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