Mathematical Sciences: Spectral Geometry and Index Theory on Foliated Manifolds

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This project continues mathematical research on foliation theory. The main emphasis will be the application of the methods of algebraic topology (secondary characteristic classes) and spectral geometry. Work will be done investigating the index and spectrum of transversal Dirac operators on a foliated space, associated Cheeger-Chern-Simons classes, eta invariant and families of transversal Dirac operators parametrized by suitable moduli spaces. Applications to mathematical physics, in particular to gauge theory on foliated spaces, and global chiral anomalies will be further pursued. The overall objective of the project is to establish via heat equation methods and the theory of spectral asymptotics on singular spaces, an index theorem for transversally elliptic operators on the basic section of a Riemannian Foliation. The main requirement is that the theorem and its extensions would be explicitly given in terms of data relative to the transversal geometry of the foliation and of the spectral properties of the associated zeta-function. The study of foliations in geometric analysis is a natural generalization of domains for differential operators. These operators can be shown to act in particularly simple ways on sets which decompose domains in a layer fashion. On the layers it is possible to analyze many operators using analogues of some of the fundamental tools applied to the study of differential equations.