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

9/26 (Optional seminar talk later in the day) Hutchings seminar talk on obstruction bundle gluing.  Reference: https://arxiv.org/abs/math/0701300

9/28 Brief discussion of obstruction bundle gluing following up on Hutchings' seminar talk.  Preliminaries for the proof of the fundamental theorem of linear elliptic equations: Sobolev spaces, the Sobolev embedding theorem, statement of the elliptic inequalities, statement of the existence of a parametrix.  Proof, assuming preliminaries.   Reference: Donaldson, Chapter 3

10/3 Rigorous construction of Sobolev spaces.  Proof of some Sobolev embedding theorems; statement and proof of Relich compactness.  Construction of a parametrix for a constant coefficient operator acting on sections of the trivial bundle over the torus.  Reference: Roe, Chapter 5.1.  Donaldson, Chapter 3 

10/5  Completion of the proof of the fundamental theorem.  Applications: the Poisson equation.  The Bochner technique using the hodge decomposition; the Weitzenboch formula for the Laplacian; manifolds of positive Ricci curvature.  Reference: Donaldson, Chapter 3

10/10 Examples: a) Of a linear PDE for which elliptic regularity fails; b) Of an ellipitic operator for which there is a topological formula for the index.  Background for the uniformization theorem for Riemann surfaces: complex differential forms, Dolbeault cohomology, idea of the proof in the genus 0 and genus 1 cases.  Reference: Donaldson, Chapter 3.  Donaldson's notes on Riemann surfaces, Chapters 5, 6 and 8

10/12 The "Hodge diamond" for Riemann surfaces.  Uniformization of genus 0 and genus 1 Riemann surfaces: a genus 0 Riemann surface has a meromorphic function with a unique pole; a genus 1 Riemann surface has a nowhere vanishing holomorphic 1-form.  Statement of uniformization in higher genus.  References:  Donaldson's notes on Riemann surfaces, Chapters 6 and 8; https://static1.squarespace.com/static/5a409a83a803bbafedd09784/t/5ab853faf950b75093399ac5/1522029563781/hodge_notes.pdf 

10/17 Uniformization, part 1.    Reference: Donaldson's notes on Riemann surfaces, Chapter 10.  see http://staff.ustc.edu.cn/~wangzuoq/Courses/16F-Manifolds/Notes/Lec22.pdf for a reference about degree theory in the noncompact case

10/19 Uniformization, part 2.  Reference: Donaldson's notes on Riemann surfaces, Chapter 10.   

10/24 Uniformization, part 3.  The Atiyah-Singer index theorem.  Statement for twisted Dirac operators.  More about Dirac operators. References: Donaldson's notes on Riemann surfaces, Chapter 10.  Introduction to Chapter 3 of https://www3.nd.edu/~stolz/2020S_Math80440/Index_theory_S2020.pdf.  https://web.math.ucsb.edu/~dai/book.pdf.  Donaldson, Chapter 4.

10/26 Characteristic classes.  Rokhlin's theorem on the signature of spin manifolds.  An unsmoothable 4-manifold.  References: Section 1.3 of https://web.ma.utexas.edu/users/a.debray/lecture_notes/u17_characteristic_classes.pdf is essentially what I was saying, and we were just sketching the definitions anyways, although I would probably say things slightly differently; https://www3.nd.edu/~andyp/notes/Rochlin.pdf; Donaldson, Chapter 4

10/31 More about the Atiyah-Singer theorem: twisted Dirac operators, genera, the Chern character, the A-hat genus.  Pontryagin classes.  The signature operator; grading operators.  The index of the spin Dirac operator.  References: Chapter 3 and 13 of https://www.maths.ed.ac.uk/~v1ranick/papers/roeindex.pdf.  1.4.1 of https://www3.nd.edu/~stolz/2020S_Math80440/Index_theory_S2020.pdf.

11/2 The index of the signature operator.  The Lichnerowicz formula.  There are no positive scalar curvature metrics on the four-torus.  [end of material about linear equations]  Linearization and the continuation method; the inverse function theorem.  Constant negative curvature metrics on surfaces.  References: Donaldson, Chapters 4 - 6.

11/7 A priori bounds for the negative curvature metrics on surfaces problem.  The inverse function theorem machinery in more detail.  Computing linearizations.   Analyzing linearizations through the fundamental theorem of elliptic PDE.  Examples.  References: Donaldson, Chapters 5 - 6.

11/9 The Seiberg-Witten equations.  Some applications (statement only) and their context: i) an exotic structure on R^4; ii) CP3 # CP3 # CP3 does not admit a symplectic form.  Preliminaries: spin-c structures, spin-c connections, imaginary valued 1-forms, the Pauli matrices.  References: Donaldson, Chapter 7; https://math.berkeley.edu/~hutching/pub/tn.pdf, "Monopoles and Three-Manifolds" (Kronheimer-Mrowka), Chapter 1.

11/14 More about the Seiberg-Witten equations:  The gauge group.  Reducible solutions.  Based gauge transformations.  The compactness theorem.  The transversality theorem.  Reducibles and b_2+.  References:  https://math.berkeley.edu/~hutching/pub/tn.pdf, Chapter 3.  Donaldson, Chapter 7.

11/16 The dimension of the moduli space.  Relating the moduli spaces for different choices of data.  Definition of the Seiberg-Witten invariants.  The simple type conjecture.  References:  https://math.berkeley.edu/~hutching/pub/tn.pdf, Chapter 3.  Donaldson, Chapter 7.  https://arxiv.org/pdf/2009.06791.pdf