Geometry of Quiver Varieties
Graduate Study Seminar
12-2pm Wednesdays, Fall 2000
939 Evans Hall

Schedule

(LNA=LECTURE NOTES AVAILABLE)
  • September 6 : Tamás Hausel, Why quiver varieties? (LNA)
  • September 13 : Marco Zambon, Holonomy groups, Kähler, Calabi-Yau and hyperkähler manifolds (LNA)
  • September 20 : Allen Knutson, Hyperkähler reductions, ALE spaces and McKay correspondence
  • September 27 : Matthew Szczesny, Kac-Moody algebras, quivers and representations
  • October 4 : Tamás Hausel, Instantons on ALE spaces: generalized ADHM constructions (LNA)
  • October 11 : Valerio Toledano, Classical representation theory of quivers (LNA)
  • October 18 : Kevin Purbhoo, Geometric Invariant Theory and moduli spaces of representations of quivers (LNA)
  • October 25 : Noam Shomron, Representations of algebras using quivers (LNA)
  • November 1 : Allen Knutson, Honeycombs and "multiple flag varieties of finite type" (LNA)
  • November 8 : Mark Haiman, Hilbert schemes as quiver varieties (LNA)
  • November 15 : Bernd Sturmfels, Toric quiver varieties (LNA)
  • November 22 : Raphael Rouquier, Springer representations of Weyl groups (LNA)
  • November 29 : Matthew Szczesny, The fermionic form
  • December 13 : Tamás Hausel, Cohomology ring of quiver varieties

    News

  • (Added 2006: A general formula for Betti numbers of quiver varieties has been found in here.)
  • The scanned notes of the 12th seminar are available too.
  • The scanned notes of 7th talk and the texed slides of the 8th are also available.
  • The texed notes of the 11th seminar are already available.
  • The texed lecture notes of the 9th and the scanned notes of 10th are also available.
  • The scanned lecture notes of the 5th and 6th talks are here and here, respectively.
  • The scanned lecture notes and slides of the second seminar are also available.
  • The scanned lecture notes of the first introductory seminar are available here.
  • There exists a mailing list of the seminar. If you want to get onto it please contact me.
  • Lecturers are needed for giving seminars! Check out the schedule. If you are interested please contact me.

    About the seminar

    This seminar is intended to be a graduate study seminar (that is the lectures are presented mostly by graduate students), but open to anyone interested. The aim would be to read current research articles in subjects explained below, all related to the geometry and topology of Nakajima's quiver varieties and to establish the fundamental knowledge needed to do research in these areas. These may include elements from: Riemannian, symplectic, complex and algebraic geometry, algebraic and differential topology and global analysis and possibly representation theory and low dimensional topology. The proposed research problems many times are related to contemporary theoretical high energy physics, namely quantum field theories and string theory. The subject is chosen to be as narrow as possible to be attackable by a graduate student but broad enough to provide a wealth of worthy research problems.

    The problems

    The problems all concern the geometry and topology of Nakajima's quiver varieties [95i:53051] which provide a huge number of examples of non-compact, complete, smooth hyperkähler manifolds (or holomorphic symplectic algebraic varieties in the language of algebraic geometry) related to a combinatorial object called quiver.

    1. Calculate the L^2 cohomology of quiver varieties! This problem is motivated by Sen's conjecture [ hep-th/9402032] which predicts the L^2 cohomology of the hyperkähler moduli space of magnetic monopoles on R^3 using S-duality in string theory. The problem involves understanding the metric structure of the quiver variety near infinity, and probably related to intersection cohomology of Goresky and MacPherson [90e:55013].

    2. Calculate the Poincaré polynomial of quiver varieties! There is a recent conjecture by Lusztig [math.QA/0005010] which states that these Poincaré polynomials can be described with an expression coming from the representation theory of quantum groups. There are many possible ways to attempt to calculate the Poincaré polynomial of the hyperkähler quotient, but there is no general theory like for the symplectic quotients [ 86i:58050]. (Added 2006: A general formula for Betti numbers of hyperkahler quotients and in particular quiver varieties has been found in here.)

    3. Calculate (U(1)-equivariant) intersection numbers of quiver varieties! The idea of the calculation appeared in a paper by physicists [hep-th/9712241]. These numbers seem to be fundamental invariants of quiver varieties and the methods should be - yet unexplored - hyperkähler analogues of the Duistermaat-Heckman-type localizations in symplectic geometry. An interesting outcome of these calculations could involve calculations of (U(1)-equivariant) Rozansky-Witten invariants [hep-th/9612216] of quiver varieties, which would give a rather interesting connection to 3-manifold topology.

    Links

  • Hiraku Nakajima
  • SUMMER SCHOOL on Geometry of quiver-representations and preprojective algebras, 10 - 17 September 2000 Sussex - England
  • Online papers on quiver varieties

    Contact

    If you are interested and/or have questions or suggestions please contact Tamás Hausel in 837 Evans Hall, e-mail to hausel@math.berkeley.edu, phone 1-510-642-4573 or come to the first organizing seminar at 12pm on Wednesday, September 6 in 939 Evans.
    You are visitor number to this page since December 1 2000.