Algebraic Varieties with Group Actions
Org: Jaydeep Chipalkatti (Manitoba)
- MEGUMI HARADA, McMaster University, Department of Mathematics and
Statistics, 1280 Main Street West, Hamilton, Ontario,
An introduction to the topology of symplectic quotients
This talk will be an overview of modern symplectic-geometric
techniques which compute the topology of symplectic quotients. There
are many parallels with the algebraic-geometric theory of GIT
(Geometric Invariant Theory) quotients, and I will mention these as
time allows. First, I will give a very brief account of the
construction of symplectic quotients, starting from the data of a
Hamiltonian compact group action on a symplectic manifold. I will
then review the pioneering work of Kirwan, as well as that of her
collaborators and followers (Jeffrey-Kirwan, Tolman-Weitsman,
Goldin, to name a few) which allows us to compute the cohomology of
these symplectic quotients using equivariant techniques "upstairs"
on the original Hamiltonian space. The original work in this area
uses rational Borel-equivariant cohomology. Time permitting, I will
mention my recent work with collaborator Greg Landweber which
generalizes this "Kirwan package" to the case of integral
topological K-theory, thus incorporating torsion considerations.
- COLIN INGALLS, University of New Brunswick, Fredericton, NB, E3B 5A3
Central extensions of finite subgroups of GL2
We link the classification of two dimensional tame orders by Reiten
and Van den Bergh using AR quivers to the classification of M. Artin
with ramification data by using a third classification: Central
extensions of finite subgroups of GL2. This third classification
also classifies local smooth Deligne-Mumford stacks of dimension two
with cyclic generic stabilizer.
- KIUMARS KAVEH, Dept. of Math., Univ. of Toronto, 40 St. George St.,
String polytopes, G-C polytopes and geometry of flag and
The classical results in toric geometry (e.g. results of Bernstein,
Kushnirenko and Khovanskii) relate the geometry/topology of a toric
variety and the complete intersections inside it to the combinatorics
of the associated Newton polytopes. In this talk we show how one can
obtain the same results for the flag variety and the string polytopes.
More specifically we give formula for the intersection numbers of
divisors, (arithmetic and geometric) genus of complete intersections
as well as the Euler char. of a complete intersection in the flag
variety, in terms of the number of integral points and the volume of
the corresponding polytopes. The formula for the intersection numbers
of divisors is due to Brion-Alexeev. These results (partially)
generalize to the bigger class of spherical varieties.
This is joint work with A. G. Khovanskii.
String polytopes are the generalization of the classical
Gelfand-Cetlin polytopes assocaited to an irreducible representation
of GL(n,C), to any reductive group.
- JOCHEN KUTTLER, University of Alberta
On the Luna stratification of quotients
If X is an affine variety on which a reductive group G acts, the
invariant theoretic quotient X//G admits a finite stratification by
stabilizer subgroups. A natural questions is, whether this
stratification is connected to the geometry of X//G, e.g. whether
it is preserved by any automorphism of X//G. We study this question
for interesting families of representations of G, giving an
affirmative answer for example in the case of more than three copies
of the adjoint representation.
This is joint work with Zinovy Reichstein.
- HUGH THOMAS, University of New Brunswick, Fredericton, NB, E3B 5A3
(Co)minuscule Schubert calculus
I will discuss a root system uniform, concise combinatorial rule for
Schubert calculus of minuscule and cominuscule flag varieties G/P,
generalizing Schützenberger's jeu de taquin formulation of Schubert
calculus in type A. I will spend some time discussing the
underlying tableau combinatorics of cominuscule dual equivalence,
which extends a concept introduced by Haiman. I will also give a
conjecture for K-theory in the minuscule types.
This is all joint work with Alex Yong.
- MATTHIEU WILLEMS, University of Toronto, Department of Mathematics, 100 St
George Street, Toronto, Ontario, Canada, M4S 2E4
A Chevalley formula in Equivariant K-theory
In this talk, I will give a Chevalley formula in equivariant
K-theory. First, I will decompose the class of a line bundle in the
equivariant K-theory of a Bott-Samelson variety. Then I will use
this result to give a formula to multiply the class of a line bundle
by the class of a Schubert variety in the equivariant K-theory of a