Moscow Arnold's seminar

14.03.2000
Maxim Kazarian
Characteristic Classes of Lagrange Singularities

A Russian translation of V.A.Vassiliev's book "Lagrange and Legendre
Characteristic classes" is to appear this year. In this book the
theory of classes dual to cycles formed by certain type of
singularities of Lagrangian (resp. Legendrian) projection is
developed.

In the talk we present a new point of view to the problem based on the
notion of Classifying Space of singularities. This approach provides a
universal method for computation of all possible Thom polynomials. In
particular we compute classes dual to singularities A7, A8, D8, E8,
P9, X9, which was missed in Vassiliev's book.

We consider also the complexification of the problem. As always, the
Thom polynomials of the Real problem can be obtained from the
corresponding polynomials of Complex problem by replacing all Chern
classes by Stiefel-Whitney classes and reducing the coefficients
modulo 2. The complexification of the problem looks rather natural, It
explains, in particular, why the same sequence 1,1,2,2,3,4,5,...
serves as the number of singularity classes in each codimension and
also as the ranks of cohomology groups of Lagrange Grassmannian.

As a byproduct we obtain a Giambelli type formula for Schubert cells
on Lagrange Grassmannian. 

21.03.00 (continuation)
R.Rimanyi's idea works in case of Lagrange and Legendre
singularities as well. This also will be explained.