Math 595 Intro to Nakajima Quiver Varieties, Fall 2018
MWF, 12-12:50 p.m., 243 Altgeld Hall
357 Altgeld Hall
nevins AT illinois DOT edu
Office hours: To be announced; and by appointment.
Some Suggested Texts:
- Quiver varieties and Kac-Moody algebras by H. Nakajima, here. [Note: this and its predecessor are the fundamental references but they can be challenging to read!]
- Quiver representations and quiver varieties by A.A. Kirillov Jr., here.
- Lectures on Nakajima's quiver varieties by V. Ginzburg, here.
- Moduli of representations of finite dimensional algebras by A. King, here.
- On the geometry of symplectic resolutions by V. Ginzburg, here.
There will be some assigned reading and homework but it will be optional. Grading will be very gentle, but if you want to receive an A+ you should plan to complete some homework.
Tentative Course Plan:
- A basic example. Rough outline of topics of the course.
- Affine and GIT quotients of affine varieties.
- Hamiltonian reduction.
- Combining the two (Hamiltonian reduction and GIT). A few basic examples.
- Symplectic singularities and symplectic varieties. Poisson deformations??
- Definition of Nakajima quiver varieties. A bunch of examples.
- Quivers, the path algebra, and the preprojective algebra. Moduli of representations.
- Common features of Nakajima quiver varieties (conical action, symplectic resolution, paving, recursive structure?).
- Tautological bundles/classes. Cohomology of quiver varieties. Kirwan surjectivity.
- Nakajima operators, Nakajima action. Basic theorems.
- Hyperkahler quotients and hyperkahler metric?
- A host of other possibilities, if there is any time! Categorical actions; or multiplicative quiver varieties; or Cherkis bow varieties; or ...
Lecture Notes: [wk1][wk2][wk3][wk4][wk5][wk6]
Homework: [hw1 due 9/17/18] [hw2 due 10/23/18]
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