Electronic ISBN:  9781470406103 
Product Code:  MEMO/211/993.E 
List Price:  $70.00 
MAA Member Price:  $63.00 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 211; 2011; 77 ppMSC: Primary 35; 49; Secondary 53;
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

Chapters

1. Introduction

2. The topology on $\mathcal {M}$ and a differential calculus of curves

3. The calculus of curves, revisited

4. Tangent and cotangent bundles

5. Calculus of pseudo differential 1forms

6. A symplectic foliation of $\mathcal {M}$

7. The symplectic foliation as a Poisson structure

A. Review of relevant notions of Differential Geometry


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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 \(\mathbb{R}^D\).

Chapters

1. Introduction

2. The topology on $\mathcal {M}$ and a differential calculus of curves

3. The calculus of curves, revisited

4. Tangent and cotangent bundles

5. Calculus of pseudo differential 1forms

6. A symplectic foliation of $\mathcal {M}$

7. The symplectic foliation as a Poisson structure

A. Review of relevant notions of Differential Geometry