 | Gangbo W. Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems / W.Gangbo, H.K.Kim, T.Pacini. - Providence: American Mathematical Society, 2011. - v, 77 p. - (Memoirs of the American Mathematical Society; vol.211, N 993). - Bibliogr.: p.75-77. - ISBN 978-0-8218-4939-2; ISSN 0065-9266
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Chapter 1. Introduction ........................................ 1
Chapter 2. The topology on M. and a differential calculus of
curves .............................................. 5
2.1 The space of distributions ................................. 5
2.2 The topology on  ......................................... 5
2.3 Tangent spaces and the divergence operator ................. 6
2.4 Analytic justification for the tangent spaces .............. 8
Chapter 3. The calculus of curves, revisited .................. 11
3.1 Embedding the geometry of  D into ...................... 11
3.2 The intrinsic geometry of .............................. 11
3.3 Embedding the geometry of into (Cc∞) ................... 13
3.4 Further comments .......................................... 14
Chapter 4. Tangent and cotangent bundles ...................... 17
4.1 Push-forward operations on and TM ...................... 17
4.2 Differential forms on .................................. 18
4.3 Discussion ................................................ 23
Chapter 5. Calculus of pseudo differential 1-forms ............ 25
5.1 Green's formula for smooth surfaces and 1-forms ........... 25
5.2 Regularity and differentiability of pseudo 1-forms ........ 27
5.3 Regular forms and absolutely continuous curves ............ 31
5.4 Green's formula for annuli ................................ 37
5.5 Example: 1-forms on the space of discrete measures ........ 43
5.6 Discussion ................................................ 44
Chapter 6. A symplectic foliation of ....................... 47
6.1 The group of Hamiltonian diffeomorphisms .................. 47
6.2 A symplectic foliation of .............................. 49
6.3 Algebraic properties of the symplectic distribution ....... 53
Chapter 7. The symplectic foliation as a Poisson structure .... 57
7.1 Review of Poisson geometry ................................ 57
7.2 The symplectic foliation of , revisited ................. 59
Appendix A. Review of relevant notions of Differential
Geometry ........................................... 63
A.l. Calculus of vector fields and differential forms ......... 63
A.2. Lie groups and group actions ............................. 66
A.3. Cohomology and invariant cohomology ...................... 69
A.4. The group of diffeomorphisms ............................. 71
Bibliography ................................................... 75
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