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