Hanukkah STAR mini-workshop 2025 

• Ilya Dumanski (MIT)

Perverse coherent sheaves on symplectic singularities

Abstract: Perverse constructible sheaves are ubiquitous in algebraic geometry and geometric representation theory. Bezrukavnikov introduced their coherent analog, called perverse coherent sheaves. For technical reasons, there are essentially two interesting examples when this notion is well-behaved: the nilpotent cone and the affine Grassmannian. In both these cases, this category is very meaningful and well-studied.

We will present a generalization of this construction to an arbitrary symplectic singularity. This may be seen as a step towards building the Kazhdan—Lusztig theory in this setting.

• Boris Feigin (HUJI)

Toroidal shifted algebras and extensions of vertex algebras

Abstract: Shifted affine algebras are the relatively new class of algebras which is extensively studied now. I plan to discuss the affinizations of such algebras .We get  the new class of algebras which has many applications. The interesting and important one - we can construct vertex algebras with big group of symmetries. Such algebras appear in geometric Langlands.

• Michael Finkelberg (HUJI)

Relative Langlands duality for osp(2n+1|2n)

Abstract: This is our joint work with A. Braverman, D. Kazhdan and R. Travkin. We prove that the S-dual of SO(2n+1) x Sp(2n) acting on the tensor product of their tautological representations U and V, is the symplectic mirabolic space V x T*Sp(2n) equipped with the action of two copies of Sp(2n). 

• Maria Gorelik (WIS)

Affine Kac-Moody superalgebras and their root systems

• Victor Kac (MIT)

Modular invariant vertex operator algebras

Abstract: Vertex operator algebra (VOA) V with a Virasoro element L is called modular invariant if its normalized character ch_V (tau):=tr_V e^{2pi i tau (L_0 -c/24)} is a modular invariant function. The basic examples are rational VOA and affine VOA at admissible levels.

In my talk I will present a number of examples of modular invariant VOA beyond the rational ones, and state several conjectures and open problems about them. In conclusion I will discuss the quasi-invariant VOA, for which  the characters are quasi-modular functions. The examples include the simply-laced affine VOA  at negative integer level \geq -b, where b is the length of the longest leg in the affine Dynkin diagram.

• Vasily Krylov (Harvard)

Graded traces on quantized Coulomb branches and 3D mirror symmetry  

Abstract: Higgs and Coulomb branches of three-dimensional quantum field theories form two rich families of Poisson varieties that are expected to be exchanged by 3D mirror symmetry. One important approach to study modules over quantizations of functions on Higgs and Coulomb branches is by analyzing their graded traces. Graded traces generalize the notion of characters and are closely related to q-characters introduced by Frenkel and Reshetikhin. The quantum Hikita conjecture, formulated by Kamnitzer, McBreen, and Proudfoot, provides a conjectural framework for understanding graded traces of Coulomb branches via the enumerative geometry of Higgs branches. In this talk, I will present our refined version of this conjecture and outline its proof in the case of ADE quiver gauge theories. Based on joint work with Dinkins and Karpov.

• Shifra Reif (BIU)

The Harish-Chandra Theorem for Ghost Distributions

Abstract: The universal enveloping algebra of a Lie superalgebra admits an anti-center. Gorelik introduced this object in 1999 and computed its Harish-Chandra image. We shall recall these notions, present the classical results and generalize them to symmetric spaces.

Joint work with Siddhartha Sahi, Vera Serganova and Alexander Sherman.

• Vera Serganova (UC Berkeley)

Representation Type of a Quasireductive Supergroup

Abstract: A celebrated theorem of Drozd states that any finite-dimensional associative algebra A over an algebraically closed field K has either finite, tame, or wild representation type. Let A=K[G] be the group algebra of a finite group G over a field K of positive characteristic p. One may ask for which groups G the algebra K[G] has finite or tame representation type. Such groups were classified by Higman, Brenner, Bondarenko, Drozd, and Ringel, and the answer depends solely on the structure of a Sylow p-subgroup of G.

An analogous question arises for algebraic supergroups G whose underlying algebraic group is reductive, over an algebraically closed field K of characteristic zero. In view of the recently developed theory of Sylow subgroups for algebraic supergroups, one may further ask whether the representation type of G is determined by that of its Sylow supergroup. I will review old and recent results addressing these questions.

If time permits, I will also discuss the stable category and the semisimplification of representation categories for tame supergroups.

• Ran Tessler (WIS)

Plabic tangles and Cluster structures in planar N=4 SYM amplitudes

Abstract: I will start by describing the nonnegative Grassmannian, the amplituhedron and their connection to the planar N=4 Super Yang Mills QFT. I will then describe cluster structures  that were discovered in amplitudes of this theory.

I will then explain the source of these phenomena, leading to the definition of plabic tangles and their operad structure.