Tangents and rectifiability of Lipschitz differentiability spaces

 In 1999, Cheeger gave a vast generalization of Rademacher's theorem, providing a way of differentiating Lipschitz functions on some abstract metric measure spaces. We will discuss some recent work on geometric properties of metric spaces that support such a "differentiable structure", focusing on the case of Ahlfors regular and doubling spaces. We will discuss conditions implying that such a space has uniformly rectifiable tangents, in the sense of David (who is not the present speaker!) and Semmes, and, conversely, conditions that imply strong unrectifiability of the space. Applications to bi-Lipschitz non-embedding problems and quasiconformal geometry will also be mentioned.