Harmonic forms on compact manifolds

Abstract: In this talk I will discuss the classical Hodge theorem, which states that given a compact oriented Riemannian manifold, every de Rham cohomology class has a unique harmonic representative. The statement of the theorem and a sketch of its proof motivate various structures in geometric analysis, such as the Hodge Laplacian and Sobolev spaces of tensor fields over manifolds. This expository talk should be accessible to anyone who has taken the first year real analysis and manifolds sequences.