Monday, November 13, 2017

3:10 p.m.

BA6183

PhD Candidate: Nikita Nikolaev

Supervisor: Marco Gualtieri

Thesis title: Abelianisation of Logarithmic Connections

******

Abstract:

This thesis studies an equivalence between meromorphic connections of higher rank and abelian connections. Given a complex curve $X$ and a spectral cover $\pi : \Sigma \to X$, we construct a functor $\pi^\textup{ab} : \mathsf{Conn}_X \to \mathsf{Conn}_\Sigma$, called the \textit{abelianisation functor}, from some category of connections on $X$ with logarithmic singularities to some category of abelian connections on $\Sigma$, and we prove that $\pi^\textup{ab}$ is an equivalence of categories. At the level of the corresponding moduli spaces $\mathbb{M}_X, \mathbb{M}_\Sigma$, which are known to be holomorphic symplectic varieties, this equivalence recovers a symplectomorphism constructed by Gaiotto, Moore, Neitzke in their work on Spectral Networks (2013). Moreover, the moduli space $\mathbb{M}_\Sigma$ is a torsor for an algebraic torus, so in fact $\pi^\textup{ab}$ provides a Darboux coordinate system on $\mathbb{M}_X$, known as the \textit{Fock-Goncharov coordinates} constructed in their work on higher Teichm\”uller theory (2006). To prove that $\pi^\textup{ab}$ is an equivalence of categories, we introduce a new concept called the \textit{Voros class}. It is a canonical cohomology class in $H^1$ of the base $X$ with values in the nonabelian sheaf $\mathcal{Aut} (\pi_\ast)$ of groups of natural automorphisms of the direct image functor $\pi_\ast$. Any $1$-cocycle $v$ representing the Voros class defines a new functor $\mathsf{Conn}_\Sigma \to \mathsf{Conn}_X$ by locally deforming the pushforward functor $\pi_\ast$; the result is an explicit inverse equivalence to $\pi^\textup{ab}$, called a \textit{deabelianisation functor}.

We generalise the abelianisation equivalence to the case of \textit{quantum connections}: these are $\hbar$-families of meromorphic connections restricted to a sectorial neighbourhood in $\hbar$ with prescribed asymptotic regularity. The Schr\”odinger equation is a quintessential example. The most important invariant of a quantum connection $\nabla$ is the Higgs field $\nabla^{\tiny(0)}$ obtained by restricting $\nabla$ to $\hbar = 0$ (the so-called \textit{semiclassical limit}). Then abelianisation may be viewed as a natural extension to an $\hbar$-family of the spectral line bundle of $\nabla^{\tiny(0)}$. That is, we show that for a given quantum connection $(\mathcal{E}, \nabla)$, the line bundle $\mathcal{E}^\textup{ab}$ obtained from $\mathcal{E}$ by abelianisation $\pi^\textup{ab}$ restricts at $\hbar = 0$ to precisely the spectral line bundle of the Higgs field $\nabla^{\tiny(0)}$.

Finally, in this thesis we explore the relationship between abelianisation and the WKB method, which is an asymptotic approximation technique for solving differential equations developed by physicists in the 1920s and reformulated by Voros in 1983 using the theory of Borel resummation. We give an algebro-geometric formulation of the WKB method using vector bundle extensions and splittings. We then show that the output of the WKB analysis is precisely the data used to construct the abelianisation functor $\pi^\textup{ab}$.

A copy of the thesis can be found here: https://www.dropbox.com/s/7u89i9j27y5ivuo/PhDThesis.pdf?dl=0

]]>Wednesday, November 29, 2017

3:10 p.m.

BA6183

PhD Candidate: Rosemonde Lareau-Dussault

Supervisor: Robert McCann

Thesis title: Coupled Education and Labour Market Models

******

Abstract:

This study addresses the dynamics of the distribution of skills in a population over many generations. Two overlapping generation models are proposed: the first assumes complete information, which allows (and requires) all generations to be solved simultaneously, while the second assumes incomplete information, forcing the competitive equilibrium at each subsequent generation to be found iteratively. Both models combine a labour and an education matching problem. The skill distribution for each generation of adults is determined from that of the previous generation by the educational matching market.

We present conditions for the sequence of adult skills to converge. Then we study the asymptotic which is specific to each model. For the incomplete information model, we prove that, if the sequence of wage functions over the generation converges, the limiting steady state solves the steady state model of Erlinger, McCann, Shi, Siow, and Wolthoff (2015), which allows us to get an explicit formulation for solutions of the model of Erlinger et al. To study the limiting solution of the complete information model, we introduce a new steady state model, which includes a discounting factor to reduce the impact of future generations relative to how far in the future they are.

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

]]>**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

]]>**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)

**3) Canadian Mathematical Society Graduate Scholarship**:

Alexander Mangerel (student of John Friedlander)

**4) Coxeter Graduate Scholarship**:

Evan Miller (student of Robert McCann)

**5) International Graduate Student Scholarship**:

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

]]>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

******

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

]]>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

******

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

]]>*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.

]]>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

******

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

]]>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

******

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

]]>Friday, May 19, 2017

11:10 a.m.

BA6183

PhD Candidate: Tracey Balehowsky

Co-Supervisors: Spyros Alexakis, Adrian Nachman

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

******

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 $p$.

]]>