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Speaker: Simon Bortz, University of Alabama

Abstract: In the 1980’s harmonic analysis was being rapidly adapted to `rough’ settings. Among the important early developments was the proof of the L^2 boundedness of the (principal value) Cauchy integral operator on a Lipschitz graph due to Coifman, McIntosh and Meyer. Soon after Coifman, David and Meyer extended this result to higher dimensions and more general singular integral operators. A natural question arose: Is there a characterization of sets for which all nice singular integral operators are L^2 bounded? David and Semmes found many characterizations of these sets and called them uniformly rectifiable. More recently these sets have played a significant role in potential theory.

In this talk, I will discuss the history of uniformly rectifiable sets and some of my recent work on their parabolic analogs. The parabolic theory presents some obstacles in the form of `non-local’ smoothness and, in fact, many of the characterizations of uniform rectifiability fail to have parabolic analogs.