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。特别是局部几何的相互作用及其下流形的拓扑性质是一个值得研究的领域。这延伸到曲率和奇异空间的综合概念（由亚历山德罗夫几何构成）。这个项目基于三个支柱，这些支柱改变了这些问题（特别是，对截面曲率及其推广的看法）：一方面，我们将研究亚历山德罗夫空间（和轨道等），这些空间允许低同质性的紧李群的作用及其上同调性质；另一方面，等变k理论的不同方法将用于在合适的流形（如双商）上装备具有非负截面曲率直至稳定的度量的向量束。最后，特别是驯服同伦理论，将应用于将通过有理不变量获得的不同结果和技术扩展到有限特征的设置。除了曲率性质的讨论，这些问题的进一步相互依赖关系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.