This is a preliminary syllabus for the Summer School in Geometric Representation Theory. As Eisenhower once said: “Plans are worthless, but planning is everything,” so don’t take this too literally, but I hope it will be a useful guide to what we’ll discuss.
Day 1 (Mon, June 19):
9:15-10:15 Lecture 1: General orientation: QFT, symplectic resolutions, knot invariants (BW) [Braverman & Finkelberg, Sec. 1.8, 4, Notes, Ch. 0, Webster-Yoo]
slides (pre-talk version) (after-talk version)
10:45-11:45 Lecture 2: Symplectic singularities and resolutions (Danilenko) [Kamnitzer, Section 2]
Lunch
2-3 Tutorial (equivariant cohomology, Poisson brackets, deformation quantization)
3:30-4:30 Lecture 3a: Convolution algebras and Springer theory (Hilburn) [Ginzburg, Sections 1 and 2, Notes, Ch. 2-3]
4:30-5:30 Office Hours
Day 2 (Tues, June 20):
9:15-10:15 Lecture 3b: Convolution algebras and Springer theory (Hilburn) [Ginzburg, Sections 1 and 2, Notes, Ch. 2-3]
10:45-11:45 Lecture 4: The affine Grassmannian and its equivariant cohomology (Krylov) [Notes, Ch. 5, BFM]
Lunch
2-3 ****Lecture 5: The BFN definition of Coulomb branches (Hilburn) [Kamnitzer, Sec. 7, Braverman & Finkelberg, Sec. 5, Notes, Ch. 6]
3:30-4:30 Tutorial (more fun with BFN)
4:30-5:30 Office Hours
Day 3 (Weds, June 21):
9:15-10:15 Lecture 6: Examples of Coulomb branches (BW)
slides (pre-talk version) (after-talk version)
10:45-11:45 Lecture 7: Planar KLRW algebras as generalized Springer theory (BW) [Notes, Ch. 4]
slides (pre-talk version) (after-talk version)
Lunch
2-3 Tutorial (Steinberg algebras)
3:30-4:30 Lecture 8: Cylindrical KLRW algebras as Coulomb branches of quiver gauge theories (BW) [Notes, Ch. 5,7]
slides (after-talk version)
4:30-5:30 Office Hours
Day 4 (Thurs, June 22):
9:15-10:15 Lecture 9: Tilting generators, localization for sheaves of algebras (Danilenko) [Kaledin, Sec. 3.1]
10:45-11:45 Lecture 10: Char p quantizations of Coulomb branches and tilting bundles (BW) [Coherent sheaves and quantum Coulomb branches I]
slides (pre-talk version) (after-talk version)
Lunch
2-3 Tutorial (cylindrical KLRW algebras and tilting bundles)
3:30-4:30 Lecture 11: Knot homology introduction (BW)
slides (pre-talk version) (after-talk version)
4:30-5:30 Office Hours
Day 5 (Fri, June 23):
9:15-10:15 Lecture 12: Aganagic approach to knot homology overview (LePage) [Aganagic ICM]
10:45-12 Lecture 13: Interpretation of Aganagic’s homology in terms of tilting bundles (BW) [Coherent sheaves and quantum Coulomb branches II]
slides (pre-talk version) (after-talk version)
Goodbye!
These are roughly ranked from most to least essential:
Equivariant cohomology, mainly for tori and GL_n (Notes, Ch. 1)
Convolution in Borel-Moore homology (Ginzburg, Sections 1 and 2, Notes, Ch. 2)
Homological algebra and derived categories (general notions, doesn’t have to be deep) (for a quick introduction, this paper of Thomas; if you want lots of details, Yekuteli’s book)
Symplectic forms and Poisson brackets (Ch. 1, Loja Fernandes and Marcut)
GIT quotients (Proudfoot, Sec. 4)
Notes from a previous seminar, currently being edited for this summer school as well:
Here on Overleaf. Ambitious students are welcome to edit or to add comments/chats
More on 3-d mirror symmetry and connections to QFT:
A survey article written by Webster and Yoo for the Notices of the AMS: Webster-Yoo
Notes from a mini-course on Coulomb branches by Braverman-Finkelberg: Braverman-Finkelberg