On the relation between classical and quantum dynamics --- a mathematician's perspective

Classical mechanics and quantum mechanics give very different descriptions of the laws of 

evolution of a physical system which at  high energies should be quite similar. I will discuss

 one aspect of this in a very simple system: one spineless particle constraint to  lie on a closed 

and bounded manifold with no external force. In this case the quantum unique ergodicity conjecture 

states that if the  classicaldynamics is uniformly hyperbolic ("chaotic" in a strong sense) the steady 

states of the Schroedinger evolution (which are just  the eigenfunctions of the Laplace-Beltrami operator) 

become equidistributed in the high energy limit. I will present some of the rigorous results in this direction, and 

focus on the case of arithmetic manifolds which have a subtle but very rich symmetry whichcan be used to

aid the analysis.