The Khovanov–Rozansky homology categorifies the classical Jones and HOMFLY-PT polynomials. In this talk, we will explore how the Khovanov-Rozansky homology of the (m, n)-torus knot can be derived from the finite-dimensional representation of the rational Cherednik algebra at slope m/n, equipped with the Hodge filtration. This result confirms a conjecture by Gorsky, Oblomkov, Rasmussen, and Shende. Our approach involves the geometry of Hilbert schemes of points and character D-modules. Numerous examples will be provided to introduce and clarify the main concepts. 

The full Hecke algebra of a p-adic group G is complicated, but at Iwahori level, it is the specialization of an affine Hecke algebra at v = q. Lusztig defined the asymptotic Hecke algebra, which is a "limit" of the affine Hecke algebra as v goes to infinity. Although the "limiting process" does not make sense for the full Hecke algebra, Braverman and Kazhdan were able to extend the asymptotic Hecke algebra to arbitrary level. I will explain a proof of a conjecture by Bezrukavnikov, Braverman, and Kazhdan that the cocenter of the Hecke algebra is isomorphic to the cocenter of the asymptotic Hecke algebra.

A natural problem in the study of local systems on complex varieties is to characterize those that arise in a family of varieties. We refer to such local systems as motivic. While a classification of motivic local systems is evidently out of reach, Simpson conjectured that for a reductive group G, rigid G-local systems with suitable finiteness conditions at infinity are motivic. This was proven for curves when G = GL_n by Katz, who classified such rigid local systems. In this talk, we discuss our generalization of Katz's theorem to a general reductive group. Our proof goes through the (tamely ramified) categorical geometric Langlands program in characteristic zero.

I will explain how folding Khovanov—Rozansky’s SL(2n) homology gives a new approach to categorifying the spin colored SO(2n + 1) Reshetikhin—Turaev link polynomial. To develop this new approach I will mention: skew Howe duality, categorical braid group actions, i-quantum groups, and graphical calculus for SO(2n + 1) centralizer algebras. 

In the talk I’ll describe the various contexts in which the geometric Langlands conjecture can be formulated, and indicate the main ideas that go into its proof. This is a joint project with D. Arinkin, D. Beraldo, J. Campbell, L. Chen, J. Faergeman, K. Lin, N. Rozenblyum & you know who.

Consider a holomorphic fibration P : X → B whose general fiber is a torus. Its Shafarevich–Tate group parametrizes fibrations that are isomorphic to P locally over the base, i.e., fibers are the same but are glued in a different way. The fibrations with this property are called Shafarevich–Tate twists. I’ll describe the Shafarevich–Tate group in the case when P is a Lagrangian fibration on a compact hyperkähler manifold X. Then we’ll figure out which twists are projective, which are Kähler, and which are non-Kähler. In particular, I’ll show how to obtain the Bogomolov–Guan manifold, which is the only known example of a non-Kähler holomorphic symplectic manifold, as a Shafarevich-Tate twist of a Kähler manifold. 

Continuous cohomology classes of the group GL_n(Z_p) with coefficients in Q_p give rise to a theory of characteristic classes for étale Z_p-local systems on algebraic varieties, valued in (absolute) étale cohomology with Q_p-coefficients. These classes can be thought of as p-adic analogs of Chern–Simons characteristic classes of complex vector bundles with a flat connection.

By a theorem of Reznikov, Chern–Simons classes of all complex local systems on a smooth proper algebraic variety are torsion in degrees > 1. The same turns out to be true for p-adic characteristic classes on smooth varieties over algebraically closed fields (at least for p large enough, as compared to the rank of the local system). But for varieties over number fields and local fields these p-adic characteristic classes happen to be non-trivial even rationally, and for local systems coming from cohomology of a family of algebraic varieties they can be partially expressed in terms of the Chern classes of the corresponding Hodge bundles. 

This computation of p-adic characteristic classes relies on the notion of a Chern class for pro-étale vector bundles, and on the Hodge–Tate filtration in relative p-adic Hodge theory. This is joint work with Lue Pan.

