The principal investigator proposes to study various aspects of index theory on singular spaces. The PI and his collaborators will, for example, apply their recent analysis of the resolution of the orbit space of a group action to the index theory of a family of equivariant operators. A similar resolution, but of an arbitrary pseudomanifold, will be used in a project to understand the properties of the signature operator and extend the class of spaces known to satisfy the Novikov conjecture from topology. A related project aims more generally to extend many of the methods of geometric microlocal analysis to the setting of stratified spaces. The PI will also work with his collaborators to understand the spectral geometry of non-compact or singular spaces, with projects to understand isospectrality for non-compact surfaces and extend the class of spaces where the topological Reidemeister torsion can be expressed analytically. Finally, the PI is involved in various efforts to extend the number of mathematicians aware of these results and working in this field, including the production of lecture notes and survey articles as well as the organization of conferences and seminars.
Singular spaces arise naturally in mathematics and physics, even when one is interested in studying smooth phenomena. For instance, recent models in cosmology and particle physics require an understanding of spaces with singularities such as corners and edges. However, many of the techniques used in geometric analysis do not apply to singular spaces. One aspect of this proposal is to contribute to the development and application of new techniques to problems arising in index theory and spectral geometry. These problems are in the intersection of analysis with other fields, such as topology, algebraic geometry, and mathematical physics, a confluence which is a source of both problems and applications.
