Title: Birational geometry of quiver varieties

Abstract: In this talk I will report on joint work in progress with A. Craw and T. Schedler on the birational geometry of quiver varieties. Using the surjectivity of the linearization map, as shown by McGerty-Nevins, we are able to describe explicitly the ample and moveable cones of a quiver variety in terms of GIT chambers. This shows that all projective symplectic resolutions of a quiver variety (with the exception of the (2,2) case) are given by variation of GIT. One can then give an explicit local description of the birational transformations that occur under VGIT. I will explain what our results mean in two concrete classes of examples. Namely, for framed affine Dynkin quivers (corresponding to wreath product quotient singularities) and star shape quivers (corresponding to hyperpolygon spaces).

Title: Blowing down noncommutative cubic surfaces

Abstract: Let A = A(E,σ) be the Sklyanin algebra associated to an elliptic curve E and a translation automorphism σ of E. It is well-known that A can be considered as the coordinate ring of a “noncommutative P2”, and there are analogues of classical results on birational transformations of (commutative) surfaces. In particular, Van den Bergh, and separately Rogalski, have given methods to blow up A at a collection of points on the curve E. Further, Van den Bergh has shown that the blowup of A at 6 points is isomorphic to the coordinate ring of a cubic surface in a noncommutative P3, giving a noncommutative version of a well-known classical result.

We consider the converse problem: is a noncommutative cubic surface always isomorphic to the 6-point blowup of a Sklyanin algebra? Using noncommutative intersection theory and the Castelnuovo contraction criterion of the author, Rogalski, and Stafford, we show that this is generically true. We further discuss the geometry of various moduli spaces associated to this situation.

This is joint work with Ingalls, Okawa, and Van den Bergh.

Title: One-dimensional topological quantum field theories with zero-dimensional defects and finite state automata

Abstract: Quantum groups are related to 3-dimensional topological quantum field theories. Downsizing from three dimensions to one and from a ground field to a semiring, I will explain a surprising relation between topological theories for one-dimensional manifolds with defects and values in the Boolean semiring and finite-state automata and their generalizations. This is joint with Mikhail Khovanov.

Title: Functions on commuting stacks via mirror symmetry

Abstract: For a complex reductive group G, its commuting stack parametrizes pairs of commuting group elements up to conjugacy. One can also interpret the commuting stack as G-local systems on a torus.

I’ll explain joint work with Penghui Li and Zhiwei Yun that calculates global functions on the commuting stack via mirror symmetry, in particular Betti geometric Langlands.

Title: D-modules and rings of differential operators in singular and noncommutative settings

Abstract: This is a report on work of my PhD student Haiping Yang. D-modules, the algebraic approach to differential equations, are usually defined on smooth varieties, where the theory is very well-behaved. On the other hand, in the singular setting, various definitions which were equivalent in the smooth setting become quite different. Seminal work of Ben-Zvi and Nevins explained that, when the singularities are mild enough, then these definitions nonetheless coincide. In general, however, they only coincide if one uses derived algebraic geometry. In this talk, I will explain how to construct a natural differential graded ring of differential operators which recovers this derived category; it is a derived correction of the usual ring of differential operators. I will give examples. Finally, I will explain some analogous D-modules in the setting of quantizations of Poisson varieties, and some applications to Hochschild homology.

Title: Morava E-theories and quantum groups

Abstract: I will talk about a family of quantum groups labelled by a prime number and a positive integer constructed using the Morava E-theories. This is motivated by Lusztig’s 2015 reformulation of his conjecture from 1979 on character formulas for algebraic groups over a field of positive characteristic. I will also explain the quantum Frobenius homomorphisms among these quantum groups. The main ingredient in constructing these Frobenii is the transchromatic character map of Hopkins, Kuhn, Ravenal, and Stapleton. This talk is based on my joint work with Gufang Zhao.

Title: Invariant holonomic systems for symmetric spaces

Abstract: Fix a complex reductive Lie group G with Lie algebra ℊ and let V be a symmetric space over ℊ with ring of differential operators (V). A fundamental class of 𝒟(V)-modules consists of the admissible modules (these are natural analogues of highest weight g-modules).

In this lecture I will describe the structure of some important admissible modules. In particular, when V = ℊ these results reduce to give Harish-Chandra’s regularity theorem for G-equivariant eigendistributions and imply results of Hotta and Kashiwara on invariant holonomic systems. If I have time I will describe extensions of these results to the more general polar -representations. A key technique is to relate (the admissible module over) invariant differential operators 𝒟(V)^G on V to (highest weight modules over) Cherednik algebras. This research is joint with Bellamy, Levasseur, and Nevins.