D-Modules

Course Description:

D-modules provide an algebraic language for studying systems of linear partial differential equations. In some sense the idea goes back to Riemann: rather than studying some complicated function directly, we study the equations which it satisfies and try to relate these equations on different spaces. D-module have found major applications in representation theory, algebraic analysis, algebraic geometry and mathematical physics. This course will introduce the basics of the theory of algebraic D-modules (initially following some famous notes of Bernstein). The localization theorem of Beilinson and Bernstein will be discussed in detail. We will then move on to other applications in representation theory. Namely we hope to explain how certain old results of Harish-Chandra and Langlands become quite transparent in the language of D-modules.

Course Info

Time: 2.30 – 4.30ish each Friday until the end of November.

Location: S223 (main quad, Sydney uni).

Lecturers: Dragan Milicic, Geordie Williamson and (possibly) others.

Zoom link, password is XAlg where X is the last name of lie theorist Hermann W.

There is also a slack channel for the course, please join via the following link.

Each lecture will be recorded and added to the YouTube playlist on the SMRI YouTube channel.

We will also have a short workshop in the semester break featuring (last week of September) lectures by Kari Vilonen and Dougal Davis, who have recently used D-modules and mixed Hodge modules to make progress on fundamental problems in the representation theory of real Lie groups.

Notes:

Lecture 1: Introduction (Williamson). Note there is a typo on page 3, should be for lambda not in Z.

Lecture 2: Sheaves of differential operators (Stapledon).

Lecture 3: Connections and functors (Williamson).

Lecture 4: Pullbacks, lefts and rights (Williamson, Hone).

Lecture 5 (Chapter 2): Borel-Weil-Bott and Localisation (Milicic).

Lecture 6 (Chapter 2): The Localisation Theorem (Milicic).

Lecture 7: Borel-Weil-Bott and Localisation (Milicic).

Lecture 8: Borel-Weil-Bott Theorem (Hone).

Lecture 9: Symplectic stuff, Singular support and Holonomicity (Williamson).

Resources:

Kashiwara, D-modules and microlocal calculus, Transl. Math. Monogr., 217 Iwanami Ser. Mod. Math. American Mathematical Society, Providence, RI, 2003, xvi+254 pp.

Bernstein, Algebraic Theory of D-modules.

Milicic, Lectures on Algebraic Theory of D-Modules.

Milicic, Localisation and Representation theory of Reductive Lie groups.

``Topics in algebraic geometry": D-modules course by Christian Schnell

https://www.math.stonybrook.edu/~cschnell/mat615/

``D-modules and singularities": D-modules course by Mircea Mustata

http://www-personal.umich.edu/~mmustata/DmodulesNotes.pdf

``The rising sea: foundations of algebraic geometry" by Ravi Vakil

http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf

Conference information:

The week of 26-28th of September there was a small D-modules conference.

The speakers and some abstracts are as follows:

Dougal Davis (University of Melbourne) and Kari Vilonen (University of Melbourne):

In these talks, we will explain some very recent results on mixed Hodge modules and the unitary dual of a real reductive Lie group. (With a little luck, the ink will have dried and there will be a preliminary version on the arXiv by the time the workshop starts.) The main idea behind our work is to upgrade Beilinson-Bernstein localisation from D-modules to mixed Hodge modules, following a proposal made by Schmid and Vilonen over 10 years ago. When it applies, this endows everything in sight with a canonical filtration, the Hodge filtration, which we prove has some extremely nice properties, such as cohomology vanishing and global generation. In the context of real groups, we also prove that the Hodge filtration “sees” exactly which representations are unitary. We hope that this will lead to new progress on the very old problem of determining the unitary dual of a real group. We’ll do our best to put this problem in context, explain what our theorems say, and give the main ideas behind the proof.

Dragan Milicic (University of Utah)

Behrouz Taji (University of NSW):

My aim in this talk is to discuss how various discoveries in the theory of variation of Hodge structures, including Saito’s Hodge modules, can be used to establish a striking geometric fact (originally due to Arakelov, Migliorini and Kovács): Over C* the only smooth projective family of curves of genus higher than 1, or more generally canonically polarized complex manifolds, is the isotrivial one. This talk is roughly based on works of Popa-Schnell and myself joint with Kovács.

The schedule was:

Tuesday

1-2: Dougal I.

2-3: Tea at SMRI

3-4: Dragan

Wed:

1.30-2.30: Dougal 2.

2.30-3: quick tea at SMRI

3-4: Kari

Thursday:

1-2: Dougal 3.