## Masterclass on## Randomness: classical and quantum## The University of Copenhagen## November 4 - 8, 2013 |

This masterclass will focus on recent development in various aspects of randomness. In particular the relation between dynamical systems in quantum unique ergodicity, scattering theory, and microlocal analysis, and the application of homogeneous dynamics to problems in kinetic theory.

The masterclass is primarily aimed at graduate students and postdocs.

## Venue and dates:

The lectures will be held at the Department of Mathematical Sciences at the University of Copenhagen and is hosted by the reseach group "Geometric Analysis and Mathematical Physics". The masterclass will run from Monday, November 4th to Friday, November 8th, 2013.

## Lecturers:

A series of lectures will be given by:

Nalini Anantharaman (Université Paris-Sud and ENS Paris): Entropy and localization of eigenfunctions

Abstract. This course will start with an introduction to the Quantum Ergodicity theorem (Shnirelman theorem) and the Quantum Unique Ergodicity conjecture. On compact Riemannian manifolds of negative curvature, this conjecture predicts the asymptotic equidistribution of eigenfunctions of the laplacian, in the limit of large eigenvalues. After that, the course will be devoted to the proof of a result by Anantharaman & Nonnenmacher (2006) according to which the eigenfunctions have asymptotically large entropy. This proves, in particular, that the eigenfunctions cannot localize on sets of small dimensions - which may be considered as a step towards Quantum Unique Ergodicity. The course is intented to be enjoyable without the need of any prerequisite. However, knowledge of the following notions will help : - some basic notions of Riemannian geometry (Riemannian metric, geodesic) and symplectic geometry (symplectic form); - from the theory of dynamical systems, we will use the notion of entropy. Its definition and needed properties will be recalled; - the constructions of microlocal analysis will be defined and used as a blackbox.

Maciej Zworski (University of California, Berkeley): Microlocal approach to dynamical zeta functions

Abstract. Dynamical zeta functions of Selberg, Smale and Ruelle are analogous to the Riemann zeta function with the product over primes replaced by products over closed orbits of Anosov flows. In 1967 Smale conjectured that these zeta functions should be meromorphic but admitted "that a positive answer would be a little shocking". Nevertheless the continuation was proved in 2012 by Giulietti-Liverani-Pollicott. In my lectures I will present a proof of this result obtained by Dyatlov and myself and inspired by a trace formula of Guillemin and by recent work of Faure-Sjöstrand. It is based on a simple idea involving wave front sets and propagation of singularities: we apply methods of microlocal analysis to the generator of the flow. The goal of the lectures is to use this problem in classical dynamics as an introduction to microlocal analysis, and in particular to results on propagation of singularities due to Duistermaat-Hörmander, Melrose and Vasy.

Jens Marklof (University of Bristol): Kinetic limits of dynamical systems

Abstract. Since the pioneering work of Maxwell and Boltzmann in the 1860s and 1870s, a major challenge in mathematical physics has been the derivation of macroscopic evolution equations from the fundamental microscopic laws of classical or quantum mechanics. Macroscopic transport equations lie at the heart of may important physical theories, including fluid dynamics, condensed matter theory and nuclear physics. The exercise that establishes their consistency with the fundamental laws of physics: the possibility of finding deviations and corrections to classical evolution equations makes this subject both intellectually exciting and relevant in practical applications. The plan of these lectures is to develop a renormalization technique that will allow us to derive transport equations for the kinetic limits of certain dynamical systems, including the Lorentz gas and kicked Hamiltonians (linked twist maps). The technique uses the ergodic theory of flows on homogeneous spaces (homogeneous flows for short), and is based on joint work with Andreas Strömbergsson, Uppsala. I will explain the basic steps of the renormalization approach, give a gentle introduction to the ergodic theory of homogeneous flows, and discuss key properties of macroscopic transposrt equations that emerge in the kinetic limit. The lectures are aimed at a broad mathematical audience.

## Schedule:

Mon | Tues | Wed | Thur | Fri | |

10:15-12:00 |
Registration (11:15-12:15) |
Marklof Aud 4 | Zworski Aud 6 | Anantharaman Aud 4 | Zworski Aud 8 |

12:30-13:30 | Lunch | Lunch | Lunch | Lunch | Lunch |

13:45-15:30 |
Marklof (14:15-16:00) Aud 5 |
Zworski Aud 5 | Anantharaman Aud 6 | Marklof Aud 4 | |

15:30-16:30 | 16:00-17 Coffee and discussion | Coffee and discussion | Coffee and discussion | Coffee and discussion |

## Participants:

Nilin Abrahamsen | Copenhagen University |

Nalini Anantharaman | Université Paris-Sud |

Mikkel Brynildsen | Aalborg University |

Giacomo Cherubini | Copenhagen University |

Horia Cornean | Aalborg University |

Tyrone Crisp | Copenhagen University |

Jonas Dahlbæk | Copenhagen University |

Wen Deng | University of Lund |

Bergfinnur Durhuus | Copenhagen University |

Fredrik Ekström | University of Lund |

Flemming von Essen | Copenhagen University |

Søren Fournais | Aarhus University |

Jory Griffin | University of Bristol |

Heiko Gimperlein | Edinburgh University |

Petri Hynek | KTH Stockholm |

Alexei Iantchenko | Malmö University |

Rostyslav Kozhan | KTH Stockholm |

Mathias Makedonski | Copenhagen University |

Jens Marklof | University of Bristol |

Steffen Højris Pedersen | Aarhus University |

Tomas Persson | University of Lund |

Robin Reuvers | Copenhagen University |

Morten Risager | Copenhagen University |

Elmar Schroe | University of Hannover |

Jan Philip Solovej | Copenhagen University |

Anders Södergren | Copenhagen University |

Morten Hein Tiljeset | Aarhus University |

Tatyana Turova Schmeling, | University of Lund |

Petri Tuisku | University of Helsinki |

Izbassar Zaurbek | University of Lund |

Maciej Zworski | UC Berkeley |

Organised by Viviane Baladi, Bergfinnur Durhuus, Morten S. Risager, and Jan Philip Solovej (Copenhagen).