We will define what is a Laplacian on a smooth Riemannian Manifold. We will then define what are harmonic forms and state the Hodge Decomposition Theorem. In particular, we will explain how it implies Poincare Duality. We will then go on to define the notion of a weak solution and explain how elliptic regularity of the Laplacian, implies the Hodge Decomposition Theorem. We will review the basic notions of smooth manifolds and keep prerequisites to a minimum. Those who are interested in Analysis and PDE are encouraged to attend.
School of Mathematical Sciences
NISER, PO- Bhimpur-Padanpur, Via- Jatni, District- Khurda, Odisha, India, PIN- 752050