This award for a long term visit to the United Kingdom will support collaborative research in geometric analysis and differential geometry between Steven Rosenberg of Boston University, K.D. Elworthy of the University of Warwick, and T. Lyons of the University of Edinburgh, Scotland. The investigators propose to apply probabilistic and classical geometric methods to problems in differential geometry. They will study Riemannian manifolds and their applications to topology and geometry of manifolds. Three problems will be investigated: (1) The Hopf conjecture for the Euler characteristic of negatively curved manifolds; (2) a method for extending Bochner's and Myers' theorems for positively curved manifolds; and (3) a method for yeilding soft eigenvalue estimates for the Laplacian using a modification of the Fukaya method. The later problem will be developed with Dr. T. Lyons of the University of Edinburgh. Manifolds are a fundamental concept in many areas of mathematics and physics. This collaboration offers an innovative approach to studying these problems by combining the complementary expertise of the U.S. and British investigators in classical differential geometry and in probability respectively.