Abstract: 

The features of Laplace eigenfunctions and their eigenvalues on a compact Riemannian manifold reflect structures present in the geometry of the space.  Inversely, the geometry of the space constrains the possible solutions to the Laplace eigenvalue equation on a manifold.

One of the key geometrical considerations is the ergodic or measure-theoretic features of the (classical) dynamics that occurs on a Riemannian manifold.  Hyperbolic surfaces provide a simple model with uniformly hyperbolic dynamics which generates a strongly mixing, ergodic classical geodesic flow.

Motivated by the principle that classical physics is an approximation of quantum physics in the semi-classical limit, we expect the probability density of an eigenfunction on a compact hyperbolic surface to approach the uniform density as the eigenvalue (or energy) approaches infinity.  Remarkably, while conjecture of this kind (and others) have been made for arbitrary compact hyperbolic surfaces,  the main class of spaces where significant progress has been made in recent years is in the context of arithmetic hyperbolic surfaces, where the hyperbolic surface admits certain arithmetic symmetries.

In our thesis, we explore a possible approach to generating results for arbitrary hyperbolic surfaces via a study of the so called Anantharman-Zelditch intertwining operator.  We introduce some significant modifications to the operator which improves its boundedness properties on natural symbol spaces, without destroying its key features. We make a further study of its properties as an oscillatory integral operator and we situate the operator as a key piece towards making progress in quantum chaos on hyperbolic surfaces. We shall discuss some of these ideas in our talk.