Departmental PhD Thesis Exam – John Enns

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

Monday, May 28, 2018
11:10 a.m.
BA6183

PhD Candidate:  John Enns
Supervisor:   Florian Herzig
Thesis title: On mod p local-global compability for unramified GL3

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Abstract:

Let $K$ be a $p$-adic field. Given a continuous Galois representation $\bar{\rho}: \mathrm{Gal}(\overline{K}/K)\rightarrow \mathrm{GL}_n(\overline{\mathbb{F}}_p)$, the mod $p$ Langlands program hopes to associate with it a smooth admissible $\overline{\mathbb{F}}_p$-representations $\Pi_p(\bar{\rho})$ of $\mathrm{GL}_n(K)$ in a natural way.  When $\bar{\rho}=\bar{r}|_{G_{F_w}}$ is the local $w$-part of a global automorphic Galois representation $\bar{r}:\mathrm{Gal}(\overline{F}/F)\rightarrow \mathrm{GL}_n(\overline{\mathbb{F}}_p)$, for some CM field $F/F^+$ and place $w|p$, it is possible to construct a candidate $H^0(\bar{r})$ for $\Pi_p(\bar{r}|_{G_{F_w}})$ using spaces of mod $p$ automorphic forms on definite unitary groups.

Assume that $F_w$ is unramified. When $\bar{r}|_{G_{F_w}}$ is semisimple, it is possible to recover the data of $\bar{r}|_{G_{F_w}}$ from the $\mathrm{GL}_n(\mathcal{O}_{F_w})$-socle of $H^0(\bar{r})$ (also known as the set of Serre weights of $\bar{r}$). But when $\bar{r}|_{G_{F_w}}$ is wildly ramified this socle does not contain enough information. In this thesis we give an explicit recipe to find the missing data of $\bar{r}|_{G_{F_w}}$ inside the $\mathrm{GL}_3(F_w)$-action on $H^0(\bar{r})$ when $n=3$ and $\bar{r}|_{G_{F_w}}$ is maximally nonsplit, Fontaine-Laffaille, and generic.  This generalizes work of Herzig, Le and Morra who found analogous results when $F_w=\mathbb{Q}_p$ as well as work of Breuil and Diamond in the case of unramified $\mathrm{GL}_2$.

Departmental PhD Thesis Exam – Zackary Wolske

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

Monday, June 18, 2018
2:10 p.m.
BA6183

PhD Candidate:  Zackary Wolske
Supervisor:   Henry Kim
Thesis title:   Number Fields with Large Minimal Index

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Abstract:

The index of an integral element alpha in a number field K with discriminant D_K is the index of the subring Z[alpha] in the ting of integers O_K. The minimal index m(K) is taken over all alpha in O_K that generate the field. This thesis proves results of the form m(K) << |D_K|^U for all Galois quartic fields and composites of totally real Galois fields with imaginary quadratic fields, and of the form m(K) >> |D_K|^L for infinitely many pure cubic fields, both types of Galois quartic fields, and the same composite fields, with U and L depending only on the type of field. The upper bounds are given by explicit elements and depend on finding a factorization of the index form, while the lower bounds are established via effective Diophantine approximation, minima of binary quadratic forms, or norm inequalities. The upper bounds improve upon known results, while the lower bounds are entirely new. In the case of imaginary biquadratic quartic fields and the composite fields under consideration, the upper and lower bounds match.

A copy of the thesis can be found here:  ZWolskePhDThesisJune14

Departmental PhD Thesis Exam – Shuangjian Zhang

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

Tuesday, May 15, 2018
11:10 a.m.
BA6183

PhD Candidate:  Shuangjian Zhang
Supervisor:   Robert McCann
Thesis title: Existence, Uniqueness, concavity and geometry of the monopolist’s problem facing consumers with nonlinear price preferences

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Abstract:

