I will introduce systems of conservation laws and their stability. In particular, I will highlight the Kruzkov theory, which gives $L^1$ stability for scalar laws, and the Dafermos-DiPerna relative entropy method, which gives $L^2$ stability for systems as long as the solution in question is Lipschitz. We then turn to the theory of a-contraction with shifts, a method developed by Alexis Vasseur and collaborators to extend these results. To this point, most of the work of a-contraction has been done with scalar laws, systems of two equations, and the extremal families of systems (consisting of the shocks with the highest speeds.) I will motivate the difficulties of extending these results the intermediate shock case and present necessary and sufficient conditions for a-contraction to hold for these families. *This is a mock candidacy talk.*

Singular integrals have played a fundamental role in the development of the modern theory of harmonic analysis. At the same time, the subject’s technical nature can make it seem off-puttingly arcane without motivation. The goal of this talk is to describe the motivation before giving an overview of the off-puttingly arcane technicalities. 

The attention of the wave turbulence community has been mainly focused on derivations of wave kinetic equations (WKE) from dynamics governed by nonlinear dispersive equations which model the evolution of a system of interacting waves. The inhomogeneous six-wave kinetic equation is one such effective equation derived from the quintic nonlinear Schrödinger equation in the mesoscopic limit. However, the exploration of this WKE has started just recently. In that context, we will show global well-posedness and existence of nonnegative solutions to the WKE in exponentially weighted $L^\infty$ spaces. Moreover, these arguments motivate our analysis of long-time behavior of solutions and we show that they scatter and that the corresponding wave operators are bijective. This is based on a joint work with N. Pavlović and M. Tasković.

Stochastic Differential Equations (SDEs) are an important probabilistic object, but they are also a good modeling tool, and they exhibit deep connections with partial differential equations. In this talk, we will try to cover some preliminary material on stochastic analysis and SDEs; and then explore some questions related to interacting diffusions, such as convergence to a McKean – Vlasov limit in the case of mean field interactions, propagation of chaos, and derivation of the Fokker Planck equation.

I will give an introductory talk to Weyl’s law, regarding the asymptotic distribution of eigenvalues of certain self-adjoint operators. We will mostly talk about its application to study the semi-classical limit of Schrödinger operators. At the end we will also touch upon the connection with Thomas-Fermi theory in many-particle fermonic systems.

We study stability properties of solutions to $1-d$ scalar conservation laws with a class of non-convex fluxes. Using the theory of a-contraction with shifts, we show $L^2$-stability for shocks among a class of large perturbations, and give estimates on the weight coefficient a in regimes where the shock amplitude is both large and small. Then, we use these estimates as a building block to show a uniqueness theorem under reduced entropy conditions for weak solutions to the conservation law via a modified front tracking algorithm. The proof is inspired by an analogous program carried out in the $2\times 2$ system setting by Chen, Golding, Krupa, and Vasseur. *Note* Jeffrey actually did not present this and instead discussed the regularizing effect of the nonlinearity for scalar conservation laws.

In this talk, we will explore continued fraction algorithms as dynamical systems. We will discuss a proof of why regular continued fractions are ergodic and how this proof can be extended to a certain continued fraction algorithm on 2D Minkowski space. Time permitting, we will also discuss the extended even continued fractions and their higher dimensional analog.

Sets of finite perimeter are useful in solving certain geometric variational problems. We will develop the theory of sets of finite perimeter up to the first and second variation of perimeter and time permitting use that to solve some of these problems.

The aim of this talk is not to dive deep into any particular topic or theorem, but to motivate the study of optimal transport (OT) by outlining its connections with partial differential equations (PDEs). We will begin by reviewing basic OT theory, including the Monge problem, the Kantorovich problem, the dual problem, and Brenier’s theorem. We will then briefly discuss the use of PDEs to study the regularity of optimal solutions to the Monge problem. Finally, I will outline some applications of OT theory to the study of PDEs. If time permits, I will talk about the appearance of the continuity equation in Wasserstein spaces and the JKO scheme.

The space of Riemannian manifolds satisfying a lower bound on the Ricci curvature and an upper bound on the dimension is pre-compact in the Gromov-Hausdorff convergence. A natural question is: what are the limit spaces, and how can we characterize them? One way to do this is to search for properties which are “stable” with respect to Gromov-Hausdorff convergence. In particular, in this talk I will explain how the Sobolev spaces converge in a certain sense, and the limiting space coincides with a typical way to define Sobolev spaces on metric measure spaces.