Organizers: Ionut Ciocan-Fontanine, Adeel Khan, Y.P. Lee
Institute of Mathematics, Academia Sinica
For a hyperkahler manifold X, I will formulate a version of Kapustin's conjectural duality between type A and B-branes on X, relating the Fukaya category of I-holomorphic Lagrangians to deformation quantisation. On the other hand, a conjecture of Smith and Solomon-Verbitsky predicts the formality of the Fukaya category associated to a suitable collection of I-holomorphic Lagrangians. The aim of this series of 3(?) talks will be to explain a proof of (the B-side analogue of) this conjecture.
The log etale topology is a natural analogue of the etale topology for log schemes. Unfortunately, very few things satisfy log etale descent -- not even vector bundles or the structure sheaf. We introduce a new rhizomic topology that sits in between the usual and log etale topologies and show most things do satisfy rhizomic descent! As a case study, we look at tropical abelian varieties and give some exotic examples.
The first part of this talk provides a brief and gentle introduction to the theory of two-dimensional cohomological Hall algebras of quivers and surfaces, focusing on the construction of these algebraic structures and the main results concerning them.
The second part focuses on the introduction of the nilpotent cohomological Hall algebra COHA(S, Z) of coherent sheaves on a smooth quasi-projective complex surface S that are set-theoretically supported on a closed subscheme Z. This algebra can be viewed as the "largest" algebra of cohomological Hecke operators associated with modifications along a subscheme Z of S. When S is the minimal resolution of an ADE singularity and Z is the exceptional divisor, I will describe how to characterize COHA(S, Z) in terms of the Yangian of the corresponding affine ADE quiver (based on arXiv:2502.19445, joint work with Emanuel Diaconescu, Mauro Porta, Oliver Schiffmann, and Eric Vasserot).
For a hyperkahler manifold X, I will formulate a version of Kapustin's conjectural duality between type A and B-branes on X, relating the Fukaya category of I-holomorphic Lagrangians to deformation quantisation. On the other hand, a conjecture of Smith and Solomon-Verbitsky predicts the formality of the Fukaya category associated to a suitable collection of I-holomorphic Lagrangians. The aim of this series of 3(?) talks will be to explain a proof of (the B-side analogue of) this conjecture.
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.