A monopolist wishes to maximize her profits by finding an optimal price menu. After she announces a menu of products and prices, each agent will choose to buy that product which maximizes his own utility, if positive.  The principal’s profits are the sum of the net earnings produced by each product sold. These are determined by the costs of production and the distribution of products sold, which in turn are based on the distribution of anonymous agents and the choices they make in response to the principal’s price menu.
In this thesis, two existence results will be provided, assuming each agent’s disutility is a strictly increasing but not necessarily affine (i.e.\ quasilinear) function of the price paid. This has been an open problem for several decades before the first multi-dimensional result given by N\”oldeke and Samuelson in 2015.
Additionally, a necessary and sufficient condition for the convexity or concavity of this principal’s (bilevel) optimization problem is investigated.  Concavity when present, makes the problem more amenable to computational and theoretical analysis;  it is key to obtaining uniqueness and stability results for the principal’s strategy in particular.  Even in the quasilinear case, our analysis goes beyond previous work by addressing convexity as well as concavity,  by establishing conditions which are not only sufficient but necessary,  and by requiring fewer hypotheses on the agents’ preferences. Moreover, the analytic and geometric interpretation of certain condition that equivalent to concavity of the problem has been explored.
Finally, various examples has been given, to explain the interaction between preferences of agents’ utility and monopolist’s profit to concavity of the problem. In particular, an example with quasilinear preferences on $n$-dimensional hyperbolic spaces was given with explicit solutions to show uniqueness without concavity. Besides, similar results on spherical and Euclidean spaces are also provided. What is more, the solutions of hyperbolic and spherical converges to those of Euclidean space as curvature goes to 0.

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

Departmental PhD Thesis Exam – Benjamin Briggs

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

Friday, May 18, 2018
2:10 p.m.
BA6183

PhD Candidate:  Benjamin Briggs
Co-Supervisors:   Joel Kamnitzer, Srikanth Iyengar
Thesis title:  Local Commutative Algebra and Hochschild Cohomology Through the
Lens of Koszul Duality

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Abstract:

This thesis splits into two halves, the connecting theme being Koszul duality. The first part concerns local commutative algebra. Koszul duality here manifests in the homotopy Lie algebra. In the second part, which is joint work with Vincent G\’elinas, we study Hochschild cohomology and its characteristic action on the derived category.

We begin by defining the homotopy Lie algebra $\pi^*(\phi)$ of a local homomorphism $\phi$ (or of a ring) in terms of minimal models, slightly generalising a classical theorem of Avramov. Then, starting with work of F\'{e}lix and Halperin, we introduce a notion of Lusternik-Schnirelmann category for local homomorphisms (and rings). In fact, to $\phi$ we associate a sequence $\cat_{0}(\phi)\geq \cat_1(\phi)\geq \cat_2(\phi)\geq \cdots$ each $\cat_i(\phi)$ being either a natural number or infinity. We prove that these numbers characterise weakly regular, complete intersection, and (generalised) Golod homomorphisms. We present examples which demonstrate how they can uncover interesting information about a homomorphism. We give methods for computing these numbers, and in particular prove a positive characteristic version of F\'{e}lix and Halperin’s Mapping Theorem.

A motivating interest in L.S. category is that finiteness of $\cat_2(\phi)$ implies the existence of certain six-term exact sequences of homotopy Lie algebras, following classical work of Avramov. We introduce a variation $\pic(\phi)$ of the homotopy Lie algebra which enjoys long exact sequences in all situations, and construct a comparison $\pic(\phi)\to \pi^*(\phi)$ which is often an isomorphism.
This has various consequences; for instance, we use it to characterise quasi-complete intersection homomorphisms entirely in terms of the homotopy Lie algebra.

In the second part of this thesis we introduce a notion of $A_\infty$ centre for minimal $A_\infty$ algebras. If $A$ is an augmented algebra over a field $k$ we show that the image of the natural homomorphism $\chi_k:\HH(A,A)\to {\rm Ext}^*_A(k,k)$ is exactly the $A_\infty$ centre of $A$, generalising a theorem of Buchweitz, Green, Snashall and Solberg from the case of a Koszul algebra. This is deduced as a consequence of a much wider enrichment of the entire characteristic action $\chi:\HH(A,A)\to {\sf Z}(D(A))$. We give a number of representation theoretic applications.

A copy of the thesis can be found here:  Benjamin_Briggs_201811_PhD_thesis

Departmental PhD Thesis Exam – Anup Dixit

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

Wednesday, March 21, 2018
2:10 p.m.
BA6183

PhD Candidate:  Anup Dixit
Supervisor:   Kumar Murty
Thesis title:  The Lindelof class of L-functions

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Meromorphic functions, called L-functions, play a vital role in number theory.  In 1989, Selberg defined a class of L-functions that serves as an axiomatic model for L-functions arising from geometry and arithmetic. Even though the Selberg class successfully captures many characteristics common to most L-functions, it fails to be closed under addition. This creates obstructions, in particular, not allowing us to interpolate between L-functions. To overcome this limitation, V. K. Murty defined a general class of L-functions based on their growth rather than functional equation and Euler product. This class, which is called the Lindelof class of L-functions, is endowed with the structure of a ring.

