Abstract: The talk is based on joint work with A.Smirnov. We obtain a factorization theorem about the limit of elliptic stable envelopes to a point on a wall in H^2(X,R), which generalizes the result of M.Aganagic and A.Okounkov. This approach allows us to extend the action of quantum groups, quantum Weyl groups, R-matrices, etc., to actions on the K-theory of the symplectic dual variety. In the case of the Hilbert scheme of points in the plane, our results imply the conjectures of E.Gorsky and A.Negut. As another application of this technique, we gain a better geometric understanding of the wall crossing operators and the quantum difference equations.

Abstract: The moduli spaces of representations of Galois groups of p-adic fields with p-adic Hodge theoretic conditions play a pivotal role in the study of arithmetic properties of automorphic forms. Despite this, the geometry of these spaces is poorly understood, perhaps for good reason: conjecturally their complexity is bounded below by the modular representation theory of finite groups of Lie type. In this talk, I will survey some progress on the study of these moduli spaces in some special but important cases, where it turns out that they are closely related to degenerations of flag varieties into certain (deformed) affine Springer fibers. This is based on joint work with (various subsets of) D. Le, B. Levin and S. Morra.

Abstract: Non-topological twists of supersymmetric gauge theories have played an increasingly important role in mathematics in part due to relationships to vertex algebras and quantum groups. On the other hand, motivated by the higher genus B-model, twists of 10-dimensional theories of supergravity have also been introduced. In this talk, we give a complete description of the maximally non-topological twist of 11-dimensional supergravity, the low energy limit of M-theory. I will explain the unexpected result that the global symmetry algebra of the model is equivalent to an infinite-dimensional exceptional super Lie algebra known as E(5,10). I will also explain the relationship between other exceptional algebras and extended objects such as M2 and M5 branes in the twisted setting.

Abstract：I will talk about the positive part of a certain affine Springer fiber studied by Goresky, Kottwitz, and MacPherson, and a certain interesting open subvariety. The Hilbert series of their Borel-Moore homology turn out to be related to reproducing kernels of the Bergeron-Garsia nabla operator. This operator is easy to define in the basis of modified Macdonald polynomials, but producing explicit combinatorial evaluations of this operator is usually difficult and (conjecturally) relates to interesting Hilbert series associated to various moduli spaces. Our work is motivated by the nabla positivity conjecture of Bergeron, Garsia, Haiman, and Tesler that predicts that nabla evaluated on a Schur function is sometimes positive, sometimes negative. We categorify this conjecture and reduce it to a vanishing conjecture for the interesting open variety. It turns out, each irreducible S_n representation mysteriously prefers to live in certain degrees and weights in the cohomology. This is a joint work with Erik Carlsson.

Abstract: This talk will discuss two closely-related geometric questions about the moduli space of vector bundles on an algebraic curve and its cotangent bundle, the moduli space of Higgs bundles. One involves computing some intersection-theoretic invariants of the nilpotent cone and the other involves understanding the stalks of the Hecke eigensheaves on this moduli space defined in the geometric Langlands program. The motivation for these problems comes from classical number theory and the relationship between the two passes through a new characteristic p version of a general theorem of Massey. The talk will give an overview of all these aspects.

Abstract: We prove an equivalence between the equivariant derived category of constructible sheaves on the loop space of n-by-n matrices and the category of perfect complexes on a certain formal dg algebra. This equivalence is compatible with actions of the derived Satake category. I will explain how our result is related to conjectural dualities in the Langlands program and in physics. The proof involves certain invariant theoretic properties of a certain "dual" cotangent bundle. This is joint work with Tsao-Hsien Chen (in preparation).

Abstract: Let G and G’ be Langlands dual reductive groups (e.g. SL(n) and PGL(n)). According to a theorem by Donagi-Pantev, the generic fibres of the moduli spaces of G-Higgs bundles and G’-Higgs bundles are dual abelian varieties and are therefore derived equivalent. It is an interesting open problem to prove existence of a derived equivalence over the full Hitchin base. I will report on joint work in progress with Shiyu Shen, in which we construct a K-theoretic shadow thereof: natural equivalences between complex K-theory spectra for certain moduli spaces of Higgs bundles (in type A).

Abstract: The notion of a "t-structure on a triangulated category" was introduced around 1980. The notion of "co-t-structure" could have been defined back then as well, but it didn't receive much attention until the past 10 years or so, probably because it wasn't clear what it was good for. I'll explain these notions and give some elementary examples.

I will then discuss some "modern" examples of co-t-structures in geometric representation theory. In particular, I will explain a remarkable new co-t-structure on the derived category of coherent sheaves on the nilpotent cone of a reductive group. The study of this co-t-structure leads to the proof of the Humphreys conjecture on tilting modules for a reductive group. This is joint work with W. Hardesty.

Abstract: Let X be a smooth projective curve over a finite field k, and let G be a reductive group. The unramified part of the theory of automorphic forms for the group G and the field k(X) studies functions on the k-points on the moduli space of G-bundles on X and the eigen-functions of the Hecke operators (to be reviewed in the talk!) acting there. The spectrum of the Hecke operators has continuous and discrete parts and it is described by the global Langlands conjectures (which in the case of functional fields are essentially proved by V.Lafforgue).

After recalling the above notions and constructions I will discuss what happens when k is replaced by a local field. The corresponding Hecke operators were essentially defined by myself and Kazhdan about 10 years ago, but the systematic study of eigen-functions has begun only recently. It was initiated several years ago by Langlands when k is archimedian and then Etingof, Frenkel and Kazhdan formulated a very precise conjecture describing the spectrum in terms of the dual group. Contrary to the classical case only discrete spectrum is expected to exist. I will discuss what is is known in the case when k is a local non-archimedian field (joint work in progress with D.Kazhdan).

Abstract: Kazhdan-Lusztig polynomials are fascinating! In the 80s Lusztig and Dyer independently noticed that the Kazhdan-Lusztig polynomial for a pair x,y of elements in a Coxeter group appears to only depend on the isomorphism type of the interval [x,y] in Bruhat order. This statement became known as the combinatorial invariance conjecture. I will review this conjecture, and discuss what is known. I will present a conjecture which should lead to a proof when W is the symmetric group.

Abstract: We give a description of the Namikawa-Weyl groups of affinizations of smooth Nakajima quiver varieties based on combinatorial data of the underlying quiver, and compute some explicit examples. This extends a result of McGerty and Nevins for quiver varieties associated to Dynkin quivers.

Abstract: After discussing the notion of temperedness arising in the geometric Langlands program, I’ll sketch a proof of a version of the Ramanujan conjecture in that setting. Essential ingredients for the definition and the proof are the derived Satake equivalence and the Deligne-Lusztig (or Alvis-Curtis) duality functors. I will then explain the role of the Ramanujan conjecture in the geometric Langlands program for the group SL_2.