Memoirs of the American Mathematical Society 2011; 77 pp; softcover Volume: 211 ISBN10: 0821849395 ISBN13: 9780821849392 List Price: US$66 Individual Members: US$39.60 Institutional Members: US$52.80 Order Code: MEMO/211/993
 Let \(\mathcal{M}\) denote the space of probability measures on \(\mathbb{R}^D\) endowed with the Wasserstein metric. A differential calculus for a certain class of absolutely continuous curves in \(\mathcal{M}\) was introduced by Ambrosio, Gigli, and Savaré. In this paper the authors develop a calculus for the corresponding class of differential forms on \(\mathcal{M}\). In particular they prove an analogue of Green's theorem for 1forms and show that the corresponding first cohomology group, in the sense of de Rham, vanishes. For \(D=2d\) the authors then define a symplectic distribution on \(\mathcal{M}\) in terms of this calculus, thus obtaining a rigorous framework for the notion of Hamiltonian systems as introduced by Ambrosio and Gangbo. Throughout the paper the authors emphasize the geometric viewpoint and the role played by certain diffeomorphism groups of \(\mathbb{R}^D\). Table of Contents  Introduction
 The topology on \(\mathcal{M}\) and a differential calculus of curves
 The calculus of curves, revisited
 Tangent and cotangent bundles
 Calculus of pseudo differential 1forms
 A symplectic foliation of \(\mathcal{M}\)
 The symplectic foliation as a Poisson structure
 Review of relevant notions of differential geometry
 Bibliography
