TCC Course (Winter 2013)

Dynamics and Quantum Chaos on hyperbolic surfaces

Course Description:

The aim of this course is to develop, from scratch, classical and quantum dynamics in the setting of hyperbolic surfaces. Flows on hyperbolic surfaces can be introduced with minimal prerequsites, but at the same time they exhibit surprisingly rich chaotic properties such as ergodicity and mixing. From the quantum point of view these chaotic properties are reflected in the asymptotic behaviour of high-energy eigenstates of the Laplace operator which are of fundamental importance in mathematical physics and the theory of automorphic forms.

In this course we only assume basic knowledge of analysis and start with an elementary discussion of hyperbolic geometry and constructions of hyperbolic surfaces. Then we introduce the geodesic and horocycle flows and investigate distribution of their orbits. In the second part of the course we turn to discussion of quantum phenomena. This involves the study of eigenfunctions φλ for the Laplace operator Δ. It turns out that the chaotic properties of the geodesic flow are reflected in the asymtotic behaviour of the eigenfunctions φλ as λ→∞. We explain, in details, such phenomena as the quantum ergodicity and the quantum unique ergodicity in the setting of arithmetic surfaces.

The highlight of the course would be an outline Lindenstrauss' proof of arithmetic quantum unique ergodicity.

We plan to cover the following topics:

Time: Friday 10-12am; See TCC Timetable

Lecture notes:

Homework Problems:

Course Assessment: The course is assessed by the above problem sheets. Solutions for 10 problems have to be submitted by April 22.