Algebraic Geometry Seminar

I shall explain my very recent work which resulted in a proof of the result that has been around as a conjecture for several decades, and is perhaps best known as the rational hit conjecture (named by Reg Wood in the 1990s); it states that a certain space of coinvariants for the rational Steenrod algebra is of dimension n!. The argument uses multiarrangements of hyperplanes, and various algebraic and geometric objects attached to them, such as their tangent Lie-Rinehart algebras and twisted logarithmic de Rham complexes.

A number of interesting cohomology theories in algebraic geometry that are encompassed by the setting of motivic stable homotopy theory can be equipped with multiplicative norms along finite étale maps. These norms can be encoded as an algebraic structure on a motivic spectrum, called a normed algebra by Bachmann-Hoyois. I will explain how normed algebras can be described as algebras for a canonical monad in an (∞,2)-category of "bispans", given by an algebro-geometric analogue of the usual formula for free commutative algebras. This is joint work with Elden Elmanto.

We study standard and costandard modules of the cohomological Hall algebra associated with a quiver with potential. We will discuss their properties, such as equivariant formality and highest weight generation. As an application, we prove a Kirwan surjectivity statement for the nilpotent Grothendieck-Springer fiber in the Grassmannian.

We then use these representation-theoretic results to construct a triangulated category equipped with a monoidal structure, which we call the category of lines. For a quiver of ADE Dynkin type, we establish a monoidal equivalence between the category of lines and the category of 1/2 BPS line operators in 4d, N=2 pure gauge theory, as constructed by Braverman, Finkelberg, and Nakajima and further developed by Cautis and Williams.

This talk is based on ongoing joint work with Ryo Fujita, Yan Soibelman, and Gufang Zhao.

Elliptic cohomology of a topological space (possibly equipped with a group action) naturally takes values in coherent sheaves on an abelian variety. In this talk, we study its Fourier–Mukai transform, and explain applications to Kähler moduli spaces, in particular, to the monodromy of quantum K-theory, and to a new conjectural duality of convolution algebras arising in 3d mirror symmetry. A central ingredient is an interpretation of stable envelopes of Maulik and Okounkov in terms of the Fourier–Mukai transform. This is based on ongoing work in collaboration with Khan and Sibilla.

Let $G$ be a semisimple algebraic group and $G/B$ its flag variety. For $G = SL_n$, Givental and Lee showed that the quantum $K$-theoretic $J$-function is an eigenfunction of a $q$-difference Toda Hamiltonian. They also proposed a construction of commuting Hamiltonians for general $G$. It has since been pointed out that this construction requires modification in non-simply-laced types.

The general problem is to embed the Weyl-invariant representation ring into an algebra of $q$-difference operators so that the $J$-function is a joint eigenfunction, and to determine the resulting Hamiltonians explicitly. In this talk, I will discuss the current status of this problem and present an explicit formula for the Hamiltonian associated with a minuscule fundamental weight. This is joint work with Koushik Brahma, Yicen Huang, Takafumi Kouno, and Kohei Yamaguchi.

We study orbifold Calabi-Yau hypersurfaces in weighted projective stacks from the point of view of the gauged linear sigma model. We give a very brief introduction to the gauged linear sigma model which is a physics version of quasimap theory and review some general conjectures. We explain how these conjectures allow for determination of the orbifold Gromov-Witten invariants without using Gromov-Witten or quasimap theory.

There are two natural ways to deform Joyce's construction of vertex algebras. One is to include tautological insertions, and the other is to work equivariantly. I introduce T-deformed vertex algebras, which unify these two refinements. For the purpose of wall-crossing, they must be treated differently because the associated Lie algebras use different expansions. I will explain this behavior explicitly by studying tautological stable pair wall-crossing and the equivariant 2-loop quiver. Using the calculus of T-deformed vertex algebras developed in an upcoming joint work with E. Bouaziz, we derive 7 new Virasoro algebras for the affine plane.