In this thesis, we study further properties of this class, specifically, its ring structure and topological structure. We also study the zero distribution and the a-value distribution of elements in this class and prove certain uniqueness results, showing that distinct elements cannot share complex values and L-functions in this class cannot share two distinct values with any other meromorphic function. We also establish the value distribution theory for this class with respect to the universality property, which states that every holomorphic function is approximated infinitely often by vertical shifts of an L-function. In this context, we precisely formulate and give some evidence towards the Linnik-Ibragimov conjecture.

A copy of the thesis can be found here: Anup-Dixit-Thesis

Departmental PhD Thesis Exam – Fulgencio Lopez

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

Friday, March 16 , 2018
2:30 p.m.
Fields Institute – 210

PhD Candidate:  Fulgencio Lopez
Supervisor:   Stevo Todorcevic
Thesis title: Construction schemes and their applications

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We study capturing construction schemes, a new combinatorial tool introduced by Todorcevíc to build uncountable structures. It consists of a ranked family of finite sets that provides a framework to do recursive constructions of uncountable objects by working with finite amal-gamations of finite isomorphic substructures, the uncountable substructures of the final object can be further study using capturing.

In this Thesis we study the consistency of capturing construction schemes, and related defi-
nitions, we prove results of consistency, and give several applications of this tool both to infinite combinatorics and Banach space theory. For example, we show weaker forms of capturing, such as n-capturing, form a strict hierarchy which is related to the m-Knaster Hierarchy.  We also show how capturing construction schemes can be used in constructing Suslin trees and Haus-dorff gaps of a special kind in an intuitive manner. And give some applications to the theory of nonseparable Banach spaces.

A copy of the thesis: Thesis Fulgencio Lopez

Departmental PhD Thesis Exam – Yuan Yuan Zheng

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

Friday, March 16 , 2018
1:30 p.m.
Fields Institute – 210

PhD Candidate:  Yuan Yuan Zheng
Supervisor:   Stevo Todorcevic
Thesis title:  Parametrizing topological Ramsey spaces

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We prove a general theorem indicating that essentially all infinite-dimensional Ramsey-type theorems proven using topological Ramsey space theory can be parametrized by products of infinitely many perfect sets. This theorem has applications in several known spaces, showing that certain ultrafilters are preserved under both side-by-side and iterated Sacks forcing. In particular, the well-known result of `selective ultrafilters on the natural numbers are preserved under Sacks forcing’ is extended to the corresponding ultrafilters on richer structures. We also characterize ultrafilters in topological Ramsey spaces in an abstract
setting. The technique of combinatorial forcing is crucial in the proof of the general parametrization theorem, and ultra-Ramsey theory plays an important role in the applications.

A copy of the thesis can be found here: Yuan Yuan thesis

Departmental PhD Thesis Exam – Nan Wu

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

Thursday, March 15 , 2018
5:10 p.m.
BA1180

PhD Candidate:  Nan Wu
Co-Supervisors:  Alex Nabutovsky, Hau-tieng Wu
Thesis title:  Differential Geometry Approach for Unsupervised Machine Learning Algorithms

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Abstract:

Since its introduction in 2000, locally linear embedding (LLE) algorithm has been widely applied in data science.

In this thesis, we provide an asymptotical analysis of LLE under the manifold setup. First, by study the regularized barycentric problem, we derive the corresponding kernel function of LLE. Second, we show that when the point cloud is sampled from a general closed manifold, asymptotically LLE algorithm does not always recover the Laplace-Beltrami operator, and the result may depend on the non-uniform sampling. We demonstrate that a careful choosing of the regularization is necessary to ensure the recovery of the Laplace-Beltrami operator. A comparison with the other commonly applied nonlinear algorithms, particularly the diffusion map, is provided. Moreover, we discuss the relationship between two common nearest neighbor search schemes and the relationship of LLE with the locally linear regression. At last, we consider the case when the point cloud is sampled from a manifold with boundary.

We show that if the regularization is chosen correctly, LLE algorithm asymptotically recovers a linear second order differential operator with “free” boundary condition. Such operator coincides with Laplace-Beltrami operator in the interior of the manifold. We further modify LLE algorithm to the Dirichlet Graph Laplacian algorithm which can be used to recover the Laplace-Beltrami operator of the manifold with Dirichlet boundary condition.

A copy of the thesis can be found here: Nan Wu’s thesis

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

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.