## Departmental PhD Thesis Exam – Beatriz Navarro Lameda

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

Wednesday, April 8, 2020
11:00 a.m.
BA6183

PhD Candidate:  Beatriz Navarro Lameda
Supervisor:   Kostya Khanin
Thesis title:  On Global Solutions of the Parabolic Anderson Model and Directed Polymers in a Random Environment

***

This thesis studies global solutions to the semidiscrete stochastic heat equation and the associated Cauchy problem known as Parabolic Anderson Model. Via a Feynman-Kac formula, it is linked with the analysis of directed polymers in random environment, and this thesis establishes a number of results for the corresponding partition function.
We consider a continuous-time simple symmetric random walk on the integer lattice $\Z^d$ in dimension $d \geq 3$, subject to a random potential given by two-sided Wiener processes. In the high-temperature regime, we prove the existence of the $L^2$- and almost sure limit of the partition function as time $t \to \pm \infty$. We show that the $L^2$-convergence rate is at least polynomial and that the limiting partition function is positive almost surely. Furthermore, we show that this limiting partition function defines a global stationary solution to the semidiscrete stochastic heat equation which is unique up to a rescaling, and which in some sense attracts solutions to the Parabolic Anderson Model for any subexponentially growing initial data. One of the primary tools in the proof of this uniqueness and attraction result is a factorization formula for the point-to-point partition function, which is related to the ones obtained by Sinai (1995) and Kifer (1997) for other polymer models, but valid not only on the diffusive scale but up to any sub-ballistic scale. This factorization formula allows us to obtain a uniqueness result for physical invariant probability measures of a certain skew product that can be naturally associated with the semidiscrete stochastic heat equation, which in turns gives uniqueness of global stationary solutions.

A copy of the thesis can be found here: Navarro-Lameda_PhDThesis

## Departmental PhD Thesis Exam – Debanjana Kundu

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

Wednesday, April 8, 2020
2:00 p.m. (sharp)
BA6183

PhD Candidate:  Debanjana Kundu
Supervisor:   Kumar Murty
Thesis title:  Iwasawa Theory of Fine Selmer Groups

****

Iwasawa theory began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa as part of the theory of cyclotomic fields.In the early 1970’s, Barry Mazur considered generalizations of Iwasawa theory to Selmer groups of elliptic curves (Abelian varieties in general).  At the turn of this century, Coates and Sujatha initiated the study of a subgroup of the Selmer group of an elliptic curve called the \textit{fine} Selmer group.

The focus of this thesis is to understand arithmetic properties of this subgroup. In particular, we understand the structure of fine Selmer groups and their growth patterns. We investigate a strong analogy between the growth of the $p$-rank of the fine Selmer group and the growth of the $p$-rank of the class groups. This is done in the classical Iwasawa theoretic setting of (multiple) $\ZZ_p$-extensions; but what is more striking is that this analogy can be extended to non-$p$-adic analytic extensions as well, where standard Iwasawa theoretic tools fail.

Coates and Sujatha proposed two conjectures on the structure of the fine Selmer groups. Conjecture A is viewed as a generalization of the classical Iwasawa $\mu=0$ conjecture to the context of the motive associated to an elliptic curve; whereas Conjecture B is in the spirit of generalising Greenberg’s pseudonullity conjecture to elliptic curves. We provide new evidence towards these two conjectures.

A copy of the thesis can be found here: Debanjana_thesis

## Departmental PhD Thesis Exam – Yvon Verberne

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

Monday, March 30, 2020
10:00 a.m.
BA2139

PhD Candidate:  Yvon Verberne
Supervisor:   Kasra Rafi
Thesis title:  Pseudo-Anosov homeomorphisms constructed using positive Dehn twists

****

The mapping class group is the group orientation preserving homeomorphisms of a surface up to isotopy. The mapping class group encodes information about the symmetries of a surface. We focus on studying the pseudo-Anosov mapping classes, which are the elements of the group that mix the underlying surface in a complex way. These maps have applications in physics, notably in fluid dynamics, since we can stir a disk of fluid to create topological chaos, and in the study of magnetic fields since pseudo-Anosov maps create odd magnetic fields. Pseudo-Anosov maps also appear in industrial applications such as food engineering and polymer processing.

