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.

本研究项目的主要目的是研究 低维和高维流形的拓扑和几何。 将特别注意流形的对称性。 将要研究的问题涵盖了一个很大的范围， 问题，从非常具体的问题，涉及低， 维流形到一般几何问题 正弯曲的高维流形。 主要 本研究将采用的技术包括： 理论，手术，同伦理论，变换群和 微分几何 流形是非常自然的研究对象。 他们是 局部类似于欧几里得空间的空间。 比如说 我们生活的空间（忽略时间），有三个维度， 使之成为一个三维流形，局部的，但不是 就像欧几里得三维空间一样。 花时间 考虑到这一点，物理空间就变成了一个四维流形。 无论它是全局欧几里得四维空间还是其他空间， 可能有界的四维流形是一个深刻的问题， 宇宙学 讨论这个问题不是一个问题， 数学本身，但探索各种可能性是一个 重要的和适当的数学问题。