Curvature, Symmetry, and Periodic Cohomology

曲率、对称性和周期上同调

基本信息

批准号：
1904354
负责人：
Lee Kennard
金额：
$ 8.66万
依托单位：
Syracuse University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-17 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1904354&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.

在理解超越边界的物理世界和数学时，几何和对称性无处不在。该项目旨在促进我们对具有两个属性的抽象形状的了解：积极的曲率和全球对称性。首先是要求形状以类似于球曲线表面的方式弯曲，在各个方向上都相同。第二个意味着旋转时物体看起来相同，类似于以完美精神抛出足球的表现。首席研究者研究通过与其他研究人员合作，与博士生合作以及半手体和会议的组织来形成类似的形状。此外，PI此外，通过他与俄克拉荷马州的高中宣讲，PI致力于成长和多样化，使STEM领域中的学生和研究人员的团体多样化。该项目的第一个方面是PI的延续，是PI在Grove对称方案上的努力，最近几年来，该计划在近年进行了多种积极的态度和对中的实例，以实现良好的态度，以实现良好的态度，以实现良好的效果，使其逐渐实现，以实现良好的效力，使其能够实现，并实现了良好的效果。歧管。第二个方面是对拓扑实现问题的系统研究，涉及周期性的同种学条件，尤其是在对称性的情况下。这些方面已经有一些结果，该提案旨在使该研究计划正式化，并找到促进该领域理解的新途径。