A quasimap from a curve to a GIT quotient is a map to the stack quotient that is generically stable. The geometry of Laumon spaces (an open subset of quasimaps from P^1 to the flag variety) is closely related to the representation theory of gl_n. It has been shown that one can construct an action of gl_n on the cohomology of Laumon spaces via geometric correspondences, and this cohomology can be identified with dual Verma modules of gl_n under this action. The full moduli space of quasimaps provides a natural compactification of Laumon spaces. I will explain how to construct an action of gl_n on the equivariant cohomology of these moduli spaces and explore its relation to tilting modules in Category O.

The de Rham-Hitchin system is a deformation of the Hitchin system in prime characteristic. It plays a main role in the prime characteristic Non Abelian Hodge Theory, relating Higgs bundles and connections. In this talk, I will survey what I know about the geometry and topology of the de Rham-Hitchin system, emphasizing how it relates to complex geometry and number theory. We start by motivating some correspondences on the category level using p-curvatures. Then we do some moduli theory to get a correspondence at the moduli space level. Finally, we deduce some consequences on the cohomologies of the moduli spaces.

The microlocal point of view was introduced by M. Sato in the 1960s for studying partial differential equations. It was then adopted by M. Kashiwara and P. Schapira and developed into a systematic theory in the context of sheaves on manifolds. The theory has since had applications in many fields, including partial differential equations, symplectic geometry, geometric Langlands, and exponential sums. In this talk, I will explain the basic ingredients of this theory, and discuss recent development of its analogues in the contexts of étale sheaves on algebraic varieties and rigid analytic varieties.

 If A is a cluster algebra then, by the Laurent phenomenon, every cluster determines an open torus in the cluster variety Spec(A) called a cluster torus. In general, the union of cluster tori only covers Spec(A) up to codimension 2, and the complement of the union of cluster tori in Spec(A) is called the deep locus. Any "bad" (e.g. singular) point in the cluster variety must belong to the deep locus, but the deep locus may be nonempty even when Spec(A) is nice. In joint work with Marco Castronovo, Mikhail Gorsky, and David Speyer, we conjecture that the deep locus may be characterized as those points with nontrivial stabilizer under a natural action of a group on Spec(A). We are able to prove this conjecture for algebras of finite cluster type and for algebras associated to Grassmannians Gr(3,n), that are typically of infinite cluster type. An essential tool in our approach is the realization of the corresponding cluster varieties as braid varieties. In particular, for braid varieties the geometry of the deep locus should be related to properties of the link obtained when closing the braid. I won't assume previous knowledge of cluster algebras or braid varieties.

The cohomology of moduli spaces of 1-dimensional sheaves on del Pezzo surfaces carries a perverse filtration, from which curve counting invariants of CY 3-folds can be extracted. Conjecturally, the perverse filtration matches a second filtration, defined in terms of tautological classes, called the Chern filtration. I will explain this conjecture, what is currently known, and some consequences. In the second part of the talk I will explain some new ideas to prove properties of the Chern filtration by using parabolic sheaves. In particular, I will discuss the top Chern degree and chi-independence phenomena. The talk is based on joint work in progress with W. Lim and W. Pi.

The affine Hecke category, defined using affine Soergel bimodules, categorifies the affine Hecke algebra. I will compare the derived horizontal trace of the affine Hecke category with the elliptic Hall algebra, and with the derived category of the commuting stack. In particular, I will describe certain explicit generators for the trace category and some categorical commutation relations between these. This is a joint work with Andrei Negut.