Hamiltonian Group Actions and Equivariant Cohomology, 1st ed. 2019 SpringerBriefs in Mathematics Series

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.

Self-contained treatment of equivariant cohomology

Treatment of moduli spaces of flat connections (a topic of considerable current interest)

The only background required is a course on differential manifolds (a standard offering at the advanced undergraduate or introductory graduate level)