**abstract:**
Gromov introduced a good definition of limits of sequences of pointed metric spaces. Hence, we have a notion of tangent metric spaces. The general theme of the talk are metric spaces that have the same metric tangent at every point.

First, we shall characterize among finite dimensional spaces what are such tangents: for geodesic spaces we only have subFinsler Carnot groups; in general, we still have graded nilpotent Lie groups.

Second, we discuss some properties that one can deduce on the initial metric space from its tangents, possibly assuming that the tangents are `uniformly close’, in the sense that the convergence of the dilated metric spaces converge uniformly on the point that is chosen as base point. In this case, for example, we have that the doubling dimension is locally given by the dimension of its metric tangents.

The main discussion will be whether there are good maps between the tangents and the original space. After the work of G.David & T.Toro and of G.C. David, we have that every n-regular metric space with uniformly close tangents isometric to Euclidean n-space is uniformly rectifiable. The situation for non-Abelian tangents is different. Indeed, on the one hand there exist subRiemannian Lie groups for which there are no quasiconformal maps between their open subsets and the tangents. On the other hand, we shall describe a recent result stating that if M is a sub-Riemannian manifold with tangent N at every point, then all of M except for a null set can be covered by countably many bilipschitz maps defined on subsets of N.

We shall end with open problems on other settings:
a) spaces with normed vector spaces as tangents
b) submanifolds of Carnot groups that are intrinsically C^{1.}

The presented results comes from a collection of works obtained in collaborations with G.C. David, S. Li, S. Nicolussi Golo, A. Ottazzi, T. Rajala, B. Warhurst, and R. Young.

Tue 16 Apr, 9:30 - 10:20, Aula Dini

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