Cohomogeneity, Curvature, Cohomology

理解局部定义的概念(如曲率)的全局含义仍然是黎曼几何的中心任务。特别是流形的局部几何和拓扑性质之间的相互作用是一个值得研究的领域。这延伸到了曲率和奇异空间的综合概念(由Alexandrov几何构成)。这个项目基于三个支柱，它们改变了这样的问题(特别是从截面曲率及其推广的角度来看)：一方面，我们将研究Alexandrov空间(和奥比洛夫空间等)。它允许低上齐性的紧致李群的作用及其上同调性质；另一方面，等变K-理论的不同方法将被用来为适当流形(如双商)上的向量丛配备非负截面曲率直到稳定化的度量。最后，我们将应用Tame同伦理论，将有理不变量得到的不同结果和技巧推广到有限特征的设定上。除了曲率性质的讨论外，这些问题的进一步的相互依存性可以在等变上同调和有理同伦理论的概念的推广和应用中找到。

It remains a central task in Riemannian Geometry to understand global implications of locally defined concepts like curvature. Especially the interactions of the local geometries and the topological properties of the underlying manifolds are a worthwhile field of study. This extends to synthetic notions of curvature and singular spaces (as constituted by Alexandrov Geometry).This project is based on three pillars, which vary such questions (in particular, with a view towards sectional curvature and its generalisations): on the one hand we shall investigate Alexandrov spaces (and orbifolds, etc.) which admit actions of compact Lie groups of low cohomogeneity and their cohomological properties; on the other hand different approaches to equivariant K-theory will be used to equip vector bundles over suitable manifolds (like biquotients) with metrics of non-negative sectional curvature up to stabilisation. Finally, tame homotopy theory, in particular, will be applied in order to extend different results and techniques obtained via rational invariants to the setting of finite characteristic.Beside the discussion of curvature properties, further interdependencies of these questions can be found in generalisations and applications of concepts from equivariant cohomology and rational homotopy theory.