Faculty Area Research (FARS)

Interacting systems of particles and waves are foundational in many natural phenomena. This talk will outline mathematical approaches for deriving effective, statistical descriptions of such many-body dynamics by connecting them to solutions of nonlinear partial differential equations. Key…

Counting special lattices inside the "moduli space" of all lattices with extra symmetry leads to interesting invariants e.g. L-functions and orbital integrals, and interesting questions e.g. the fundamental lemma. I will explain my related research on a generalization of this toy model to "…

Abstract: In this talk, I will give an overview of statistical physics and give an introduction to spin glasses.

Unlike many classical models , like the Ising model, which has structurally regular properties, spin glasses

are pattern-less with an irregular distribution.…

I will talk about the analytic aspects of black holes, especially waves and stability questions, particularly in the context of Kerr-de Sitter spaces (positive cosmological constant).

Abstract: I will explain how the  count of algebraic curves in the complex projective plane can be reduced to a solution of a Hamilton-Jacobi equation.

Abstract: I will discuss Boltzmann's celebrated H theorem for collisional kinetic equations such as the Boltzmann equation or the Landau equation. I will explain what it does and does not imply for the nonlinear flow and describe some open problems.

Abstract: The "area" functional takes a submanifold of a Riemannian manifold and returns its area. A natural idea is to try to use Morse theory to find critical points of the area functional (of considerable interest to geometers, these are called minimal submanifolds). I will describe what we…

Inspired by physicist, Atiyah and Segal initiated an area of mathematics which is the sometimes called "topological field theory." While this can be a useful way of thinking about algebra using pictures, it is essentially impossible to produce genuine new examples using only topology. The…

In this talk, based on joint works with Nicholas Cook, Huy Tuan Pham and Sohom Bhattacharya, I will discuss recent developments in the study of the upper tails for counts of several fixed subgraphs in a large sparse random graph (such as Erdős–Rényi or uniformly d- regular). These…

This is the story of understanding 'things' by asking 'what does a typical thing 'look like'. The things can be finite (permutations, elements of a finite group, graphs, or integers between 1 and N). They can also be infinite (random matrix theory asks about the eigenvalues of 'typical…