## Panel discussion: What can you do with a PhD in math anyway?

A career panel for graduate students in mathematics

When: April 20, 2018
Where: BA6183
Time: 3:00-4:00 p.m.

Panelists

Alex Bloemendal:
Alex  is a computational scientist at the Broad Institute of MIT and Harvard and at the Analytic and Translational Genetics Unit of Massachusetts General Hospital. As a member of Broad institute member Ben Neale’s lab, Alex leads a group in developing new methods to analyze genetic data, harnessing its unprecedented scope and scale to discover the genetic causes of disease. He also co-founded and directs the Models, Inference & Algorithms initiative at the Broad, bridging computational biology, mathematical theory, and machine learning. Alex is an institute scientist at the Broad.  Alex was previously a research scientist in the Program for Evolutionary Dynamics and a Simons Fellow in the Department of Mathematics at Harvard University. His research in probability theory and random matrices focused on questions of signal and noise in high-dimensional data; he proved an open conjecture with wide-reaching applications for fields including population genetics. He also earned a teaching award for an advanced course on probability.  Alex received an Hon. B.Sc., M.Sc., and Ph.D. in mathematics from the University of Toronto

Aaron Chow:
Aaron is a Senior Information Security Consultant – Security Engineering at CIBC. He graduated from our doctoral program in 2014.

Dorian Goldman:
Dorian develops mathematical models using modern methods in machine learning and statistics for Conde Nast. He’s also an Adjunct Professor of data science at Columbia University, where he’s teaching a course on using data science in industry which has received overwhelmingly positive reviews. He completed his MSc degree in mathematics at UofT and his PhDs at the Courant Institute (NYU) and UPMC (Paris VI) and worked full time as a research-only fellow and instructor of mathematics at DPMMS at the University of Cambridge. He worked in Germany, France, England and the USA over the past several years while completing his degrees and gained considerable experience in variational methods, differential equations and applied analysis. He transitioned into data science and machine learning three years ago, and became very passionate about the mathematical sophistication and significant impact that the field has.

Diana Ojeda:
Diana got her PhD in set theory at Cornell University and was a postdoc at U of T from 2014 to 2017.  She now works as a SoC Engineer at Intel, developing modelling and analysis tools for FPGAs.

Ben Schachter:
Benjamin Schachter is a Consultant at the Boston Consulting Group, based in the Toronto office. He joined BCG full time in January 2018, after previously working at BCG as a summer Consultant in 2016. Ben has primarily worked in the technology, media, and telecommunications (TMT) practice area.  Ben completed his PhD in mathematics at the University of Toronto in 2017; his research focused on optimal transport and the calculus of variations.  Ben also holds an MSc in mathematics from the University of Western Ontario and an MA and BA (hons.), both in economics, from the University of Toronto.

## Yuri Cher – Posthumous Degree Award Ceremony

The Department of Mathematics will be holding a posthumous degree award ceremony for

Yuri Cher

on Tuesday, March 28, 2017
at
10:30 a.m.

in

BA6183
40 St. George St.

~
Everyone is welcome to attend.  Refreshments will be served in the Graduate Lounge.

Yuri Cher memorial invitation

## Memorial Service for Yuri Cher

The Department of Mathematics will hold a Memorial Service for our late graduate student, Yuri Cher, on Tuesday, March 28, 2017 at 10:30 a.m. in the Math lounge.

A remembrance book will be available in the main office the week of March 20.

## 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

*****

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

********

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

****

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