*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

Exam PhD
*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

Exam PhD
*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

Exam PhD
*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-june19

Exam PhD
*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

Co-Supervisors: Spyros Alexakis, Adrian Nachman

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 $p$.

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

Friday, March 24, 2017

4:10 p.m.

BA6183

PhD Candidate: Benjamin Schachter

Supervisor: Almut Burchard

Thesis title: An Eulerian Approach to Optimal Transport with Applications to the Otto Calculus

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Abstract:

This thesis studies the optimal transport problem with costs induced by Tonelli Lagrangians. The main result is an extension of the Otto calculus to higher order functionals, approached via the Eulerian formulation of the optimal transport problem. Open problems 15.11 and 15.12 from Villani’s Optimal Transport: Old and New are resolved. A new class of displacement convex functionals is discovered that includes, as a special case, the functionals considered by Carrillo-Slepčev. Improved and simplified proofs of the relationships between the various formulations of the optimal transport problem, first seen in Bernard-Buffoni and Fathi-Figalli, are given. Progress is made towards developing a rigourous Otto calculus via the DiPerna-Lions theory of renormalized solutions. As well, progress is made towards understanding general Lagrangian analogues of various Riemannian structures.

A copy of the thesis can be found here: DraftThesisSchachter

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

Tuesday, April 25, 2017

2:10 p.m.

BA6183

PhD Candidate: Yiannis Loizides

Supervisor: Eckhard Meinrenken

Thesis title: Norm-square localization for Hamiltonian LG-spaces

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Abstract:

In this thesis we prove norm-square localization formulas for two invariants of Hamiltonian loop group spaces: twisted Duistermaat-Heckman measures and a K-theoretic `quantization’. The terms in the formulas are indexed by the components of the critical set of the norm-square of the moment map. These results are analogous to results proved by Paradan in the case of Hamiltonian G-spaces. An important application of the norm-square localization formula is to prove that the multiplicity of the fundamental level k representation in the quantization is a quasi-polynomial function of k. This is closely related to the [Q,R]=0 theorem of Alekseev-Meinrenken-Woodward for Hamiltonian loop group spaces.

A copy of the thesis can be found here: YLThesis

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

Thursday, February 23, 2017

2:10 p.m.

BA6183

PhD Candidate: Jack Klys

Supervisor: Jacob Tsimerman

Thesis title: Statistics of class groups and related topics

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Abstract:

We prove several results concerning class groups of number fields and function fields.

Firstly we compute all the moments of the $p$-torsion in the first step of a filtration of the class group defined by Gerth \cite{gerthprank} for cyclic number fields of degree $p$, unconditionally for $p=3$ and under GRH in general. We show that it satisfies a distribution which Gerth conjectured as an extension of the Cohen-Lenstra-Martinet conjectures. In the $p=3$ case this gives the distribution of the $3$-torsion of the class group modulo the Galois invariant part. We follow the strategy used by Fouvry and Kl$\ddot{\text{u}}$ners in their proof of the distribution of the $4$-torsion in quadratic fields \cite{fk1}.

Secondly, we compute all the moments of a normalization of the function which counts unramified $H_{8}$-extensions of quadratic number fields, where $H_{8}$ is the quaternion group of order $8$, and show that the values of this function determine a constant distribution. Furthermore we propose a similar modification to the non-abelian Cohen-Lenstra heuristics for unramified $G$-extensions of quadratic fields for several other 2-groups $G$, which we conjecture will give finite moments which determine a distribution. These are all cases in which the unnormalized average is known or conjectured to be infinite. Our method additionally can be used to determine the asymptotics of the unnormalized counting function, which we also do for unramified $H_{8}$-extensions. This part of the thesis is joint work with Brandon Alberts.

Thirdly we present several new examples of reflection principles which apply to both class groups of number fields and picard groups of of curves over $\mathbb{P}^{1}/\mathbb{F}_{p}$. This proves a conjecture of Lemmermeyer \cite{franz1} about equality of 2-rank in subfields of $A_{4}$, up to a constant not depending on the discriminant in the number field case, and exactly in the function field case. More generally we prove similar relations for subfields of a Galois extension with group $G$ for the cases when $G$ is $S_{3}$, $S_{4}$, $A_{4}$, $D_{2l}$ and $\mathbb{Z}/l\mathbb{Z}\rtimes\mathbb{Z}/r\mathbb{Z}$. The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.

