In the local classification of differential equations of one complex variable, torsors under a certain sheaf of algebraic groups (the Stokes sheaf) play a central role. On the other hand, Deligne defined in positive characteristic a notion of skeletons for l-adic local systems on a smooth variety, constructed an algebraic variety parametrizing skeletons and raised the question wether every skeleton comes from an actual l-adic local system. We will explain how to use a variant of Deligne's skeleton conjecture in characteristic 0 to prove the existence of an algebraic variety parametrizing Stokes torsors. We will show how the geometry of this moduli can be used to prove new finiteness results on differential equations.[-]

Homological mirror symmetry asserts that the connection, discovered by physicists, between a count of rational curves in a Calabi-Yau manifold and period integrals of its mirror should follow from an equivalence between the derived Fukaya category of the first manifold and the derived category of coherent sheaves on the second one. Physicists' observation can be reformulated as, or rather upgraded to, a statement about an isomorphism of certain Hodge-like data attached to both manifolds, and a natural first step towards proving the above assertion would be to try to attach similar Hodge-like data to abstract derived categories. The aim of the talk is to report on some recent progress in this direction and illustrate the approach in the context of what physicists call Landau-Ginzburg B-models.[-]

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.[-]

Given a perverse sheaf or a holonomic D-module on an abelian variety there are two ways to associate a set of holomorphic one forms on it one via the singular support and one via the generic vanishing theory. In this talk I will present a joint work with Feng Hao and Yongqiang Liu where we connect these two sets. On a smooth projective irregular variety our results relates to a conjecture proposed by Kotschick and studied by Schreieder and shows that their conjecture can be reinterpreted as follows: the existence of nowhere vanishing holomorphic one forms is equivalent to the non-existence of components given by conormal space of varieties of general type in the decomposition theorem for the albanese morphism. Using some known results we show that the condition is necessary.[-]

The intimate relation between the arithmetic properties of varieties varying in families and the properties of the associated Picard-Fuchs differential is subject with a long and rich history that can be traced back to Deuring, Igusa, Dwork, Honda, Katz from which the notion of crystals emerged. A particular nice situation arises from families of Calabi-Yau motives, which can arise via various constructions, most notably via Mirror-Symmetry. In the two talks I will try to give a rough overview of this field, and illustrate it with specific examples. In particular, I will indicate how Calabi-Yau operators can be used to realise certain rank 4 motives attached Siegel paramodular forms by specific Calabi-Yau threefolds.[-]

There is a very general story, due to Joyce and Kontsevich-Soibelman, which associates to a CY3 (three-dimensional Calabi-Yau) triangulated category equipped with a stability condition some rational numbers called Donaldson-Thomas (DT) invariants. The point I want to emphasise is that the wall-crossing formula, which describes how these numbers change as the stability condition is varied, takes the form of an iso-Stokes condition for a family of connections on the punctured disc, where the structure group is the infinite-dimensional group of symplectic automorphisms of an algebraic torus. I will not assume any knowledge of stability conditions, DT invariants etc.[-]

I will sketch the construction - following ideas of Kontsevich and Nori - of a Tannakian category of exponential motives over a subfield of the complex numbers. It is a universal cohomology theory for pairs of varieties and regular functions, whose de Rham and Betti realizations are given by twisted de Rham and rapid decay cohomology respectively. The upshot is that one can attach to any such pair a motivic Galois group which conjecturally generalizes the Mumford-Tate group of a Hodge structure and, over number fields, governs all algebraic relations between exponential periods. This is a joint work with Peter Jossen (ETH).[-]

The classical Riemann-Hilbert correspondence establishes an equivalence between the triangulated categories of regular holonomic D-modules and of constructible sheaves. In a joint work with Masaki Kashiwara, we proved a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular. The construction of our target category is based on the theory of ind-sheaves by Kashiwara-Schapira and is influenced by Tamarkin's work on symplectic topology. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya.[-]

In their preprint about the Shafarevich conjecture for hypersurfaces on abelian varieties, Lawrence and Sawin prove a big monodromy theorem for families of hypersurfaces by reducing it to a similar result for Tannaka groups of perverse intersection complexes. A large part of their work is an intricate combinatorial argument about Hodge numbers, which is used to exclude that the Tannaka group acts via wedge powers of the standard representation of SL(n). We explain a simple geometric proof of the analogous result when hypersurfaces are replaced by subvarieties of high codimension; this is joint work in progress with Ariyan Javanpeykar, Christian Lehn and Marco Maculan.[-]

I will give a general introduction to the study of the Hodge filtration on local cohomology sheaves associated to closed subschemes of smooth complex varieties, using techniques from both D-module theory and birational geometry. In the case of hypersurfaces, this is essentially the theory of Hodge ideals, which I will recall. This study has applications to various topics, like local vanishing, local cohomological dimension, the Du Bois complex, minimal exponents of singularities, etc. I will discuss a few, and more will appear in M. Mustaja's lecture.[-]