Obstructions to positive curvature and symmetry

摘要奖：DMS 1404670，首席研究员：Lee Kennard，郭方伟流形的曲率是从欧几里得平面开始，试图量化空间可能弯曲的方式，欧几里得平面的平坦性是通过宣布平面为零曲率来捕捉的。三维空间中的二维圆形球体具有正曲率，其大小反映了这样一种感觉：当一个人穿过一个大半径的球体时，它的切面在空间中的转动速度比与一个小半径球体的切面更慢：半径为R的球体的曲率为1/R^2。曲率大于或等于零的流形多年来一直是一个活跃的研究领域，大多数已知的例子都是通过施加一定的对称性而产生的，这些研究项目将同伦理论工具引入了Grove和Ziller研究非负曲率黎曼流形的方法中。要研究的一些问题遵循阿德姆、亚当斯和陈的著名定理和猜想。特别地，推广了单生成上同调环上的Adams定理，并考虑了正曲线流形的基本群上的陈氏问题在对称性存在时的推广。

AbstractAward: DMS 1404670, Principal Investigator: Lee Kennard, Guofang WeiThe curvature of a manifold is an attempt to quantify the ways that a space may bend, beginning with the Euclidean plane, whose flatness is captured by declaring the plane to have zero curvature. The two-dimensional round sphere in three-dimensional space has positive curvature, of a size that reflects the sense that as one traverses a sphere of large radius, its tangent plane turns more slowly in space than the tangent plane to a sphere of smaller radius: the sphere of radius R has curvature 1/R^2. Manifolds of curvature greater than or equal to zero have been an active area of study for many years, with most known examples produced by imposing some amount of symmetry, with the sphere and its large group of rotations being the most symmetric example.These research projects bring homotopy-theoretic tools into the approach of Grove and Ziller to the study of Riemannian manifolds of non-negative curvature. Some of the questions to be studied follow on well-known theorems and conjectures of Adem, Adams, and Chern. In particular, a generalization is suggested of Adams' theorem on singly-generated cohomology rings and generalizations in the presence of symmetry for Chern's problem on the fundamental groups of positively curved manifolds are considered.