Geometric Analysis and Topology Seminar

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Uniformly rectifiable metric spaces

Speaker: Raanan Schul, Stony Brook University

Location: Warren Weaver Hall 512

Date: Friday, January 26, 2024, 11 a.m.

Synopsis:

Abstract:
In their 1991 and 1993 foundational monographs, David and Semmes
characterized uniform rectifiability for subsets of Euclidean space in
a multitude of geometric and analytic ways. The fundamental geometric
conditions can be naturally stated in any metric space and it has long
been a question of how these concepts are related in this general
setting. In joint work with D. Bate and M. Hyde, we prove their
equivalence. Namely, we show the equivalence of Big Pieces of
Lipschitz Images, Bi-lateral Weak Geometric Lemma and Corona
Decomposition in any Ahlfors regular metric space. Loosely speaking,
this gives a quantitative equivalence between having Lipschitz charts
and approximations by nice spaces. After giving some background, we
will explain the main theorems and outline some key steps in the proof
(which will include a discussion of  Reifenberg parameterizations). We
will also  mention some open questions.