Kurvature, Kohomology and K-Theory

黎曼几何的一个重要课题是理解几何结构的拓扑含义。当人们想知道全局定义的拓扑对象如何阻碍局部定义的属性时，这变得特别吸引人——在我们的例子中，例如非负截面曲率。在这个项目中，我们想研究这类具体问题。一方面，我们的目标是了解齐次空间上哪些向量束及其许多推广允许非负截面曲率的度量-导致关于束上非负弯曲度量的模空间的各种问题。另一方面，我们对一些奇异空间的问题感兴趣，如轨道、Alexandrov空间等，它们推广了流形上已知的结果。封闭测地线的表征、满足某些里奇曲率条件的度量的构造以及从同伦理论的观点出发对比轨道和流形，都是这一方向的具体问题。从代数拓扑的角度来看，我们希望主要通过（等变）k理论和上同调以及理性同伦理论的结合来解决这些问题-从而希望在工具和应用上都有协同作用。

An important subject in Riemannian geometry is to understand topological implications of geometric structures. This becomes particularly appealing when one wonders how globally defined topological objects obstruct locally defined properties - in our case for example non-negative sectional curvature.In this project we want to investigate concrete questions of this kind. On the one hand we aim to understand which vector bundles over homogeneous spaces and their many generalisations permit metrics of non-negative sectional curvature - leading to various questions on moduli spaces of non-negatively curved metrics on bundles.On the other hand we are interested in questions on several singular spaces like orbifolds, Alexandrov spaces, etc., which generalise known results on manifolds. The characterisation of closed geodesics, the construction of metrics satisfying certain Ricci curvature conditions and contrasting orbifolds and manifolds from a certain homotopy theoretic viewpoint, are concrete problems in this direction.From the point of view of algebraic topology we want to tackle these questions mainly by a combination of (equivariant) K-theory and cohomology, as well as rational homotopy theory - thereby hoping for synergies both in tools and applications.