Introduction to spectral geometry

Part I

Winter-Spring 2016

About this course

This course is an introduction to spectral theory and spectral geometry. Spectral theory is a mathematical theory to study the eigenvalue problem for linear operators. Spectral geometry is the study of the relations between the geometry of a manifold and the spectrum of the Laplace operator defined over the manifold. The first part of the course covers basic notions of spectral theory. The second part of the course is dedicated to some examples and theorems in spectral geometry. The course is focused on the study of the Laplace operator over a compact manifold.

Prerequisites

Basic linear algebra and analysis. Some idea of the notions of manifold, metric, tangent vector, etc. is helpful but not required. Essentially all the notions used in the course are introduced, though not necessarily discussed in detail.

Topics

Part I

  1. Fundamental notions of spectral theory.
  2. The Laplacian on Riemannian manifolds.
  3. The spectrum of the Laplacian on compact Riemannian manifolds.

Part II

  1. Direct problems in spectral geometry.
  2. Inverse problems in spectral geometry.

Bibliography

Grading

Problem sets

  1. Symmetry properties of the Laplacian [pdf] (Due Thu, Jan 28) [solutions]
  2. Operators on Hilbert spaces [pdf] (Due Thu, Feb 4) [solutions]
  3. Compact operators and spectrum [pdf](Due Fri, Feb 26) [solutions]
  4. Manifolds and the Laplacian [pdf] (Due Fri, Mar 18) [solutions]

Articles

Lectures

Date Topic Reading
Lecture 1 – Introduction to spectral geometry Sections 1-4 of Kac's article (Pages 1-4) [html], Chapter 1
Lecture 2 – Operators on Hilbert spaces Harrell's article [html], Sections 2.1-2.3
Lecture 3 – Spectrum of operators Section 2.4
Lecture 4 – Compact operators Section 2.5-2.6.1 and 2.7.1
Lecture 5 – Spectrum of compact operators
Lecture 6 – Manifolds, tangent vectors, derivative Section 3.1.1
Lecture 7 – Forms, connections, Riemannian manifolds Section 3.1.2
Lecture 8 – Geodesics, completeness, Laplacian Section 3.1.3
Lecture 9 – Integration on manifolds Section 3.1.5
Lecture 10 – Lp spaces, distributions, Sobolev spaces Section 3.2