Mathematical Sciences: Symmetries of Manifolds, Surgery and Scalar Curvature

数学科学：流形的对称性、外科手术和标量曲率

• 批准号: 9101575
• 负责人: Slawomir Kwasik
• 金额: $ 5.13万
• 依托单位: Tulane University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-06-01 至 1993-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9101575&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Symmetries; Manifolds; Surgery

The main purpose of this research project is to study the topology and geometry of low and higher dimensional manifolds. Particular attention will be paid to symmetries of manifolds. The problems which will be studied cover a large spectrum of questions, ranging from very specific problems concerning low- dimensional manifolds to general problems concerning the geometry of positively curved higher dimensional manifolds. The main techniques which will be employed in this research include: gauge theory, surgery, homotopy theory, transformation groups and differential geometry. Manifolds are very natural objects to study. They are the spaces which are locally like Euclidean spaces. For example, the space in which we live (ignoring time), has three dimensions, making it a three-dimensional manifold, locally, but not necessarily globally, like Euclidean three-space. Taking time into account, physical space becomes a four-dimensional manifold. Whether it is globally Euclidean four-space or some other, possibly bounded, four-dimensional manifold is a deep question of cosmology. Answering this question is not a matter for mathematics alone, but exploring the various possibilities is an important and properly mathematical matter.

该研究项目的主要目的是研究低维歧管和更高维歧管的拓扑和几何形状。 特别关注流形的对称性。 将研究的问题涵盖了各种各样的问题，范围从有关低维流形的非常具体的问题到有关正面弯曲的较高维歧管的几何形状的一般问题。 这项研究中将采用的主要技术包括：仪表理论，手术，同义理论，转化组和差异几何形状。 歧管是非常自然的研究对象。 它们是当地的空间，就像欧几里得空间一样。 例如，我们居住的空间（忽略时间）具有三个维度，使其成为本地但不一定在全球的三维流形，例如欧几里得三空间。 考虑到时间，物理空间成为一个四维的多种多样。 无论是全球欧几里得四空间还是其他可能有限的四维流形都是宇宙学的深刻问题。 回答这个问题并不是数学上的问题，而是探索各种可能性是重要且正确的数学问题。