The principal investigator will analyze various problems reflecting the interaction between partial differential equations and the geometry and topology of manifolds. Specifically, he will extend his existing work on L^2-cohomology and derive topological invariants for L^p-cohomology spaces. In addition, the problem of small eigenvalues of negatively curved manifolds will be investigated. L^p-cohomology has links with number theory, algebraic geometry, group representations, and combinatorial topology. The principal investigator will use his knowledge of the L^2 theory to solve two difficult but natural problems which arise from considering the more general L^p case.
