Daniel Freed will continue his work in differential geometry concentrating on areas having significance in mathematical physics. In fact this is an area which in recent years has had great influence in topology. Ji-Ping Sha will investigate the topological consequences of various curvature restrictions on manifolds. He will also investigate applications of the Atiyah- Singer index theorem in Riemannian geometry. Freed's previous work on the geometry of the determinant bundle associated to the Dirac operator has been very influential. He will continue to study this and in particular calculate the first Chern class of the determinant line bundle. He will also work to gain geometric understanding of the various new cohomology theories which are currently being used to great effect in topology. Sha will continue his work on the topology and geometry of Riemannian manifolds with non-negative curvature. Together with Yang, he has established some quite striking results and will now work towards a topological classification of simply connected closed four and five dimensional manifolds with positive Ricci curvature.
