Lectures on D-modules

These are lecture notes of a course given at the University of Chicago in Winter 1998. The purpose of the lectures is to give an introduction to the theory of modules over the (sheaf of) algebras of algebraic differential operators on a complex manifold. This theory was created about 15-20 years ago in the works of Beilinson-Bernstein and Kashiwara, and since then had a number of spectacular applications in Algebraic Geometry, Representation theory and Topology of singular spaces. We begin with defining some basic functors on D-modules, introduce the notion of characteristic variety and of a holonomic D-module. We discuss b-functions, and study the Riemann-Hilbert correspondence between holonomic D-modules and perverse sheaves. We then go on to some deeper results about D-modules with regular singularities. We discuss D-module aspects of the theory of vanishing cycles and Verdier specialization, and also the problem of "gluing" perverse sheaves. We also outline some of the most important applications to Representation theory and Topology of singular spaces. The contents of the lectures has effectively no overlapping with Borel's book " Algebraic D-modules".