Ana Balibanu (LSU)
Whittaker reduction for group-valued moment maps
Abstract: Let G be a semisimple complex group and let M be a Hamiltonian G-space. Whittaker reduction is a type of Hamiltonian reduction along Slodowy slices that encodes the Poisson geometry of M in the direction transverse to the action of G. We will review this procedure, which produces many interesting Poisson varieties in geometric representation theory. Then we will construct a multiplicative analogue of Whittaker reduction in the setting of Poisson-Lie groups, where the moment map takes values in the group G (rather than in the dual of its Lie algebra). This multiplicative reduction occurs along a class of transversal slices to unipotent orbits in G which generalize the Steinberg cross-section and are indexed by conjugacy classes in the Weyl group.
Roman Bezrukavnikov (MIT)
On the Grothendieck group of modular representations
Abstract: The talk is based on two papers, with P. Boixeda Alvarez, M. McBreen and Z. Yun and with C. Morton-Ferguson, M. Finkelberg and D. Kazhdan. A representation of a reductive group over a characteristic p field can be restricted to the Frobenius kernel G_1 or to the finite group G(F_q). In the first case the result can be described via top homology of an affine Springer fiber, in the second one via functions on the fixed points of the p-th power map on T/W. This leads to results motivated, respectively, by homological mirror symmetry and by Lusztig's conjecture on restriction of unipotent representations to the finite Chevalley group.
Elijah Bodish (MIT)
Categorified quantum symmetric pairs
Abstract: The purpose of the first talk will be to introduce quantum symmetric pairs/iquantum groups via examples and concrete realizations. During the second talk, I will discuss some instances in which we are beginning to understand how these algebras can be categorified.
Tommaso Botta (Columbia)
3d mirror symmetry and bow varieties
Abstract: As advocated by Aganagic and Okounkov, mirror symmetry in three dimensions admits an enumerative interpretation in terms of quasimap counts to mirror dual symplectic varieties. Specifically, the generating series of the counts, known as vertex functions, are expected to match up to some distinguished class in elliptic cohomology, known as the elliptic stable envelope. The latter also satisfies a remarkable mirror symmetry statement. After reviewing this general picture, I will focus on the case of bow varieties, which can be thought of as the type A version of mirror symmetry, and discuss the main ingredients of its proof (joint with subsets of R. Rimanyi and H. Dinkins).
Tom Gannon (UCLA)
Talk 1. Coulomb branches
Abstract: After the discovery of symplectic duality, it was soon realized that many examples of symplectic dual pairs arise as the Higgs and Coulomb branches associated to a finite-dimensional representation of a complex reductive group. While the Higgs branch corresponding to this data is straightforward to define, the Coulomb branch is more subtle, and its precise mathematical definition appeared only years later in a seminal paper by Braverman, Finkelberg, and Nakajima. In this talk, we will present the mathematical definitions of these Higgs and Coulomb branches, compute some examples, and highlight key geometric properties. Time permitting, we will also discuss a more elementary construction, due to Teleman, of Coulomb branches for certain representations of reductive groups known as massive representations.
Talk 2. Functoriality of Coulomb branches
Abstract: 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. We will moreover explain how one can use this result to recover the Coulomb branch associated to any quiver without loops from Coulomb branches of quivers with exactly two vertices. Time permitting, we will also 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.
Nicolle Gonzalez (UC Berkeley)
Khovanov-Rozansky homology of Coxeter Knots and the Oblomkov-Rasmussen-Shende conjecture
Abstract: Computing the Khovanov-Rozansky homology of algebraic links remains a difficult and highly sought after open problem. Part of the appeal lies in the myriad of conjectures that relate this homology theory to various algebro-geometric and combinatorial objects. One particular conjecture due to Oblomkov-Rasmussen-Shende states that for algebraic knots, the KR-homology coincides with the homology of the compactified Jacobian of the associated curve. Until recently, this and other related conjectures were only known in the special case of torus knots T(m,n). Indeed, although algebraic knots arise as cabled torus knots, understanding how these constructions behave under cabling has been a major stumbling block. In a recent preprint, arXiv: 2407.18123, Caprau-Hogancamp-Mazin and I made considerable progress towards this goal by computing this homology theory for a family of cabled torus.
