TITLE: Down-To-Earth Uses of de Rham Cohomology to Convince a Wide Audience of its Usefulness
QUESTION [46 upvotes]: I'm soon giving an introductory talk on de Rham cohomology to a wide postgraduate audience. I'm hoping to get to arrive at the idea of de Rham cohomology for a smooth manifold, building up from vector fields and one-forms on Euclidean space. However, once I've got there I'm not too sure how to convince everyone that it was worth the journey. What down-to-earth uses could one cite to prove the worth of the construct?

REPLY [4 votes]: Here is my  attempt to explain DeRham cohomology and  Hodge theory to a group of first year grad students that  have no knowledge of manifolds.