Triangle Seminars

Abstract:
Many systems in nature are chaotic, i.e., their classical time evolution depends sensitively on the initial conditions.
This has important consequences for their quantum behaviour. In particular the energy levels of chaotic systems
display universal statistical properties, and have a tendency to repel each other. I will give an introduction into
recent research to explain this universal behaviour. A crucial ingredient is the discovery of correlations between
classical periodic orbits. For example if a periodic orbit contains crosses itself with a small angle there is also
a partner orbit narrowly avoiding that crossing. In a semiclassical approach one sees that contributions of such correlated
orbits can interfere constructively. I will show how this gives rise to universal spectral statistics.
I will also discuss connections to random matrix theory (which gives predictions for spectral statistics) as well as
the combinatorics of permutations.