As one of the deepest and most beautiful theorems in geometry, the
Hodge theorem builds a bridge between Riemannian metric and
topological invariants. It gives an isomorphism between the space of
harmonic p forms on a Riemannian manifold and the pde Rham cohomology group of a smooth structure. By the de Rham theorem, we see the
isomorphism between the space of harmonic p forms and p real singular cohomology group.

The Hodge theorem is a good example of how PDEs help us understand geometric structure and even topological structure. In this talk, we
will give an introduction to this theorem, explain the idea behind it, and give some applications in Riemannian geometry.