Recently, there has been an increasing interest from the community in real-life complex system modeling. The most popular example is provided by systems where the number of agents is so large, that only a statistical description (reminiscent to the statistical mechanics description of systems in thermodynamics) turns out to be viable. The usual way to…

Several topics will be presented in this talk. In the first part, we consider some portfolio optimization problems with stochastic dividends, stochastic volatility or delays. The Hamilton-Jacobi-Bellman (HJB) equations are derived, which are second order nonlinear PDEs. We then establish the existence results of the HJB equations and prove the verification theorems. In the second…

The regulation of cell division, cell sizes and cell arrangements is central to tissue morphogenesis. To study these processes, we develop a mechanistic hybrid continuum-discrete mathematical model of cell dynamics that has advantages over previous approaches. This model borrows ideas from statistical physics, materials science and applied mathematics and follows the framework of dynamic density…

Reception to follow in SAS Hall Atrium.   Recent advances in large scale sequencing of microbial DNA without culturing or isolation gives us easy and direct access to “wild” microbial metagenomes that are otherwise virtually impossible to study.  However, because these data sets are large and relatively unstructured, analyzing them requires advancing data science techniques…

Tensors are multiway arrays, and these occur naturally in many data analysis. Consider a series of experiments tracking multiple sensors over time, resulting in a three-way tensor of the form experiment-by-sensor-by-time. Tensor decompositions are powerful tools for data analysis that can be used for data interpretation, dimensionality reduction, outlier detection, and estimation of missing data.…

Many contemporary problems in the calculus of variations involve describing the limit of singular perturbations of variational problems. These problems arise naturally in a number of fields, for example in materials science (phase transition problems) and mathematical statistics (regularized empirical risk minimization). This talk will give an introduction to these problems, their applications, and recent work…

A wide range of natural and engineering systems exhibit extreme events; i.e., spontaneous intermittent behavior manifested through sporadic bursts in the time series of their observables. Examples include ocean rogue waves, intermittency in turbulence, extreme weather patterns and epileptic seizure. Because of their undesirable impact on the system or the surrounding environment, the real-time prediction and mitigation of extreme events is of great interest.…

Reaction networks taken with mass-action kinetics arise in many settings, from epidemiology to population biology to systems of chemical reactions. This talk focuses on certain biological signaling networks, namely, phosphorylation networks, and their resulting dynamical systems. For many of these systems, the set of steady states admits a rational parametrization (that is, the set is the image of a map…

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…