Spectral Geometry

Spring 2016 

Michael Magee 

418 Dunham Labs

michael.magee@yale.edu

Relevant Textbooks (in progress)

M. Reed and B. Simon, Methods of Modern Mathematical Physics. I: Functional Analysis. 

N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac Operators. 

J.H. Conway, The Sensual (Quadratic) Form. 

P. Buser, Geometry and Spectra of Compact Riemann Surfaces. 

H. Iwaniec, Spectral Methods of Automorphic Forms: Second Edition. 

Planned topics

The spectral theory of the Laplacian on a complete Riemannian manifold. 

Heat Kernel and Weyl law. 

Flat tori. 

Spherical harmonics. 

Compact Riemann surfaces. 

The modular curve. 

Schedule

We meet for Lecture in LOM 200, usually on Monday and Wednesday from 10.20am to 11.35am.

The following is the Schedule for the class. 

Lecture 01. Laplacian type operators.  25th Jan. 

Lecture 02. Laplacian type operators II.  27th Jan. 

Class won't meet on Monday 1st Feb

Lecture 03. Elliptic regularity and densely defined operators.  3rd Feb. 

Lecture 04. The spectral theorem.  8th Feb. 

Lecture 05. The resolvent.  10th Feb. 

Lecture 06. Flat tori I: The spectrum.  15th Feb. 

Lecture 07. Flat tori II: Weyl law, theta functions, heat trace.  17th Feb. 

Lecture 08. 22nd Feb. 

Lecture 09. 24th Feb. 

Lecture 10. 29th Feb. 

Lecture 11. 2nd Mar. 

Lecture 12. 7th Mar. 

Lecture 13. 9th Mar. 

Spring Break

Lecture 14. 28th Mar. 

Lecture 15. 30th Mar. 

Lecture 16. 4th Apr. 

Lecture 17. 6th Apr. 

Lecture 18. 11th Apr. 

Lecture 19. 13th Apr. 

Lecture 20. 18th Apr. 

Lecture 21. 20th Apr. 

Lecture 22. 25th Apr. 

Lecture 23. 27th Apr. 

Reading Period