The principal investigators will continue their investigation of a broad range of problems in differential geometry and global analysis. The problems will focus on the relationship between geometric invariants and topological invariants. In particular they will study knots and linking, structures on four manifolds, quantum groups, and Chern-Simons path integrals. This award will support research in the general area of differential geometry and global analysis. Differential geometry is the study of the relationship between the geometry of a space and analytic concepts defined on the space. Global analysis is the study of the overall geometric and topological properties of a space by piecing together local information. Applications of these areas of mathematics in other sciences include the structure of complicated molecules, liquid-gas boundaries, and the large scale structure of the universe.