### Math 595 Intro to Nakajima Quiver Varieties, Fall 2018

MWF, 12-12:50 p.m., 243 Altgeld Hall

__My information__:

Thomas Nevins

357 Altgeld Hall

217.265.6762

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]
[wk7]
[wk8]

**Homework:** [hw1 due 9/17/18] [hw2 due 10/23/18]

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