Abstract: Let G, G^L be Langlands dual reductive groups. The geometric Satake equivalence is the wonderful fact that one can realize the monoidal category of G^L-representations as G[[t]]-equivariant perverse sheaves on the affine Grassmannian of G.  

The affine Grassmannian is a basic example of a semi-infinite partial flag manifold Fl_P,  associated to the parabolic P = G. For a general parabolic P, we will explain how G[[t]]-equivariant perverse sheaves on Fl_P are equivalent to Rep(P^L). Moreover, one can obtain not only Rep(P^L) but Rep(M^L), where M^L is the Levi quotient of P^L, by considering G[[t]]-equivariant perverse sheaves which are factorization modules for the semi-infinite intersection cohomology sheaf on Fl_P. 

Time permitting, we will indicate the derived equivalences these abelian statements fit into, and how relaxing G[[t]]-equivariance to smaller parahorics yields realizations of the `regular blocks' for various quantum groups, including small and mixed quantum groups, as well as analogues with positive characteristic coefficients. 

The contents of this talk builds on work and conjectures of many people, notably Arkhipov, Bezrukavnikov, Braverman, Feigin, Finkelberg, Frenkel, Gaitsgory, Lusztig, Mirkovic, and Raskin, and is the subject of works in progress with Campbell, Chen, Lysenko, and Achar--Riche. 

Abstract: In recent years, it has been observed that symplectic and algebraic invariants of algebraic symplectic stacks are related by some remarkable dualities, particularly symplectic duality and relative Langlands duality. The 3-dimensional topological field theory perspective suggests that these relations ought to have their origins in an equivalence of 2-categories, and I will explain how this works in the abelian case. The new ingredient is a 2-category of perverse sheaves of categories, which are expected to play the same role in categorical representation theory that perverse sheaves have played in geometric representation theory. This is based on joint work with Justin Hilburn and Aaron Mazel-Gee, and continuing work with Justin Hilburn.

Abstract:  The notion of asymptotic Hecke algebra originates to the works of Josef and Lusztig. However, no geometric description of this object has been found up to the very recent time.

In my talk, I will proof such a description (conjectured by Qiu and Xi), and explain some applications. In particular, the description of the cocenter of usual affine Hecke algebra will naturally appear.

This is a joint work with Roman Bezrukavnikov and Vasily Krylov.

Abstract: I will give a (biased) overview of the program, known as the Bethe/Gauge correspondence, identifying integrability structures (e.g., representations of Yangians) in supersymmetric gauge theories. Geometric approach to the subject was initiated by Maulik and Okounkov, and reinterpreting their ideas in terms of quantum field theory will play a major role in this talk. 

Abstract: We will explain and prove the following result: the (oo,2)-category of categories equipped with loop group actions can be fully faithfully embedded into the (oo,2)-category of categories equipped with *factorization* affine Grassmannian actions, i.e., D(LG)-mod \to D(Gr_G)-factmod is fully faithful. This is a joint work with Yuchen Fu, Dennis Gaitsgory and David Yang. 

Abstract: Recent work of Ben-Zvi, Sakellaridis, and Venkatesh (BZSV) proposes some conjectures about an analogue of the derived geometric Satake equivalence for spherical varieties, where the spectral side is related to Hamiltonian varieties for the dual group. If X is an affine spherical G-variety, this conjecture is concerned with describing the category of G[[t]]-equivariant sheaves of vector spaces over X((t)). In this talk, I will describe a homotopy-theoretic approach to this conjecture when X is an affine homogeneous spherical variety. When X is of rank 1, this leads to a derived Satake theorem as conjectured in BZSV. Along the way, we observe that the same techniques also allow a study of sheaves with coefficients in "complex connective K-theory"; this leads to a "grouplike deformation" of the derived Satake theorem. For instance, the adjoint quotient stack g^/G^ appearing on the spectral side of the usual derived geometric Satake equivalence is replaced by the canonical degeneration of the conjugation quotient G^/G^ into g^/G^.

Abstract:  The Breuil-Mezard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" that should govern congruences between mod p automorphic forms on a reductive group G. Most of the progress thus far has been concentrated on the case G = GL_2, which has several special features. I will talk about joint work with Bao Le Hung on a new approach to the Breuil-Mezard Conjecture, which applies for arbitrary groups (and in particular, in arbitrary rank). It is based on the intuition that the Breuil-Mezard conjecture is analogous to homological mirror symmetry. 

