Math 614 — Topics in Analysis

Spring 2003

Instructor: Bent Ørsted

Time: TR 10:10-11:25

Room: MT 205

This course will be an introduction to spectral geometry, with an emphasis on finding local and global invariants of compact Riemannian manifolds. These invariants arise from studying the spectrum of Laplace-type operators on the manifolds as well as the corresponding heat and wave equations. This is in the spirit of Kac's famous question: Can you hear the shape of a drum?

We shall give the necessary background from Riemannian geometry and pseudodifferential operators, and discuss examples in low dimensions. Spectral geometry has several applications in other parts of mathematics (index theory, characteristic classes, differential geometry) and in physics (string theory, gauge theory). Depending on the interests of the audience, we shall give some applications, for example in connection with the study of determinants of elliptic operators.

Prerequisites: Basic knowledge of manifolds.

Topics:

A. Pseudodifferential operators
B. Elliptic operators and Fredholm operators
C. Spectral theory and asymptotics of the heat kernel
D. Zeta and eta functions associated elliptic operators
E. Spectral geometry and some applications.

Literature: P.B. Gilkey: Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, CRC Press 1995.