Activities organised by SÉDIGA

November 2011

Workshop at ENS Paris
Date and duration: November 14-15, 2011
Topic: All topics of the ANR SÉDIGA program
Where?: ENS Paris, 45 rue d'Ulm, salle W
Organizers & Contacts: Phil Boalch, Alexandru Dimca & Claude Sabbah

Program:

Monday, November 14

14h30-15h30 Thomas Reichelt (FSMP, ENS Paris): Laurent polynomials, hypergeometric systems and mirror symmetry
Abstract: In this talk I will explain a close relationship between the Gauss-Manin systems of families of Laurent polynomials and the A-hypergeometric systems of Gelfand, Kapranov and Zelevinsky. As an application I will discuss the computation of the mirror Landau-Ginzburg model of a smooth, toric, Fano variety.

16h-17h Carlos Simpson (CNRS, Nice): Functoriality in nonabelian Hodge theory
Abstract: The goal of this work in progress with R. Donagi and T. Pantev, is to try to understand the parabolic structure of the higher direct image Higgs bundle in terms of the parabolic structure on a Higgs bundle on the domain variety of a map.

17h30-18h30 Hélène Esnault (Essen and ENS Paris): On a variant in equal characteristic p of the conjecture on p-curvatures
Abstract: This is a joint work with Adrian Langer (pdf)
This conjecture is false as such (the corresponding one in equal characteristic zero is true, after Yves André). We correct the formulation and we show a part of what can be supposed to hold.

19h45 SÉDIGA Dinner

Tuesday, November 15

9h-10h Christian Sevenheck (Mannheim): Gauß-Manin systems and free divisors
Abstract: I will present some results (a joint work with D. Mond and I. de Gregorio) concerning Gauß-Manin systems of hyperplane sections of Milnor fibers of some non isolated singularities, namely, the linearly free divisors. These singularities appear in the theory of representations of quivers. Our construction is a generalization of the mirror of the quantum cohomology of the projective space. If time permits, I will also speak of the Bernstein polynomial of these singularities, and of some very speculative ideas on the mirror objects of the linearly free divisors.

10h30-11h30 Michel Granger (Angers): Partial normalizations of Coxeter arrangements and their discriminants
Abstract: In a joint work with D. Mond and M. Schultze, we study a partial normalization of a Coxeter arrangement and its discriminant. The ring structures occurring come from the Frobenius manifold structure on space of orbits which contains the discriminant and from a lifting (without a unit) to the space of the arrangement. We also give another description by using a duality on fractional maximal Cohen-Macaulay ideals, and this produces a differential condition of order 3 on the Coxeter invariants. These normalizations also allow us to construct new free divisors by a kind of adjunction.

12h-13h Antoine Douai (LJD, Nice): Mirror symmetry and quantum differential systems : example(s) and application(s)
Abstract: We describe an aspect of mirror symmetry (correspondence between the A side (quantum cohomology) and the B side (singularities)) by using bundles with meromorphic connections. The starting idea is that the latter may be computed on the B side (in some cases), by using "elementary" methods. This correspondence can be used to define rational structures on the A side, and this generalizes a well-known method. In this context, the conformal solutions of Dubrovin play a central role.

13h-14h30 Lunch

14h30-15h30 Phil Boalch (CNRS, ENS Paris): Fission varieties
Abstract: I'll recall the quasi-Hamiltonian approach to moduli spaces of flat connections on Riemann surfaces, as a finite dimensional algebraic version of operations with loop groups, such as fusion. Recently, whilst extending this approach to meromorphic connections, a new operation arose, which we will call fission. This operation enables the construction of many new algebraic symplectic manifolds, going beyond those we were trying to construct, and suggests some conjectures about the irregular analogue of the Deligne-Simpson problem.

15h45-16h45 Alexandru Dimca (LJD, Nice): Hodge theory for projective hypersurfaces and applications
Abstract: We will discuss recent results concerning the relation between the Hodge filtration and the filtration by the order of the pole on the cohomology of the complement of a complex projective hypersurface D. Applications concern bounds on the degree of syzygies between the partial derivatives of the defining equation for D.

17h-18h Viktoria Heu (Strasbourg): Rank two connections on genus two curves
Abstract: We will present the construction of a moduli space of suitable rank two vector bundles on genus two curves, which allow to study unstability phenomena along analytic families of such bundles. This result, which is a joint work in progress with Frank Loray, relies on a symmetry, discovered by William Goldman, of rank two connections, which is specific to the genus two case.