Hamiltonian Group Actions and Equivariant Cohomology

Description

This monograph could be used for a graduate course on symplectic geometry as well as for independent study. The monograph starts with an introduction of symplectic vector spaces, followed by symplectic manifolds and then Hamiltonian group actions and the Darboux theorem. After discussing moment maps and orbits of the coadjoint action, symplectic quotients are studied. The convexity theorem and toric manifolds come next and we give a comprehensive treatment of Equivariant cohomology. The monograph also contains detailed treatment of the Duistermaat-Heckman Theorem, geometric quantization, and flat connections on 2-manifolds. Finally, there is an appendix which provides background material on Lie groups. A course on differential topology is an essential prerequisite for this course. Some of the later material will be more accessible to readers who have had a basic course on algebraic topology. For some of the later chapters, it would be helpful to have some background on representation theory and complex geometry. Front Matter ....Pages i-xi Symplectic Vector Spaces (Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk)....Pages 1-7 Hamiltonian Group Actions (Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk)....Pages 9-15 The Darboux–Weinstein Theorem (Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk)....Pages 17-20 Elementary Properties of Moment Maps (Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk)....Pages 21-26 The Symplectic Structure on Coadjoint Orbits (Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk)....Pages 27-29 Symplectic Reduction (Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk)....Pages 31-46 Convexity (Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk)....Pages 47-60 Toric Manifolds (Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk)....Pages 61-70 Equivariant Cohomology (Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk)....Pages 71-88 The Duistermaat–Heckman Theorem (Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk)....Pages 89-102 Geometric Quantization (Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk)....Pages 103-112 Flat Connections on 2-Manifolds (Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey, Theo van den Hurk)....Pages 113-122 Back Matter ....Pages 123-132