Program
Monday, May 18
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: |
Tuesday, May 19
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/ |
Wednesday, May 20
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. |
Thursday, May 21
9:30-10:30 | Okounkov | Enumerative geometry of rational curves in equivariant symplectic resolutions |
Abstract: | ||
10:30-10:50 | Coffee Break | |
10:50-11:50 | Rains | Derived equivalences of noncommutative rational surfaces |
Abstract: A typical rational elliptic surface (with section) admits a large group of derived autoequivalences (with structure $Z\rtimes \Lambda_{E_8}^2\rtimes SL_2(Z)$). Commutative deformations of such surfaces have much more rigid derived categories, but it turns out that if we allow noncommutative deformations, that the entire group survives. I'll describe a couple of constructions of an 11-dimensional family of noncommutative rational surfaces on which the above group acts as (essentially the only) derived equivalences, and explain some of the consequences, ranging from new Lax pairs for Painlevé equations to new deformations of Hilbert schemes of rational surfaces. | ||
11:50-12:00 | Break | |
12:00-1:00 | Rouquier | Higher representations |
Abstract: I will survey some recent directions in 2-representation theory of Lie algebras. |