# Lecture summaries

8/29 Welcome; the syllabus; what is geometric analysis?; some examples of PDEs in geometry: isometric embeddings, enumerative invariants, applications to smooth structures; the Poisson equation. Reference: Donaldson, Chapter 1

8/31 Geometry preliminaries: smooth manifolds, compactness, Riemannian metrics, differential forms; Stokes' theorem. The gradient and the divergence on a Riemannian manifold; the integral of the Laplacian of a function over a compact Riemannian manifold is zero. Reference: Donaldson, beginning of Chapter 2; For preliminaries, see Lee and Guillemin/Pollack;

9/5 Solving the Poisson equation in some special cases: the flat torus, R^n. Review: Fourier series, convolutions, distributions. The strategy for solving the equation over a general compact Riemannian manifold. Reference: Donaldson, Chapter 2

9/7 Completions and Hilbert spaces. The Riesz representation theorem. The Poincare inequality. Reference: Donaldson, Chapter 2

9/12 Rest of the proof of the Poincare inequality. Isothermal co-ordinates. Weak solutions are smooth: proof in flat co-ordinates or isothermal co-ordinates on surfaces. Reference: Donaldson, Chapter 2

9/14 Rest of the proof that weak solutions are smooth. Compact operators. Beginning of the proof of the compactness theorem. Reference: Donaldson, Chapter 2

9/19 Rest of the proof that the inclusion of L^2_1 into L^2 is compact. Eigenvectors for the Laplace operator. Statement of the fundamental theorem of linear elliptic operators. Reference: Donaldson, Chapters 2 - 3

9/21 More geometry preliminaries: Vector bundles and linear differential operators. Examples. The symbol and the principal symbol. Ellipticity. Reference: Donaldson, Chapter 3, many other options, e.g. Lee (Introduction to Smooth Manifolds) Chapters 10 - 14

9/26 Examples and non-examples of elliptic operators. The principal symbol and the cotangent bundle. The Hodge decomposition. Reference: Donaldson, Chapter 3; for more about principal symbols, see https://mathoverflow.net/questions/3477/what-is-the-symbol-of-a-differential-operator