### Summary

## Courses

In Spring 2019 I am teaching a TCC course "Symplectic resolutions and singularities", meeting Thursdays 2-4 PM for eight weekly meetings commencing 17 Jan 2019.

Lecture notes written by me: Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lectures 5 and 6 (unedited), Lecture 7, Lecture 8.

Actual smartboard notes: Lecture 1, Lecture 2, Lecture 4, Lecture 5, Lecture 6 (after the CleverMaths crash near the end), Lecture 7, Lecture 8.

Here are the problems: Problem Sheet 1, Problem Sheet 2.

There will be a half-hour of office hour after lectures, Thursdays 4-4:30 PM, and let me know if you'd like to Skype in.

Click here to access the Slack discussion forum for this course.

This course will have (semi-)weekly coursework, mostly non-assessed, with a few exercises to be collected twice during the term. Here you may find the lecture notes and exercises.

Lecture 1: Introduction to geometric representation theory and the Borel--Weil and Beilinson--Bernstein theorems.

Lecture 2: (tentative syllabus:) Background on symplectic resolutions and singularities, D-modules, Poisson varieties, and quantization. Equivariant vector bundles and cohomology. Beilinson--Bernstein theorem and applications. D-affineness. Geometry of the nilpotent cone: nilpotent orbit closures, Kostant--Slodowy slices, finite W-algebras.

Tentative list of possible topics for remaining lectures: Cherednik and symplectic reflection algebras, Hilbert schemes of surfaces, Haiman's n! theorem. Symplectic reduction, hypertoric and quiver varieties, character varieties. Classification of symplectic resolutions. Deformations and quantizations of symplectic varieties, the period map, twistor quantizations, general localization theorems. Finiteness theorems and D-modules. Category O and Koszul duality. Symplectic duality. Coulomb branch varieties. Twisting and shuffling functors and autoequivalences. Quantum cohomology and quantum groups.