Faculty Area Research (FARS)

Fixing integers q,d>2, denote by Q(n,T,B) the ferromagnetic q-Potts measures on graphs G(n), at temperature T>0 and non-negative external field strength B, where as n grows the uniformly sparse G(n) of n vertices converge locally to the infinite d-regular tree. I will review a…

I will discuss a combinatorial conjecture which states that a "core" (or "stable") solution exists for any instance of an election where each voter presents a set of approved candidates (without ranking). Some of the concepts surrounding this conjecture go back to the work of Thiele on elections…

Furstenberg and Katznelson's multidimensional Szemer\'edi theorem says that any subset of $\mathbf{Z}^d$ having positive upper density must contain a homothetic copy of any fixed finite collection of points. For example, any such subset must contain the four vertices of an axis-aligned square.…

Abstract: Black hole thermodynamics is a celebrated analogy between the laws of classical thermodynamics and black hole mechanics governed by the theory of general relativity. In this talk, I will give an introduction to the theory of black holes, their thermodynamics, and explain how recent…

Abstract: We consider high-frequency limits of solutions to the Einstein vacuum equations. There are known explicit examples showing that these limits need not be vacuum. I will explain a conjecture describing what is expected to happen in general, and how it is…

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.