Departmental PhD Thesis Exam – Nikita Nikolaev

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

Monday, November 13, 2017
3:10 p.m.
BA6183

PhD Candidate:  Nikita Nikolaev
Supervisor:  Marco Gualtieri
Thesis title:  Abelianisation of Logarithmic Connections

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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

Departmental PhD Thesis Exam – Rosemonde Lareau-Dussault

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

BA6183

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

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Abstract:

 

A copy of the thesis can be found here:

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

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
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 $p$.

 

Departmental PhD Thesis Exam – Benjamin Schachter

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

Departmental PhD Thesis Exam – Yiannis Loizides

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

Departmental PhD Thesis Exam – Jack Klys

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