Algebraic Geometry Seminar

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.

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.

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 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 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).

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.

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.

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 seminal works of Kapustin–Witten and Gaiotto–Witten propose that various "Langlands type" correspondences can be interpreted as S-duality for a family of 4D gauge theories. Beginning with an exposition of relative Langlands duality in the sense of Ben-Zvi–Sakellaridis–Venkatesh along these lines, we discuss a program in which we explore and realize some features of the hypothetical Langlands TQFT in the Dolbeault twist. This talk involves joint work in progress with Emilio Franco, Enya Hsiao, and Mengxue Yang.

The theory of electromagnetism --- also known as Yang-Mills theory for the group U(1) --- admits a systematic generalization involving connections on higher U(1)-bundles. We will articulate these theories using sheaves and derived stacks, offering a synthesis of differential (or Deligne) cohomology and the BV formalism. This framework allows us to formulate in easy terms the physical idea of electromagnetic, or abelian, duality: such abelian theories on n-dimensional manifolds come in pairs, swapping gauge theory for p-bundles with gauge theory for (n-p-2)-bundles. Given time, we may discuss how these methods apply to the 6d self-dual 2-form gauge theory that appears as part of the 6d N=(1,0) and (2,0) superconformal theories, which are supposed to be the origins of the geometric Langlands correspondence. This is joint work in progress with Chris Elliott, Ingmar Saberi, and Brian Williams.

In the first talk, I will introduce the Newton stratification of the loop group LG. I will start by an introduction to the classical results on the parametrization of these points. Then I will explain how to construct a finitely presented sub ind-scheme of LG associated to each Newton point. Finally, I will explain a result that relates the adjoint quotient of a single arbitrary Newton stratum to that of a basic Newton stratum for a Levi subgroup of G.

In the second talk, I will present the formal constructions that allow for an extension of the theory of (ind-)constructible $\ell$-adic étale sheaves from schemes of finite type to the geometric objects that include $\frac{LG}{LG}$. Then I will give an idea of how using geometry one is able to identify the categorical trace of the affine Hecke category with a subcategory of LG-equivariant sheaves on LG. Finally, I will put this together with the stratification and reduction result of the first talk to sketch a second definition of affine character sheaves.