We introduce a construction of pseudo-Anosov homeomorphisms on $n$-times punctured spheres and surfaces with higher genus using only sufficiently many positive half-twists. These constructions can produce explicit examples of pseudo-Anosov maps with various number-theoretic properties associated to the stretch factors, including examples where the trace field is not totally real and the Galois conjugates of the stretch factor are on the unit circle.

We construct explicit examples of geodesics in the mapping class group and show that the shadow of a geodesic in mapping class group to the curve graph does not have to be a quasi-geodesic. We also show that the quasi-axis of a pseudo-Anosov element of the mapping class group may not have the strong contractibility property. Specifically, we show that, after choosing a generating set carefully, one can find a pseudo-Anosov homeomorphism $f$, a sequence of points $w_k$ and a sequence of radii $r_k$ so that the ball $B(w_k, r_k)$ is disjoint from a quasi-axis $a$ of $f$, but for any projection map from the mapping class group to $a$, the diameter of the image of $B(w_k, r_k)$ grows like $\log(r_k)$.

A copy of the thesis can be found here: Verberne_Yvon_ML_202006_PhD_thesis

## Departmental PhD Thesis Exam – Travis Ens

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

Wednesday, February 19, 2020
1:00 p.m.
BA1210

PhD Candidate:  Travis Ens
Supervisor:   Dror Bar-Natan
Thesis title:   On Braidors: An Analogue of the Theory of Drinfel’d Associators for Braids in
an Annulus

***

We develop the theory of braidors, an analogue of Drinfel’d’s theory of associators in which braids in an annulus are considered rather than braids in a disk.  After defining braidors and showing they exist, we prove that a braidor is defined by a single equation, an analogue of a well-known theorem of Furusho [Furusho (2010)] in the case of associators. Next some progress towards an analogue of another key theorem, due to Drinfel’d [Drinfel’d (1991)] in the case of associators, is presented. The desired result in the annular case is that braidors can be constructed degree be degree. Integral to these results are annular versions \textbf{GT}$_a$ and \textbf{GRT}$_a$ of the Grothendieck-Teichm\”uller groups \textbf{GT} and \textbf{GRT} which act faithfully and transitively on the space of braidors.

We conclude by providing surprising computational evidence that there is a bijection between the space of braidors and associators and that the annular versions of the Grothendieck-Teichm\”uller groups are in fact isomorphic to the usual versions potentially providing a new and in some ways simpler description of these important groups, although these computations rely on the unproven result to be meaningful.

A copy of the thesis can be found here:  ens_thesis

## Departmental PhD Thesis Exam – Justin Martel

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

Friday, August 9, 2019
2:10 p.m.
BA6183

PhD Candidate:  Justin Martel
Supervisor:   Robert McCann
Thesis title:  Applications of Optimal Transport to Algebraic Topology: A Method for Constructing Spines from Singularity

***

Our thesis describes new applications of optimal transport to algebraic topology. We use a variational definition of singularity based on semicouplings and Kantorovich duality, and develop a method for building Spines (Souls) of manifolds from singularities. For example, given a complete finite-volume manifold X we identify subvarieties Z of X and construct continuous homotopy-reductions from X onto Z using the above variational definition of singularities.

The main goal of the thesis is constructing compact Z with maximal codimension in X. The subvarieties Z are assembled from a contravariant functor arising from Kantorovich duality and solutions to a semicoupling program.

The program seeks semicoupling measures from a source (X,σ) to target (Y, τ) which minimize total transport with respect to a cost c. Best results are obtained with a class of anti-quadratic costs we call “repulsion costs”. We apply the above homotopy-reductions to the problem of constructing explicit small-dimensional EΓ classifying space models, where Γ is a finite-dimensional Bieri-Eckmann duality group.