In general, the approach for proving the known cases of these conjectures has been to compute both sides separately and compare the results. Recently, Gorsky, Mazin, and Oblomkov made progress on the geometric side by computing the homology of the compactified Jacobian of the curve associated to a particular family of cabled torus knots, T(m,n)(d,dmn+1). In particular, they proved that its Poincare polynomial is given by a specialization of the (q,t)-Catalan polynomial, a very well-studied and ubiquitous object in algebraic combinatorics. Separately, Galashin and Lam recently provided a combinatorial description of so-called Coxeter knots, of which T(m,n)(d,dmn+1) is a special case. Alongside Caprau, Mazin, and Hogancamp, we combined this combinatorial description with previous methods developed by Hogancamp-Mellit and Gorsky-Mazin-Vazirani, to compute a closed combinatorial formula for the graded dimensions of the KR-homology for a large subfamily of Coxeter knots containing all torus knots and the cables T(m,n)(d,dmn+1) as special cases. This formula yields several important consequences. On the one hand, our polynomials compute the hook components of the Schur-expansion of the Hikita polynomials in the celebrated Shuffle Theorems, thus providing a direct link with the classical action of the elliptic Hall algebra on symmetric functions. On the other hand, our polynomials specialize to (q,t)-Catalan polynomials. Thus, combined with the work of Gorsky-Mazin-Oblomkov, we obtain a proof of the Oblomkov-Rasmussen-Shende conjecture for the (d,dmn+1)-cables of (m,n)-Torus knots.
Joel Kamnitzer (McGill)
MV cycles and Hamiltonian reduction
Abstract: Via the geometric Satake correspondence, MV cycles provide bases for representations of semisimple Lie algebras. However the Lie algebra action on these MV cycles is a bit mysterious. Using work of Krylov, we can study MV cycles as irreducible components of attracting sets in generalized affine Grassmannian slices. By our previous work with Pham and Weekes, these affine Grassmannian slices are related to each other by Hamiltonian reduction. I will explain how to use the Hamiltonian reductions to define action of Chevalley generators E_i, F_i on MV cycles.
Vasya Krylov (Harvard/CMSA)
Quantum Hikita conjecture and graded traces via enumerative geometry
Abstract: One important approach to study modules over quantizations of symplectic singularities 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. Kamnitzer, McBreen, and Proudfoot introduced a particular D-module, known as the D-module of graded traces, whose solutions are these graded traces.
The quantum Hikita conjecture predicts an isomorphism between the D-module of graded traces for the quantized Coulomb branch and certain specialization of the quantum D-module of the resolved Higgs branch. In particular, it predicts a canonical identification between their solutions. We will discuss the general conjecture relating graded traces of Verma modules for quantized Coulomb branches with certain specializations of so-called vertex functions coming from the quasimap counts to Higgs branches. We will discuss the proof of this conjecture for ADE quivers and deduce the quantum Hikita conjecture as a corollary. Our proof uses recent results of Kamnitzer, Kalmykov, Leroux-Lapierre, Pinet and Weekes.
Time permitting, we will discuss applications of our results to the classification of graded traces. Based on joint work with Hunter Dinkins and Ivan Karpov.
Hiraku Nakajima (IPMU, Tokyo)
Involutions on Higgs/Coulomb branches of quiver gauge theories of type A
Abstract: We consider three types (Q,N,G) of involutions on quiver varieties of type A, and more generally on bow varieties. The fixed point sets are varieties of classical (BC)D type (Q)uiver varieties/(N)ilpotent slices/affine (G)rassmannian respectively. The symplectic duality exchanges (Q) and (G), fixes (N). In the latter half, following an idea of Yiqiang Li, we consider realization of K-matrices by stable envelope. It is expected to realize the Yangian for quantum symmetric pairs introduced by Olshanski and Sklyanin, Molev-Ragoucy.
Raphael Rouquier (UCLA)
Talk 1. Higher representations
I will introduce the notion of 2-representations of Kac-Moody algebras and give algebraic and geometric examples.
