## Departmental PhD Thesis Exam – Rosemonde Lareau-Dussault

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

Friday, September 1, 2017
1:10 p.m.
BA6183

PhD Candidate:  Rosemonde Lareau-Dussault
Supervisor:  Robert McCann
Thesis title:

******

Abstract:

A copy of the thesis can be found here:

## 2017 Award Winners

We will celebrate the significant contributions of this year’s award winners during our reception in the Math lounge on Wednesday, June 7 starting at 3:10 p.m.

F. V. Atkinson Teaching Award for Postdoctoral Fellows

• Payman Eskandari (working with Kumar Murty)

Daniel B. DeLury Teaching Assistant Awards

• Andrew Colinet, student of Robert Jerrard
• Ozgur Esentepe, student of Ragnar Buchweitz
• Yvon Verberne, student of Kasra Rafi

Ida Bulat Teaching Awards for Graduate Students

• Ivan Khatchatourian, student of Stevo Todorcevic
• Fabian Parsch, student of Alex Nabutovsky
• Asif Zaman, student of John Friedlander

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

The awards committee received many praiseful comments about our TAs and CIs. Outstanding work is being done by them; we can take pride in their performance.

Congratulations to Payman, Andrew, Ozgur, Yvon, Ivan, Fabian, and Asif!  Graduation-Awards-Invitation

Over the last few years, the generosity of faculty, alumni and friends of the Department have allowed us to create a number of significant scholarships to support graduate students. This year’s winners are listed below.

1) The Inaugural Ida Bulat Memorial Graduate Fellowship:

Mykola Matviichuk (student of Marco Gualtieri)

2) Vivekananda Graduate Scholarship for International students:

Leonid Monin (student of Askold Khovanskii)

Alexander Mangerel (student of John Friedlander)

Evan Miller (student of Robert McCann)

Ivan Telpukhovskiy (student of Kasra Rafi)

6) Margaret Isobel Elliott Graduate Scholarship:

Yvon Verberne (student of Kasra Rafi)

7) Irving Kaplansky Scholarship:

Selim Tawfik (student of Eckhard Meinrenken)

Please join me in congratulating all of them on their work and extending our best wishes for continued success.

Regards,

Kumar

## Departmental PhD Thesis Exam – David Reiss

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

Wednesday, June 28, 2017
3:10 p.m.
BA6183

PhD Candidate:  David Reiss
Co-Supervisors:  Jim Colliander, Catherine Sulem
Thesis title: Global Well-Posedness and Scattering of Besov Data for the Energy-Critical Nonlinear Schr\”{o}dinger Equation

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Abstract:

We examine the Defocusing Energy-Critical Nonlinear Schr\”{o}dinger Equation in dimension 3.  This equation has been studied extensively when the initial data is in the critical homogeneous Sobolev space $\dot{H}^1,$ and a satisfactory theory is given in the work of Colliander, Keel, Sataffilani, Takaoka and Tao.  We extend the analysis of this equation to include infinite energy data $u_0 \in \dot{B}^1_{2,q}$ ($2 \leq q \leq \infty$) that can be decomposed as a finite energy component (a part  in $\dot{H}^1$) and a small Besov part, with the size of the energy part depending on the size of the Besov part.  If $2 \leq q < \infty,$ the solution is shown to scatter.  For $q = \infty$, the solution is shown to be globally well-posed.  Traditionally, the well-posedness theory has been studied in Strichartz spaces, but we use more subtle spaces to deal with the high frequencies that arise from the Besov data, $X^q(I)$.  These spaces are variants of bounded variation spaces and satisfy a duality that allows us to recover the traditional multilinear estimate along with a Strichartz variant that allows for extracting smallness by shrinking the time interval.

We also discuss a conjecture that all data $u_0 \in \dot{B}^1_{2,q}$ for $2 \leq q < \infty$ evolve to a global solution that scatters and we discuss the next steps to proving this.

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

## Departmental PhD Thesis Exam – James Lutley

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

Tuesday, June 13, 2017
3:10 p.m.
BA6183

PhD Candidate:  James Lutley
Supervisor:  Georges Elliott
Thesis title: The Structure of Diagonally Constructed ASH Algebras

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Abstract:

We introduce a class of recursive subhomogeneous algebras which are constructed using a type of diagonal map similar to those previously defined for homogeneous algebras. We call these diagonal subhomogeneous (DSH) algebras.
Using homomorphisms that also exhibit a kind of diagonal structure, we study certain limits of DSH algebras. Our first result is that a simple limit of DSH algebras with diagonal maps has stable rank one.  As an application we show that whenever $X$ is a compact Hausdorff space and $\sigma$ is a minimal homeomorphism thereof, the crossed product algebra $C^*(\mathbb{Z},X,\sigma)$ has stable rank one. We also define mean dimension in the context of these limits. Our second result is that mean dimension zero implies $\mathcal{Z}$-stability for simple separable limits of DSH algebras with diagonal maps. We also show that the tensor product of any two simple separable limit algebras of this kind is $\mathcal{Z}$-stable.

