TOPICS FOR FINAL PROJECTS (1) Stationary phase formula with degenerate phase functions. (Hormander Vol 1, §7.3.) (2) The Stone-Von Neumann Theorem (Guillemin-Sternberg, Chapter 16) (B. Hall, Chapter 14) (3) Fefferman-Phong inequality. (Microlocal version: Hormander Vol 3, §18.6) (Microlocal version: Fefferman-Phong, On positivity of pseudo-differential operators, Proceedings of the National Academy of Sciences 75 (1978), 4673- 4674.) (4) Damped wave equation. (Zworski, §5.3) (5) Quasimodes, pseudospectra (Zworski, §7.4, §10.4, §12.5) (Dencker-Sj¨ostrand-Zworski, Pseudospectra of semiclassical differential operators, Comm. Pure Appl. Math. 57(2004), 384-415.) (6) The F.B.I. transform (Zworski 13.1-3) (Martinez chapter 3) (7) Toeplitz quantization (Zworski 13.4-5) (8) Weyl’s law: Dirichlet-Neumann bracketing proof (Reed-Simon, Volume 4, §8.15) (Zworski §6.2, 6.4) (9) Various estimates for solutions of PDE (Agmon and Carleman, vanishing order) (Zworski Chapter §7.1-3) Date: today. 1
2 TOPICS FOR FINAL PROJECTS (10) Semiclassical L p estimates (Zworski Chapter §10.3-4) (H. Koch, D. Tataru and M. Zworski, Semiclassical L p estimates, Annales Henri Poincare 8 (2007), 885-916.) (11) PsDOs acting on sections of vector bundles. For microlocal versions, c.f.: (Erik van den Ban and Marius Crainic, Analysis on Manifolds, https://webspace.science.uu.nl/∼ban00101/geoman2017/AS-2017rev.pdf) (Liviu I. Nicolaescu, Pseudo-differential operators and some of their geometric applications, https://www3.nd.edu/∼lnicolae/Pseudo.pdf ) (12) Other topics? You propose and talk to me