Kurvature, Kohomology and K-Theory

黎曼几何中的一个重要课题是理解几何结构的拓扑蕴涵。当人们想知道全局定义的拓扑对象如何阻碍局部定义的性质时，这就变得特别有吸引力。在我们的例子中，非负截面曲率。在这个项目中，我们想要研究这类具体问题。一方面，我们的目的是了解齐次空间上的向量丛及其许多推广允许非负截面曲率的度量--从而引出关于丛上的非负曲线度量的模空间的各种问题；另一方面，我们感兴趣的是几个奇异空间上的问题，如orbillold，Alexandrov空间等，这些问题推广了流形上的已知结果。闭测地线的刻画，满足一定Ricci曲率条件的度量的构造，以及从某种同伦理论的观点来对比Orbbon流形和流形，都是这一方向的具体问题。从代数拓扑学的角度来看，我们希望主要通过(等变)K-理论和上同调理论以及有理同伦理论的结合来解决这些问题，从而希望在工具和应用方面产生协同作用。

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.