Monodromy invariants in symplectic topology

Abstract

This text is a set of lecture notes for a series of four talks given at I.P.A.M., Los Angeles, on March 18-20, 2003. The first lecture provides a quick overview of symplectic topology and its main tools: symplectic manifolds, almost-complex structures, pseudo-holomorphic curves, Gromov-Witten invariants and Floer homology. The second and third lectures focus on symplectic Lefschetz pencils: existence (following Donaldson), monodromy, and applications to symplectic topology, in particular the connection to Gromov-Witten invariants of symplectic 4-manifolds (following Smith) and to Fukaya categories (following Seidel). In the last lecture, we offer an alternative description of symplectic 4-manifolds by viewing them as branched covers of the complex projective plane; the corresponding monodromy invariants and their potential applications are discussed.