## Algebraic Analysis and GeometryMonday, 20 June 2022 - Padova - Italy |

In room 2BC30

11:00-12:00 | Giordano Cotti | Borel-Laplace multi-transform, and integral representations of solutions of qDEs |
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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 |

**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)