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Wiki Markup 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.
Wiki Markup Conley-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.
Wiki Markup 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. Ignore all issues about grading.Wiki Markup Maximum principle for pseudo-holomorphic maps. Review the statement of the Hopf maximum principle. Prove the Discuss the maximum principle for ordinary holomorphic curves (e.g. Lemma 1.4 in \[Oancea\]). Then do the more general example appearing in the first half of Section (3c) of \[Seidel\] (up to the paragraph that starts with “Analogously, one can consider solutions”).
Wiki Markup 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).Wiki Markup Quantitative symplectic (co)-homology: This is discussed in Section 6.6 of \[Hofer-Zehnder\]. It might help to also take a look at Section 2 of \[Wendl\], and Section 1.3.2 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.*
Wiki Markup Definition of Symplectic (co)-homology as a direct limit over linear Hamiltonians as discussed in Section 3d of \[Seidel\].