Mathematical Sciences: Cohomological Problems in Differential Equations and Mathematical Physics

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Ian Anderson will continue his investigations of the variational bicomplex. This is a topic which has rapidly moved into the forefront of research into the calculus of variations. Its applications range from conservation laws of partial differential equations to Riemannian structures and variational problems in differential geometry. The variational bicomplex is a bicomplex of differential forms which derives its name from the fact that one of its differentials may be identified with the Euler-Lagrange operator from the calculus of variations. Anderson's work will focus principally on the calculation of the cohomology of the variational bicomplex. When applied to systems of differential equations the importance of this cohomology arises from the fact that it is generated, in part, by the conservation laws of the equations. In this context Anderson will investigate the case of over-determined systems of equations. Group actions on a fiber bundle also give rise to the problem of calculating equivariant cohomology. Here Anderson will carry out the computations in cases of interest to mathematical physics.