We will assign the talks.
From the works of K. Saito and Dubrovin, the base of the miniversal deformation of an isolated hypersurface singularity germ carries a Frobenius manifold structure. In the equivariant setting, this structure admits a refinement compatible with the group action, giving rise to the notion of atomic decomposition for such singularities. In this talk, I will present these ideas, following Section 2 of [KKPY]. Towards the end, I will briefly indicate how this connects to the notion of A-model F-bundles introduced in the previous talk.
We will introduce non-archimedean k-analytic F-bundles and discuss their basic properties. We will then recall the Gromov-Witten cycle classes associated with a pair of a smooth projective variety X over a field of characteristic zero together with a Weil cohomology theory, their basic properties, and how these properties facilitate the construction of a certain k-analytic F-bundle, referred to as the A-model F-bundle of X. The algebraic origins of Gromov-Witten cycles endow these F-bundles with important additional equivariant structure, and we will end by discussing one such enhancement.
We will introduce G-atoms of projective varieties, as well as the simpler notion of G-atomic F-bundles and their relation to decompositions of F-bundles into overmaximal pieces, making their role as F-bundle building blocks more explicit. Our focus will be on the particular example of Hodge atoms, which arise from the A-model F-bundles introduced in the previous talk. We will discuss some numerical invariants of Hodge atoms, and how they give rise to a non-rationality criterion for smooth projective varieties. We will also state Iritani's blowup formula in terms of F-bundles, and describe its relevance to the theory of atoms.
Continue the previous lecture and state Iritani's results on quantum D-modules and their relevance.
I will survey the relationship between T-equivariant quantum cohomology and the quantum cohomology of GIT quotients by T.
We will state and explain Iritani's results on the Quantum D-module of a blow-up. The goal is to explain the main idea of the proof and show how the Fourier transforms from last lecture are used to construct the relevant maps.
I will sketch work with C. Woodward on a gauged version of Gromov-Witten invariants and its relation to GW theory of GIT quotients. using Woodward's quantum Kirwan map.
The goal of this talk is to give a quick overview of what was known about cubic 4-folds before KKPY, focusing on rationality. I'll start with some special families and the Fano variety of lines, sketch a proof of the Torelli theorem (and deduce the one classical fact needed by KKPY), and then discuss Hassett and Kuznetsov's rationality conjectures and the relationship between them.
Finish proving the main theorem and discuss other applications, e.g., agreement of Hodge numbers of birational Calabi-Yau varieties.
Discuss enhancements of atoms, including pairings, integral structures, and deeper motivic aspects.