## On differentiability in the Wasserstein space of probability measures

In 2008 Gangbo, Nguyen and Tudorascu showed that certain variational solutions of the Euler-Poisson system in 1D can be regarded as optimal paths for the value-function giving the viscosity solution of some Hamilton-Jacobi equation whose phase-space is the Wasserstein space of Borel probability measures with finite second moment. Lasry, Lions, and others became interested in such Hamilton-Jacobi equations (HJE) in connection with their developing theory of Mean-Field Games. A different approach (less intrinsic than ours) to the notion of viscosity solution was preferred, one that made an immediate connection between HJE in the Wasserstein space and HJE in Hilbert spaces (whose theory was well-studied and fairly well-understood). At the heart of the difference between these approaches lies the choice of the sub/supper-differential in the context of the Wasserstein space (i.e. the interpretation of cotangent space'' to this pseudo-Riemannian'' manifold). The connections between these various notions will be the topic of my talk. Based on joint work with W. Gangbo.