Lectures on Symplectic Geometry

- Symplectic Manifolds
- Symplectic Forms
... Symplectic Form on the Cotangent Bundle
... Symplectomorphisms
- Lagrangian Submanifolds
... Generating Functions
... Recurrence
... Local Forms
- Preparation for the Local Theory
... Moser Theorems
... Darbox-Moser-Winstein Theory
... Weinstein Tubular Neighborhood Theorem
... Contact Manifolds
- Contact Forms
... Contact Dynamics
... Compatible Almost Complex Structures
- Almost Complex Structures
... Compatible Triples
... Dolbeault Theory
... Kaehler Manifolds
- Complex Manifolds
... Kaehler Forms
... Compact Kaehler Manifolds
... Hamiltonian Mechanics
- Hamiltonian Vector Fields
... Variational Principles
... Legendre Transform
... Moment Maps
- Actions
... Hamiltonian Actions
... Symplectic Reduction
- The Marsden-Weinstein-Meyer Theorem
... Reduction
... Moment Maps Revisited
- Moment Map in Gauge Theory
... Existence and Uniqueness of Moment Maps
... Convexity
... Symplectic Toric Manifolds
- Classification of Symplectic Toric Manifolds
... Delzant Construction
... Duistermaat-Heckman Theorems.

From the reviews of the first printing

"I find this to be both the best introduction to symplectic geometry as well as a model for how to introduce any field of study. ... one feels the hand of a master in the text's homework sets: concrete, illustrative, and enhancing the material presented. ... For an upper-level undergraduate or beginning graduate student, Lectures on Symplectic Geometry remains, in my opinion, an ideal starting point into an exciting, active and growing area of mathematics." (Andrew McInerney, MAA Reviews, June, 2018)





From the reviews of the first printing



Over the years, there have been several books written to serve as an introduction to symplectic geometry and topology, [...] The text under review here fits well within this tradition, providing a useful and effective synopsis of the basics of symplectic geometry and possibly serving as the springboard for a prospective researcher.

The material covered here amounts to the "usual suspects" of symplectic geometryand topology. From an introductory chapter of symplectic forms and symplectic algebra, the book moves on to many of the subjects that serve as the basis for current research:symplectomorphisms, Lagrangian submanifolds, the Moser theorems, Darboux-Moser-Weinstein theory, almost complex structures, Kãhler structures, Hamiltonian mechanics, symplectic reduction, etc.

The text is written in a clear, easy-to-follow style, that is most appropriate in mathematical sophistication for second-year graduate students; [...].

This text had its origins in a 15-week course that the author taught at UC Berkeley. There are some nice passages where the author simply lists some known results and some well-known conjectures, much as one would expect to see in a good lecture on the same subject. Particularly eloquent is the author's discussion of the compact examples and counterexamples of symplectic, almost complex, complex and Kähler manifolds.

Throughout the