Dragan Milicic and Geordie Williamson present this course which ran in Semester Two, 2023
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, study the equations which it satisfies and try relate these equations on different spaces. D-modules 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 (following some famous notes of Bernstein). The localisation 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.
Seminar information and recordings can be found below and in the D-Modules YouTube playlist.
Lecture One ‘Introduction’
Geordie Williamson, 11 August 2023
Lecture Two ‘Sheaves of differential operators’
Alan Stapledon, 18 August 2023
Lecture Three ‘Connections and Functors’
Geordie Williamson, 25 August 2023
Lecture Four ‘Pullbacks, lefts and rights’
Part one: Geordie Williamson, Part two: Christ Hone, 1 September 2023
Lecture Five ‘Borel-Weil-Bott and Localisation’
Dragan Milicic, 8 September 2023
Lecture Six ‘The Localisation Theorem’
Dragan Milicic, 15 September 2023
Lecture Seven ‘Borel-Weil-Bott and Localisation’
Dragan Milicic, 22 September 2023
Lecture Eight ‘Borel-Weil-Bott Theorem’
Chris Hone, 22 September 2023
Lecture Nine ‘Symplectic stuff, Singular support and Holonomicity’
Geordie Williamson, 22 September 2023
Lecture Ten
Dragan Milicic, 20 October 2023
Lecture Eleven
Dragan Milicic, 27 October 2023
Lecture Twelve
Dragan Milicic, 3 November 2023
Lecture Thirteen
Dragan Milicic, 10 November 2023
Lecture Fourteen
Dragan Milicic, 17 November 2023
Lecture Fifteen
Dragan Milicic, 24 November 2023
Lecture Sixteen
Dragan Milicic, 24 November 2023
D-Modules Workshop Day One
Dougal Davis & Dragan Milicic, 26 September 2023
Abstract: 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. Watch the recording of Session 1 and Session 2.
D-Modules Workshop Day Two
Dougal Davis, 27 September 2023
D-Modules Workshop Day Three
Dougal Davis & Behrouz Taji, 28 September 2023
Abstract: 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. Watch the recording of Session 1 and Session 2.