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Curvature, Symmetry, and Periodic Cohomology

曲率、对称性和周期上同调

基本信息

批准号：
1708493
负责人：
Lee Kennard
金额：
$ 16.38万
依托单位：
University of Oklahoma Norman Campus
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2019-03-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1708493&HistoricalAwards=false
关键词：
Curvature Symmetry Periodic Cohomology

项目摘要

In understanding both the physical world and the mathematics that lies beyond its boundaries, geometry and symmetry are everywhere. This project seeks to advance our knowledge of abstract shapes that share two properties: positive curvature and global symmetry. The first of these is a requirement that the shape is curved in a manner similar to how the surface of a ball curves, the same way in all directions. The second means that the object looks the same when spinning, similar to how a football appears when thrown in a perfect spiral. The principal investigator studies shapes likes these through collaborations with other researchers, working with PhD students, and the organization of seminars and conferences. In addition, through his outreach to high schools in Oklahoma and his involvement with undergraduates, the PI is committed to growing and diversifying the body of students and researchers in STEM fields.The first aspect of this project is a continuation of the PI's work on the Grove symmetry program, which, in recent years, has exposed numerous instances of how positive curvature and symmetry can come together to force periodicity in the cohomology of the underlying manifold. The second aspect is a systematic study of topological realization problems involving the condition of periodic cohomology, especially in the presence of symmetry. There are already instances of results along these lines, and the proposal seeks to formalize this research program and find new paths toward advancing understanding in this area.

在理解物理世界和超越其边界的数学时，几何和对称性无处不在。该项目旨在增进我们对具有两个属性的抽象形状的了解：正曲率和全局对称性。第一个要求是形状的弯曲方式类似于球表面的弯曲方式，并且在所有方向上都以相同的方式弯曲。第二个意味着物体在旋转时看起来是一样的，类似于足球以完美螺旋抛出时的样子。主要研究者的研究通过与其他研究人员的合作、与博士生的合作以及组织研讨会和会议来形成这些内容。此外，通过他对俄克拉荷马州高中的推广以及对本科生的参与，PI 致力于发展 STEM 领域的学生和研究人员队伍并使其多样化。该项目的第一个方面是 PI 在 Grove 对称性项目上的工作的延续，该项目近年来暴露了许多实例，说明正曲率和对称性如何结合在一起，以强制基础上同调中的周期性。歧管。第二个方面是系统研究涉及周期上同调条件的拓扑实现问题，特别是存在对称性的情况。已经有一些这样的结果实例，该提案旨在使该研究计划正式化，并找到增进对该领域理解的新途径。