Here is where I will be posting the very short lecture summaries.

8/31: Introduction: syllabus, expectations for the course. Three motivating problems: exotic R^4, Gromov’s non-squeezing theorem, Weyl’s law.

Part I) What can you hear about a drum?

9/2: Weyl’s Law (for domains in R^n), Part 1: the case of rectangles; some preliminaries about Sobolev spaces, weak solutions, and regularity; domain monotonicity (statement). References: see the note I wrote on our supplements page

9/7: Weyl’s Law, Part 2: Variational characterization of eigenvalues; proof of domain monotonicity; proof of the theorem. References: see the note I wrote on our supplements page

9/9: Introduction to Sunada’s method. You can’t hear the shape of a drum; the method of transplantation. References: Gordon and Webb; Brooks (from supplemental notes page)

Part II) Weyl’s law on a general Riemannian manifold, Dirac operators, and related topics

9/14: Weyl’s Law on a general Riemannian manifold: the heat kernel method and statement of existence results; idea of the proof and heuristics; proof of Karamata’s Tauberian Theorem. Reference: Roe, Chapters 7 and 8. (If you compare with Roe, a very important point that we haven’t covered in lecture yet is that our Laplacian is the square of a Dirac operator, that is \Delta = D^2 for a particular D, in Roe’s notation.)

9/16: The Heat Equation on a Riemannian manifold, part 1: small time heat kernel asymptotics (statement); calculation of the heat operator applied to the naive guess of the fundamental solution

9/21: The Heat Equation, part 2: construction of an approximate heat kernel. Reference: Roe, Chapter 7

9/23: The Heat Equation, part 3: how well does an approximate heat kernel approximate the true heat kernel?; L2 energy decay for solutions and uniqueness; the non-homogeneous heat equation and the difference between an approximate heat kernel and the true heat kernel. Reference: Roe, Chapter 7

9/28: The Heat Equation, part 4: The solution operator; Duhamel’s principle; statement of the Sobolev embedding theorem and Rellich’s Theorem. Reference: Roe, Chapters 5 and 7; Evans, Chapter 5

9/30: Sobolev spaces on manifolds; the Hodge-de Rham operator; Garding’s inequality (statement). Reference: Roe, Chapters 1 and 5; note that my treatment of Sobolev spaces is slightly different

10/5. The Hodge-de Rham operator is self adjoint; bootstrapping via Garding’s inequality; application to the solution operator to the heat equation; vector buundles and connections. Reference: Roe, Chapters 1 and 5

10/7 Proof of Garding’s inequality assuming the Weitzenboch formula; constructions involving vector bundles and inner products; the connection Laplacian. Reference: Roe, Chapters 1 and 5

10/12 Operators without eigenvalues; unbounded operators; the spectral theorem (statement); the extension of the Hodge-de Rham operator; eigenbases for the Hodge-de Rham operator, part 1. Reference: Roe, Chapter 5

10/14 Eigenbases for the Hodge-de Rham operator, part 2; Clifford algebras and Clifford modules; factoring the Laplacian on flat space; introduction to Dirac operators. Reference: Roe, Chapter 3

10/19 Clifford bundles; proof of the Weitzenboch formula for any Dirac operator; recollection of curvature; the Hodge-de Rham operator. Reference: Roe, Chapter 3

10/21 More examples of Dirac operators; mollifiers; discussion of other possible topics. Reference: Roe, Chapters 3 and 5

10/26 Hodge theory. Reference: Roe, Chapter 6

10/28 Some applications of Hodge theory; the Bochner technique; A manifold with a metric of positive Ricci curvature must have b_1 = 0. Reference: Roe, Chapters 3 and 6

Part III) An interlude

11/2 Guest lecture on curvature flow by Rich Schwartz. Reference: https://arxiv.org/abs/2106.09213

Part IV) The Atiyah-Singer index theorem

11/4 Statement of the Atiyah-Singer index theorem; an example; basic idea of the proof; the McKean-Singer trick. Reference: https://www.mathematik.hu-berlin.de/~wendl/Winter2017/AtiyahSinger/talk1_notes.pdf

11/9 Characteristic classes and Chern-Weil theory. Reference: Roe, Chapter 2

11/11 Genera; the super trace; the heat kernel method; the asymptotic expansion of the heat kernel; comparison with Weyl’s law; introduction to the Getzler rescaling. References: Roe, Chapters 2 and 11; https://diracoperat.org/wp-content/uploads/2013/09/Heat-Kernel-Proof-of-Index-Theorem.pdf

11/16 The Getzler rescaling; outline of the rest of the argument. Reference: https://diracoperat.org/wp-content/uploads/2013/09/Heat-Kernel-Proof-of-Index-Theorem.pdf

11/18 Representation theory of Clifford algebras; spin and the spin representation; the super trace and Clifford multiplication. Reference: Roe, Chapter 4, and https://diracoperat.org/wp-content/uploads/2013/09/Heat-Kernel-Proof-of-Index-Theorem.pdf

11/23 The superalgebra structure on the Clifford algebra; calculating supertraces; the limit under the Getzler rescaling. Reference: Roe, Chapters 4 and 11, and https://diracoperat.org/wp-content/uploads/2013/09/Heat-Kernel-Proof-of-Index-Theorem.pdf

Thanksgiving break

11/30 Mehler’s formula. Reference: Roe, Chapters 9 and 12, and https://diracoperat.org/wp-content/uploads/2013/09/Heat-Kernel-Proof-of-Index-Theorem.pdf

12/2 End of the proof of the Atiyah-Singer index theorem (for graded Dirac operators). Start of applications: positive scalar curvature metrics, Chern-Gauss-Bonnet. References: https://diracoperat.org/wp-content/uploads/2013/09/Heat-Kernel-Proof-of-Index-Theorem.pdf, https://www3.nd.edu/~lnicolae/ind-thm.pdf

12/7 Some nonexistence results about positive scalar curvature metrics; four-manifolds and their intersection form; statement of Rokhlin’s theorem. More about Chern-Gauss-Bonnet References: Roe, Chapter 13; https://www3.nd.edu/~andyp/notes/Rochlin.pdf; https://www3.nd.edu/~lnicolae/ind-thm.pdf

12/9 The signature theorem; Proof of Rokhlin’s theorem. Reference: Roe, Chapter 13