Mathematical Sciences: Group Actions, Scalar Curvature and Surgery-Theoretic Problems

研究流形的对称性和流形上具有弱曲率特性的黎曼度量的存在性问题。对称性研究将包括关于低维流形的问题，以及外科理论和同伦理论在高维流形问题中的应用；后者的大部分将涉及其奇异集具有相对较大维度的对称性，并将这些行为的可能分类与奇异集维度相对较小的情况进行比较。所有这些问题都将通过代数和几何拓扑方法的结合，以及稳定和不稳定同伦理论、流形手术理论、拓扑和可微变换群以及低维拓扑方法（包括规范理论）的输入来研究。这项工作的目的是提高对正弯曲度量的存在性和某些对称的存在性和分类的一些问题的理解；这些问题通常与几何拓扑学的最新进展有关。流形在数学中是无处不在的，作为高维空间中方程组的解自然产生。对于低维流形，我们生活在一维，三维或四维中，这取决于是否考虑时间维度以及三个空间维度。因此，更多关于流形的对称性、曲率和其他性质的信息必然是有用的。//

The investigator intends to study questions involving the symmetry properties of manifolds and the existence of riemannian metrics with weak curvature properties on manifolds. The symmetry investigations will include questions about low- dimensional manifolds and the applications of surgery theory and homotopy theory to problems involving higher-dimensional manifolds; much of the latter will involve symmetries whose singular sets have relatively large dimension and comparing the possible classifications of such actions with cases where the singular set dimension is relatively small. All these problems will be studied by a combination of methods from algebraic and geometric topology, with input from stable and unstable homotopy theory, the theory of surgery on manifolds, topological and differentiable transformation groups, and low-dimensional topological methods, including gauge theory. The objective of the work is an improved understanding of some questions about the existence of positively curved metrics and the existence and classification of certain symmetries; these questions have generally arisen in connection with recent advances in geometric topology. Manifolds are ubiquitons in mathematics, arising naturally as the solutions of systems of equations in higher dimensional spaces. As to low-dimensional manifolds, we live in one, three-dimensional or four-dimensional, depending on whether the time dimension is considered as well as the three space dimensions. More information about symmetries of manifolds, their curvature, and other properties is thus bound to be useful. //