Curvature, Symmetry, and Periodic Cohomology

在理解物理世界和超越其边界的数学时，几何和对称无处不在。该项目旨在提高我们对具有两个特性的抽象形状的认识：正曲率和全局对称性。第一个要求是形状弯曲的方式与球的表面弯曲的方式相似，在所有方向上都是相同的。第二个意思是物体在旋转时看起来是一样的，类似于一个足球在一个完美的螺旋中抛出时的样子。首席研究员通过与其他研究人员合作、与博士生合作以及组织研讨会和会议来研究这些形状。此外，通过他在俄克拉何马州的高中外展和他与本科生的参与，PI致力于发展和多样化STEM领域的学生和研究人员。该项目的第一个方面是PI在Grove对称计划上的工作的延续，该计划近年来已经揭示了许多实例，说明了正曲率和对称如何在基础流形的上同调中结合在一起来强制周期性。第二方面是系统地研究周期上同条件下的拓扑实现问题，特别是在存在对称性的情况下。在这方面已经有了一些结果的实例，该提案寻求将这一研究项目正式化，并找到促进这一领域理解的新途径。

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.