Organizers: Ionut Ciocan-Fontanine, Adeel Khan, Y.P. Lee
Institute of Mathematics, Academia Sinica
In this talk, I will explain joint work with Tasuki Kinjo (Kyoto), whose goal is to establish a quantitative version of the decomposition theorem (in the sense of Beilinson-Bernstein-Deligne-Gabber) for the morphism from a symmetric algebraic stack to its good moduli space.
This result provides a formula allowing one to reconstruct the cohomology of smooth stacks from the intersection cohomology of the stacks of graded points. It has applications to the study of the Borel-Moore homology of 0-shifted symplectic stacks and to the critical cohomology of (-1)-shifted symplectic stacks. I will discuss consequences for the purity of Hodge structures and a Kirwan surjectivity theorem for the restriction morphism to the semistable locus.
For a smooth surface S, the Hilbert scheme of n points on S, Hilb(S,n), is smooth of dimension 2n. The Hilbert-Chow morphism Hilb(S,n)\to S^n/S_n is a crepant resolution of the (singular) n-fold symmetric product variety S^n/S_n associated to S. The so-called crepant resolution conjecture in Gromov-Witten theory predicts in this case explicit equalities between generating functions of Gromov-Witten invariants of Hilb(S,n) and generating functions of Gromov-Witten invariants of the symmetric product stack Sym(S,n). In this talk, we discuss the formulation of this conjecture and proofs in known cases.
Organizer: Cheng-Chiang Tsai
Higgs bundles on algebraic varieties have many important applications, including the proof of the fundamental lemma and non-abelian Hodge theory. In this talk, I introduce a notion of Higgs bundles on the Fargues–Fontaine curve, a fundamental object in the local Langlands program that is not itself an algebraic variety. I will state a version of the BNR correspondence, which relates Higgs bundles with line bundles on suitable curves. I will then describe an action of a Picard stack on the moduli stack of Higgs bundles and show that, after taking the quotient by this action, the resulting stack factorises as a product of B_dR-affine Grassmannians. Finally, I will discuss connections with number-theoretic objects and outline several directions for future research.