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

## Departmental PhD Thesis Exam – Li Chen

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

Friday, July 26, 2019
2:10 p.m.
BA6183

PhD Candidate:  Li Chen
Supervisor:   Michael Sigal
Thesis title:  Macroscopic Electrostatics at Positive Temperature from the Density Functional Theory

***

The purpose of this thesis is to study local perturbations of equilibrium crystalline states of the density functional theory (DFT) at positive temperature through the Kohn-Sham equations with local-density approximation (LDA). Under suitable scaling and at low temperature, we prove an existence result for the Kohn-Sham equations and show that local macroscopic perturbations from periodic equilibrium states gives rise to the Poisson equation as an effective equation.

A copy of the thesis can be found here:  LiChen_thesis

## Departmental PhD Thesis Exam – Jia Ji

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

Monday, June 17, 2019
12:10 p.m.
BA6183

PhD Candidate:  Jia Ji
Supervisor:   Lisa Jeffrey
Thesis title:  Volume Formula and Intersection Pairings of N-fold Reduced Products

***

Let $G$ be a semisimple compact connected Lie group. An $N$-fold reduced product of $G$ is the symplectic quotient of the Hamiltonian system of the Cartesian product of $N$ coadjoint orbits of $G$ under diagonal coadjoint action of $G$. Under appropriate assumptions, it is a symplectic orbifold. Using the technique of nonabelian localization and the residue formula of Jeffrey and Kirwan, we investigate the symplectic volume and the intersection pairings of an $N$-fold reduced product of $G$. In 2008, Suzuki and Takakura gave a volume formula of $N$-fold reduced products of $\mathbf{SU}(3)$ via Riemann-Roch.

We compare our volume formula with theirs and prove that up to normalization constant, our volume formula completely matches theirs in the case of triple reduced products of $\mathbf{SU}(3)$.

The draft of the thesis can be found here:  ut-thesis_Ji_draft_v1_1

## Departmental PhD Thesis Exam – Jihad Zerouali

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

Friday, July 5, 2019
11:10 a.m.
BA6183

Supervisor:   Eckhard Meinrenken
Thesis title:  Twisted conjugation, quasi-Hamiltonian geometry, and Duistermaat-Heckman measures

***

Let $G$ be a Lie group, and let $\kappa\in\mathrm{Aut}(G)$. Let  $G\kappa$ denote the group $G$ equipped with the $\kappa$-twisted conjugation action, $\mathrm{Ad}_{g}^{\kappa}(h)=gh\kappa(g^{-1})$. A twisted quasi-Hamiltonian manifold is a triple $(M,\omega,\Phi)$, where $M$ is a $G$-space, the equivariant map $\Phi:M\to G\kappa$ is called the moment map, and $\omega$ is a certain invariant 2-form with properties generalizing those of a symplectic structure.

The first topic of this work is a detailed study of $\kappa$-twisted conjugation, for $G$ compact, connected, simply connected and simple, and for $\kappa$ induced by a Dynkin diagram automorphism of $G$. After recovering the classification of $\kappa$-twisted conjugacy classes by elementary means, we highlight several properties of the so-called \textit{twining characters} $\tilde{\chi}^{(\kappa)}:G\rightarrow\mathbb{C}$.

We show that as elements of $L^{2}(G\kappa)^{G}$, the twining characters generalize several properties of the usual characters in a natural way. We then discuss $\kappa$-twisted representation and fusion rings, in relation to recent work of J. Hong. The second topic of this work is the study of the Duistermaat-Heckman (DH) measure $\mathrm{DH}_{\Phi}\in\mathcal{D}'(G\kappa)^{G}$ of a twisted quasi-Hamiltonian manifold $(M,\omega,\Phi)$. After developing the necessary background, we prove a localization formula for the Fourier coefficients of the measure $\mathrm{DH}_{\Phi}$, and we illustrate the theory with several examples of twisted moduli spaces. These character varieties parametrize a class of local systems on bordered surfaces, for which the transition functions take values in $G\rtimes\mathrm{Aut}(G)$ instead of $G$.

