Departmental PhD Thesis Exam – Duncan Dauvergne

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

Monday, March 25, 2019
10:10 a.m.

PhD Candidate:  Duncan Dauvergne
Supervisor:   Balint Virag
Thesis title:  Random sorting networks, the directed landscape, and random polynomials


The first part of this thesis is on random sorting networks. A sorting network is a shortest path from $12 \cdots n$ to $n \cdots 21$ in the Cayley graph of the symmetric group $S_n$ generated by adjacent transpositions. We prove that in a uniform random $n$-element sorting network $\sigma^n$, all particle trajectories are close to sine curves with high probability. We also find the weak limit of the time-$t$ permutation matrix measures of $\sigma^n$. As a corollary, we show that if $S_n$ is embedded into $\mathbb{R}^n$ via the map $\tau \mapsto (\tau(1), \tau(2), \dots \tau(n))$, then with high probability, the path $\sigma^n$ is close to a great circle on a particular $(n-2)$-dimensional sphere. These results prove conjectures of Angel, Holroyd, Romik, and Vir\’ag. To prove these results, we find the local limit of random sorting networks and prove that the local speed distribution is the arcsine distribution on $[-\pi, \pi]$.

The second part of this thesis is on last passage percolation. The conjectured limit of last passage percolation is a scale-invariant, independent, stationary increment process with respect to metric composition. We prove this for Brownian last passage percolation. We construct the Airy sheet and characterize it in terms of the Airy line ensemble. We also show that the last passage geodesics converge to random functions with H\”older-$2/3^-$ continuous paths. To prove these results, we develop a new probabilistic framework for understanding the Airy line ensemble.

The third part of this thesis is on random sums of orthonormal polynomials. Let $G_n = \sum_{i=0}^n \xi_i p_i$, where the $\xi_i$ are i.i.d. non-degenerate complex random variables, and $\{p_i\}$ is a sequence of orthonormal polynomials with respect to a regular measure $\tau$ supported on a compact set $K$. We show that the zero measure of $G_n$ converges weakly almost surely to the equilibrium measure of $K$ if and only if $\mathbb{E}\log(1 + |\xi_0|) < \infty$. We also show that the zero measure of $G_n$ converges weakly in probability to the equilibrium measure of $K$ if and only if $\mathbb{P}(|\xi_0| > e^n) = o(n^{-1})$. Our methods also work for more general sequences of asymptotically minimal polynomials in $L^p(\tau)$, where $p \in (0, \infty]$.

A copy of the thesis can be found here: MainThesisPhD

Departmental PhD Thesis Exam – Chia-Cheng Liu

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

Wednesday, December 5, 2018
11:10 a.m.

PhD Candidate:  Chia-Cheng Liu
Co-Supervisors:   Joel Kamnitzer/Alexander Braverman
Thesis title:  Semi-innite Cohomology, Quantum Group Cohomology, and the Kazhdan-Lusztig

The Kazhdan-Lusztig tensor equivalence is a monoidal functor which sends modules over ane
Lie algebras at a negative level to modules over quantum groups at a root of unity. A positive
level Kazhdan-Lusztig functor is dened using Arkhipov-Gaitsgory’s duality between ane Lie
algebras of positive and negative levels. We prove that the semi-innite cohomology functor
for positive level modules factors through the positive level Kazhdan-Lusztig functor and the
quantum group cohomology functor with respect to the positive part of Lusztig’s quantum
group. This is the main result of the thesis.

Monoidal structure of a category can be interpreted as factorization data on the associated
global category. We describe a conjectural reformulation of the Kazhdan-Lusztig tensor equivalence
in factorization terms. In this reformulation, the semi-innite cohomology functor at
positive level is naturally factorizable, and it is conjectured that the factorizable semi-innite
cohomology functor is essentially the positive level Kazhdan-Lusztig tensor functor modulo the
Riemann-Hilbert correspondence. Our main result provides an important technical tool in a
proposed approach to a proof of this conjecture.

A copy of the thesis can be found here: thesis_chiachengliu-1

Departmental PhD Thesis Exam – Krishan Rajaratnam

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

Friday, October 19, 2018
2:10 p.m.

PhD Candidate:  Krishan Rajaratnam
Supervisor:   Michael Sigal
Thesis title: Abrikosov lattice solutions of the ZHK Chern-Simons equations


In this thesis we study the ZHK Chern-Simons equations which occur in the study of the fractional quantum hall effect of condensed matter physics. After stating basic properties of these equations, we first prove the existence of Abrikosov lattice solutions of them.  Among these solutions, we find the physically interesting one whose lattice shape minimizes the average energy per lattice cell. In addition to the Abrikosov lattice solutions, we find solutions of the ZHK Chern-Simons equations on Riemann surfaces of higher genus $g$, by utilizing similar results for the Ginzburg-Landau equations.

Finally, we study the orbital stability of the Abrikosov lattice solutions under perturbations which preserve the lattice.

A copy of the thesis can be found here:

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.

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


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.

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



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.

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



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.

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



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

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.

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



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.

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



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.

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



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