This proposal consists of seven items. The first item is a joint project of the two principal investigators and returns to previous joint work on the wave trace for a manifold with boundary, in order to sharpen and simplify the earlier results by defining the wave trace, not as a trace on the space of initial data of the wave equation, but as a trace on the space of boundary data. The second item is a joint project of Guillemin and K. Okikiolu and involves refinements of a recent result on ``two-tiered'' asymptotics for Szego estimates. The third item is a joint project of Guillemin with C. Zara, and concerns the interplay between graph theory and topology on GKM manifolds. The fourth item is a joint project of Melrose and P. Loya extending the Atiyah-Patodi-Singer index theorem to manifolds with corners. The fifth item involves on-going work of Melrose and R. Mazzeo and the beginning of a project with D. Grieser to analyze the Laplacian on singular algebraic varieties by blow-up and pseudodifferential methods. The sixth project involves Melrose, A. Hassell and A. Vasy in the description of scattering by potentials which are smooth up to infinity. The final item is a joint project of Melrose and J. Wunsch in which the propagation of singularities for waves on manifolds with conic singularities is investigated.
A common theme of the items above is the wave equation and related techniques. For instance, one of the important applications of the joint project of the two principal investigators will be to the understanding of the reflection of waves in a domain in the plane. For a convex planar domain the results of this project should shed light on the celebrated problem: "Can one hear the shape of a drum", that is, do the frequencies of vibration of a planar domain, corresponding to the head of a drum, determine its shape? Earlier work of the investigators showed that these frequencies of vibration determine the so-called "length spectrum" of the domain, namely the lengths of inscribed polygons of minimal circumference. One result of the investigation above should be a determination of the SHAPES of these polygons as well.
