Course notes will be continuously updated as the class progresses.
- Final projects:
- Jonier Antunes, Witten deformation and Morse theory
- Simone Cecchini, The strong Novikov conjecture and scalar curvature
- Brian Hepler, What is L2 cohomology and why should you care?
- Monika Pichler, Representation theory for groups of unitary operators and an application in acoustic obstacle scattering
- Pengshuai Shi, Boundary value problems for Dirac-type operators and an application
- Saif Sultan, Analytic torsion
- Fri 4/22: Index formula and applications.
- Fri 4/15: Mehler's formula.
- Mon 4/11: Getzler rescaling continued.
- Fri 4/8: Getzler rescaling.
The deadline for final projects will be Monday April 25.
- Mon 4/4: Lincherowicz formula, Getzler rescaling.
- Fri 4/1: Bochner/Weitzenbock formulas.
As a reminder, the final project is an expository paper (10-20 pages) on a topic related to the course material. Depending on how focused your topic is, you may want either want to go into some technical detail, or present a survey.
Here is a list of possible topics: (not meant to be exhaustive!)
- Spectral functions of elliptic operators: resolvent, zeta, and eta functions and relations between these (and the heat kernel). Residue trace and meromorphic continuation.
- Analytic torsion: zeta regularized determinants of elliptic complexes, Cheeger-Mueller theorem relating analytic torsion and Riedemeister torsion.
- Witten deformation and Morse theory (has some relation to the previous topic).
- Improved Weyl asymptotic formula via wave trace, Duistermaat-Guillemin formula.
- Properties of eigenfunctions of the Laplacian, global harmonic analysis, quantum chaos.
- Equivariant index theorem.
- Scattering theory on some non-compact manifolds (say R^n, H^n or similar spaces): parameterization of continuous spectrum, scattering matrix, resonances.
- L^2 cohomology on noncompact manifolds: relation to intersection cohomology in special cases.
- Mon 3/28: Spinors.
- Fri 3/25: Dirac operators and spin structures.
- Mon 3/21: Class cancelled.
- Fri 3/18: Clifford algebras and Dirac operators.
- Mon 3/14: Weyl Asymptotics.
- Fri 3/4: Heat kernel for Laplace-type operators.
- Mon 2/29: Heat space for parametrices on a compact manifold.
- Fri 2/26: No class
- Mon 2/22: Trace-class operators, continued, heat kernel on R^n, radial blow-up.
- Fri 2/19: Sketch of PsiDOs, spectral theory and trace-class operators.
- Fri 2/12: Sobolev spaces, spectral theory of self-adjoint elliptic operators.
- Mon 2/8: Classes cancelled due to snow.
- Fri 2/5: L^2 boundedness and Sobolev spaces.
- Mon 2/1: Fredholm properties, continued. Hodge Theory.
- Fri 1/29: Elliptic operators, regularity and Fredholm properties.
- Fri 1/22: Distributions on manifolds and pseudodifferential operators.
- Tue 1/19: Differential operators and principal symbols.