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Algebraic Analysis and Geometry

Monday, 20 June 2022 - Padova - Italy

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Schedule

In room 2BC30

11:00-12:00 Giordano Cotti Borel-Laplace multi-transform, and integral representations of solutions of qDEs
lunch break
14:30-15:30 Andreas Hohl Moderate and rapid decay nearby cycles via enhanced ind-sheaves
15:30-16:30 Teresa Monteiro Fernandes An application of the relative Riemann-Hilbert correspondence

Abstracts

Giordano Cotti (Universidade de Lisboa)
Borel-Laplace multi-transform, and integral representations of solutions of qDEs
The quantum differential equation (qDE) is a rich object attached to a smooth projective variety X. It is an ordinary differential equation in the complex domain which encodes information of the enumerative geometry of X, more precisely its Gromov-Witten theory. Furthermore, the monodromy of its solutions conjecturally rules also the topology and complex geometry of X. These differential equations were introduced in the middle of the creative impetus for mathematically rigorous foundations of Topological Field Theories, Supersymmetric Quantum Field Theories and related Mirror Symmetry phenomena. Special mention has to be given to the relation between qDE's and Dubrovin-Frobenius manifolds, the latter being identifiable with the space of isomonodromic deformation parameters of the former. The study of qDE’s represents a challenging active area in both contemporary geometry and mathematical physics. In this talk I will introduce some analytic integral multitransforms of Borel-Laplace type, and I will use them to obtain Mellin-Barnes integral representations of solutions of qDEs.

Andreas Hohl (IMJ Paris)
Moderate and rapid decay nearby cycles via enhanced ind-sheaves
Enhanced ind-sheaves were introduced by A. D'Agnolo and M. Kashiwara as a topological target category of a Riemann-Hilbert functor for (not necessarily regular) holonomic D-modules. They proved that a holonomic D-module $\mathcal{M}$ can be reconstructed from the enhanced ind-sheaf $\mathcal{DR}^\mathrm{E}_X(\mathcal{M})$ associated to it. Inspired by some constructions by D'Agnolo-Kashiwara for enhanced ind-sheaves in dimension one and Sabbah on nearby cycles for holonomic D-modules in higher dimensions, we define moderate growth and rapid decay objects associated to any enhanced ind-sheaf. We show that these objects recover the classical moderate growth and rapid decay De Rham complexes of a holonomic D-module if applied to the object $\mathcal{DR}^\mathrm{E}_X(\mathcal{M})$. The proof of this statement works along the lines of a duality result between tempered and Whitney holomorphic functions established by Kashiwara-Schapira. Once proved, one derives in particular some well-known duality results for connections. In this talk, I will motivate and describe these constructions of moderate and rapid decay nearby cycles as well as the duality between them. This is joint work with Brian Helper.

Teresa Monteiro Fernandes (Universidade de Lisboa)
An application of the relative Riemann-Hilbert correspondence
Let $X$ be a complex analytic manifold and let $S$ be a complex manifold which plays the role of a parameter space. The relative Riemann-Hilbert (denoted by $RH^S$) correspondence is now well established in previous work by Luisa Fiorot, myself and Claude Sabbah. In this talk, using the sheaf $\mathcal{D}_{X\times S/S}^{\infty}$ of relative differential operators of infinite order, we apply $RH^S$ and Schapira-Schneiders (Elliptic pairs I) to generalize a classical theorem by Kashiwara-Kawai (On Holonomic systems of micro-differential equations III): given a holonomic $\mathcal{D}_{X\times S/S}$-module $\mathcal{M}$, denoting by $\mathcal{M}^{\infty}$ the tensor product of $\mathcal{M}$ by $\mathcal{D}_{X\times S/S}^{\infty}$, there exists a regular holonomic $\mathcal{D}_{X\times S/S}$-submodule $\mathcal{M}_{reg}$ of $\mathcal{M}^{\infty}$ such that $\mathcal{M}^{\infty}$ and $\mathcal{M}_{reg}^{\infty}$ are isomorphic. We shall give examples to ilustrate.

(Last update: Sunday June 5, 2022)


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