Differential Forms on Wasserstein Space and Infinite-Dimensional Hamiltonian Systems.

Intro -- Contents -- Abstract -- Chapter 1. Introduction -- Chapter 2. The topology on M and a differential calculus of curves -- 2.1. The space of distributions -- 2.2. The topology on M -- 2.3. Tangent spaces and the divergence operator -- 2.4. Analytic justification for the tangent spaces -- Chapter 3. The calculus of curves, revisited -- 3.1. Embedding the geometry of RD into M -- 3.2. The intrinsic geometry of M -- 3.3. Embedding the geometry of M into (Cc)* -- 3.4. Further comments -- Chapter 4. Tangent and cotangent bundles -- 4.1. Push-forward operations on M and TM -- 4.2. Differential forms on M -- 4.3. Discussion -- Chapter 5. Calculus of pseudo differential 1-forms -- 5.1. Green's formula for smooth surfaces and 1-forms -- 5.2. Regularity and differentiability of pseudo 1-forms -- 5.3. Regular forms and absolutely continuous curves -- 5.4. Green's formula for annuli -- 5.5. Example: 1-forms on the space of discrete measures -- 5.6. Discussion -- Chapter 6. A symplectic foliation of M -- 6.1. The group of Hamiltonian diffeomorphisms -- 6.2. A symplectic foliation of M -- 6.3. Algebraic properties of the symplectic distribution -- Chapter 7. The symplectic foliation as a Poisson structure -- 7.1. Review of Poisson geometry -- 7.2. The symplectic foliation of M, revisited -- Appendix A. Review of relevant notions of Differential Geometry -- A.1. Calculus of vector fields and differential forms -- A.2. Lie groups and group actions -- A.3. Cohomology and invariant cohomology -- A.4. The group of diffeomorphisms -- Bibliography.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.