Floer Homology

Arnold’s conjecture. The action functional. The L^2-gradient of the action functional: Floer equation. Compactness of the space of solutions. The Conley-Zehnder index. Linearization of Floer equation and transversality. Somewhere injective solutions. The Fredholm property. Computation of the Fredholm index of the Floer operator. Exponential decay. Broken trajectories. Gluing of trajectories. Elliptic regularity of the Floer operator. The Floer homology of an autonomous C^2-small Hamiltonian is isomorphic to the Morse homology. Invariance of Floer homology. Additional topics (time permitting): Floer homology of cotangent bundles, symplectic homology, equivariant symplectic homology, symplectic field theory.

Referências:
M. Audin, M. Damian, “Morse theory and Floer homology”, Universitext. London: Springer; Les Ulis: EDP Sciences (ISBN 978-1-4471-5495-2/pbk; 978-1-4471-5496-9/ebook). xiv, 596 p. (2014).
D. Salamon, “Lectures on Floer homology”, Eliashberg, Yakov (ed.) et al., Symplectic geometry and topology. Lecture notes from the graduate summer school program, Park City, UT, USA, June 29-July 19, 1997. Providence, RI: American Mathematical Society. IAS/ Park City Math. Ser. 7, 145-229 (1999).
F. Laudenbach, “Symplectic geometry and Floer homology”, pp. 1–50. Sociedade Brasileira de Matemática (2004).
M. Schwarz, “Morse homology”, Progress in Mathematics (Boston, Mass.). 111. Basel: Birkhäuser Verlag. ix, 235 p. (1993).
D. McDuff, D. Salamon, “J-Holomorphic Curves and Symplectic Topology”. American Mathematical Society Colloquium Publications, vol. 52. Am. Math. Soc., Providence (2004).