Floer homology in symplectic geometry
In 1965 Arnold posed his famous conjecture on the number of fixed points of Hamiltonian diffeomorphisms. Based on Gromov’s pseudoholomorphic curve (1985) and Witten’s QFT interpretation of Morse theory (1982), Floer (1988) invented the Floer homology and provided the most influential idea towards the Arnold conjecture. This idea soon lead to a series of landmark developments in different fields, including symplectic geometry, gauge theory, and low-dimensional topology. In this course, we will go through the rigorous construction of the Hamiltonian version of Floer homology. At the same time, you will learn the detailed techniques related to pseudoholomorphic curves, including Gromov compactness, transversality, and gluing.
Main reference: D. Salamon, Lectures on Floer homology. Supplementary reference: McDuff-Salamon, J-holomorphic curves and symplectic topology,
Differential Geometry 1 (532), Differential Topology (548), Partial Differential Equations I (517), or instructor’s approval.