## 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

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

• 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

********

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:

********

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

## 2016 Course Instructors Teaching Excellence Award Winner

Congratulations to our graduate student, Peter Crooks, winner of the 2016 CI Teaching Excellence Award.

In 2015, the Teaching Assistants’ Training Program’s (TATP) Teaching Excellence Award launched the first-ever TATP award specifically for graduate student Course Instructors.  This is a university-wide award that “recognizes one graduate student whose outstanding work as a sole-responsibility Course Instructor shows evidence of educational leadership, meaningful contributions to course and curriculum development, and impact on student learning.”

Peter is co-supervised by Lisa Jeffrey and John Scherk and is working in the area of Lie Theory and Equivariant Geometry.  Peter is defending his thesis this Wednesday, May 4, 2016.

## 2016 Daniel B. DeLury Teaching Assistant Awards winners

We are happy to announce that this year’s winners of the Daniel B. DeLury Teaching Assistant Awards for graduate students in mathematics are:

• Tracey Balehowsky
• Beatriz Navarro Lameda
• Nikita Nikolaev
• Asif Zaman

The selection committee consisted of Mary Pugh, Abe Igelfeld and Peter Crooks.  Nominations were made by faculty members, course instructors, and undergraduate students.

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 Tracey, Beatriz, Nikita and Asif!

## Ida Bulat Teaching Award for Graduate Students

The Department of Mathematics invites nominations for the inaugural:

Ida Bulat Teaching Award

This prize honors outstanding teaching by graduate students.  It is named in memory of Ida Bulat, former Graduate Administrator in the Department of Mathematics at the University of Toronto.

Nominations can be submitted online through this link, no later than Friday, May 13, 2016:

http://blog.math.toronto.edu/forms/nomination-form-2016/

## 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)

********

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

## Departmental PhD Thesis Exam – Jennifer Vaughan

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

Monday, May 9, 2016
11:10 a.m.
BA6183

PhD Candidate:  Jennifer Vaughan
Co-Supervisors:  Yael Karshon
Thesis title:  Quantomorphisms and Quantized Energy Levels for Metaplectic-c Quantization

*******

Abstract:

Metaplectic-c quantization was developed by Robinson and Rawnsley as an alternative to the classical Kostant-Souriau quantization procedure with half-form correction.  This thesis extends certain properties of Kostant-Souriau quantization to the metaplectic-c context.  We show that the Kostant-Souriau results are replicated or improved upon with metaplectic-c quantization.

We consider two topics:  quantomorphisms and quantized energy levels.  If a symplectic manifold admits a Kostant-Souriau prequantization circle bundle, then its Poisson algebra is realized as the space of infinitesimal quantomorphisms of that circle bundle.  We present a definition for a metaplectic-c quantomorphism, and prove that the space of infinitesimal metaplectic-c quantomorphisms exhibits all of the same properties that are seen in the Kostant-Souriau case.

Next, given a metaplectic-c prequantized symplectic manifold $(M,\omega)$ and a function $H\in C^\infty(M)$, we propose a condition under which $E$, a regular value of $H$, is a quantized energy level for the system $(M,\omega,H)$.  We prove that our definition is dynamically invariant:  if two functions on $M$ share a regular level set, then the quantization condition over that level set is identical for both functions.  We calculate the quantized energy levels for the $n$-dimensional harmonic oscillator and the hydrogen atom, and obtain the quantum mechanical predictions in both cases.  Lastly, we generalize the quantization condition to a level set of a family of Poisson-commuting functions, and show that in the special case of a completely integrable system, it reduces to a Bohr-Sommerfeld condition.

The draft to the thesis can be found here: Vaughan-Draft