Graduate course, Spring 2017

20 hours

**Location:** Howard House, 4th floor seminar room

**Time:** Monday 11am - 1pm

Quantum chaos can be thought as the study of quantum systems for which the corresponding classical dynamics is chaotic. In this course we will introduce this theory via the study of the spectrum and eigenfunctions of the Laplacian on hyperbolic surfaces. This setting connects ergodic theory, number theory (automorphic forms) and mathematical physics and is at the centre of current active research. The ultimate goal would be to understand the Arithmetic Quantum Unique Ergodicity theorem of Lindenstrauss, and its connection to eigenfunctions on discrete regular graphs. There will be minimal prerequisites, basic analysis and algebra should be sufficient.

- Hyperbolic surfaces (hyperbolic plane, Fuchsian groups)
- Geodesic and horocyclic flows, mixing
- Spectral decomposition of the Laplacian
- Selberg trace formula, Weyl law
- Quantum ergodicity
- Hecke operators, Eigenfunctions on regular graphs
- Arithmetic Quantum Unique Ergodicity

- S. Katok: Fuchsian groups, University of Chicago press, 1992.
- N. Bergeron: The spectrum of hyperbolic surfaces, Springer, 2016.
- M. Einsiedler and T. Ward: Arithmetic quantum unique ergodicity. Lecture notes, 2010.
- E. Lindenstrauss: Invariant measures and arithmetic quantum unique ergodicity. Annals of Mathematics (2006): 165-219.
- S. Brooks and E. Lindenstrauss: Joint Quasimodes, Positive Entropy and Quantum Unique Ergodicity, Invent. math. (2014) 198(1):219-259.
- S. Brooks and E. Lindenstrauss, Non-localization of eigenfunctions on large regular graphs, Isr. J. Math. (2013) 193.
- P. Sarnak: Spectra of hyperbolic surfaces. Bulletin of the AMS 40.4 (2003): 441-478.
- A. Gorodnik: Dynamics and Quantum Chaos on Hyperbolic Surfaces. Lecture notes 2013.

- 30.01.2017: Introduction and hyperbolic geometry.
- 06.02.2017: End of the chapter on hyperbolic geometry.
- 13.02.2017: Mixing of the geodesic and horocycle flows.
- 20.02.2017: Spectral decomposition of the Laplacian.
- 27.02.2017: Construction of the heat kernel.
- 06.03.2017: Selberg trace formula.
- 13.03.2017: Microlocal lift.
- 20.03.2017: Quantum ergodicity, Hecke operators.
- 27.03.2017: Arithmetic QUE: Hecke recurrence
- 03.04.2017: Arithmetic QUE: Positive entropy