talk will not be rigorous and should be accessible for new graduate students. Knowledge of undergraduate topology and algebra recommended.

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

2016 award winners and graduate scholarships recipients

Congratulations to the 2016 awards winners and graduate scholarships recipients.

F. V. Atkinson Teaching Award for Postdoctoral Fellows

• André Belotto da Silva (working with Ed Bierstone)
• Anton Izosimov (working with Boris Khesin)

Daniel B. DeLury Teaching Assistant Award

• Tracey Balehowsky, student of Spyros Alexakis and Adrian Nachman
• Beatriz Navarro Lameda, student of Kostya Khanin
• Nikita Nikolaev, student of Marco Gualtieri
• Asif Zaman, student of John Friedlander

Inaugural Ida Bulat Teaching Award for Graduate Students

• Payman Eskandari, student of Kumar Murty
• Tyler Holden, student of Lisa Jeffrey
• Beatriz Navarro Lameda, student of Kostya Khanin

CI Teaching Excellence Award

• Peter Crooks, student of Lisa Jeffrey and John Scherk

Vivekananda Graduate Scholarship for international students

• Huan Vo, student of Dror Bar-Natan

• Zhifei Zhu, student of Rina Rotman

• Anne Dranovski, student Joel Kamnitzer