## EVENT DETAILS AND ABSTRACT

**Analysis and Probability Seminar**
**Title:** Critical points of smooth processes and suprema distributions

**Speaker:** Professor Jonathan Taylor

**Speaker Info:** Stanford University

**Brief Description:**

**Special Note**:

**Abstract:**

We discuss the (well-studied) problem of approximating the distribution of the supremum of a stochastic process focusing our attention on smooth processes defined on piecewise smooth manifolds -- a "canonical" example being a smooth isotropic process restricted to $[0,1]^2$. Based on the trivial observation that the supremum of the process exceeds a level u, if and only if, there exists a critical point above the level u, an obvious question is: "How much can we learn /compute about the distribution of the supremum solely based on the critical points of the process?" Through a "Morse-theoretic" point processes on the parameter space of the process, we discuss the expected Euler characteristic approximation to the suprema distribution and its connection to classical integral geometry. Further, based on a point process representation of the global maxima we describe an exponentially sharp bound on the relative error of the expected Euler characteristic approximation.

**Date:** Monday, March 15, 2004

**Time:** 4:10pm

**Where:** Lunt 105

**Contact Person:** Prof. Elton P. Hsu

**Contact email:** elton@math.northwestern.edu

**Contact Phone:** 847-491-8541

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