Mathematical Sciences: Topological Aspects of Group Actions and Scalar Curvature

研究者计划研究涉及代数拓扑在流形曲率和对称性研究中的应用的三类问题。一类问题涉及具有正标量曲率的黎曼度量的存在性。第二类问题涉及相对低维流形上的群作用，第三类问题涉及同伦理论、手术理论和高维流形上的群作用之间的相互作用。特别强调的主题包括非线性球面空间形式的曲率特性和4流形和球体上的循环群作用。这些问题中有几个以拓扑和几何之间的相互作用为特征，不直接或明显涉及测量的几何性质在多大程度上实际上需要对曲率等度量性质进行限制。强大的现代方法已经揭示了越来越多这样的情况，与物理学家关于我们生活的世界的理论有着深刻的联系。

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.