This is the first lecture in a longer, somewhat open-ended series in which various aspects of quasimaps and their applications to enumerative geometry questions will be discussed. The intention is to use it to fill the empty slots in the Cohomology seminar, interrupting it whenever other speakers are available and resuming it afterwards, as long as there is interest. For the first lecture, an appropriate subtitle might be "A review of Givental's proof of the mirror theorems", the purpose being to help put the genesis of the ideas behind the theory in an appropriate context. The actual introduction to the general theory, with formal definitions etc. will begin in the second lecture.

I will speak about how the standard theory of Grothendieck D-modules generalizes to non-smooth varieties by interpreting the standard definition of differential operators in a derived way.

In this second lecture I will wrap-up the discussion about Givental's work from the mid-nineties, then proceed to introduce the notion of quasimaps to (a certain large class of) GIT quotients. Next, I will discuss stability conditions, the resulting moduli spaces, and their general properties.

**No seminar on Oct 4.**

I’ll continue about moduli of quasimaps and introduce their associated wall-crossing.

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.