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。这个项目旨在提高我们对抽象形状的认识，这些形状具有两个属性：正曲率和全局对称性。其中第一个是要求形状以类似于球的表面如何弯曲的方式弯曲，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，类似于一个足球在完美的螺旋中被抛出时的样子。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，通过他在俄克拉荷马州的高中和他与本科生的接触，PI致力于发展和多样化的学生和研究人员在STEM领域的身体。该项目的第一个方面是PI的格罗夫对称计划，其中，近年来，揭示了许多正曲率和对称性如何结合在一起，在基础流形的上同调中产生周期性的例子。第二个方面是系统地研究了涉及周期上同调条件的拓扑实现问题，特别是在对称性存在的情况下。已经有沿着这些路线取得成果的实例，该提案旨在使这一研究计划正规化，并找到推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.