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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.
http://cvgmt.sns.it/seminar/892/