Math 242: Symplectic geometry
UC Berkeley, Spring 2014
Instructor Michael Hutchings. [My last name with the last
letter removed]@math.berkeley.edu Tentative office hours: Tuesday
2:00-5:00, 923 Evans.
The following two books are recommended but not required:
- A. Cannas da Silva, Lectures
on Symplectic Geometry, 2006. This book very nicely explains the basic structures of symplectic geometry. It is available online at the above link, and also as a printed book published by Springer.
D. McDuff and D. Salamon, Introduction to symplectic
topology, 2nd edition, 1998 (called "little McDuff-Salamon" below). This is a very nice introduction to some of
the more topological aspects of symplectic geometry. There have been futher developments since this book was written, some of which we will see in the course.
The course has two basic parts (which will not be completely
separate from each other). The first part introduces the basic
structures of symplectic geometry (roughly corresponding to the
official course description), as well as some basic notions of
contact geometry (the odd dimensional cousin of symplectic geometry). The second part introduces holomorphic curves and some of their applications to symplectic topology (skipping hard analysis proofs).
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). I can help you find a topic when the time comes.
Lecture summaries and references
(1/23) Started talking about automorphisms of symplectic manifolds.
- Definition of a symplectic manifold and basic examples. See e.g. Cannas da Silva, Chapters 1 and 2.
- Statement of Gromov nonsqueezing. See e.g. McDuff-Salamon, J-holomorphic Curves and Symplectic Topology, second edition, (which below I will call "big McDuff-Salamon"), Section 9.3.
- Statement of McDuff's theorem on symplectic embeddings of four-dimensional ellipsoids. See e.g. this survey article.
- Discussed the functor from differential topology to symplectic geometry. See e.g. this paper by Abouzaid.
- Symplectomorphisms, Hamiltonian vector fields, Hamiltonian isotopy, symplectic isotopy. (See any symplectic geometry textbook.)
- How the equations of classical mechanics can be interpreted as a Hamiltonian vector field via the Legendre transform. (Will finish explaining this next time. What I am explaining is similar to little McDuff-Salamon, section 1.1, but with more details.)
- Finished explaining how classical mechanics can be reinterpreted in symplectic geometry, and gave a survey of why this is useful.
- Started to discuss the question of when a Hamiltonian vector field has a periodic orbit on a given regular level set.
Introduced contact manifolds, contact type hypersurfaces in symplectic manifolds, and the statement of the Weinstein conjecture. The first few pages of this survey may be helpful.
- Discussed contact manifolds more. Digressed to review connections on principal S^1-bundles.
- Review of the first Chern class. For details of the obstruction theory definition see these lecture notes. For the definition in terms of curvature see the appendix to Characteristic Classes by Milnor and Stasheff (although they use affine connections, while I was using connections on principal bundles).
- Statement of (two versions of) the Arnold conjecture on fixed points of Hamiltonian symplectomorphisms.
- Definition of Lagrangian submanifolds and basic examples. Statement of Arnold's "nearby Lagrangian conjecture".
- More about the nearby Lagrangian conjecture.
- Legendrian knots.
A bit more of an introduction to contact geometry including the notions of tight and overtwisted contact structures and strong symplectic cobordisms.
- Introduction to the Moser method.
- The "relative Moser theorem" and applications: Darboux's theorem, the symplectic neighborhood theorem, and the Lagrangian neighborhood theorem. (I didn't have time to prove the latter, but it is explained in the textbooks.)
Applications of the Moser method to contact geometry: Gray's stability theorem and the Darboux theorem for contact forms.
- Started on Hamiltonian S^1-actions.
- Hamiltonian torus actions and Delzant's theorem.
- Symplectic blowups and blowdonws. The approach I used for this is a special case of "symplectic cutting", which you might want to look up.
- Started discussing almost complex structures.
- Showed that the space of almost complex structures compatible with a given symplectic form is contractible.
- Started discussing holomorphic curves, showed that their Riemannian area is equal to their "symplectic area".
- Showed that a holomorphic curve minimizes area in its homology class.
- Explained how Gromov's nonsqueezing theorem follows from the existence of a certain holomorphic curve, via the monotonicity lemma for minimal surfaces.
- Stated some basic facts about holomorphic curves, which imply the existence of the holomorphic sphere needed to prove Gromov nonsqueezing. Will go into more (but not total) detail about some of these facts starting next time.
- Defined what it means for a holomorphic curve (with fixed domain) to be regular, and sketched why regularity implies that the moduli space is an oriented manifold and what the dimension is.
- Sketched the proof that all somewhere injective holomorphic curves (with a fixed domain) are regular if J is generic. For a detailed proof see big McDuff-Salamon, Chapter 3, and see the appendices for proofs of the various analytic facts that are needed.
- Proved an automatic regularity criterion, and said more about the proof of Gromov nonsqueezing.
- We will start on some new applications of holomorphic curves to symplectic geometry after spring break.
- Stated intersection positivity and adjunction formula for holomorphic curves in four dimensions, and proved easy cases. See big McDuff-Salamon for the general case.
- Stated Gromov's theorem on recognition of R^4 and started proving it.
- Finished discussing the theorem on the recognition of R^4 and some of the facts that go into its proof.
- Discussed Gromov's theorem that there does not exist a compact exact Lagrangian submanifold of C^n.
- Brief review of Morse homology.
- Introduced the symplectic action functional, whose critical points correspond to fixed points of a Hamiltonian symplectomorphism.
Talked about gradient flow lines of the symplectic actional functional, and outlined the strategy for proving the Arnold conjecture for fixed points of nondegenerate Hamiltonian symplectomorphisms.
Introduced the "index=spectral flow" principle.
- Explained the linearization of Floer's equation and the Conley-Zehnder index.
- Outlined the definition of Hamiltonian Floer homology when pi_2(M)=0. For more details about Hamiltonian Floer homology, see D. Salamon, Lectures on Floer homology (available on his webpage).
- Explained the Salamon-Zehnder proof that Hamiltonian Floer homology agrees with Morse homology, which implies the Arnold conjecture on fixed points of nondegenerate Hamiltonian symplectomorphisms.
- Started explaining how to generalize Hamiltonian Floer homology to monotone symplectic manifolds.
- Finished outlining how Hamiltonian Floer homology generalizes to the monotone case, and briefly indicated the difficulties with the general case.
Started talking about cylindrical contact homology.
- Introduction to cylindrical contact homology. The introduction was much too brief due to lack of time, but next semester I will be organizing a 290 seminar where we will have plenty of time to discuss this (and other kinds of contact homology) and their applications in detail.