9 Oct 2013; IMS (UiTø), room U7=A152. 12:15-13:30.

Juha Pohjanpelto (Oregon State University, USA and Aalto University, Finalnd)

“Variational Bicomplex, Cohomology and Groups Actions”

I will describe various new techniques based on the moving frames method and Lie algebra cohomology for analyzing the cohomology of a variational bicomplex and the associated edge, or Euler-Lagrange, complex invariant under a continuous pseudo-group action G.

I will first show that under general assumptions the interior rows of the G-invariant variational bicomplex are locally exact, which allows one to analyze the cohomology of the G-invariant Euler-Lagrange complex in terms of the de Rham cohomology of the full G-invariant bicomplex. Next, I will introduce moving frames, our basic tool for the study of invariant bicomplexes, and discuss how these can be used to produce complete sets of differential invariants and invariant coframes and to study the algebraic structure of the invariant quantities. In particular, in various situations the moving frames method allows one, at least in principle, to reduce the computation of the local cohomology of the G-invariant variational bicomplex to an algebraic problem.

As an example, I will describe an analogue of the Chern-Weil construction for the variational bicomplex associated with foliations. Time permitting, I will also discuss the relationship between the Lie algebra cohomology of an infinite dimensional pseudogroup G and the cohomology of the G-invariant variational bicomplex in examples pertaining to completely integrable equations.