|9:00 - 10:15
||JP Brasselet||JP Brasselet||
|10:45 - 12:00
||Tatsuo Suwa||Tatsuo Suwa||
|13:30 - 14:45
||Paolo Aluffi||Paolo Aluffi||
|15:15 - 16:30
||Jörg Schürmann||Jörg Schürmann||Jörg Schürmann||
|Coffee||JP Brasselet||Tatsuo Suwa||Matilde Marcolli|
In my lectures I will stress the "computability" of several characteristic classes, and in particular of the functorial Chern-Schwartz-MacPherson classes. I will emphasize the role of Segre classes, and relations with standard and not-too-standard commutative algebra.
I will also present two constructions of Chern-Schwartz-MacPherson classes relying on the factorization theorem of Abramovich et al. and on taking limits of Chow groups.
A tentative plan for the lectures is as follows:
---Lecture 1: Characteristic classes and playing with blocks
---Lecture 2: Segre and Schwartz-MacPherson, Fulton, Fulton-Johnson classes
---Lecture 3: The hypersurface case, and some commutative algebra
---Lecture 4: CSM classes through limits
A short programme of my lectures:
1. Poincaré-Hopf Theorem
- smooth case, with (and without) boundary,
- case of singular varieties (radial vector fields).
2. Characteristic classes by obstruction theory in the smooth case,
3. Singular varieties: triangulations, stratifications.
4. Characteristic classes in the singular case
- by obstruction theory (Schwartz classes),
- MacPherson classes,
- brief survey of other classes.
I will talk about joint work with Alain Connes where we describe the Riemann-Hilbert correspondence underlying the Connes-Kreimer theory of renormalization in perturbative quantum field theory. The resulting Galois group is universal with respect to renormalizable physical theories and can be identified with the motivic Galois group of a category of mixed Tate motives.
Lecture 1: Constructible functions and sheaves.
Introduction to the calculus of constructible functions and sheaves on stratified spaces, including Poincar'e-Verdier duality together with Grothendieck and Witt groups of constructible sheaves.
Lecture 2: Characteristic classes of Lagrangian cycles.
Introduction to stratified Morse theory for constructible functions and Lagrangian cycles. Applications to Poincar'e-Hopf index theorems and indices of 1-forms on stratified spaces. Stiefel-Whitney and Chern classes of (selfdual) constructible functions.
Lecture 3: L-classes of selfdual constructible sheaves.
Intersection cohomology and L-classes of stratified Witt spaces after Goresky-MacPherson and Siegel. The L-class transformation of Cappell-Shaneson for selfdual constructible sheaves.
Lecture 4: Motivic characteristic classes of singular spaces.
Survey of recent developments in the theory of motivic characteristic classes of singular spaces
In my lectures, I discuss such characteristic classes of singular varieties as the Schwartz-MacPherson class and the Fulton-Johson class. They will be treated in the framework of localization theory using the Chern-Weil theory adapted to triangulated spaces. Then the Milnor classe naturally appears as the difference of these two classes, which is a priori localized at the singular set of the variety.
I also discuss, as another type of characteristic classes on singular varieties, the homology Chern characters and classes of coherent sheaves, in particular the tangent sheaf, on singular varieties.