### Quantitative rectifiability in metric spaces

Analysis and Geometry Seminar

25th May 2023, 3:30 pm – 4:30 pm

Fry Building, 2.04

The theory of quantitative rectifiability for Ahlfors regular subsets of Euclidean space was developed extensively by David and Semmes in the early 1990s, partly motivated by questions arising in harmonic analysis. They proved, among many other things, the equivalence of Uniform Rectifiability (UR) and the Bi-lateral Weak Geometric Lemma (BWGL). The first condition being a natural quantitative version of rectifiability, the second, a quantitative condition measuring local Hausdorff approximations by affine subspaces. Their result can be seen as quantification of the equivalence between rectifiability and the almost everywhere existence of approximate tangent planes. In this talk we discuss the equivalence of UR and BWGL for Ahlfors regular metric spaces. While the definition of UR makes sense in this context, BWGL does not. Instead, the BWGL condition is stated in terms of local Gromov-Hausdorff approximations by *n*-dimensional Banach spaces. This is based on joint work with David Bate and Raanan Schul.

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