Mathematics Department Colloquium: Geometric Measure Theory and the fine structure of harmonic measure
Colloquium | March 12 | 4:10-5 p.m. | 60 Evans Hall
Alexander Volberg, Michigan State University
Singular integrals are ubiquitous objects that play an important part in Geometric Measure Theory. The simplest ones are called Calderon-–Zygmund operators. The theory of these operators was completed in the 1950s by Zygmund and Calderon---or so it seemed. The last twenty years have seen the need to consider CZ operators in very bad environments, where kernels are still very good, but the ambient set has no regularity whatsoever. Initially the study of such situations was motivated by the wish to solve some outstanding problems in complex analysis: problems of Painlevé, Ahlfors, Denjoy, and Vitushkin and others.
The analysis of CZ operators on very bad sets is also very fruitful in that part of Geometric Measure Theory that deals with rectifiability. It can be viewed as the study of very-low-regularity free boundary problems. As such, this analysis illuminates the geometry of harmonic measure. Lennart Carleson, Nikolai Makarov, Jean Bourgain, Peter Jones and Tom Wolff obtained important results on metric properties of harmonic measure in the 1980s and 1990s. But most of the results concerned the structure of harmonic measure of planar domains. As an example of the use of non-homogeneous harmonic analysis, we will show how it allows us to understand very fine properties of harmonic measure of any domain in any dimension and to find the answer to several problems of C. Bishop, dated from 1991.