Institut de Mathématiques de Luminy


Rencontre 2012 du GDR Singularités et Applications

GDR Singularities and Applications 2012 Annual Meeting


Programme scientifique - Schedule

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Le rencontre sera organisée autour d'un « mini-cours » introductif au sujet, donné par Jean-Paul Brasselet (IML-Marseille) et de trois cours (tous en anglais) :
La rencontre comprendra aussi des conférences sur les thèmes proches.

 

programme GDR Singularités 2012

COURSES


- Paolo Aluffi (Thalahassee, USA)
Title: Chern classes of singular varieties, string theory, and Feynman integrals

Abstract:
We will describe an elementary approach to the study of Chern classes for (possibly singular) algebraic varieties, providing the needed background from algebraic geometry. We will also discuss points of contact of the theory of characteristic classes with an analogue of motivic integration, leading to the definition of a `stringy' Chern class, and emphasize parallels between computation of Chern classes and computations of classes in the Grothendieck ring of varieties. Examples will include results on Euler characteristics of elliptic fibrations, motivated by string theory, and computations involving `graph hypersurfaces', motivated by the study of Feynman integrals.

- Adam Parusinski (Nice, France)
Title: Characteristic classes and characteristic cycles

Abstract:
We give an introduction to the characteristic classes of real or complex algebraic sets, or more generally complexes of constructible sheaves, via the characteristic cycle construction. The characteristic cycle is a Lagrangian cycle defined in the conormal space. It encodes the vanishing Euler characteristic and can be used to define various characteristic cycles of singular spaces as the Chern-Mather, Chern Mac-Pherson classes and Stiefel-Whitney classes. We discuss several aspects and insights this construction gives: stratifications, microlocal analysis, Milnor fibers and Milnor classes, Chern-Gauss-Bonet and the limit of curvature measures.

- Jörg Schürmann (Münster, Allemagne)
Title: Hirzebruch classes of singular spaces

Abstract:
The most important characteristic classes of singular spaces are formulated as natural transformtions to some homology theory: MacPherson's Chern classes, Baum-Fulton- MacPherson's Todd classes and Goresky-MacPherson's resp. Cappell-Shaneson's L-classes. Recently Brasselet-Schuermann-Yokura introduced a unifying theory of a Hirzebruch class transformation, defined on relative Grothendieck groups of algebraic varieties as studied in the context of motivic integration. Recent joint work of Cappell-Maxim-Schuermann- Shaneson studies Hirzebruch classes of hypersurfaces and its relation to motivic nearby and vanishing cycles, as well as equivariant versions of thèse characteristic classes like equivariant Chern-, Atiyah-Singer- and Hirzebruch classes for algebraic actions of finite groups. These are then used to study generating functions of characteristic classes of symmetric products as well as Hilbert schemes of points.


LECTURES


- Omid Amini (ENS, Paris)
Title: Linear Series in Algebraic Geometry and Combinatorics

Abstract:
I will give a survey of recent results on the behavior and applications of linear series on singular spaces (mostly, singular curves), and the interesting combinatorics behind.

- Nicolas Dutertre (AMU, Marseille)
Title: On the topology of real Milnor fibrations

Abstract:
ABSTRACT Nicolas Dutertre
- Francisco-Jesús Castro-Jiménez (Universidad de Sevilla, Spain)
Title: Recent and less recent results in A-hypergeometric systems

Abstract:
A-hypergeometric systems, which are systems of linear partial differential equations, have been studied by several authors and namely by Gelfand, Graev, Kapranov and Zelevinsky since the '80. Each of these systems is associated with an integer matrix A and a complex parameter vector. We will review some properties of these systems from the point of view of D-module theory

- Alexandru Dimca (Université de Nice)
Title: Log-convex and log-concave sequences: algebra and topology

Abstract:
Some 40 years ago the Teissier numbers of an isolated hypersurface were introduced, either using an algebraic approach (mixed multiplicities of ideals) or a topological view point (Milnor numbers of generic linear sections of the singularity). They form a log-convex sequence.
Using a similar pattern, June Huh has introduced in a récent paper (J. Amer.Math.Soc 25 (2012), 907-927) a log-convex sequence related to the topology of an arbitrary projective hypersurface. This led to a proof of a long standing conjecture on the chromatic polynomial of a graph, via the theory of hyperplane arrangements.
In our talk we'll try to report on this new development.


 
Mise à jour : 25 septembre 2012, EL.