Connected Isotropy Groups in the Grove Symmetry Program

PI旨在促进对称为流形的光滑物体的局部几何性质和全局拓扑性质之间相互作用的理解。这项工作可以追溯到150多年前黎曼几何的起源，它涉及几十年前提出的问题，仍然没有答案。幸运的是，在过去的25年里，通过对称性的透镜来看待这些问题，已经取得了重大进展，开发了新的工具，并建立了新的例子。PI广泛合作，并在将世界人民及其经济联系在一起方面发挥作用。着眼于未来，PI还积极参与未来美国机构的成长，教育和多样化。以STEM为中心的研究人员。在过去的十年里，PI和他的合作者在格罗夫对称性计划中证明了惊人的新结果。这个程序是在20世纪90年代开发的，为黎曼几何中围绕正曲率的基本但困难的问题提供了一个立足点。这些新成果需要新的工具。例如，PI和他的合作者最近获得了令人惊讶的结果与连接各向同性群的环面表示。证明只使用初等表示理论。特别是，它们不涉及曲率假设，因此可能在其他几何领域有更广泛的应用。完成这项工作并进一步调查其应用是该项目期间的首要任务。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The PI seeks to advance understanding of the interaction between the local, geometric properties and the global, topological properties of smooth objects called manifolds. This continues work that goes back over 150 years to the origins of Riemannian geometry, and it involves questions asked decades ago that still have no answers. Fortunately viewing some of these problems through the lens of symmetry in the last 25 years has led to major advances, the development of new tools, and the construction of new examples. The PI collaborates widely and plays his part in tying together the world’s people and their economies. With an eye to the future, the PI is also actively involved in the growth, education, and diversification of tomorrow’s body of U.S.-based, STEM-focused researchers. In the past ten years, the PI and his collaborators have proved striking new results in the Grove symmetry program. This program was developed in the 1990s to provide a foothold into the basic but difficult questions in Riemannian geometry around positive curvature. These new results have required new tools. For example, the PI and his collaborators have recently obtained surprising results for torus representations with connected isotropy groups. The proofs use only elementary representation theory. In particular, they do not involve curvature assumptions and might therefore have wider applications in other areas of geometry. Completing this work and further investigating its applications are top priorities for this project period.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.