## Departmental PhD Thesis Exam – Steven Amelotte

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

Tuesday, June 26 2018
1:10 p.m.
BA6183

PhD Candidate:  Steven Amelotte
Supervisor:   Paul Selick
Thesis title: Unstable Homotopy Theory Surrounding the Fibre of the $p^\text{th}$ Power Map on Loop Spaces of Spheres

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In this thesis, we study the fibre of the $p^\text{th}$ power map on loop spaces of spheres with a view toward obtaining homotopy decompositions. For the prime $p=2$, we give an explicit decomposition of the fibre $\Omega^3S^{17}\{2\}$ of the squaring map on the triple loop space of the $17$-sphere, building on work of Campbell, Cohen, Peterson and Selick, who showed that such decompositions are only possible for $S^5$, $S^9$ and $S^{17}$. This induces a splitting of the mod $2$ homotopy groups $\pi_\ast(S^{17}; \mathbb{Z}/2\mathbb{Z})$ in terms of the integral homotopy groups of the fibre of the double suspension $E^2 \colon S^{2n-1} \to \Omega^2S^{2n+1}$ and refines a result of Cohen and Selick, who gave similar decompositions for $S^5$ and $S^9$. For odd primes, we find that the decomposition problem for $\Omega S^{2n+1}\{p\}$ is equivalent to the strong odd primary Kervaire invariant problem. Using this unstable homotopy theoretic interpretation of the Kervaire invariant problem together with what is presently known about the $3$-primary stable homotopy groups of spheres, we give a new decomposition of $\Omega S^{55}\{3\}$.

We relate the $2$-primary decompositions above to various Whitehead products in the homotopy groups of mod $2$ Moore spaces and Stiefel manifolds to show that the Whitehead square $[i_{2n},i_{2n}]$ of the inclusion of the bottom cell of the Moore space $P^{2n+1}(2)$ is divisible by $2$ if and only if $2n=2,4,8$ or $16$. As an application of our $3$-primary decomposition, we prove two new cases of a longstanding conjecture which states that the fibre of the double suspension is a double loop space.

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

## Departmental PhD Thesis Exam – Vincent Gelinas

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

Monday, June 18,  2018
11:10 a.m.
BA6183

PhD Candidate:  Vincent Gelinas
Co-Supervisors:   Joel Kamnitzer, Colin Ingalls
Thesis title:  Contributions to the Stable Derived Categories of Gorenstein Rings

***

Abstract:

The stable derived category ${\rm D}_{sg}(R)$ of a Gorenstein ring $R$ is defined as the Verdier quotient of the bounded derived category ${\rm D}^b(\modsf R)$ by the thick subcategory of perfect complexes, that is, those with finite projective resolutions, and was introduced by Ragnar-Olaf Buchweitz as a homological measure of the singularities of $R$. This thesis contributes to its study, centered around representation theoretic, homological and Koszul duality aspects.

In Part I, we first complete (over $\C$) the classification of homogeneous complete intersection isolated singularities $R$ for which the graded stable derived category ${\rm D}^{\Z}_{sg}(R)$ (respectively, ${\rm D}^b(\coh X)$ for $X = \proj R$) contains a tilting object. This is done by proving the existence of a full strong exceptional collection of vector bundles on a $2n$-dimensional smooth complete intersection of two quadrics $X = V(Q_1, Q_2) \subseteq \mathbb{P}^{2n+2}$, building on work of Kuznetsov. We then use recent results of Buchweitz-Iyama-Yamaura to classify the indecomposable objects in ${\rm D}_{sg}^{\Z}(R_Y)$ and the Betti tables of their complete resolutions, over $R_Y$ the homogeneous coordinate rings of $4$ points on $\mathbb{P}^1$ and $4$ points on $\mathbb{P}^2$ in general position.

In Part II, for $R$ a Koszul Gorenstein algebra, we study a natural pair of full subcategories whose intersection $\mathcal{H}^{\mathsf{lin}}(R) \subseteq {\rm D}_{sg}^{\Z}(R)$ consists of modules with eventually linear projective resolutions. We prove that such a pair forms a bounded t-structure if and only if $R$ is absolutely Koszul in the sense of Herzog-Iyengar, in which case there is an equivalence of triangulated categories ${\rm D}^b(\mathcal{H}^{\mathsf{lin}}(R)) \cong {\rm D}_{sg}^{\Z}(R)$. We then relate the heart to modules over the Koszul dual algebra $R^!$. As first application, we extend the Bernstein-Gel’fand-Gel’fand correspondence beyond the case of exterior and symmetric algebras, or more generally complete intersections of quadrics and homogeneous Clifford algebras, to any pair of Koszul dual algebras $(R, R^!)$ with $R$ absolutely Koszul Gorenstein. In particular the correspondence holds for the coordinate ring of elliptic normal curves of degree $\geq 4$ and for the anticanonical model of del Pezzo surfaces of degree $\geq 4$. We then relate our results to conjectures of Bondal and Minamoto on the graded coherence of Artin-Schelter regular algebras and higher preprojective algebras; we characterise when these conjectures hold in a restricted setting, and give counterexamples to both in all dimension $\geq 4$.

