Lipschitz decompositions of the complements of bilaterally flat sets

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.