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

数学科学：群作用、标量曲率和外科理论问题

• 批准号: 9102711
• 负责人: Reinhard Schultz
• 金额: $ 10.56万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-07-01 至 1995-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9102711&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Group; Actions; Scalar

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. //

研究人员打算研究涉及以下方面的问题： 流形的对称性与黎曼流形的存在性 流形上的弱曲率度量 的 对称性研究将包括低- 三维流形和外科理论的应用， 高维同伦理论 流形;后者的大部分将涉及对称性， 奇异集具有相对较大的维数， 可能对此类行动进行的分类， 奇异集维数相对较小。 所有这些问题 将通过代数和 几何拓扑，输入来自稳定和不稳定同伦 理论，理论的外科手术流形，拓扑和 可微变换群和低维 拓扑方法，包括规范理论。 的目标 这项工作是对一些问题的进一步理解， 正曲度量的存在性， 某些对称的分类;这些问题有 一般出现在与最近的进展，几何 topology. 流形在数学中无处不在， 作为高维方程组的解 空间. 至于低维流形，我们生活在一个， 三维或四维，这取决于 时间维度被认为是以及三个空间 尺寸. 更多关于流形对称性的信息， 因此它们的曲率和其它性质必然是有用的。 //