Reading Seminar:

Geometric Invariant Theory and Nakajima's Quiver Varieties

The seminar takes place on Fridays from 10am - 12am in the seminar room at the MPIM.

List of talks:

The current distribution of talks is given below.

Invariant Theory:

1. Classical Invariant Theory: Marc

  Goal: Present examples of classical invariant theory of matrices and quiver representations, [Schmitt, Section 1.3]

2. Quotients: Thorge

  Goal: Definition and Constructions of quotients, [Schmitt, Section 1.4]

3./4. Stability and Hilbert-Mumford Criterium: Jacinta / Joanna

  Goal: Definition of stability and proof of the criterium, [Schmitt, Section 1.5.1]

5. Quiver Moduli: Gustavo

  Goal: Present quiver moduli following King, [Schmitt, Section 1.5.1], [King]

6. Classical Moduli Problems: Hans

  Goal: Explain one or two classcial moduli problems, [Mumford, Chapter 4]

Quiver Varities:

7. Symplectic Geometry: David

  Goal: Explain hamiltonian reduction and relation to GIT quotients, [Ginzburg, Section 4.1], [Kirwan, Chapters 6-8]

8. Lusztig's Quiver Varieties I: Johannes

  Goal: Explain definitions and first properties, see Theorem 4.5.6 in Ginzburg, [Ginzburg, Section 4.2-4.5]

9. Lusztig's Quiver Varieties II: Catharina / Sheng

  Goal: Explain the construction of quantum groups using quiver varieties, [Lusztig]

10./11. Nakajima's Quiver Varieties I: Bea / Arik

  Goal: Explain definitions and first properties, see Theorem 5.2.2 in Ginzburg, [Ginzburg, Section 5.1-5.3]

12. Nakajima's Quiver Varieties II: Dyhan

  Goal: Explain the construction of the lagrangian subvarieties and their properties, [Ginzburg, Section 5.4]

13. Convolution: Tomasz

  Goal: Explain convolution both of functions and Borel-Moore homology, [Ginzburg, Section 6]

14. Nakajima's Quiver Varieties III: Michael

  Goal: Explain constructions of Lie algebras and highest weight modules using quiver varieties, [Ginzburg, Section 7]

References:

The following is a list of references for the seminar.

[Ginzburg] V. Ginzburg, Lectures of Nakajima's Quiver Varieties

[King] A.D. King, Moduli of Representations of Finite Dimensional Algebras

[Kirwan] F.C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry

[Lusztig] G. Lusztig, Constructible functions on varieties attached to quivers

[Schmitt] A.H.W. Schmitt, Geometric Invariant Theory and Decorated Principal Bundles