The investigator will study problems arising in the classification of foliated manifolds (as represented by the homotopy type of foliation classifying spaces), and how the analytic structures (the analytic K-theory and spectral theory) associated to elliptic operators along the leaves of a foliation are related to the algebraic topology of the ambient manifold. This project will study the role of global topology in the spectral properties of foliation operators, especially regarding: leafwise eta-invariants and odd analytic K-theory; Chern characters and asymptotic methods for transverse operators; gap phenomenon for leafwise Schrodinger operators based on generalizing Brillouin zone theory of solid state physics via Fourier integral operator methods. Recent work of T. Tsuboi has introduced a new method for the study of the algebraic topology of the Haefliger classifying spaces of foliations. The investigator hopes to extend the approach of Tsuboi to make a comprehensive study of how the cohomology of a classifying space is related to the dynamics of the foliation groupoids that it classifies, and to their transverse differentiability. He intends to continue his research into the application of the transverse cyclic cohomology invariants of Anosov foliations to providing rigidity phenomena for Riemannian manifolds with negative sectional curvatures. The classical foliation problem is whether one can comb the hair on a cocoanut (without whorls). Vast generalizations of this have been conceived and elaborate algebraic and analytic machinery evolved for treating them. Still the underlying ideas have a natural appeal to geometric intuition which gives a special flavor to the entire subject and inclines one to bear with its many technicalities.
