| 9:15 |
Opening Remarks |
| 9:30-10:30 |
Bezrukavnikov |
Commutative and noncommutative symplectic resolutions and perverse sheaves |
|
Abstract:
I will outline an old program (joint with A. Okounkov) of constructing canonical
basis in K-groups of symplectic resolutions (esp. quiver varieties). Time permitting I will
offer a speculation on relating these bases to perverse sheaves based on a joint work in
progress with M. Kapranov.
|
| 10:30 |
Coffee Break |
| 11:00-12:00 |
Ginzburg |
Indecomposable objects and potentials over a finite field |
|
Abstract:
We prove a formula for a exponential sum over the set of absolutely indecomposable
objects of a category (satisfying a number of conditions) over a finite field in
terms of the geometry of the cotangent bundle on the moduli stack of (all) objects
of that category. Our formula, inspired by the work of Hausel, Letellier, and
Rodriguez-Villegas as well as an earlier work by Crawley-Boevey and van den Bergh,
provides a new approach for counting absolutely indecomposable quiver representations,
vector bundles with parabolic structure on a projective curve, and irreducible
$\ell$-adic local systems (via a result of Deligne).
|
| 12:00-1:45 |
Lunch |
| 1:45-2:45 |
Yun |
Simple cases of the topological Langlands correspondence |
|
Abstract:
In joint work in progress with David Nadler, we consider a
topological version of the geometric Langlands correspondence with
ramifications, in which the conformal structure of the curve is not
expected to play a role. I will formulate this correspondence as a
precise conjecture, and report on the simplest known cases of this
correspondence.
|
| 2:45-3:00 |
Break |
| 3:00-4:00 |
Shan |
On the center of quiver Hecke algebras |
| Abstract:
I will explain how to relate the center of a cyclotomic quiver Hecke algebras
to the cohomology of Nakajima quiver varieties using a current algebra action.
This is a joint work with M. Varagnolo and E. Vasserot.
|
| 4:00-4:30 |
Coffee Break |
| 4:30-5:30 |
Braverman |
Towards a mathematical definition of Coulomb branches of
3-dimensional gauge theories |
| Abstract:
|
| 9:30-10:30 |
Proudfoot |
Symplectic duality |
|
Abstract:
I will give an introduction to certain aspects of the symplectic duality
program of Braden, Licata, Webster, and myself. My goal will be to
explain how this program can be used to relate Nakajima’s construction
of irreducible representations of a simply laced Lie algebra to the
construction arising via the geometric Satake equivalence.
|
| 10:30 |
Coffee Break |
| 11:00-12:00 |
Braden |
Ringel Duality for perverse sheaves on hypertoric varieties |
|
Abstract:
(Joint work with Carl Mautner) Hypertoric varieties provide a laboratory
for testing conjectures for all symplectic resolutions, since many
questions can be answered by explicit combinatorial methods. We prove a
result for hypertoric varieties that is analogous to results for the
type A nilcone proved by Mautner and Achar-Mautner: the category of
perverse sheaves on an affine hypertoric variety with coefficients in a
field of arbitrary characteristic is highest weight and is Ringel dual
to perverse sheaves on the symplectic dual hypertoric variety. Our
result is obtained by means of explicit combinatorial descriptions of
the category of perverse sheaves and its tilting objects.
|
| 12:00-1:45 |
Lunch |
| 1:45-2:45 |
Gorsky |
Stable bases and q-Fock space |
|
Abstract:
Maulik and Okounkov defined a family of K-theoretic stable bases for the
Hilbert scheme of points on the plane. These bases depend on a single
parameter and interpolate between the (modified) Schur and Macdonald
polynomials. We give a conjectural explicit formula for the transition
matrix between bases for different parameter values in terms of certain
combinatorial operators on the q-Fock space introduced by Leclerc and
Thibon. Joint work with Andrei Negut.
|
| 2:45-3:00 |
Break |
| 3:00-4:00 |
Nevins |
Morse theory of D-modules |
|
Abstract:
Categories of equivariant D-modules on varieties possess internal structure that
mirrors, one categorical level higher, the Morse-theoretic structure of the spaces
on which they live. I will discuss such structure and some applications to
compact generation questions and linear invariants.
|
| 4:00-4:30 |
Coffee Break |
| 4:30-5:30 |
Nakajima |
Towards a mathematical definition of Coulomb
branches of 3-dimensional gauge theories (continued)
|
|
Abstract:
Let N be a representation of a complex reductive group G.
