# Chris Gartland

University of Illinois, Urbana-Champaign

**Talk Title: **Thick Families of Geodesics and Differentiation

**Abstract: **The differentiation theory of Lipschitz functions taking values in a Banach space with the Radon-Nikodym property (RNP), originally developed by Cheeger-Kleiner, has proven to be a powerful tool to prove non-biLipschitz embeddability of metric spaces into these Banach spaces. Important examples of metric spaces to which this theory applies include nonabelian Carnot groups and Laakso spaces. In search of a metric characterization of the RNP, Ostrovskii found another class of spaces that do not biLipschitz embed into RNP spaces; spaces containing thick families of geodesics. In this talk, we will give a brief overview of these results, and discuss a recent result of the speaker: any metric space containing a concatenation closed thick family of geodesics also contains a subset which satisfies a weakened form of RNP Lipschitz differentiability. This gives an alternate proof of the non-biLipschitz embeddability of such metric spaces into RNP Banach spaces.