Singularities of DIfferential Equations in Algebraic Geometry

ANR Programme blanc N° ANR-08-BLAN-0317-01/02

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Description of the project (January 2009 - June 2013)

The aim of the SÉDIGA project is to make progress on the following topics:

1. Irregular singularities of linear differential systems in any dimension and over various base fields,and their algebraic de Rham cohomology.

2. Structures on the deformation spaces of Gauss-Manin systems and Brieskorn lattices ofsingularities of holomorphic functions (Hodge structures, flat structures, tt* structures, families of (a,b)-modules,...).

3. The fundamental group in algebraic geometry and nonabelian Hodge theory with tame or wild ramification.

One of the original aspects of the project consists in obtaining results in each topic by exhibiting relations between these topics through the use of various tools and methods (p-adic differential equations, non-archimedean p-adic geometry, algebraic geometry, complex topology, singularity theory and D-modules, differential geometry) with, in the background, motivations and conjectures formulated by physicists.

The project is built around two partners (École polytechnique and Université de Nice). Partner 1 has a decentralized character: Palaiseau (coordinator), Angers, Paris-ENS, Nancy, while Partner 2 is mainly centered in Nice. Teams of Mannheim (C. Hertling, Ch. Sevenheck) and Padua (F. Baldassarri) are associated to the project.