# Introduction to spectral geometry

Part I

Winter-Spring 2016

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

## 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 et cetera is helpful but not required. Essentially all the notions used in the course will be introduced, though not necessarily discussed in detail.

## Topics

### Part I

- Fundamental notions of spectral theory.
- The Laplacian on Riemannian manifolds.
- The spectrum of the Laplacian on compact Riemannian manifolds.

### Part II

- 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.
- Steven Rosenberg, The Laplacian on a Riemannian Manifold, Cambridge University Press, 1997.
- Isaac Chavel, Eigenvalues in Riemannian Geometry, Academic Press, 1984.

## Grading

- Problem sets

## Problem sets

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