Abstract
In the 1920s, a mathematical theory (the diagonalization of matrices) and a physical question (the determination of the spectrum of atoms), born independently, came together to give rise to quantum mechanics, and to the branch of mathematics known as " spectral theory ".
Spectral theory comes into play whenever we need to study a linear evolution equation : it allows us to decompose the equation's solutions as a superposition of stationary solutions, called " eigenmodes ", vibrating at certain frequencies known as " eigenfrequencies ". The eigenfrequencies make up the " spectrum ". This is how a sound breaks down into a superposition of harmonics, or how light is a superposition of colors.
A question that is always at the heart of spectral theory is how to distinguish the discrete spectrum from the continuous spectrum, and determine where the eigenmodes are located. Spectral theory is a field of mathematical analysis in which we must always work in infinite-dimensional spaces, which makes the calculations very abstract. However, for the needs of physics, or simply because we need to retain a geometric intuition of phenomena, we seek to understand the link between the initial geometry of the problem (the shape of a musical instrument, the planetary description of the atom, etc.) and the spectrum of the object. This is the raison d'être of Spectral Geometry.
The lesson will explain the history of the field, some major research themes past and present, and my contributions.
