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Lipschitz decompositions of the complements of bilaterally flat sets

Jared Krandel, Stony Brook University
Tuesday, May 2, 2023 - 1:30am to 3:30pm
PDL C-401
In the course of proving the Analysts's Traveling Salesman Theorem, Peter Jones proved that any simply connected domain in the plane with finite boundary length can be decomposed into a disjoint collection of Lipschitz domains with controlled total boundary length. In this talk, we will explore higher-dimensional analogues of this theorem. 
In the first part, I will give a short introduction to quantitative geometric measure theory and some of its key ideas such as stopping time arguments and the corona decomposition. In the second part, I will show how to use these ideas to give geometric sufficient conditions on higher-dimensional domain boundaries for the existence of Jones-type decompositions. The notion of a set being "often" bilaterally close to a d-plane, as quantified by Reifenberg flatness or uniform rectifiability, is vital.
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