# Helices, Quantum cohomology and (a,b)-Laplace multi-transform

In this talk I will report some very recent results about a conjecture, originally due to B. Dubrovin (arXiv:math/9807034v2), relating the quantum cohomology of a wide class of smooth projective varieties and exceptional collections in their derived categories. After stating a refined version of this conjecture (arXiv:1811.09235v1) and briefly reviewing its proof for Grassmannians of type A, I will claim its validity for a half of Hirzebruch surfaces (namely $\mathbb F_{2k}$, those diffeomorphic to $\mathbb P1\times \mathbb P1$). Details about the geography of the helices appearing at points of small quantum cohomology will be given. Furthermore, I will introduce an integral transform, the $(\alpha,\beta)$-Laplace multi-transform, and I will show that it plays a key-role for the integration of the quantum differential equations (QDE) of a wide class of Fano varieties. In particular, I will explain how to obtain explicit Pincherle-Mellin-Barnes integral representations of the solutions of the QDE of the remaining half of Hirzebruch surfaces ($\mathbb F_{2k+1}$, those diffeomorphic to the blow-up of $\mathbb P^2$ in a point), a crucial step towards the proof of the conjecture. If time allows, I will also exemplify the usefulness of the $(\alpha,\beta)$-Laplace multi-transform in order to solve the QDE of the isotropic Grassmannian IG(2,6), whose small quantum cohomology is non-semisimple at all points. Based on joint works with B. Dubrovin, D. Guzzetti, and work in progress, partially joint with M. Smirnov.

This talk is part of the Geometry and Mathematical Physics seminar series.