A copy of the thesis can be found here:

Thesis_AJ_Zerouali_Draft_V20190702

## Departmental PhD Thesis Exam – Val Chiche-Lapierre

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

Monday, July 15, 2019
1:10 p.m.
BA6183

PhD Candidate:  Val Chiche-Lapierre
Supervisor:   Jacob Tsimerman
Thesis title:   Length of elements in a Minkowski basis for an order in a number field
(or a ring of integers of a number field)
Exam type:    One-defense

***
Abstract:

Suppose K is a number field of degree n, and R is an order in K with discriminant D. If K has r real embeddings and s pairs of complex embeddings then we can look at R as a lattice in \R^r x \C^s. We call the length of elements of R their Euclidean length in \R^r x \C^s and denote it by |.|. Let v1=1,v2,…,vn be a Minkowski basis for R. We are interested in the asymptotic lengths of these vi’s for a family or orders with arbitrarily large discriminant D. By the theory of Minkowski bases we have that 1\leq |v2| \leq … \leq |vn| and \prod |v_i| \asymp |D|^{1/2} and by \cite{J}, we also know that |v_n| << |D|^{1/n}.

We say a family of orders in number fields have Minkowski type \delta_2,…,\delta_n if the members of the family have arbitrarily large discriminant and each have a Minkowski basis of the form v1=1,v2,…,vn with |vi| \asymp |D|^{\delta_i} for each i, where D is the discriminant.

In the thesis, we are interested in possible Minkowski types. The first question is: Can we find sufficient and necessary bounds on some rational numbers \delta_2,…,\delta_n such that there is a family of orders in number fields having Minkowski type \delta_2,…,\delta_n?

We already know the following necessary conditions: \delta_2 \leq … \leq \delta_n and \delta_2+…+\delta_n=1/2 by Minkowski basis theory, and \delta_n \leq 1/n by \cite{J}. We prove that bounds of the form \delta_k << \delta_i+\delta_j for each i+j=k are sufficient bounds, and if K has no non trivial subfield, we conjecture that these bounds are actually necessary. We can prove this in some cases (of n,i,j,k). In particular, for n=3,4,5,6, we prove that all these bounds are necessary.

The second question is: For some fixed \delta_2,…\delta_n, “how many” orders in number fields have Minkowski type \delta_2,…,\delta_n. We will make sense of what we mean by “how many” using the Delone-Faddeev correspondence (n=3), and the correspondence of Bhargava (n=4,5). Using these correspondences and counting, we are also able to give a sieving argument to count those orders that are maximal (and therefore are ring of integers of number fields).

A copy of the thesis can be found in this link: val_chichelapierre_thesis

## Departmental PhD Thesis Exam – Francis Bischoff

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

Thursday, June 13, 2019
11:10 a.m.
BA6183

PhD Candidate:  Francis Bischoff
Supervisor:   Marco Gualtieri
Thesis title: Morita Equivalence and Generalized Kahler Geometry

***

Generalized Kahler (GK) geometry is a generalization of Kahler geometry, which arises in the study of super-symmetric sigma models in physics. In this thesis, we solve the problem of determining the underlying degrees of freedom for the class of GK structures of symplectic type. This is achieved by giving a reformulation of the geometry whereby it is represented by a pair of holomorphic Poisson structures, a holomorphic symplectic Morita equivalence relating them, and a Lagrangian brane inside of the Morita equivalence.

We apply this reformulation to solve the longstanding problem of representing the metric of a GK structure in terms of a real-valued potential function. This generalizes the situation in Kahler geometry, where the metric can be expressed in terms of the partial derivatives of a function. This result relies on the fact that the metric of a GK structure corresponds to a Lagrangian brane, which can be represented via the method of generating functions. We then apply this result to give new constructions of GK structures, including examples on toric surfaces.

Next, we study the Picard group of a holomorphic Poisson structure, and explore its relationship to GK geometry. We then apply our results to the deformation theory of GK structures, and explain how a GK metric can be deformed by flowing the Lagrangian brane along a Hamiltonian vector field. Finally, we prove a normal form result, which says that locally, a GK structure of symplectic type is determined by a holomorphic Poisson structure and a time-dependent real-valued function, via a Hamiltonian flow construction.

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