University of Wisconsin-Madison

Schubert puzzles are combinatorial gadgets used to compute Schubert calculus, that is, to find the structure constants in the cohomology ring of the Grassmannian. We generalize Schubert puzzles, which are classically triangular, to now include puzzles with convex polygonal boundary shapes. We will present recent results characterizing the commutative properties of these polygonal Schubert puzzles, which generalize the basic commutative property of classical triangular puzzles.

In Lusztig's papers of character sheaves that were published in the time period 1985-1986, he proved a remarkable property of cuspidal perverse Q-sheaves on the nilpotent variety: they are "clean," meaning that their stalks vanish outside a single orbit. This property is crucial in making character sheaves computable by an algorithm, and it plays an important role in "block decompositions" of generalized Springer correspondence both in characteristic 0 and positive. About 10 years ago, Mautner conjectured that cuspidal perverse sheaves remain clean after reduction modulo p (with some exceptions for small p). In this talk, I will discuss the history and context of the cleanness phenomenon, along with recent progress on Mautner's conjecture. This is joint work with P. Achar.

The Coulomb branch associated to a representation of a complex reductive group G is an algebraic variety whose construction is motivated by considerations in physics and was precisely defined in a seminal paper of Braverman-Finkelberg-Nakajima in 2018. After giving an overview of these varieties, we will discuss a result, joint with Ben Webster, on the functoriality of Coulomb branches. Specifically, given a map H to G of complex reductive groups and a representation of G satisfying an assumption we call gluable, we will explain a procedure by which one can recover the Coulomb branch for the induced representation of H from the corresponding Coulomb branch for G. Time permitting, we will discuss how this result can be used to prove a conjecture of Bourget-Dancer-Grimminger-Hanany-Zhong that the ring of functions on T^*(SL_n/U_P) can be identified with the ring of functions of a Coulomb branch, where U_P is the group of block strictly upper triangular matrices.

Kleinian singularities are remarkable singular affine surfaces. They arise as quotients of C^2 by finite subgroups of SL_2(C). The exceptional loci in the minimal resolutions of Kleinian singularities are in 1-to-1 correspondence with the simply-laced Dynkin diagrams. In this talk, I will introduce certain singular Lagrangian subvarieties in the minimal resolutions of Kleinian singularities that are related to the classification of irreducible Harish-Chandra (g,K)-modules annihilated by certain unipotent ideals. The irreducible components of these singular Lagrangian subvarieties are P^1's and A^1's. I will describe how they intersect with each other through the realization of Kleinian singularities as Nakajima quiver varieties.

In this talk, we first discuss a definition of Lie algebra in the higher Verlinde category Ver_4^+ and PBW theorem that provides a notion of Lie superalgebra in characteristic 2 and some of its properties. We then define the notion of Lie superalgebra in Ver_4^+, which will unify both a pre-existing notion of Lie superalgebra in characteristic 2 as a Z/2-graded Lie algebra with squaring map (Bouarroudj et. al) and a Lie algebra in Ver_4^+, and prove a PBW theorem for such a notion. Such a notion has a mixed characteristic deformation theory via a natural lift to characteristic 0 (for perfect k), which we call a mixed Lie superalgebra over a ramified quadratic extension of the ring of Witt vectors W(k). This is joint work with Pavel Etingof.

We use the Bondal-Thomsen collection to promote the Coherent Constructible Correspondence to a simultaneous correspondence across all chambers in a toric GIT problem. This is achieved by studying structural properties of microlocal sheaves along a Lagrangian constructed by applying equivariant mirror symmetry to the multigraded Bondal-Thomsen free modules over the Cox ring. Our result achieves a direct A-side gluing of FLTZ mirrors of all birational models into a sheaf of categories over the real GIT parameter space, complementing results in recent work arXiv:2501.00130 on the B-side. The talk is based on joint work with David Favero.

In recent work, Aganagic proposed a categorification of quantum link invariants based on a category of A-branes. The theory is a generalization of Heegaard-Floer theory from gl(1|1) to arbitrary Lie algebras. This theory is solvable explicitly and can be used to compute homological link invariants associated to any representation of a simple Lie algebra. By considering functors between these categories of A-branes, one can upgrade this construction to realize categorified representations of quantum groups. In this talk, I will introduce the relevant categories of A-branes, review Aganagic’s construction of homological link invariants, and explain how categorified quantum groups arise in this setting. This talk is based on 2305.13480 with Mina Aganagic and Miroslav Rapcak and work in progress with Mina Aganagic, Ivan Danilenko, Yixuan Li, and Peng Zhou.

In this talk, we will explore how the Khovanov-Rozansky homology of the (m,n)-torus knot can be extracted from the finite-dimensional representation of the rational Cherednik algebra at slope m/n, equipped with the Hodge filtration. Our approach involves constructing a family of coherent sheaves on the Hilbert scheme of points on the plane, arising from cuspidal character D-modules.

Let W be a Weyl group. In 2008, Tymoczko defined the dot action of W on the cohomology of a discrete family of smooth varieties. For the symmetric group, the dot action links the combinatorics of chromatic symmetric functions to the geometry of these varieties, called Hessenberg varieties. This talk will give a brief overview of that story and discuss how the dot action representation can be computed for the hyperoctahedral group.

In this talk, we give an explicit construction of a Bridgeland stability condition associated with a non-nef divisor. This is the image of a standard Bridgeland stability condition associated with an ample divisor under the autoequivalence of spherical twist. This is based on joint work with Tristan Collins, Jason Lo, and Shing-Tung Yau.

Given an object V in a symmetric tensor category C, one can define the affine group scheme GL(V) in C and the corresponding category of representations D=Rep_C GL(V). This category admits a so-called categorical type A action through translation functors. In many cases, this action defines a representation of the Lie algebra sl_infty on the Grothendieck ring of D in the characteristic zero zero case, or of affine sl_p in characteristic p case. Coulembier, Etingof, and Ostrik proved that over a field of char p, any Frobenius exact symmetric tensor category of moderate growth admits a symmetric tensor functor to the Verlinde category Ver_p. It then follows from the reconstruction theorem that for any such category C and an object V of C, the category of representations of GL(V) is equivalent to the category of representations of GL(X) for an object X in Ver_p. I will talk about the categories of representations of general linear group schemes in Ver_p, the corresponding categorical type A action, and related computations.