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.
Follow [I1] to discuss quantum GIT reduction conjecture from the point view of D-modules and why it matters for studying the quantum cohomology of blowups and projective bundles.
First lecture about the blowup formula: perhaps focus on Fano blowups of toric Fano varieties.
Second lecture about the blowup formula: perhaps focus on Fano blowups of toric Fano varieties.
Follow [V] and other references to discuss inputs from classical birational geometry.
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.