Speaker
Description
The geometric tangent cone to the Wasserstein space is known to be isometric to a set of plans since the work of Gigli (2008). It is a nonlocal object, in that restricting the measure on a subset might change its tangent cone substantially. However we show that the subset of centred elements, those with barycenter 0 in each fiber, is stable by restriction. Following this we compute a decomposition of a measure by the dimension of its centred tangent elements. This resemble existing decompositions following the dimension of the so-called tangent/differentiability bundle introduced independently by Bouchitté-Champion-Jimenez (2005) and Alberti-Marchese (2016), but does not coincide. In the last part of the talk, we focus on measures for which this tangent cone differs from another rather natural construction as a quotient of curves, and characterize the most pathological cases.