TOPOLOGY SEMINAR

2/24/2005

Martin Bendersky
CUNY

A spectral sequence approach to Normal Forms

This is joint work with Rick Churchill. An incomplete list of applications of normal forms are to vector fields, Hamiltonians at equilibria, differential equations and singularity theory. In general one tries to modify a given element in a Lie algebra into a particularly useful form. The algorithm that performs the conversion (the normal form algorithm) can be a formidable computation. We generalize the notion of normal form to that of an initially linear group representation. In this general setting we are able to interpret the normal form algorithm as a calculation of a particularly simple spectral sequence. As a consequence we can show that various vector spaces that appear in the process of carrying out the normal form algorithm are invariants of the orbit of the group representation. There will be plenty of examples.