Math 242: Symplectic geometry

UC Berkeley, Spring 2009

Instructor

Michael Hutchings. [My last name with the last letter removed]@math.berkeley.edu Tentative office hours: Tuesday 2:00-4:00, 923 Evans.

Textbooks

The main text for this course is Introduction to symplectic topology, 2nd edition, by McDuff and Salamon. Much of what we will discuss is covered in this book, although I will not follow it closely and will give additional references as we go along. For the last part of the course, An introduction to contact topology by Geiges is a good reference.

Course outline

The rough plan for the course is to discuss the following topics (not necessarily in linear order):

Basic definitions; motivation from physics, geometry, and low-dimensional topology. (cf McDuff-Salamon, Ch. 1)
Basic facts about symplectic manifolds and symplectic linear algebra (M-S Ch. 2, 3, 4)
Symplectic group actions (M-S Ch. 5)
Symplectomorphisms and fixed points (M-S Ch. 8, 9, 10, 11)
The Hofer-Zehnder capacity, Gromov nonsqueezing, and the Weinstein conjecture in R^2n (M-S Ch. 12)
Contact geometry, mainly in 3 dimensions (cf Geiges)

This course should hopefully provide good preparation for the program on Symplectic and Contact Geometry and Topology at MSRI from August 2009 to May 2010.

Final project

Each student is expected to research a topic of interest and either write a 5-10 page expository article about it, or give a short presentation to the class. The articles will be posted here (if I am given permission). The books by McDuff-Salamon and Geiges suggest a number of good starting points with many references. I can give more references. (Since McDuff-Salamon is a few years old, it does not cite the latest works.)

Lecture summaries and references

(1/20) Basic notions: symplectic manifolds, Lagrangian submanifolds, Hamiltonian vector fields, Poisson brackets. The cotangent bundle example. References: McDuff-Salamon, chapters 1 and 3. There is also a nice overview of symplectic geometry by Ana Cannas da Silva which you can download here.
(1/22) Continuing the introduction (cf McDuff-Salamon chapter 1), we discussed:
- The origins of symplectic geometry in classical mechanics. Reference: V. Arnold, Mathematical methods of classical mechanics.
- Introduction to the Weinstein conjecture. Here is a draft of a survey article I am writing about this.
(1/27)
- More about the Weinstein conjecture.
- Introduction to Gromov's nonsqueezing theorem and related topics. See McDuff-Salamon, sections 1.2 and 2.4.
(1/29)
- Basics of almost complex structures and holomorphic spheres. See McDuff-Salamon, Section 2.5 and Chapter 4.
- Outline of Gromov's proof of nonsqueezing. For some more details, see Ka Choi's lecture notes on the course I taught a year ago.
(2/3) More about holomorphic curves and Gromov's proof of nonsqueezing. Digression on the first Chern class. This material is discussed from a different perspective in sections 4.4 and 2.6 respectively of McDuff-Salamon. For more about obstruction theory (which I used to define the first Chern class) I have some old lecture notes here. (These notes explain the Euler class of an oriented k-sphere bundle, which in the case k=1 determines the first Chern class of a complex line bundle.)
(2/5)
- More about holomorphic curves and Gromov's proof of nonsqueezing. For many more details, see McDuff and Salamon, J-holomorphic curves and symplectic topology.
- Brief review of the Hodge decomposition. (We will use this just a little bit next time, but it's good to know anyway.)
(2/10) Darboux's theorem, Moser trick, etc. See eg McDuff-Salamon sections 3.2 and 3.3.
(2/12)
- More applications of the Moser trick to prove various "no local symplectic geometry" results.
- Introduction to contact manifolds. See McDuff-Salamon section 3.4, and Geiges chapters 1 and 2.
(2/17) More about contact manifolds, hypersurfaces of contact type, and the Weinstein conjecture.
(2/19)
- Contact structures on circle bundles.
- Introduction to three-dimensional contact geometry. There is a very nice introduction to this by John Etnyre here. This is the basic reference for the next two lectures. More details can be found in the book by Geiges.
(2/24)
- Tight and overtwisted contact structures in three dimensions.
- Legendrian knots, introduction to the Bennequin equality and adjunction inequality for tight contact structures.
(2/26)
- Introduction to symplectic fillings.
- Outline of the proof of the adjunction inequality.
(3/3)
- Statement of the Arnold conjecture on fixed points of Hamiltonian symplectomorphisms.
- Flux. (See McDuff-Salamon chapter 11)
(3/5) Review of Morse homology. See for example the book by Matthias Schwarz, or my mistake-ridden lecture notes.
(3/10) More about Morse homology.
(3/12) Introduction to Floer homology of Hamiltonian symplectomorphisms. See for example Dietmar Salamon, Lectures on Floer homology.
(3/17) Index and spectral flow. For the basic idea see chapter 2 of Salamon's lecture notes, and for the technical details see Robbin and Salamon, Spectral flow and the Maslov index.
(3/19) The Conley-Zehnder index and the grading on Floer homology. See chapter 2 of Salamon's lecture notes.
(3/31) More about the Conley-Zehnder index and Floer homology: the mapping torus picture and Gromov compactness.
(4/2) Details of the computation of the Floer homology of Hamiltonian symplectomorphisms in the monotone case. Reference: D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index.
(4/7) Floer homology for Lagrangian intersections.
(4/9)
- How Floer homology for symplectomorphisms is a special case of Floer homology for Lagrangian intersections.
- Symplectic capacities. Definition of the Hofer-Zehnder capacity, and how it leads to a proof of the Weinstein conjecture for contact type hypersurfaces in R^(2n). See McDuff-Salamon, chapter 12 and Symplectic invariants and Hamiltonian dynamics by Hofer and Zehnder.
(4/14) Examples of the Hofer-Zehnder capacity. Some analytic background.
(4/16) Outline of the proof of the hard part of the fact that the Hofer-Zehnder capacity is a capacity, following the Hofer-Zehnder book. This is explained in McDuff-Salamon using discretized (finite dimensional) loop spaces instead. This can be re-proved and generalized using versions of Floer homology that keep track of the values of the symplectic action functional; see eg Ginzburg and Gurel, Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles.
(4/21) Introduction to symplectic reduction. See McDuff-Salamon chapter 5. This material is also covered in lots of other books; some useful online references which I am partly following are by Cannas here and Cieliebak here
(4/23) Basic lemmas about moment maps.
(4/30) Marsden-Weinstein reduction.
(5/5) Some examples. Reference for the harder example: Atiyah and Bott, The Yang Mills equations over Riemann surfaces.
(5/7) Atiyah-Guillemin-Sternberg convexity.
(5/8) extra meeting, 10am-1pm in room 736 Evans, for final project talks.