This research concerns relations between the spectral theory of the Laplacian on a compact Riemannian manifold and the dynamics of the geodesic flow. The first project concerns a restriction of the isospectral problem: an assumption that the Laplacians are intertwined by a unitary Fourier integral operator. Broadly interpreted, this restriction includes almost all known examples. The second problem concerns the pair correlation function of the Laplacian. This function measures the distribution of gaps between eigenvalues on the scale of the mean consecutive gap. This function is determined by the qualitative dynamics of the geodesic flow. Recently the principal investigator determined the first known pair function on certain Zoll surfaces. This work will be continued. This project is in the area of geometric analysis and deals with the interface between spectral theory and dynamical systems. This research may have profound applications to physics and geometry.
