# Introduction to spectral geometry – Part I

Winter-Spring 2016

*Instructor:*Gustavo de Oliveira [contact info]*Start:**End:**Duration:*20 hours (1 cycle)*Location:*SISSA, Room 136*Time:*Tuesday and Thursday

*Office hours:* Tuesday 16:00-17:00

## 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

- Fundamental notions of spectral theory
- The Laplacian on Riemannian manifolds
- The spectrum of the Laplacian on compact Riemannian manifolds
- Direct problems in spectral geometry
- Inverse problems in spectral geometry

## Bibliography

- Olivier Lablée, Spectral Theory in Riemannian Geometry, EMS Textbooks in Mathematics, European Mathematical Society, 2015.
- Isaac Chavel, Eigenvalues in Riemannian Geometry, Academic Press, 1984.
- Steven Rosenberg, The Laplacian on a Riemannian manifold, Cambridge University Press, 1997.

## Grading

- Problem sets

## Problem sets

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

## Articles

- Can One Hear the Shape of a Drum? (by Mark Kac) [Amer. Math. Monthly 73(4), 1-23 (1966)]
- A Short History of Operator Theory (by Evans Harrell II) [html]

## 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 – L^{p} spaces, distributions, Sobolev spaces |
Section 3.2 |