A copy of the thesis can be found here:  thesis

## Departmental PhD Thesis Exam – Huan Vo

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

Wednesday, June 20 2018
11:10 a.m.
BA6183

PhD Candidate:  Huan Vo
Supervisor:   Dror Bar-Natan
Thesis title:  Alexander Invariants of Tangles via Expansions

****

Abstract:

In this thesis we describe a method to extend the Alexander polynomial to tangles. It is based on a
technology known as expansions, which is inspired by the Taylor expansion and the Kontsevich integral.
Our main object of study is the space of w-tangles, which contains usual tangles, but has a much simpler
expansion. To study w-tangles, we introduce an algebraic structure called meta-monoids. An expansion
of w-tangles together with a particular Lie algebra, namely the non-abelian two-dimensional Lie algebra,
gives us a meta-monoid called Γ-calculus that recovers the Alexander polynomial. Using the language
of Γ-calculus, we rederive certain important properties of the Alexander polynomial, most notably the
Fox-Milnor condition on the Alexander polynomials of ribbon knots [Lic97, FM66]. We argue that our
proof has some potential for generalization which may help tackle the slice-ribbon conjecture. In a sense
this thesis is an extension of [BNS13].

A copy of the thesis can be found here:  Thesis_HuanVo_V1

## Departmental PhD Thesis Exam – Julio Hernandez Bellon

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

Wednesday, June 6, 2018
11:10 a.m.
BA6183

PhD Candidate:  Julio Hernandez Bellon
Supervisor:   Luis Seco
Thesis title: Correlation Model Risk and Non Gaussian Factor Models

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Abstract:

Two problems are considered in this thesis. The first one is concerned with correlation model risk and the second one with non Gaussian factor modeling of asset returns.

One of the fundamental problems in the application of mathematical finance results in a real world setting, is the dependence of mathematical models on parameters (correlations) that are hard to observe in markets. The common term for this problem is model risk. The first part of this thesis aims to provide some building blocks in the estimation of the sensitivities of mathematical objects (prices) to correlation inputs. In high dimensions, computational complexities increase faster than exponentially, a typical approach to deal with this problem is to introduce a principal component approach for dimension reduction. We consider the price of portfolios of options and approximations obtained by modifying the eigenvalues of the covariance matrix, then proceed to
find analytical upper bounds of the magnitude of the difference between the price and the approximation, under
different assumptions. Monte Carlo simulations are then used to plot the difference between the price and the
approximation.

In the second part of this thesis the assumptions and estimation methods of four different factor models with
time varying parameters are discussed. These models are based on Sharpe’s single index model, the first one
assumes that residuals follow a Gaussian white noise process, while the other three approaches combine the
structure of a single factor model with time varying parameters, with dynamic volatility (GARCH) assumptions
on the model components. The four approaches are then used to estimate the time varying alphas and betas of
three different hedge fund strategies and results are compared.

A copy of the thesis can be found here:  Thesis_Julio_Hernandez_07_30

## Departmental PhD Thesis Exam – Daniel Fusca

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

Tuesday, June 12, 2018
11:10 a.m.
BA6183

PhD Candidate:  Boris Khesin
Supervisor:   Daniel Fusca
Thesis title:  A groupoid approach to geometric mechanics

***

Abstract:

In 1966 V. Arnold proved that the Euler equation for an incompressible fluid describes the geodesic flow for a right-invariant metric on the group of volume-preserving diffeomorphisms of the fluid’s domain. This remarkable observation led to numerous advances in the study of the Hamiltonian properties, instabilities, and topological features of fluid flows. However, Arnold’s approach does not apply to systems whose configuration spaces do not have a group structure. A particular example of such a system is that of a fluid with moving boundary. More generally, one can consider a system describing a rigid body moving in a fluid. Here the configurations of the fluid are identified with diffeomorphisms mapping a fixed reference domain to the exterior of the (moving) body. In general such diffeomorphisms cannot be composed, since the domain of one will not match the range of the other.

The systems we consider are numerous variations of a rigid body in an inviscid fluid. The different cases are specified by the properties of the fluid; the fluid may be compressible or incompressible, irrotational or not. By using groupoids we generalize Arnold’s diffeomorphism group framework for fluid flows to show that the well-known equations governing the motion of these various systems can be viewed as geodesic equations (or more generally, Newton’s equations) written on an appropriate configuration space.

We also show how constrained dynamical systems on larger algebroids are in many cases equivalent to dynamical systems on smaller algebroids, with the two systems being related by a generalized notion of Riemannian submersion. As an application, we show that incompressible fluid-body motion with the constraint that the fluid velocity is curl- and circulation-free is equivalent to solutions of Kirchhoff’s equations on the finite-dimensional algebroid $\mathfrak{se}(n)$.

In order to prove these results, we further develop the theory of Lagrangian mechanics on algebroids. Our approach is based on the use of vector bundle connections, which leads to new expressions for the canonical equations and structures on Lie algebroids and their duals.

The case of a compressible fluid is of particular interest by itself. It turns out that for a large class of potential functions $U$, the gradient solutions of the compressible fluid equations can be related to solutions of Schr\”{o}dinger-type equations via the $\emph{Madelung transform}$, which was first introduced in 1927. We prove that the Madelung transform not only maps one class of equations to the other, but it also preserves the Hamiltonian properties of both equations.

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

## 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

***

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 $\mathrm{GL}_2$.

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

## 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

***

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

***

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

***

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

***

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