Abstract or Additional Information
Once a smooth manifold is equipped with a Riemannian metric, we can define the Laplacian operator; on functions, and on forms. This Laplacian is very rich in both the topological and the local information that it carries about the manifold.
One way to extract this information is through Hodge decomposition theorem, which gives us the deRham cohomology groups of the manifold as the kernel of the Laplacian.
An equality with one side coming from analysis, and the other from topology can readily give us quite interesting corollaries that would not be as obvious when dealt with in the one context alone, eg. that the cohomology groups are finite-dimensional, that the index is independent of the choice of the metric...
Outline: It will be a very classical (hence accessible) topic. I will say why a metric is needed. Then I will write down explicitly what Laplacian is. Prove (maybe not completely) that in each cohomology class there is exactly one harmonic representative. Discuss a couple of the implications of the theorem.