A beginner's introduction to Fukaya categories

Abstract

The goal of these notes is to give a short introduction to Fukaya categories and some of their applications. The first half of the text is devoted to a brief review of Lagrangian Floer (co)homology and product structures. Then we introduce the Fukaya category (informally and without a lot of the necessary technical detail), and briefly discuss algebraic concepts such as exact triangles and generators. Finally, we mention wrapped Fukaya categories and outline a few applications to symplectic topology, mirror symmetry and low-dimensional topology. This text is based on a series of lectures given at a Summer School on Contact and Symplectic Topology at Universit\'e de Nantes in June 2011.