## 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

*****

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

****
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

International Graduate Student Scholarship

• Chia-Cheng Liu, student of Joel Kamnitzer and Alexander Braverman

Margaret Isobel Elliott Graduate Scholarship

• Shuangjian Zhang, student of Robert McCann

We thank the faculty members, undergraduate students and course instructors who took the time to submit their nominations for the various awards.

Congratulations once again to André, Anton, Tracey, Beatriz, Nikita, Asif, Payman, Tyler, Peter, Huan, Zhifei, Anne, Chia-Cheng, and Shuangjian!

## 2016 F. V. ATKINSON TEACHING AWARD

We are extremely happy to announce this year’s winners of the Frederick V. Atkinson Teaching Awards for Post Doctoral Fellows:

• André Belotto da Silva
• Anton Izosimov

The prize honors outstanding teaching by post-doctoral fellows and other junior research faculty not on the tenure track.

The awards recipients will receive a monetary award and a certificate during our awards/graduation reception scheduled for Thursday, May 26 at 3:10 p.m. in the Math lounge.

Congratulations to André and Anton!

## 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

********

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