Abstract : On the occasion of the 1998 ICM in Berlin, Boris Dubrovin (1950-2019) conjectured an intriguing connection between the enumerative geometry of a Fano variety $X$ with algebraic and geometric properties of exceptional collections in the derived category $D^{b}(X)$. The aim of Dubrovin's conjecture is twofold.
In its qualitative formulation, the conjecture asserts the equivalence of the semisimplicity condition of the quantum cohomology $Q H(X)$ and the existence of full exceptional collections in $D^{b}(X)$.
In its quantitative formulation, the conjecture prescribes explicit formulas for local invariants of $Q H(X)-$ the so-called "monodromy data" - in terms of characteristic classes of exceptional collections.
The central object for the study of these conjectural relations is a family of linear ODEs labeled by points of $Q H(X)$, called the "quantum differential equation" of $X$ ( $q D E$, for short).
The $q D E$ is a rich invariant of $X$. First, it encapsulates information on the Gromov-Witten theory of $X$. Second, it also defines local moduli invariants for the Frobenius manifold structure on $Q H(X)$. Moreover, the asymptotics and monodromy of its solutions conjecturally rule the topology and complex geometry of $X$. The study of $q D E$ s represents a challenging active area in both contemporary geometry and mathematical physics: it is continuously inspiring the introduction of new mathematical tools, ranging from algebraic geometry, the realm of integrable systems, the analysis of ODEs, to the theory of integral transforms and special functions.
In the first talk, the speaker will give a gentle introduction to the isomonodromic approach to Frobenius manifolds and quantum cohomology. In addition, a historical overview of Dubrovin's conjecture (from its origin to its recent refinements) will be presented.
In the second talk, after surveying known positive results on Dubrovin's conjecture, the speaker will discuss several further research directions including:
- analytical refinements of the theory of isomonodromic deformations to coalescing irregular singularity
- results evoking an equivariant analog of Dubrovin's conjecture - integral representations of solutions for the $q D E \mathrm{~s}$.
These talks will be based on several works of the speaker, partially joint with B. Dubrovin, D. Guzzetti, and A. Varchenko.

Keywords : quantum cohomology; Frobenius manifolds; derived category