Introduction to spectral geometry – Part I

Winter-Spring 2016

About this course

This course provides 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 basic analysis (for instance, familiarity with the notions of convergence, compactness, dense set, etc.). 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

  1. Fundamental notions of spectral theory
  2. The Laplacian on Riemannian manifolds
  3. The spectrum of the Laplacian on compact Riemannian manifolds
  4. Direct problems in spectral geometry
  5. Inverse problems in spectral geometry

Bibliography

Grading

Problem sets

Articles

Lectures

Date Topic Reading
Lecture 1 – Introduction to spectral geometry [pdf] Sections 1-4 of Kac's article (Pages 1-4) [html], Chapter 1
Lecture 2 – Operators on Hilbert spaces [pdf] Harrell's article [html], Sections 2.1-2.3
Lecture 3 – Spectrum of operators [pdf] 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, connection, 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