MINI-SCHOOL AT THE BANACH CENTER

  "CHARACTERISTIC CLASSES OF SINGULAR VARIETIES"

(partially supported by EAGER)

Invited speakers:

PAOLO ALUFFI (MPI Bonn) and JOERG SCHUERMANN (Uni - Muenster)

Place: Banach Center, Warsaw, Poland

Time: April 22. 2002 (arrival day) - April 28. 2002 (departure day)

Organizers: Piotr Pragacz and Andrzej Weber


PROGRAM: There will be two minicourses on various aspects of characteristic classes for singular varieties by invited speakers. These courses will be rather advanced and they will focus on the present status of research in this domain.

LECTURE NOTES

SUMMARY OF LECTURES BY P. ALUFFI:

The theme of the lectures will be the comparison between different notions of characteristic classes for singular spaces. Our initial motivation will come from concrete questions in projective and enumerative geometry, leading us to define an invariant by intersection theoretic means; this invariant will turn out to be closely related to the difference between the Chern-Schwartz-MacPherson classes and Fulton classes of hypersurfaces. We will explore different ways to realize Chern-Schwartz-MacPherson's classes, including an approach via differential forms with logarithmic poles, and an approach expressing Chern-Schwartz-MacPherson's classes of a hypersurface in terms of a new notion of `Chern class' for a coherent sheaf.

SUMMARY OF THE LECTURES BY J. SCHUERMANN:

After recalling the basic notations of the theory of constructible functions, we will explain the isomorphism between constructible functions and Lagrangian cycles (in the embedded context), together with a translation of these operations into the context of Lagrangian cycles. Here we use the language of "stratified Morse theory for constructible functions". Based on these results, we give a new approach to the theory of characteristic classes of singular spaces, including a generalized Verdier-Riemann-Roch theorem for regular embeddings.
PLAN:
(*) Introduction (History of the theory of characteristic classes of singular spaces: Stiefel-Whitney, Chern- and Todd-classes)
(*) Theory of constructible functions (the basic functorial notions, relation to constructible sheaves, but I will explain everything in the simpler language of constructible functions)
(*) Stratified Morse theory (for constructible functions) and the (functorial) theory of Lagrangian cycles (Here I will reprove the basic operations for Lagrangian cycles as in the paper of Sabbah (e.g. proper push-down, exterior products, specialisation, non-characteristic pull-back and the intersection-formula for vanishing cycles).
(*) Characteristic classes of Lagrangian cycles
(*) Verdier-Riemann-Roch and Milnor-classes.

BIBLIOGRAPHY

ADDITIONAL TALKS
Apart of the two mini-courses by Paolo Aluffi and Joerg Schoermann there is a possibility to deliver some additional talks during afternoons. We are looking forward to possible proposals. So far we announce the following PROGRAM.

MORE INFORMATION FOR PARTICIPANTS

All correspondence about the mini-school should be sent by e-mail to Piotr Pragacz pragacz@impan.gov.pl

Mathematicians from EU wishing to participate at the mini-school may apply for the support to their home nodes of EAGER.