Definition of Symplectic (co)homology

Wiki MarkupYour main references should be Section 2-3 of \ [Seidel\], skipping 2b and 3f. You will have to supplement this reading with other papers in order to be able to give a coherent series of talks. I’ve indicated some ways to do this, but you should feel free to come up with your own.

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1. Symplectic geometry of Liouville domains and Liouville manifolds. The notion of Liouville domain is essentially equivalent to that of “symplectic manifold with restricted contact boundary.” See the introduction to Section 3 of \ [Wendl\] for an evenly paced discussion. For more context, look at \ [Eliashberg-Gromov\], where you should beware of Lemma 2.4.1.
2. Wiki MarkupConley-Zehnder index. This is defined for paths in the symplectomorphism group of C^n in Section 2.4 of \ [Salamon\]. Given the choice of trivialisation of the tangent space of a symplectic manifold along a Hamiltonian orbit, define the associated index by linearisation (see the first paragraph of Section 2.6 in \ [Salamon\]). Make sure to explain that the mod 2 value of the index does not depend on the trivialisation. Given a compatible almost complex structure, show that a trivialisation of the top exterior power of the cotangent space as a complex vector bundle (i.e. a complex volume form) induces a trivialisation of the restriction of the tangent space to any loop.unmigrated-wiki-markup
3. Hamiltonian Floer theory. First, define the theory for “symplectically aspherical manifolds,” using the conventions of Section (3b) of \ [Seidel\] for action, for the Floer equation, and for the differential. One way to do this is to go through Section (1.2) of \ [Oancea\], making sure to switch some signs. For context, it might be helpful to take a look at \ [Salamon\], especially the section on Transversality. Draw lots of pictures, but don’t mistake them for proofs. Discuss the maximum principle for holomorphic curves curves (e.g. Lemma 1.4 in \ [Oancea\]).unmigrated-wiki-markup
4. Continuation maps in Floer theory. Expand on the last two paragraphs of Section 3b in \ [Seidel\]. This will probably require you to go back to Section 3.4 of \ [Salamon\], but make sure to keep using Seidel’s conventions for the Floer equation. You will need a version of the maximum principle that applies for these Equations. This is dicussed in the second part of Section (3c) of \ [Seidel\]. Alternatively, you can look at Proposition 4.1 of \ [Wendl\], but note that the sign on the s-derivative of the Hamiltonian is different (this is because of the different conventions for defining the Floer equation and the differential).
5. Wiki MarkupQuantitative symplectic (co)-homology: This is discussed in Section 6.6 of \of [Hofer-Zehnder\]. It might help to also take a look at Section 2 of \2 of [Wendl\], and Section 1.3.2 of \of [Oancea\]. Make sure you include a discussion of a sample application. * Warning: Make sure you understand the issue brought up by Oancea in the last sentence of Section 1.3.2.*
6. Wiki MarkupDefinition of Symplectic (co)-homology as a direct limit over linear Hamiltonians as discussed in Section 3d of \ [Seidel\].