Summer Graduate School — Séminaire de Mathématiques Supérieures 2023: Periodic and Ergodic Spectral Problems

This two week school will focus on spectral theory of periodic, almost-periodic, and random operators. The study of periodic problems is one of the most classical subjects of spectral theory. Many of the basic properties of periodic operators are now well understood due to the powerful method known as the Floquet-Bloch decomposition which performs a partial diagonalisation of the operator. Another type of problems mathematicians and physicists have been interested in is known under the broad name of ergodic problems; the most prominent among them being almost-periodic and random problems. These problems are much more complicated because no straightforward partial diagonalisation is known for them. The study of the spectral properties for ergodic problems developed to a large extent independently of the periodic theory, although occasionally periodic results were used in almost-periodic or even random settings. The interplay between almost-periodic and random problems has been more significant, but still, most of the methods have been specific to each of these two types. In the last few years however there emerged a number of methods which originated in one type of these problems but were successfully used to tackle problems from neighbouring areas (periodic methods used to deal with almost-periodic or random problems, etc). This suggests that these three lines of research have more in common than previously believed. The main aim of this school is to teach the students who work in one of these areas methods used in parallel problems, explain the similarities between all these areas and show them the ‘big picture’.