- Enseignant-e: Alexander Braverman
Prerequisites: basic undergraduate algebra (rings, modules etc.) for
the first half of the course; very basic algebraic geometry (algebraic
varieties and coherent sheaves) during the 2nd half.
Course description: We are going to start by discussing a very
elementary "analytic" question asked by I.M.Gelfand in the 50's: the
question is whether certain integrals depending on a parameter have a
meromorphic continuation with respect to that parameter. We shall then
explain J.Bernstein's purely algebraic approach to this question - it
is based on some deep properties of the algebra D of (linear)
differential operators in several variables with polynomial
coefficients
and modules over it. We shall introduce a very important concept of
holonomic D-module and use it to resolve Gelfand's question. After
that we'll continue to study modules over this algebra (emphasizing
the connection between D-modules and systems of linear differential
equations with polynomial coefficients) .
In the 2nd half of the course we'll generalize the above discussion to
the case when D is replaced by the sheaf of differential operators on
an arbitrary smooth complex algebraic variety. We'll study such things
as inverse and direct images of D-modules. This will require workingwith derived categories - we'll devote several lectures to the
definition and basic properties of this notion.
In the very end of the course we'll have a brief discussion of the
notion of differential equations with regular singularities and the
so-called Riemann-Hilbert correspondence.