Jacques van Wyk, Department of Pure Mathematics, University of Waterloo
"The Hodge Decomposition Theorem"
There is a generalisation of the ordinary Laplacian \(\Delta = -\Sigma_i\partial^2/\partial x^2_i\) to an operator \(\Delta\), called the Laplace-Beltrami operator, on the space of differential forms of a manifold M. The Hodge Decomposition Theorem states that the equation \(\Delta\omega=\alpha\) has a solution \(\omega\) in the smooth k-forms on M if and only if the k-form \(\alpha\) is orthogonal (in a suitable sense) to the space of harmonic k-forms (those for which \(\Delta\eta=0\)). In this talk, we present the Hodge Decomposition Theorem, derive some of its consequences, and give as much of a proof of it as time permits.
MC 5403