D-Modules, summer 2016

Organization

Time	Monday, 10:15–11:45 April 18 until July 18
Location	FU Berlin, A3/SR210
Organizers	Elena Lavanda Wouter Zomervrucht

Here is the program.

The topic of this seminar is D-modules. These can be seen as the algebraic version of partial differential equations. There is a striking dichotomy between the characteristic 0 and the characteristic p settings. In complex geometry, the Riemann–Hilbert correspondence is a dictionary between D-modules and local systems, or representations of the fundamental group. In particular, all D-modules on a proper smooth complex variety X are trivial if the fundamental group of X is trivial. We will prove and explain these statements. Then, still in characteristic 0, we make a little detour into derived categories and study some functoriality for D-modules, as well as the Gauss–Manin connection. At the end of the seminar we move to characteristic p. An analogue of the Riemann–Hilbert correspondence was conjectured by Gieseker and recently proven by Esnault and Mehta. That proof is far beyond our scope. Instead we will study various examples of the phenomenon.

Schedule

The first talk will be on Friday, April 22, at 10:15–11:45, in room A3/SR210 as usual.

April 22	Wouter Zomervrucht	Introduction (notes).
April 25	Marco d'Addezio	The sheaf of differential operators (notes).
May 2	Yun Hao	D-modules and connections (notes).
May 9	Marcin Lara	The tannakian category of D-modules (notes).
May 16		—no seminar—
May 23	Pedro Ángel Castillejo	The Riemann–Hilbert correspondence (notes).
May 30	Fei Ren	Representations of the fundamental group (notes).
June 6*	Mara Ungureanu	The derived category of D-modules (notes).
June 13	Elena Lavanda	Functoriality of D-modules.
June 20		—no seminar—
June 27	Efstathia Katsigianni	The Gauss–Manin connection (notes).
July 4	Wouter Zomervrucht	F-divided bundles (notes).
July 11	Yun Hao	The Gieseker conjecture.
July 18	Tanya Kaushal Srivastava	D-modules on K3 surfaces and unirational varieties.

*On June 6 we start at 10:00 sharp!

Literature

Here is the list of references.

Barrientos, The Gauss–Manin connection and regular singular points.
Borel et al., Algebraic D-modules.
Coutinho, A primer of algebraic D-modules.
Deligne, Équations différentielles à points singuliers réguliers.
Deligne–Milne, Tannakian categories.
Esnault–Mehta, Simply connected projective manifolds in characteristic p > 0 have no nontrivial stratified bundles.
Esnault–Sun, Stratified bundles and étale fundamental group.
Gaitsgory, Geometric representation theory.
Gieseker, Flat vector bundles and the fundamental group in non-zero characteristics.
Grothendieck, Représentations linéaires et compactification profinie des groupes discrets.
Hartshorne, Algebraic geometry.
Hotta–Takeuchi–Tanisaki, D-modules, perverse sheaves, and representation theory.
Huybrechts, Lectures on K3 surfaces.
Kindler, Two lectures on regular singular stratified bundles.
Lenstra, Galois theory for schemes.
Liu, Algebraic geometry and arithmetic curves.
Osserman, Connections, curvature, and p-curvature.
Sabbah, Introduction to the theory of D-modules.
Serre, Géométrie algébrique et géométrie analytique.
Szamuely, Galois groups and fundamental groups.
Weibel, An introduction to homological algebra.