Compressive sensing has generated tremendous amounts of interest since first being proposed by Emmanuel Candes, David Donoho, Terry Tao, and others roughly a decade ago.  This mathematical framework has its origins in (i) the observation that traditional signal processing applications, such as MRI imaging problems, often deal with the acquisition of signals which are known…

This talk will present recent works to identify the critical scales at which regularity is propagated by advection equations with rough, i.e. non-smooth, velocity fields. After reviewing the classical theory of renormalized solutions which provides qualitative arguments of regularity and well-posedness, more recent quantitative approaches will be discussed. Our goal is to use this framework…

Cluster algebras are commutative rings defined by a set of generators and relations and equipped with a rich combinatorial structure.   It turns out that coordinate rings of many important varieties from Lie theory are cluster algebras.   In this talk, we will discuss cluster structures in open Schubert varieties of the Grassmannian and their…

The theory of Mean Field Games was invented roughly a decade ago simultaneously by Lasry-Lions on the one hand and Caines-Huang-Malhamé on the other hand. The aim of both groups was to study Nash equilibria of differential games with infinitely many players. In the first half of the talk, we will introduce some basic models…

The Teichmüller space of a surface is a rich mathematical object which can be interpreted from many different perspectives. For example, Teichmüller space can be thought of as a moduli space of hyperbolic structures, Riemann surface structures, or representations of the fundamental group into PSL(2,R) which are discrete and faithful. The aim of higher Teichmüller…

The Hull-Strominger system is a system of nonlinear PDEs describing the geometry of compactification of heterotic strings with flux to 4d Minkowski spacetime, which can be regarded as a generalization of Ricci-flat Kahler metrics coupled with Hermitian Yang-Mills equation on non-Kahler Calabi-Yau 3-folds. In this talk, we present an explicit construction of smooth solutions to…

This paper examines the investment in and the valuation of power generation projects under uncertainty. The analysis incorporates the possibility of producing from alternative types of fuels, such as renewables (wind) or fossil fuels (gas), hence alternative types of plants/technologies. The model considered in this paper cannot be reduced to a single state variable. It…

 The classical Schur-Weyl duality relates the representation theory of general linear Lie algebras and symmetric groups. Drinfeld and Jimbo independently introduced quantum groups  in their study of exactly solvable models, which leads to a quantization of the Schur duality relating quantum groups of general linear Lie algebras and Hecke algebras of symmetric groups. In this…

We study Nash equilibria for inventory-averse high-frequency traders (HFTs), who trade to exploit information about future price changes. For discrete trading rounds, the HFTs' optimal trading strategies and their equilibrium price impact are described by a system of nonlinear equations; explicit solutions obtain around the continuous-time limit. Unlike in the risk-neutral case, the optimal inventories…

We consider the problem of maximizing expected utility from terminal wealth for a power investor who can allocate his wealth in a stock, a defaultable security, and a money market account. The dynamics of these security prices are governed by geometric Brownian motions modulated by a hidden continuous time finite state Markov chain. We reduce…

Fixed point theory studies the extent to which fixed points of a self map of a space are intrinsic. In many mathematical settings, the existence of a solution can be rephrased in terms of the existence of a fixed point for an appropriate map, leading to applications across mathematics. Variations of fixed point problems have…

My research has been in two broad areas namely mathematical biology and disability studies. This talk will touch upon three of my projects in mathematical biology and one project in disability studies. The mathematical biology section will cover the work I have done in investigating permanence (species in a system are at a safe threshold…

Elliptic integrands are used to model anisotropic energies in variational problems. These energies are employed in a variety of applications, such as crystal structures, capillarity problems and gravitational fields, to account for preferred inhomogeneous and directionally dependent configurations. After a brief introduction to variational problems involving elliptic integrands, I will present an overview of the…

Experts have long realized the parallels between elliptic and parabolic theory of partial differential equations. It is well known that elliptic theory may be considered a static, or steady-state, version of parabolic theory. And in particular, if a parabolic estimate holds, then by eliminating the time parameter, one immediately arrives at the underlying elliptic statement.…

Differential games, as an offspring of game theory and optimal control, provide the modeling and analysis of conflict in the context of a dynamical system. Computing Nash equilibria is one of the core objectives in differential games, with a major bottleneck coming from the notorious intractability of N-player games, also known by the curse of…

In this talk, I will discuss two different 'ranking' models: Mallows' ranking model and rank-dependent diffusions. In the first part, I will discuss the rank-dependent diffusions. I will focus on two models: Up the River model, and N-player games with fuel constraints. These problems require treating carefully the corresponding PDEs. The former is joint with…

Quantized higher Teichmüller theory, as described by Fock and Goncharov, assigns an algebra and its representation to a surface and a Lie group. This assignment is equivariant with respect to the action of the mapping class group of the surface, and is conjectured to give an analog of a modular functor, that is it should respect the operation…

The cellular cytoskeleton is essential in proper cell function as well as in organism development. These filaments represent the roads along which most protein transport occurs inside cells. I will discuss examples where questions about filament-motor protein interactions require the development of novel mathematical modeling, analysis, and simulation. In the development of egg cells into…

Fusion categories are rich mathematical structures generalizing the representation categories of finite groups. They arise in many areas of mathematics and physics. Most strikingly, they have emerged as models for particle-like excitations with exotic exchange statistics in low dimensional quantum field theories. We will provide an introduction to these ideas and discuss recent results on…

A hypersurface in a Riemannian manifold is called minimal if its mean curvature vanishes identically.  Minimal surfaces have fascinated mathematicians since the time of Euler, and tremendous progress has been made in understanding the structure of the space of embedded minimal surfaces in various ambient manifolds in the last 50 years.  After surveying the subject, I will…