Kurvature, Kohomology and K-Theory

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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.