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.Comment: 42 pages, notes of lectures given at IPAM, Los Angele