RUI: Biquotients, Cohomogeneity-One Manifolds, and Double Disk Bundles

《首席研究者》试图增加我们对数学形状的理解，这种数学形状被称为流形，这是物理学家和数学家都特别感兴趣的。一类重要的流形是到处都是正弯曲或平坦的流形。术语“正曲率”指的是像球体一样弯曲；非常大的球体大致是平的，因此具有小但正曲率。另一方面，一个非常小的球体有一个非常大的曲率。这个项目将通过国际合作以及本科生研究项目来关注这些对象。PI的工作是由Goette，Kerin和Shankar最近在寻找所谓的余维一双商叶上的非负截面曲率的度量方面取得的突破所推动的。这些空间推广了广为人知的上齐性流形，并且它们都有一个分解，即沿着它们的公共边界粘在一起的两个圆盘丛的并。PI将致力于开发一种对此类物体进行分类的方法。这包括研究非单连通双商的分类，以及在上同调水平上为流形开发障碍以承认这样的结构。该奖项反映了NSF的法定使命，并通过使用基金会的智力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 Principal Investigator seeks to increase our understanding of mathematical shapes, called manifolds, which are of particular interest to both physicists and mathematicians. One important class of manifolds are those that are everywhere positively curved or flat. The term "positive curvature" refers to curving like a sphere; a very large sphere is approximately flat, and so has a small but positive curvature. On the other hand, a very small sphere has a very large curvature. This project will focus on these objects through international collaborations, as well as through undergraduate research projects.The PI's work is motivated by the recent breakthrough of Goette, Kerin, and Shankar on finding metrics of non-negative sectional curvature on so-called codimension one biquotient foliations. These spaces generalize the well known cohomogeneity one manifolds, and all have a decomposition as a union of two disk bundles glued together along their common boundary. The PI will work on developing a method of classifying such objects. This includes studying the classification of non-simply connected biquotients, as well as developing obstructions at the level of cohomology for a manifold to admit such a structure.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.