Lecture Notes for Hodge Theory
Lecture 1: Differential Operators, Sobolev Spaces, and Functional Analysis
Lecture 2: Supports, Mollifiers(corrected), and cohomology (To read: covector fields in Lee and the weak topology in Brezis)
Lecture 3: de Rham cohomology and symbols (To read: same as before. If you can try to read Lee's chapter on Riemannian metrics and Evans' elliptic regularity) (Supplementary: Lawson and Michelsohn Spin Geometry III.3)
Lecture 4: Cech cohomology and Sobolev Spaces on Manifolds(Notes TBD)(To read: Brezis: Rellich-Kondrachov, Banach-Alaoglu) (Supplementary: Bott and Tu Differential Forms in Algebraic Topology Chapter 2, Lawson and Michelsohn Spin Geometry III.3)
Lecture 5: Riemannian Metrics ( No Notes)(To read: Lee Riemannian Metrics)
Lecture 6: Weak Convergence and Existence of Minimizers (To read: My notes: "Weak Convergence and Constraints") (Supplementary: Morrey chapters 4,7)
Lecture 7: Solving the Dirichlet Problem (To read: My notes: "Rayleigh Quotients") (Supplementary: Morrey chapters 4,7)