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

## Graduation-Awards Reception * Wednesday, June 7, 2017

The Department of Mathematics cordially invites you to the Graduation-Awards Reception

Wednesday, June 7, 2017

3:10 p.m.

Mathematics Lounge, 6th Floor

We hope you can join us as we celebrate the significant contributions of this year’s award winners and acknowledge the achievements of our graduating students.  Graduation-Awards-Invitation

Light refreshments will be served.

## Departmental PhD Thesis Exam – Jonguk Yang

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

Wednesday, May 17, 2017
11:10 a.m.
BA6183

PhD Candidate:  Jonguk Yang
Supervisor:  Michael Yampolsky
Thesis title:  Applications of Renormalization in Irrationally Indifferent Complex Dynamics

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Abstract:

This thesis comprises of two main results which are proved using renormalization techniques.

For the first result, we show that a quadratic polynomial with a fixed Siegel disc of bounded type rotation number is conformally mateable with the basilica polynomial $f_B(z) := z^2-1$.

For the second result, we study sufficiently dissipative complex quadratic Hénon maps with a semi-Siegel fixed point of inverse golden-mean rotation number. It was recently shown by Gaidashev, Radu and Yampolsky that the Siegel disks of such maps are bounded by topological circles. We investigate the geometric properties of such curves, and demonstrate that they cannot be $C^1$-smooth.

A copy of the thesis can be found here: Jonguk Yang – Thesis Draft

## Departmental PhD Thesis Exam – Kevin Luk

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

Wednesday, June 28, 2017
11:10 a.m.
BA6183

PhD Candidate:  Kevin Luk
Co-Supervisors:  Marco Gualtieri, Lisa Jeffrey
Thesis title:  Logarithmic algebroids and line bundles and gerbes

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Abstract:

In this thesis, we first introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety.  We show that such algebroids are classified by a subspace of the de Rham cohomology of the divisor complement determined by its mixed Hodge structure.  We then solve the prequantization problem, showing that under an integrality condition, a log Picard algebroid is the Lie algebroid of symmetries of what is called a meromorphic line bundle, a generalization of the usual notion of line bundle in which the fibres degenerate in a certain way along the divisor.  We give a geometric description of such bundles and establish a classification theorem for them, showing that they correspond to a subgroup of the holomorphic line bundles on the divisor complement which need not be algebraic. We provide concrete methods for explicitly constructing examples of meromorphic line bundles, such as for a smooth cubic divisor in the projective plane.

We then proceed to introduce and develop the theory of logarithmic Courant algebroids and meromorphic gerbes. We show that under an integrality condition, a log Courant algebroid may be prequantized to a meromorphic gerbe with logarithmic connection. Lastly, we examine the geometry of Deligne and Deligne-Beilinson cohomology groups and demonstrate how this geometry may be exploited to give quantization results of closed holomorphic and logarithmic differential forms.

A copy of the thesis can be found here: ut-thesis-july1

## Departmental PhD Thesis Exam – Tracey Balehowsky

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

Friday, May 19, 2017
11:10 a.m.
BA6183

PhD Candidate:  Tracey Balehowsky
Thesis title:  Recovering a Riemannian Metric from Knowledge of the Areas of Properly-Embedded, Area-Minimizing Surfaces

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Abstract:

In this thesis, we prove that if $(M,g)$ is a $C^3$-smooth, 3-dimensional Riemannian manifold with mean convex boundary $\partial M$, which is additionally either a) $C^2$-close to Euclidean or b) $\epsilon_0$-thin, then knowledge of the least areas circumscribed by a simple closed curve $\gamma \subset \partial M$ and all perturbations $\gamma(t)\subset \partial M$ uniquely determines the metric. In the case where $(M,g)$ only has strictly mean convex boundary at a point $p\in\partial M$, we prove that knowledge of the least areas circumscribed by a simple closed curve $\gamma \subset U$ and all perturbations $\gamma(t)\subset U$ in a neighbourhood $U\subset \partial M$ of $p$ uniquely determines the metric near $p$.

## 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