CIMPA SCHOOL SCHEDULE

 Monday 14 Tuesday 15 Wedns. 16 Thursday 17 Friday 18 Satur. 19 Monday 21 9:00-10:15 Registration / Coffee Gomez-Mont 2 Artal 3 Gomez-Mont 3 Dimca 4 Nemethi 3 Nemethi 4 10:15-10:30 Break Break Break Break Break Break Break 10:30-11:45 Gomez-Mont 1 Artal 2 Dimca 2 Dimca 3 Artal 4 Veys 3 Veys 4 11:45-12:30 Coffee break Coffee break Coffee break Coffee break Coffee break Coffee break Coffee break 12:30-13:45 Artal 1 Nemethi 1 Nemethi 2 Veys 1 Veys 2 Gomez-Mont 4 Talk D. Kerner. Lunch 16:00-17:15 Dimca 1 Talk M. Pe /S. Martinez Dimca Exe Talk D.S. Kumar Talk H. Cobo /  E. Gorsky Free Talk L. Maxim 17:30-18:20 Talk A. Campillo Gomez-Mont Exer. Artal Exer. Talk P. Petrov Nemethi Exer. Free Talk Pi. Cassou-Nogues

Abstracts for talks

Evgeny Gorsky: "Motivic integrals and functional equations"
Abstract: A functional equation for the motivic integral corresponding to the Milnor number of an arc is derived using the Denef-Loeser formula for the change of variables. Its solution is a function of five auxiliary parameters, it is unique up to multiplication by a constant and there is a simple recursive algorithm to find its coefficients. The method is universal enough and gives, for example, equations for the integral corresponding to the intersection number over the space of unordered tuples of arcs.

Dmitry Kerner: "Enumeration of uni-singular plane curves / hypersurfaces"
Abstract: We enumerate complex algebraic hypersurfaces in $P^n$, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equisingular strata in the parameter space of hypersurfaces. We suggest an inductive procedure, based on the classical intersection theory combined with liftings (partial resolutions) and degenerations. The procedure computes the (co)homology class in question, whenever a given singularity type is properly defined. We will start from enumeration of uni-singular plane curves (in this case the method is particularly simple and is applicable to any singularity type). If time permit we will consider the case of hypersurfaces. Here we restrict consideration to the generalized Newton-non-degenerate singularities.

Shiv Datt Kumar: "Sections of Zero Dimensional Ideals Over a Notherian Ring".
Abstract: Let $A$ be a commutative Noetherian ring and $I \subset A[T]$ be an ideal containing a monic polynomial such that $A[T]/I$ is a zero dimensional. Suppose conormal module $I/I^{2}$ is generated by $r$ elements over $A[T]/I$. Then a set of $r$ generators of $I(0) : = \{ f(0)| f \in I \}$ can be lifted to a $r$ generating set of $I$.

Sergio Martinez: "Algorithmic computation of local braid monodromy for plane singularities"
Abstract: Braid monodromy provides a complete description of the topology of plane algebraic curves. The simplest case is the study of local singularities of plane curves. In this situation a single braid is enough to describe their braid monodromy and such a braid encodes the complete topological information because its closure is the link associated with the singularity

Laurentiu Maxim: "Genera of complex varieties and singularities of proper maps"
Abstract: I will report on recent work (jointly with S. Cappell and J.Shaneson) on Hodge-theoretic genera of (possibly singular) projective varieties. These are one-parameter families of invariants extending to the singular setting the classical Hirzebruch genera of Kahler manifolds, and including as a particular case the Goresky-MacPherson signature of a complete variety. I will present formulae that relate these global invariants of a projective variety X to such invariants of singularities of proper algebraic maps defined on X. Such formulae severely constrain, both topologically and analytically, the singularities of complex maps,even between smooth varieties.

Maria Pe Pereira "On equisingularity of parametrised surfaces"
Abstract: (Joint work with J. Fernandez de Bobadilla) Let $f_t:(\mathbf{C}^3,O)\to \mathbf{C}$ be a family of reduced holomorphic germs. We say that $f_t$ is {\em equisingular at the normalisation} if the pairs given by the topological normalisation of the zero set of $f_t$ and the inverse image by the normalisation map of the singular set of $f_t^{-1}(0)$ form a topologically equisingular family. We prove that, if L\^e's conjecture holds, equisingularity at the normalisation is equivalent to topological right-equisingularity for the family $f_t$. Furthermore, if the generic transversal type of the singularities of $f_t$ are curve singularities with smooth branches then the statement does not depend on L\^e's conjecture. The case in which the above result is most meaningful is the case of parametrised surfaces. Let $h_t:(\mathbf{C}^2,O)\to (\mathbf{C}^3,O)$ be a family of finite and generically $1-1$ holomorphic germs depending holomorphically on a parameter $t$. Assuming L\^e's conjecture we prove that the family is topologically $A$-equisingular if and only if the Milnor number of the curve given by the inverse image by $h_t$ of the critical set of $h_t(\mathbf{C}^2)$ is constant. The result is independent on L\^e's conjecture in the case of finitely $A$-determined map germs. Thus we obtain as a particular case a recent result of Calleja-Bedregal, Ruas and Houston. We also prove that if the generic L\^e numbers at the origin of $h_t(\mathbf{C}^2)$ are independent on $t$ then the family is topologically $A$-equisingular. Similar results are obtained for the case of a family of maps $h_t:(V,O)\to (\mathbf{C}^3,O),$ where $(V,O)$ is a normal surface germ.