Perverse sheaves on the affine Grassmannian of a reductive group G encode a great deal of representation-theoretic information. In characteristic 0, these sheaves have been studied in depth since the 1990s. In this talk, I will discuss recent advances in the positive-characteristic case, including the proof of the Mirkovic-Vilonen conjecture and the relationship with the Springer resolution of the Langlands dual group. This is joint work with L. Rider. I will also explain the connection to closely related independent work of Mautner-Riche.
I will explain some joint work with David Ben-Zvi and Adrien Brochier, which
constructs the representation theory of the double affine Hecke algebra (DAHA)
and rational Cherednik algebras (RCA) in type A from a certain topological
invariant called factorization homology, computed on a topological torus
.
The construction fits into a "3+1-dimensional" topological field theory, which was first proposed by Jacob Lurie and Kevin Walker -- foreshadowed by work of Crane-Yetter and Kauffman -- and which computes the "anomaly" of Chern-Simons theory. It is expected to produce, in a similar fashion, knot (and 3-manifold) invariants which will be not numbers, but rather representations of the DAHA/RCA. The machinery for working with this field theory involves some "derived" algebraic geometry, and also categorified representation theory. I'll try and highlight this interaction in the talk.
Starting from solutions of the Yang-Baxter equation we construct a noncommutative bi-algebra which can be described in purely combinatorial terms using non-intersecting lattice paths. Inside this noncommutative algebra we identify a commutative subalgebra, called the Bethe algebra, which we identify with the direct sum of the equivariant quantum cohomology rings of the Grassmannian. We relate our construction to results of Peterson which describe the quantum cohomology rings in terms of Kostant and Kumar's nil Hecke ring and the homology of the affine Grassmannian. This is joint work with Vassily Gorbounov, Aberdeen.
We interpret all Maurer-Cartan elements in the formal Hochschild
complex of a small dg-category which is cohomologically bounded above
in terms of torsion Morita deformations. This solves the “curvature
problem”, i.e. the phenomenon that such Maurer-Cartan elements
naturally parameterize curved deformations. We also discuss
the more subtle situation for infinitesimal deformations. This is
joint work with Michel Van den Bergh.
We describe how an isomorphism between certain completions of affine
Hecke algebras associated to and quiver Hecke algebras of type A
lifts to an isomorphism between similar completions of Vigneras'
affine Schur algebra for
and the affine quiver Schur algebra
defined by Stroppel and Webster. This is joint work with Catharina
Stroppel.
Given a smooth surface, the generating series of Euler characteristics of its Hilbert schemes of points can be given in closed form by (a specialisation of) Goettsche's formula. I will discuss a generalisation of this formula to surfaces with rational double points, built from the representation theory of affine Lie algebras. (Joint work with Adam Gyenge and Andras Nemethi, Budapest)
I'll give a short introduction to the conjectures on symplectic duality of the speaker and collaborators, with an emphasis on the very recent progress due to an explicit construction for Coulomb branches suggested by Braverman, Finkelberg and Nakajima.