## Fall 2016 - Taming moduli problems

I will be teaching a graduate topics course on "taming moduli problems in algebraic geometry."

CourseNo: MATHGR6263_001_2016_3
Meeting Time: TR 10:10A-11:25A

Algebraic geometry has a very rich and sophisticated language for discussing moduli problems, the language of algebraic stacks. However describing a moduli problem as an algebraic stack is never the end of the story. We will cover various methods for understanding the detailed structure of moduli problems, with an emphasis on the example of the moduli of G-bundles on a curve, where G is a reductive group. We will also discuss closely related moduli problems, such as the moduli of Higgs bundles. Topics covered, in order:

1. Discussion of algebraic stacks, with an emphasis on examples
2. Geometric invariant theory, the kempf-Ness stratification, and good moduli spaces
3. The differential geometric perspective on GIT and the Kirwan-Ness theorem
4. The moduli of G-bundles over a curve and the relationship with Yang-Mills theory and loop groups
5. The verlinde formula, its deformations and generalizations

The main goal for the course is in section (5): to prove the Teleman-Woodward index formulas on the moduli of G-bundles over a curve: http://arxiv.org/abs/math/0312154. This is a pretty hard theorem, and will be the overall motivation for the background material above. We will also need some background on topological K-theory, so I will also give a crash course on topological K-theory.

Background: I will be assuming some facility with algebraic geometry (at the level of Hartshorne's book). Some familiarity with the representation theory of Lie groups and algebraic topology would also be helpful.

Mitchell Faulk will be typsetting notes, based on the lectures and my handwritten notes.

1. Lecture 09/06: overview; notes, latex notes
2. Lecture 09/08: linear algebraic groups and their representations; notes, latex notes
3. Lecture 09/13: functor of points formalism, group actions on schemes, Sumihiro; notes_a, notes_b, latex notes
4. Lecture 09/15: sumihoro, Bialynicki-Birula; notes, latex notes
5. Lecture 09/20: quotients, algebraic spaces; notes, latex notes
6. Lecture 09/22: more algebraic spaces, principal bundles; latex notes
7. Lecture 09/27: groupoids and equivariant quasi-coherent sheaves; notes, latex notes
8. Lecture 09/29: fibered categories, descent, and stacks; notes, latex notes
9. Skipped (for now): Artin's criteria, mapping stacks
10. Skipped (for now): deformation theory
11. Lecture 10/4: more on stacks; notes, latex notes
12. Lecture 10/6: geometric stacks; notes_a, notes_b, latex notes
13. Lecture 10/20: pushforward of quasi-coherent sheaves, good moduli spaces; latex notes
14. Lecture 10/25: properties of good moduli spaces; latex notes
15. Lecture 10/27: construction of good moduli spaces; latex notes
16. Lecture 11/01: Hilbert-Mumford criterion; latex notes
17. Lecture 11/03: Hilbert-Mumford examples; notes, latex notes
18. Lecture 11/10: Kempf Ness theorem; notes, latex notes
19. Lecture 11/15: main theorem of GIT revisited, cohomology of topological stacks
20. Lecture 11/17:
1. Regular lecture: Introduction to G-bundles over a curve
2. Make up for 10/11 (Rm 622, 4:30-6PM)
21. Lecture 11/21:
1. Regular lecture
2. Make up for 10/13 (Rm 622, 3-4:30PM)