Curvature, Symmetry, and Periodic Cohomology

在理解物理世界和超越其边界的数学时，几何学和对称性无处不在。这个项目试图提高我们对抽象形状的了解，这些抽象形状共享两个属性：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，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，就像足球在完美的螺旋中投掷时的样子。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样的形状。此外，通过他对俄克拉何马州高中的接触和与本科生的参与，PI致力于培养和多样化STEM领域的学生和研究人员。这个项目的第一个方面是PI关于Grove对称性项目的工作的继续，近年来，该项目揭示了正曲率和对称性如何结合在一起，迫使潜在流形的上同调具有周期性的无数例子。第二个方面是系统地研究了周期上同调条件下的拓扑实现问题，特别是对称条件下的拓扑实现问题。已经有了这些方面的结果，该提案试图使这一研究计划正规化，并找到在这一领域促进理解的新途径。

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.