The principal investigator will study Ricci deformation of the metric on Riemannian manifolds. This technique was introduced by Hamilton in 1982 and has since become a useful and well known method in differential geometry. The goal of the research supported by this award is a better understanding of the topology of manifolds which satisfy restrictions on curvature and Ricci curvature in particular. The theory of parabolic partial differential equations will be used to obtain estimates on the solution of the Ricci deformation which should then lead to additional topological information. A physical example of parabolic deformation of curvature is the deformation a stretched rubber band would experience if it were immersed in a viscous fluid such as oil. It is possible to show that the band would asymptotically deform to a circle if the enclosed area were renormalized to one. This method can be used to study the topology of a surface or manifold in more complicated situations. The principal investigator will try to obtain precise estimates for the size of the deformations and use that information to obtain topological information.