| Ignacio de Gregorio | |
| Deformations of functions on space curves and Frobenius manifolds | | We will show how to define a (logarithmic) Frobenius structure on the base space of the miniversal deformation of a function on a space curve. This could be thought of as a sort of "limit" of the Frobenius structure on Hurwitz spaces when the curves becomes singular. |
| Luisa Fiorot | |
| Localization of categories with differential operators | I would explain one result which is included in my PhD thesis. The starting point is the article of Morihiko Saito: Modules de Hodge Polarisable (1988).
Let X be a smooth algebraic or analytic variety over the complex field. In the quoted paper Saito defines a category called DF(OX,Diff) of complexes of filtered OX-modules and differential operators localized with respect to graded qis. He proved that this category is equivalent to the category DF(DX) of complexes of filtered right DX-modules.
In a previous work Complexe de De Rham filtré d'une varieté singuliere (1981) Du Bois defined a category (which Saito) called DF1(OX,Diff) of complexes of OX_modules with differential operators of order 1 localized with respect to graded qis.
There is a canonical functor from the Du Bois category DF1(OX,Diff) to the Saito one DF(OX,Diff) which Saito asserts not to know if it is faithful. I answer to this question proving that this functor not only is fully faithful but it is also an equivalence of categories. |
| Francisco Leon Trujillo | |
| D-modules and the cohomology of an arrangement of hyperplanes | | Let A be an arrangement of hyperplanes in Cn defined by the homogeneous polynomial dA. Let Dn be the Weyl algebra of rank n over C and P=C[x1,...,xn,dA-1] the holonomic Dn-module associated to dA. Studying the structure of P as Dn-module we get a sequence of new holonomic Dn-modules. These modules allow us to define useful complexes that determine the cohomology of the variety YA, the complement of the union of all hyperplanes in A. |
| Hossein Movasati | |
| Global Brieskorn modules and Hodge cycles | | The restriction of a basis of the Brieskorn module of a polynomial f with isolated singularities to the fibers of the polynomial gives a basis of the cohomology of that fiber. Can we describe the mixed Hodge structure of the fiber using a basis of Brieskorn module? Some calculations with polynomials with only one isolated singularity at the origin, for instance f=x1m+...+xnm, support the significance of this question. Positive answers to this question will give us a new method to describe the Hodge cycles of smooth hypersurfaces. |
| Mathias Schulze | |
| The differential structure of the Brieskorn lattice | | The Brieskorn lattice H'' of an isolated hypersurface singularity defines a locally free extension of the sheaf of holomorphic sections of the cohomology bundle. The Gauss-Manin connection induces a differential structure on H''. M. Saito showed that this structure is determined by two endomorphisms A0 and A1. Important invariants of the singularity like the complex monodromy and the spectral pairs can be desribed in terms of this structure. We explain algorithmic methods to compute A0 and A1 and the above invariants. These algorithms are implemented in the computer algebra system SINGULAR. |
| Christian Sevenheck | |
| Lagrangian singularities, Lie algebroids and D-modules | | We explain the notion of Lie algebroids and of rings of generalized differential operators constructed from a Lie algebroid. To any lagrangian singularity one can canonically associate a lie algebroid. When one studies the deformation theory of lagrangian singularities, a natural generalization of the solution complex of a D-module appears. If time allows, we state a version of Kashiwara's constructibility theorem in this context. |
| Jose Maria Ucha-Enriquez | |
| Applications of Gröbner bases to D-modules | We present algorithms and implementations details for obtaining invariants associated to a D-modulule. In particular they are useful in the study of - Bernstein-Sato ideals
- Computation of the annihilator of a rational function.
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