Office: 923 Evans.

Tentative office hours: Thursday 10:00 am - noon

- A. Cannas da Silva,
*Lectures on Symplectic Geometry*, authorized free download here. This book gives a nice explanation of the basic geometric constructions and lemmas in symplectic geometry. - D. McDuff and D. Salamon,
*Introduction to Symplectic Topology*, third edition, Oxford University Press. This book is a useful reference for many of the topics that we will be discussing, as well as a good introduction to a wide range of topics of current research interest.

- V. Arnold,
*Mathematical Methods of Classical Mechanics*. This book is kind of a "prequel" to the course and discusses the use of symplectic geometry in classical mechanics. It contains a lot of rich content beyond the scope of this course. - H. Geiges,
*An Introduction to Contact Topology*. This is a good reference for contact geometry (the odd dimensional counterpart of symplectic geometry), which we will discuss a bit in the course. - D. McDuff and D. Salamon,
*J-holomorphic Curves and Symplectic Topology*. This book explains the technical details of holomorphic curves in great detail. We will discuss some applications of holomorphic curves at the end of the course, but without a lot of technical details. Chapter 9 of this book explains some of the "classical" applications which we will be discussing.

Homework exercises will occasionally be suggested, but these are optional and will not be graded.

- (Wednesday 8/28)
- Definition of symplectic manifold, basic examples. See Cannas da Silva, chapters 1 and 2.

- (Friday 8/30)
- Hamiltonian vector fields. For practice, here is some optional homework.

- (Wednesday 9/4)
- Hamiltonian and symplectic isotopies.
- Introduction to different versions of the Arnold conjecture on the minimum number of fixed points of a Hamiltonian symplectomorphism.

- (Friday 9/6)
- The flux of a symplectic isotopy.
- Introduction to Lagrangian submanifolds.

- (Monday 9/9)
- More about Lagrangians; introduction to Lagrangian intersection problems.
- For practice, the second optional homework is here.

- (Wednesday 9/11 and Friday 9/13)
- The Moser method, Darboux's theorem, and Weinstein's Lagrangian neighborhood theorem. For more details see Cannas da Silva, Part III.

- (Monday 9/16)
- Introduction to contact structures and Legendrian knots. For more about this see the book by Geiges or these lecture notes by Etnyre.

- (Wednesday 9/18)
- Connections on principal S^1 bundles, and the resulting examples of contact forms. For connections on principal bundles with more general gauge groups, see e.g. Spivak's Comprehensive Introduction to Differential Geometry, Volume II, Chapter 8.

- (Friday 9/20)
- Reeb vector fields and contact type hypersurfaces. See e.g. the first few pages of this survey article.

- (Monday 9/23)
- The Legendre transform. (Didn't quite finish.)

- (Wednesday 9/25)
- Finished the Legendre transform. See these notes.
- Started Gray stability.

- (Friday 9/27)
- Hamiltonian circle actions. See McDuff-Salamon, section 5.1. Next time we will discuss some of the more general story in Cannas da Silva sections 27-29.

- (Monday 9/30)
- Introduction to toric symplectic manifolds. See Cannas da Silva sections 27-29 and McDuff-Salamon section 5.5.
- Here is the third homework.