9404298 Gordon The research supported by this award focuses on questions related to the geometry of manifolds. It concerned with the determination of the extent to which the spectrum of the Laplace- Beltrami operator of a compact Riemannian manifold determines the geometry of the manifold. Recent examples show, for instance, that the spectrum alone does not determine manifolds up to isometry. Further work will also be done in efforts to understand how manifolds can be isospectral yet not even locally isometric. Examples growing out of these examples provide excellent opportunities to identify specific local geometric invariants which are not spectrally determined. Work related to isospectral plane domains will also continue. Constructions of such domains are surprisingly simple; they arise as underlying spaces of certain orbifolds. Using this method, work will be done to establish whether or not it is possible to construct isospectral sets of non-isometric plane domains of any finite order. Finally, studies on Laplace versus length spectra of hyperbolic manifolds and on the Schrodinger operator on tori and Heisenberg manifolds will be carried out. Analysis of domains and manifolds through the study of classical differential operators defined on these spaces is now an established field of geometric analysis. One knows that geometric properties of the domains are intertwined with properties of the spectrum of the operators. Results of the last decade show that the dependence is not as simple as once thought. This work seeks to clarify the connections and broaden the scope of the theory's applications. ***
