# Geometric Analysis and Topology Seminar

#### Vector Fields on Metric Measure Spaces, and 1-Rectifiable Structure

Speaker: Andrea Schioppa, NYU

Location: Warren Weaver Hall 517

Date: Friday, February 28, 2014, 11 a.m.

Synopsis:

In this talk we describe a correspondence between two different structures associated to a metric measure space (X,mu): Weaver derivations and Alberti representations. The module of Weaver derivations is an algebraic structure which describes, roughly speaking, the measurable vector fields on (X,mu). An Alberti representation of the measure mu is a generalized Lebesgue decomposition of mu in terms of 1-rectifiable measures. As an application of this correspondence we obtain a characterization of the differentiability spaces in the sense of Cheeger which is, roughly speaking, a quantitative version of a recent characterization due to Bate.