GSAS: Geometry of measures with Hölder density bounds in a Riemannian setting  

The geometry of measures and sets in Euclidean space can be understood through the study of various analytic quantities. One of them, the density, has played a key role in consolidating our understanding of rectifiability, as well as motivating the development of other key notions in geometric measure theory. The aim of this talk is to discuss recent quantitative problems in this direction that are motivated by the following question: what can be said about the geometry of a measure whose density ratio converges at a Hölder rate? We will explore recent results that answer this question, with emphasis on the case where the notion of density comes from a Riemannian setting.