The investigator plans to work on three classes of problems involving the applications of algebraic topology to the study of curvature and symmetry properties of manifolds. One class of questions deals with the existence of riemannian metrics with positive scalar curvature. The second class of questions concerns group actions on relatively low-dimensional manifolds, and the third class concerns the interaction between homotopy theory, surgery theory, and group actions on manifolds in higher dimensions. Topics to receive special emphasis include the curvature properties of non-linear spherical spaceforms and cyclic group actions on 4-manifolds and spheres. Several of these problems feature the interplay between topology and geometry, the extent to which geometric properties that do not directly or obviously involve measurement actually entail restrictions on such metric properties as curvature. Powerful modern methods have been revealing more and more such situations, with profound relevance for the physicists' theories about the world in which we live.
