Title of the course: Spectrum of the Laplacian Operator on Riemannian Manifolds
Instructor: Ms. Buket Can Bahadır
Institution: METU
Dates: 16-22 September 2024
Prerequisites: Functional Analysis, Differential Geometry
Level: Graduate, advanced undergraduate
Abstract: In this lecture series we want to explore how we can employ theoretical tools from functional analysis for Laplace Operator on compact Riemannian manifolds, how (and if) we can use these techniques to compute spectra of torus, sphere, real and complex projective spaces. Along the way we would like to discuss the relationship between the geometry and the spectrum of a given manifold, mainly “Given the geometry, what can we say about the spectrum of a manifold?” and “If we know its spectrum, can we say anything about the geometry of a manifold?”
SLO 1: Introduction and a (very) short summary of theoretical background and necessary tools
SLO 2: The Laplacian on a compact Riemannian manifold
SLO 3: How to calculate the spectrum: Flat Tori
SLO 4: How to calculate the spectrum: Sphere
SLO 5: How to calculate the spectrum: Projective Space
SLO 6: Can one hear the holes of a drum?/Inverse Problems
Language: TR, EN
Textbook:
1. Olivier Lablée,  – Spectral Theory in Riemannian Geometry, European Mathematical Society, 2015.
2. Piotr Hajlasz, Functional Analysis Lecture Notes, online at https://sites.google.com/view/piotr-hajasz/
3.  Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine, Riemannian Geometry (Universitext), Springer, 2004