Quantum cohomology $QH^(X)$ of a smooth projective variety $X$ is a Frobenius manifold defined by genus zero Gromov-Witten invariants. A priori, the structure constants of $QH^(X)$ are formal power series, but are expected to be convergent series in the context of mirror symmetry. It was proved by Coates, Corti, Iritani and Tseng that torus-equivariant quantum cohomology of a toric variety is convergent. It was observed by Coates, Givental and Tseng that torus-equivariant quantum cohomology $QH^_T(E)$ is formally decomposed into a direct sum of $QH^(B)$. In this talk, I will explain that, under the assumption that $QH^(B)$ is convergent, $QH^_T(E)$ is convergent and the decomposition becomes analytic. As an application, we can take the non-equivariant limit of the decomposition.
Quantum cohomology can be studied by a mirror theorem, that is, by finding an $I$-function, which is a certain generating function of genus zero Gromov-Witten invariants. In general, it is a very difficult problem explicitly describing an $I$-function. For instance, an $I$-function of a smooth semi-projective toric variety is a hypergeometric series (given by Givental, Coates, Corti, Iritani and Tseng), and that of a toric bundle constructed as a GIT quotient of a direct sum of line bundles is described as a hypergeometric modification of the $J$-function of the base space. It is natural to consider an $I$-function of a toric bundle that is a GIT quotient of a vector bundle, which is called a non-split toric bundle. However, Brown's method cannot be applied to the non-split case, so we need a completely different strategy. In this talk, I will explain how to establish a mirror theorem for non-split toric bundles. This work is partially based on joint work with Iritani.
I'll talk about Weil restriction in derived algebraic geometry, and what this has to do with blow-ups and deformation to the normal cone. Based on forthcoming joint work with Jeroen Hekking and David Rydh.
I plan to review the basics of (permutation-equivariant) quantum K-theory and discuss computations related to the quantum K-theory of a point. In particular, I'll try to explain how the quantum K-theory of a point is important in many localization computations, and why the high genus theory is particularly difficult compared to the cohomological theory.
I will talk about a virtual, relative, and orbifold version of Riemann-Roch. Based on work in progress with Charanya Ravi.
Braden's hyperbolic restriction theorem is a powerful tool for proving functorial properties of D-modules and constructible sheaves. It has a vast amount of applications in the geometric representation theory. In this talk, I will explain a proof of it following the paper of Drinfeld and Gaitsgory and illustrate its significance via examples.
The abelian/non-abelian correspondence refers to expressing invariants of a GIT quotient X//G (with X quasiprojective and G reductive) via invariants of a corresponding “abelian” quotient X//T, with T a maximal torus in G. This is well-understood at the topological level (e.g, for cohomology rings), and has also been extended to genus zero Gromov-Witten theory (i.e., to quantum cohomology). I will explain the above story in the first part of the talk, while in the second I will ruminate about my unsuccessful efforts to uncover some version of the correspondence (which I believe to exist) in higher genus GW theory.
Continuation.
I'll attempt to explain the construction of John Pardon's Grothendieck group of 1-cycles and how he uses it to give a simple proof of the MNOP conjecture.
Tropical geometry is related to compactifications and degenerations of algebraic varieties. Tropical cohomology is a tropical analog of singular cohomology, and is much simpler than singular cohomology. I believe that tropical cohomology would be helpful to study cycle class maps. I will explain this story, including an algebraicity type result and a new cohomology theory (in progress), which is much better for our purpose.
Various sheaf-theoretic incarnations of the Fourier transform have been introduced by Deligne, Sato, Brylinski-Malgrange-Verdier and others. I will discuss how these can be generalized to derived vector bundles (or perfect complexes). After briefly indicating why one might want such a thing, I will sketch a proof that the derived Fourier transform is involutive.