Tangent Fields to Sets and Measures, Their Dimension and Typical Behaviour of Lipschitz Maps

This lecture explores the concept of tangent fields to sets and measures in geometric measure theory, examining their dimension and typical behavior of Lipschitz maps. Delve into weak notions of tangent fields that can be applied to highly singular structures beyond smooth manifolds or rectifiable sets. Learn about the decomposability bundle, a field of vector subspaces that enables Lipschitz map differentiation with respect to measures, and understand how weak tangent fields attach vectors to points where Lipschitz curves follow specific directions. Discover how these concepts extend to metric spaces and their applications in solving various mathematical problems, including characterizing rectifiability and demonstrating metric-space analogues of the Besicovitch-Federer projection theorem. The presentation covers fundamental definitions and highlights significant theorems, particularly focusing on recent research about measure perturbations via Lipschitz maps, where the tangent field dimension provides sharp estimates on the Hausdorff dimension of typical perturbations.