Prob/Stat Seminar

In this talk we will first discuss some of what is known about heat kernel estimates on manifolds with ends. By this we mean a finite number of manifolds satisfying two-sided Gaussian heat kernel estimates which are glued over a compact set. In the case of recurrent (parabolic) ends or in the case of manifolds with boundary with mixed Dirichlet and Neumann condition, the estimates depend on the harmonic profiles (positive harmonic functions) of the ends. Finally, we discuss recent progress toward obtaining heat kernel estimates in the case where the gluing set is non-compact. In particular, we give some hitting probability estimates. This is joint work with Laurent Saloff-Coste.

The Ising pure p-spin model is one of the most fundamental mean field spin glasses. Its limiting free energy is characterized by a variational principle known as the Parisi formula, whose minimizer is called the Parisi measure. Understanding the structure of the Parisi measure is a central problem in spin glass theory. In this talk, I will discuss recent progress on the structure of Parisi measures for Ising pure p-spin models.

How does a given differential operator affect the distribution of a polynomial's roots? When the operator is just the derivative operator, then the Gauss--Lucas theorem guarantees that the critical points are in the convex hull of the original roots. This provides a general answer to where the roots are distributed, but not how they are distributed. It turns out answering how roots are distributed, even for just one derivative, can be a significantly more difficult task. In this talk we will discuss some motivations and partial history behind this question for both deterministic and random polynomials, and then see how random matrix theory can help us understand how the roots are distributed. Based on joint work with Jonas Jalowy (Universität Paderborn), Sean O'Rourke (CU Boulder), and David Renfrew (Binghamton University).

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