Abstract:  I will discuss the quantization of one of the simplest theories of gravity, three-dimensional Einstein gravity with negative cosmological constant. In particular I will describe a precise reformulation of AdS_3 quantum gravity in terms of a novel topological quantum field theory based on the quantization of the Teichmuller space of Riemann surfaces that we refer to as "Virasoro TQFT." This proposal clarifies the relationship between pure AdS_3 quantum gravity and (two copies of) SL(2,R) Chern-Simons theory, and resolves some well-known issues with this lore. I will argue that moreover this proposal provides a practical framework for the computation of 3d gravity partition functions, and will describe how it elucidates various aspects of the holographic dual. 

Abstract:  Borel--Laplace summation is a classical summability method that associates an analytic function to a divergent power series. Divergent power series often appear in mathematics and physics: solving ODEs with irregular singularities,  computing asymptotics,  computing perturbative expansions in QFT, in complex Chern--Simons, etc. One special feature of the Borel--Laplace sum is that it works well for integrals over Lefschetz thimbles (thimble integrals). Indeed, the Borel--Laplace sum of the asymptotics of a thimble integral is the thimble integral itself. This is part of a joint project with A. Fenyes. 

Abstract: In the 90s, Kazhdan and Lusztig established an equivalence between certain affine Lie algebra representations and representations of the Lusztig quantum group. We will begin by defining these objects, and review some ingredients of their proof. After that, we will explain some ideas from our joint work with Lin Chen, where we proved an extension of this theorem using factorization algebras. Though our methods are different from the original proof, we will try to draw some connections between the two, such as the crucial role played by conformal blocks in both cases. Time permitting, we will also explain how some of these ideas play a role in the recently announced proof (by Gaitsgory et al.) of the unramified geometric Langlands conjecture. 

Abstract: Inspired by the theory of positive depth representations of p-adic reductive groups, we study Hecke categories associated to certain open compact subgroups smaller than the Iwahori subgroup. In this talk, I will prove that in some cases these Hecke categories are monoidally equivalent to affine Hecke categories of smaller groups, therefore having applications to the local geometric Langlands correspondence. Using sheaf-function correspondence, our categorical equivalence recovers a family of Hecke algebra isomorphisms already proven by Ju-Lee Kim.  

Abstract: Given a DG category acted on by the category of quasi-coherent sheaves on LocSys_G(D^{\circ}), one can define a factorization Rep(G)-module category. Following ideas of Beilinson and Drinfeld, Gaitsgory conjectured that this construction loses no information: that it gives a fully faithful 2-functor QCoh(LocSys_G(D^{\circ}))-mod (DGCat) —> Rep(G)-mod^fact(DGCat). We will discuss this conjecture, its role in the local Geometric Langlands program, and we will prove a partial result in this direction. Namely, we will prove the fully faithfulness for QCoh(LocSys_G(D^{\circ}))-modules set-theoretically supported over the stack of local systems with restricted variation on the formal punctured disk.

Abstract: W-algebras are a family of vertex algebras obtained as Hamiltonian reductions of affine vertex algebras parametrized by nilpotent orbits.The W-algebras associated with regular nilpotent orbits enjoy the Feigin-Frenkel duality. More recently, Gaiotto and Rap\v{c}\'ak generalize this result to hook-type W-algebras as the triality for vertex algebras at the corner.

In this talk, I will present the correspondence of representation categories for the hook-type W-superalgebras and how to gain general W-algebras in type A from hook-type W-algebras. The talk is based on joint works with Creutzig, Fasquel, Genra, Linshaw, and Sato. 

 Abstract: Given a proper CY A-infinity category C, a notion closely related to that of a 2d extended TQFT, I explain how the Loday-Quillen-Tsygan map fits into a commutative diagram of dg-BV algebras. The first two corners are built from the cyclic cohomology of C, respectively from a cyclic L-infinity algebra associated to C. The other two corners are of a more geometric nature, given by chains on the moduli space of Riemann surfaces with corners, free boundaries decorated by objects of C, respectively chains on the moduli space of metric graphs. I indicate how this generalizes an observation by Kontsevich producing cocycles on M_g,n. Further I compare this to the story of closed SFT developed by Costello, Caldararu-Tu, which led to the definition of categorical Gromov-Witten invariants for categories C as above that are also smooth.