A copy of the thesis can be found here: klysthesis

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

Tuesday, April 25th, 2017

11:10 a.m.

BA6183

PhD Candidate: James Mracek

Supervisor: Lisa Jeffrey/ Clifton Cunningham

Thesis title: Applications of algebraic microlocal analysis in symplectic geometry and representation theory

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Abstract:

This thesis investigates applications of microlocal geometry in both representation theory and symplectic geometry. Accordingly, there are two bodies of work contained herein.

The first part of this thesis investigates a conjectural geometrization of local Arthur packets. These packets of representations of a $p$-adic group were invented by Arthur for the purpose of classifying the automorphic discrete spectrum of special orthogonal and symplectic groups. While their existence has been established, an explicit construction of Arthur packets remains difficult. In the case of real groups, Adams, Barbasch, and Vogan showed how one can use a geometrization of the local Langlands correspondence to construct packets of equivariant $D$-modules that satisfy similar endoscopic transfer properties as the ones defining Arthur packets. We classify the contents of these “microlocal” packets in the analogue of these varieties for $p$-adic groups, under certain restrictions, for a plethora of split classical groups.

The goal of the second part of this thesis is to find a way to make sense of the Duistermaat-Heckman function for a Hamiltonian action of a compact torus on an infinite dimensional symplectic manifold. We show that the Duistermaat-Heckman theorem can be understood in the language of hyperfunction theory, then apply this generalization to study the Hamiltonian $T\times S^1$ action on $\Omega SU(2)$. The essential reason for introducing hyperfunction theory is that the local contribution to the Duistermaat-Heckman polynomial near the image of a fixed point is a Green’s function for an infinite order differential equation. Since infinite order differential operators do not act on Schwarz distributions, we are forced to use this more general theory.

A copy of the thesis can be found here: ut-thesis (002)

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

Friday, February 10, 2017

10:10 a.m.

BA1210

PhD Candidate: Jeremy Lane

Supervisor: Yael Karshon

Thesis title: On the topology of collective integrable systems

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Abstract:

This thesis studies the topological properties of momentum maps of a large family of completely integrable systems called collective completely integrable systems.

The first result concerns the topological monodromy of a collective completely integrable system on a product of three two-spheres. This system is called a “Heisenberg spin-chain” for its connection with a quantum integrable system of the same name.

The remainder of the thesis is concerned with collective systems that are “constructed by Thimm’s trick” with the “action coordinates of Guillemin and Sternberg.” We observe that the open dense subset of a symplectic manifold where such systems define a Hamiltonian torus action is connected. This observation was absent from the literature until this point. We also prove that if a system is constructed in this manner from a chain of Lie subalgebras then the image is given by an explicit set of inequalities called branching inequalities.

When the symplectic manifold in question is a multiplicity free Hamiltonian $U(n)$-manifold, the construction of Thimm’s trick with the action coordinates of Guillemin and Sternberg yields a completely integrable Hamiltonian torus action on a connected open dense subset. If the momentum map is proper, then we are able to prove lower bounds for the Gromov width of the symplectic manifold from the classification of the connected open dense subset as a non-compact symplectic toric manifold. Convex multiplicity free manifolds of compact Lie groups have been classified by (Knop, 2010) and (Losev, 2009) and, accordingly, our lower bounds are given in terms of the combinatorics of the classifying data: the momentum set and a lattice. This result is the first estimate for the Gromov width of general multiplicity free manifolds of a nonabelian group.

This result relies crucially on connectedness of the open dense subset and the explicit description of the momentum map image. The proof is a generalization of methods used to prove lower bounds for the Gromov width of $U(n+1)$ coadjoint orbits (Pabiniak, 2014) which are an example of multiplicity free $U(n)$-manifolds.

A copy of the thesis can be found here: ut-thesis-revised-March-9

Exam PhD