Past lectures

Abstract

This course is intended to give an introduction to the subject of Floer homology for La- grangian submanifolds. This theory, introduced by Floer in the late 1980’s was originally created in order to prove the celebrated Arnold conjecture on Lagrangian intersections. However, it was gradually realized later on that Floer theory (and its later extensions), is more far reaching - it gives rise to powerful invariants of symplectic manifolds and their Lagrangian submanifolds. Consequently this theory led to various striking applications in different directions of symplectic topology and even outside of this field. Moreover, it turns out that Floer theory carries very rich algebraic structures, some of which have only recently begun to be explored.

Trying to keep the prerequisite knowledge to a minimum we shall begin the course with a rapid introduction to symplectic topology and list several motivating problems on symplectic manifolds and their Lagrangian submanifolds. We shall then go over necessary facts from Morse theory and after that introduce the main object of the course – Floer homology. We shall spend some time on the foundations and technical details involved in the construction of this homology theory, covering various geometric situations in which this theory works, especially in the presence of holomorphic disks. We shall also outline recent developments concerning new algebraic structures related to Floer theory and their relations to other symplectic invariants such as quantum homology.

The second half of the course will be devoted to applications, computations and exam- ples. In particular we shall present applications to Lagrangian intersections, to problems concerning the topology of Lagrangian submanifolds, isotopy problems, relative enumer- ative geometry as well as applications to classical algebraic geometry.

People attending the course are supposed to have some basic knowledge of algebraic topology and smooth manifolds. Knowledge of basic symplectic geometry and of Morse theory are useful but not 100% necessary.