Talk 2. Tensor structures for 2-representations
I will explain the constructions of a tensor product and a braiding for 2-representations.
Alistair Savage (Ottawa)
The iquantum Brauer category
Abstract: Symmetric pairs consist of a complex simple Lie algebra and a subalgebra fixed by an involution. Passing to enveloping algebras, the latter becomes a Hopf subalgebra. Hence, its category of representations is naturally a monoidal category. The quantum analogue of this concept is that of a quantum symmetric pair. In the quantum setting, the subalgebra, called an iquantum enveloping algebra, is not a Hopf subalgebra. Rather, it is a coideal subalgebra. This means that the category of representations of the iquantum enveloping algebra is not monoidal. Instead, it is a module category over the category of representations of the larger quantum enveloping algebra. We will explore the representation theory of iquantum enveloping algebras from the point of view of diagrammatic interpolating categories. In the first talk, we will recall the definition of the framed HOMFLYPT skein category and its connection to the representation theory of the quantum version of general linear Lie algebra. Then, in the second talk, we will see that one can obtain a presentation of the category of modules of the iquantum enveloping algebra from the framed HOMFLYPT skein category by imposing a single relation. This is joint work with Hadi Salmasian and Yaolong Shen.
Vera Serganova (UC Berkeley)
Odd orbits in the adjoint representations of Lie superalgebras
Abstract: I will present recent results on the classification of odd nilpotent orbits in the adjoint representations of Lie superalgebras with reductive even part. In particular, I will discuss the geometry of the odd nilpotent cone and generalizations of the Jacobson–Morozov theorem. This lecture is based on joint work with Inna Entova-Aizenbud.
The geometry of odd adjoint orbits appears to play a crucial role in understanding the tensor category of finite-dimensional representations of a superalgebra. I aim to substantiate this claim by presenting results on cohomological support and the Balmer spectrum, drawing on the work of Boe, Kujawa, and Nakano, as well as on my joint work with Pevtsova and Sherman.
José Simental Rodríguez (UNAM)
Talk 1. The double Dyck path algebra
Abstract: The double Dyck path algebra was introducing by Carlsson and Mellit in their proof of the shuffle conjecture in algebraic combinatorics. It is a non-unital idempotented algebra that contains all type A affine Hecke algebras, together with operators that move along these affine Hecke algebras. Following work of Carlsson-Gorsky-Mellit, I will motivate the double Dyck path algebra using geometric representation theory: it naturally acts on the equivariant K-theory of parabolic flag Hilbert schemes (that will be defined in the talk).
Talk 2. Representations of the double Dyck path algebra
Abstract: I will start outlining the relationship between the double Dyck path algebra and the elliptic Hall algebra (aka quantum toroidal algebra), namely, in joint work with Nicolle González and Eugene Gorsky, we realized the positive half of the elliptic Hall algebra as a spherical subalgebra of the double Dyck path algebra. I will then turn to studying the representation theory of the double Dyck path algebra, focusing on a class of representations where the polynomial part of all affine Hecke algebras acts in a semisimple fashion. We will see that these representations admit duals and tensor products. Motivated by these constructions, time permitting I will define a "double" of the double Dyck path algebra, that contains the full elliptic Hall algebra as a spherical subalgebra. The talk is based on joint works with Nicolle González and Eugene Gorsky (arXiv 2311.17653 and 2502.16113)
Milen Yakimov (Northeastern)
Quantum cluster algebras and quantum groups at roots of unity
Abstract: The first talk will be an overview of the idea of equipping quantized coordinate rings of interesting varieties in Lie theory with quantum cluster algebra structures. This was one of the main sources of motivation behind the introduction of cluster algebras, namely, to use those in order to study canonical bases on quantum algebras. We will focus on various results about the construction of cluster structures on quantized coordinate rings. The second talk will focus on the representation theory of root of unity quantum cluster algebras via Poisson geometry and invariant theory. The two talks are based on joint works with K. Goodearl, G. Muller, B. Nguyen, H. Oya, K. Trampel, and F. Qin.