MATH 665
“We discuss the relationship between representations of linear groups over finite and \(p\)-adic fields, a part of Lie theory, and isotopy invariants of knot and links, a part of geometric topology. The bridge is the theory of Hecke algebras and their cocenters.”
Time: TTh, 2:30–3:45 PM
Place: 17 Hillhouse Ave, Room 03 (basement)
pdf Syllabus
- pdf 0. Introduction
- pdf 1. Algebraic groups, Frobenius maps, finite groups of Lie type
- pdf 2. Borel subgroups, the flag variety
- pdf 3. Deligne–Lusztig varieties
- pdf 4. Étale cohomology, Deligne–Lusztig induction
- pdf 5. Fixed-point formulas
- pdf 6. Lusztig series
- pdf 7. Iwahori–Hecke algebras
- pdf 8. Knots & links, braid groups, link polynomials
- pdf 9. Symmetric algebras and their traces
- pdf 10. Quantum Schur–Weyl duality and Temperley–Lieb algebras
- pdf 11. The Kazhdan–Lusztig basis and Temperley–Lieb revisited
- pdf 12. Categorification, constructible sheaves
- pdf 13. Active Learning: Soergel bimodules
- pdf 14. The constructible derived category, perverse sheaves
- pdf 15. Mixed complexes
- pdf 16. The Hecke category
- pdf 17. Triply-graded link homology
- pdf 18. The horocycle correspondence, unipotent character sheaves
- pdf 19. Springer theory
- pdf 20. Markov traces via character sheaves
- 21. Affine Springer fibers
- 22. Plane algebraic curves, compactified Picard schemes
- 23. Braid varieties and their Springer theory
Problem Sets
| 0 | due 10/10 for undergraduates | |
| 1 | due 9/19 | |
| 2 | due 10/17 | |
| 3 | due 11/14 | |
| 4 | due 12/5 |