Dima Arinkin (Spring 2018):
Introduction to Algebraic D-modules
D-modules provide an algebraic formalism for the study of (systems of) linear partial differential equations. This is similar to the relation between vector spaces and linear equations. By definition, a D-module is a module over the ring of differential operators; one can study the D-modules using the methods of algebraic geometry. The theory of D-modules found numerous applications in many parts of mathematics, such as cohomology of singular spaces, Hodge theory, and (geometric) representation theory.
The course is an introduction to the theory of algebraic D-modules, including direct/inverse image of D-modules, singular support of D-modules, holonomic D-modules, and the Riemann-Hilbert correspondence. To make this abstract ideas easier to understand, I plan to start by studying the Weyl algebra (of differential operators with polynomial coefficients), and then using modules over this algebra as a source of concrete examples.
For the most part, the course should be reasonably self-contained. While we will use some algebraic geometry, most spaces that appear in the course are smooth varieties, which quite often can be assumed affine. Basic homological algebra is sometimes helpful, but I doubt we will require anything deep.