The period-index problem is a classical problem about finite-dimensional division algebras over a field. When the base field is the function field of a complex variety, there is a longstanding conjecture, which is wide open for function fields of threefolds and beyond. I will discuss recent perspectives on the conjecture from topology, Hodge theory, and moduli theory. Finally, I will explain a result showing that, roughly speaking, topological solutions to the conjecture exist for complex function fields of arbitrary dimension.

This will be a nontechnical talk about our work with David Kazhdan on the spectrum of the Laplace and Hecke operators in the span of Eisenstein series. This very classical problem in the spectral theory of automorphic forms goes back to at least Langlands, who also introduced the analytic techniques that allow to find the spectrum in principle. To replace an in-principle answer by a concrete answer, we found it very useful to have a certain geometric interpretation of the multivariate contour integrals in Langlands’ setup. My goal in this talk will be to explain this geometric interpretation.

Classifying the irreducible unitary representations of a real reductive Lie group is one of the oldest unsolved problems in representation theory. In this talk, I will discuss two conceptual approaches to this problem, Hodge theory and the orbit method, and the relationship between them. The Hodge-theoretic approach, proposed by Schmid and Vilonen in 2011, exploits links between representation theory and the topology of algebraic varieties to endow irreducible representations with canonical Hodge filtrations. In joint work with Vilonen, we have shown that these filtrations detect unitary representations. The orbit method, on the other hand, is an old idea (dating back to the 1960s) that unitary representations should arise naturally as quantisations of certain classical spaces, the co-adjoint orbits. In joint work in preparation with Lucas Mason-Brown, we show that the Hodge filtration realises this expectation for a key class of representations, called unipotent, and deduce that these representations are always unitary.

This is a continuation of the lecture from Oct 23.

The Yamaguchi–Yau finite generation conjecture predicts that the higher-genus Gromov–Witten potentials of compact Calabi–Yau threefolds are polynomials in a finite number of generators. In this talk, I will outline a recent approach to apply Givental’s quantization formalism to the Gromov–Witten theory of Calabi–Yau threefolds and explain a proof of this conjecture for smooth Calabi–Yau hypersurfaces in P(1, 1, 1, 1, 2), P(1, 1, 1, 1, 4), and P(1, 1, 1, 2, 5). 

(Vijay Higgins speaks in the Quantum Topology Seminar at 4:30 pm on Nov 11, in KT 101.)

A very important result of Lusztig asserts that characters of Deligne–Lusztig representations are obtained by the sheaf-to-function correspondence from character sheaves. The goal of my talk is to outline a proof of a generalization of this result using the categorical trace machinery. This is a joint work in progress with Dennis Gaitsgory and Nick Rozenblyum.

The double Dyck path algebra B_{q,t} is constructed from an infinite quiver of affine Hecke algebras connected via raising and lowering operators. In this talk I will discuss joint work with Eugene Gorsky and Jose Simental where we construct and classify the semisimple representations of  B_{q,t}. I will also explain how the elliptic Hall algebra arises as the spherical subalgebra of B_{q,t}. 

I will explain a formula expressing a compatibility between E_2 algebra structure on Topological Hochschild Cohomology and the cyclotomic structure on Topological Hochschild Homology. A relative analog of this formula specializes to a formula articulated by Bezrukavnikov and Kaledin in their work on quantization in positive characteristic, which expresses the conjugation of the interior product operator by the Cartier isomorphism in terms of Cartan calculus. The inspiration for the noncommutative result comes from my earlier work in equivariant symplectic geometry, which gives an elementary 'picture-proof' of the corresponding formula for symplectic invariants. By specializing this result in noncommutative geometry to certain Fukaya categories, e.g. to the Fukaya category of the quintic threefold, one can show that the p-curvature of the mod-p reduction of the quantum connection has an enumerative interpretation: it is a generating series of Z/pZ domain-equivariant Gromov–Witten invariants called the 'quantum Steenrod operation', which q-deforms the usual Steenrod operation on cohomology. This gives the first computations of such operations in any Calabi–Yau symplectic manifold admitting a nonconstant holomorphic sphere. The method suggests the existence of an 'arithmetic realization' of quantum cohomology by enhancing it with algebraically meaningful Frobenius operators, with rich connections to arithmetic aspects of geometric representation theory.