| 1 | What is potential theory? Basic properties of harmonic functions. Using Brownian Motion to solve the Dirichlet problem. Recurrence and transience. Convergence of bounded martingales. Further applications to complex analysis. |
| 2 | The Poisson kernel. No positive harmonic functions in R^n. Harmonic functions are real analytic. Harnack's principle. |
| 3 | Basic properties of subharmonic functions. Phragmen-Lindelof principle. Liouville's theorem for subharmonic functions. |
| 4 | Distributions. Weyl's lemma. Riesz decomposition. Potentials. Energy. Equilibrium measures. Polar Sets. Frostman's theorem. Brelot-Cartan theorem. Minus infinity sets. Removable singularities. |
| 5 | Capacity and transfinite diameter. Hilbert lemniscate theorem. |
| 6 | Schauder's fixed point theorem. Existence and uniqueness of solutions of the Gauss curvature equation with prescribed boundary data. Liouville's theorem on the correspondence between conformal metrics of curvature -4 and holomorphic self-maps of the unit disk. Heins theorem on prescribing finite Blaschke products by their critical points. |
| 7 | Basic properties of Sobolev spaces. Regularity of solutions of the linear Dirichlet problem with singularities. |
| 8 | Maximal functions. Sobolev capacities. Removable sets for Sobolev functions. Obstacle problems. Carleson's theorem on removable sets of Holder continuous harmonic functions. |