## Departmental PhD Thesis Exam – Alexander Mangerel

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

Wednesday, February 14, 2018
2:10 p.m.
BA6183

PhD Candidate:  Alexander Mangerel
Supervisor:  John Friedlander
Thesis title: Topics in Multiplicative and Probabilistic Number Theory

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Abstract:

A heuristic in analytic number theory stipulates that sets of positive integers cannot simultaneously be additively and multiplicatively structured. The practical verification of this heuristic is the source of a great number of difficult problems. An example of this is the well-known Hardy-Littlewood tuples conjecture, which asserts that, infinitely often, one should be able to find additive patterns of fairly general shape in the primes. Conjectures of this type are also at least morally equivalent to the expectation that a multiplicative function, unless it has a special form, behaves randomly on additively structured sets.

In this thesis, we consider several problems involving the behaviour of multiplicative functions interacting with additively structured sets. Two main topics are studied: i) the estimation of \emph{mean values} of multiplicative functions, i.e., the limiting average behaviour of partial sums of multiplicative functions along an interval whose length tends to infinity; and ii) the estimation of \emph{correlations} of multiplicative functions, i.e., the behaviour of simultaneous values of multiplicative functions at arguments that are additively related. A number of applications of the study of these topics are also addressed.

First, we prove quantitative versions of mean value theorems due to Wirsing and Hal\'{a}sz for multiplicative functions that often take values outside of the unit disc. This has a broad realm of applications. In particular we are able to extend a further theorem of Hal\'{a}sz, proving local limit theorems for vectors of certain types of additive functions. We thus confirm a probabilistic heuristic in the \emph{small deviation} regime and beyond for the functions in question.

In a different direction, we consider the collection of periodic, completely multiplicative functions, also known as Dirichlet characters. Upper bounds for the maximum size of the partial sums of these functions on intervals of positive integers is connected with the class number problem in algebraic number theory, and with I.M. Vinogradov’s conjecture on the distribution of quadratic non-residues. By refining a quantitative mean value theorem for multiplicative functions, we significantly improve the existing upper bounds on the maximum size of partial sums of odd order Dirichlet characters, both unconditionally and assuming the Generalized Riemann Hypothesis. We also show that our conditional results are best possible unconditionally, up to a bounded power of $\log\log\log\log q$.

Regarding correlations, we prove a quantitative version of the bivariate Erd\H{o}s-Kac theorem. That is, we show that the joint distribution of pairs of values of certain additive functions is asymptotically an uncorrelated bivariate Gaussian, and find a quantitative error term in this approximation. We use this probabilistic result to prove a theorem on the joint distribution of certain natural variants of the M\”{o}bius function at additively-related integers as a partial result in the direction of Chowla’s conjecture on two-point correlations of the M\”{o}bius function. We also apply our result to understanding the set of pairs of consecutive integers with the same number of divisors.

A major theme in the thesis relates to how a multiplicative function can be rigidly characterized globally by certain local properties. As a first example, we show that a completely multiplicative function that only takes finitely many values, vanishes at only finitely many primes and whose partial sums are uniformly bounded, must be a non-principal Dirichlet character. This solves a 60-year-old open problem of N.G. Chudakov. We also solve a folklore conjecture due to Elliott, Ruzsa and others on the gaps between consecutive values of a unimodular completely multiplicative function, showing that these gaps cannot be uniformly large. This is a corollary of several stronger results that are proved regarding the distribution of consecutive values of multiplicative functions. For instance, we classify the set of all unimodular completely multiplicative functions $f$ such that $\{f(n)\}_n$ is dense in $\mb{T}$ and for which the sequence of pairs $(f(n),f(n+1))$ is dense in $\mb{T}^2$. In so doing, we resolve a conjecture of K\'{a}tai.

Finally, we make some progress on some natural variants of Chowla’s conjecture on sign patterns of the Liouville function. In particular, we prove that certain natural collections of multiplicative functions $f: \mb{N} \ra \{-1,+1\}$ are such that the tuples of values they produce on \emph{almost all} 3- and 4-term arithmetic progressions equidistribute among all sign patterns of length 3 and 4, respectively. Some of the aforementioned results are joint work with O. Klurman, or with Y. Lamzouri.

A copy of the thesis can be found here: APMangerPhDThesisFeb13

## Departmental PhD Thesis Exam – Ali Feizmohammadi

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

Monday, January 29, 2018
2:10 p.m.
BA6183

Thesis title: Unique Reconstruction of a Potential from the Dirichlet to Neumann Map in Locally CTA Geometries

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Abstract:

Let $(M^3,g)$ be a compact smooth Riemannian manifold with smooth boundary and suppose that $U$ is an open set in $M$ such that $g|_U$ is Euclidean. Let $\Gamma= U \cap \partial M$ be connected and suppose that $U$ is the convex hull of $\Gamma$. We will study the uniqueness of an unknown potential for the Schr\”{o}dinger operator $-\triangle_g + q$ from the associated Dirichlet to Neumann map, $\Lambda_q$. Indeed, we will prove that if the potential $q$ is a priori explicitly known in $U^c$ then one can uniquely reconstruct $q$ from $\Lambda_q$. We will also give a reconstruction algorithm for the potential. More generally we will also discuss the cases where $\Gamma$ is not connected or $g|_U$ is conformally transversally anisotropic and derive the analogous result.

A copy of the thesis can be found here:  Ali Feiz’s Dissertation

## 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.

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

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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

## 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
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