Physicists study the Coulomb branch of the 3-dimensional gauge theory
associated with (G,N), which is a hyper-Kaehler manifold, but have no
rigorous mathematical definition. We introduce a variant of the affine
Grassmannian Steinberg variety, define convolution product on its
equivariant Borel-Moore homology group, and show that it is
commutative. We propose that it gives a mathematical definition of the
coordinate ring of the Coulomb branch. (Joint work by Braverman,
Finkelberg and Nakajima)
|
| 6:30 |
Banquet at the Royal East Restaurant:
http://www.royaleast.com/
|
| 9:30-10:30 |
Namikawa |
A finiteness theorem for symplectic singularities |
|
Abstract:
Conical symplectic varieties now play an
important role in algebraic geometry and geometric representation
theory. In this talk we discuss how many such varieties exist. If we fix
the dimension of conical symplectic varieties X and the maximal degree N of the
minimal homogeneous generators of the coordinate ring R of X, then there
are only finitely many such X up to isomorphism.
The proof consists of two ingredients: one is the
boundedness result for log Fano klt pairs with fixed
Cartier index, and the second one is the rigidity of conical symplectic
varieties under Poisson deformations.
|
| 10:30 |
Coffee Break |
| 11:00-12:00 |
Kaledin |
Symplectic resolutions, mixed Hodge
structures and hyperkaehler metrics |
|
Abstract:
Most of the symplectic resolutions that appear in nature -- e.g.
those obtained by hyperkaehler reduction -- are not only symplectic: they
also carry a natural hyperkaehler metric invariant with respect to the
$S^1$-action. It seems that so far, this much stronger structure has not
been used at all in applications to representation theory, possibly
because it looks too transcedental. However, $S^1$-hyperkaehler metric
have a very nice algebraic interpretation in terms of mixed $R$-Hodge
structures, and although it has been discovered a long time ago, it seems
not to be universally known. I am going to recall this interpretation, and
then speculate a bit about what it means in for symplectic resolutions and
representation theory.
|
| 12:00-1:45 |
Lunch |
| 1:45-2:45 |
Licata |
ADE braid groups and resolutions of Kleinian singularities |
|
Abstract:
The goal of this talk will be to describe how much of the basic structure
of Artin-Tits braid groups of type ADE can be understood by studying its
faithful action on the derived category of the corresponding resolved
Kleinian singularity, in parallel to the way the Weyl group may be understood
via its faithful action on the root lattice.
|
| 2:45-3:00 |
Break |
| 3:00-4:00 |
Negut |
Stable bases and root subalgebras |
|
Abstract:
This is a natural continuation of Eugene Gorsky's talk. I will discuss
the K-theoretic stable basis for cyclic Nakajima quiver varieties, and
study the root subalgebras that arise. The shuffle algebra allows one
to write down a distinguished set of generators of these root subalgebras,
which act in a combinatorially meaningful way in the stable basis.
|
| 4:00-4:30 |
Coffee Break |
| 4:30-5:30 |
Cautis |
K-theoretic geometric Satake |
|
Abstract:
We will discuss a quantum K-theoretic version of the usual geometric
Satake equivalence (which should be thought of as the homology version).
In this setup the representation category of G is replaced with (a quantum
version) of G-equivariant coherent sheaves on G. This is joint work with
Joel Kamnitzer.
|
