APPLIED MATH SEMINAR

Speaker: Peter Jones, Mathematics, Yale University

When: Tuesday, April 12th, 4:15 PM, AKW 200

Title: Product Formulas for Measures and Applications to Analysis

Abstract: We will discuss elementary product formalisms for positive
measures. These appeared in analysis for purposes of examining "harmonic
measures" related to elliptic equations (work of R. Fefferman, J. Pipher, C.
Kenig). We will discuss three topics where product formulas appear: applied
projects related to signal processing; SLE; and Geometric measure theory.
For the first topic we will explain some work arising in analysis of network
failures. (Joint work with D. Bassu, L. Ness, V. Rokhlin) For the second
topic (SLE) we will show the relations between some models of random
measures, and relations to SLE. (Joint work with K. Astala, A. Kupiainen, E.
Saksman) The third topic (geometric measure theory) will be a discussion of
joint work with Marianna Csörnyei. The main point here is how product
formulas can detect directionality in sets. The new result concerns Lebesgue
measurable sets E of finite measure in the unit cube (in any dimension). The
set E can be decomposed into a bounded number of sets with the property that
each (sub)set has a nice "tangent cone". This yields strong results on
points of non-differentiability for Lipschitz functions. The main technical
result needed is a d dimensional, measure theoretic version of (a geometric
form of) the Erdös-Szekeres theorem, which holds when d = 2. In what is
perhaps a small surprise, certain ideas from random measures can be used
effectively in the deterministic setting.