Abstract: The theory of quantitative rectifiability for Ahlfors regular subsets of Euclidean space was developed extensively by David and Semmes in the early 1990s. 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 tangent condition measuring local Hausdorff approximations by affine subspaces. In this talk I will discuss joint work with D. Bate and R. Schul which characterise UR metric spaces in terms of the BWGL and various other conditions related to the Euclidean theory of UR. We will introduce the problem, its history and formulate our main results.
