Research Seminar on Topology (Oberseminar Topologie)
Mondays 17-18 in 0.05 (PER 23)

Since the introduction of Ricci flow by Hamilton in 1982, it has been a fundamental question to understand the evolution of metrics and their curvature conditions under the flow. While positive scalar curvature and 2-positive curvature operator are preserved in all dimensions, there exist infinitely many dimensions where certain curvature conditions lying in between are not preserved. In this talk, I first recall some basics about homogeneous Ricci flow. Then, I present joint works with David González-Álvaro in which we examine metrics with different curvature conditions on various homogeneous spaces and discuss the evolution of their metrics under the Ricci flow.

The twisted suspension of a manifold can be seen as a smooth analogue of the classical suspension operation for topological spaces. Its construction is motivated from the spinning operation in knot theory and it is obtained by surgery on a fibre of a principal circle bundle over the given manifold. In this talk I will show that Riemannian metrics of positive Ricci curvature can be lifted along twisted suspensions. As application we obtain first examples of simply-connected manifolds of positive Ricci curvature with maximal symmetry rank in any dimension, and we obtain new examples of (rational) homology spheres with a Riemannian metric of positive Ricci curvature.

The notion of $m$-intermediate curvature interpolates between Ricci curvature and scalar curvature. In this talk we describe extensions of classical results by Bonnet-Myers and Schoen-Yau to the setting of $m$-intermediate curvature: A non-existence result for metrics of positive $m$-intermediate curvature on manifolds with topology $N^n=T^m$×$S^{n-m}$; a gluing result for manifolds with $m$-convex boundary; inheritance of spectral positivity along stable minimal hypersurfaces, and estimates for the $m$-diameter for uniform positive lower bounds. This talk is partially based on joint work with Simon Brendle and Sven Hirsch, and joint work with Aaron Chow and Jingbo Wan.

In this talk we will discuss the construction of ${\mathrm{Ric}}_2$>0 metrics on closed manifolds of dimensions 10, 11, 12, 13 and 14. These include $S^6$×$S^7$, $S^7$×$S^7$ and infinitely many homeomorphism types in dimension 13. The main new ingredient is a notion for homogeneous bundles which generalizes the concept of fatness and, together with other assumptions, ensures the existence of ${\mathrm{Ric}}_2\mathrm{>0}$ metrics on the total space of the bundle. Further examples are obtained by considering biquotients of the corresponding homogeneous spaces. This is joint work with Jason DeVito, Miguel Domínguez-Vázquez and Alberto Rodríguez-Vázquez.

In this talk we shall review a few of the many existing notions of curvature in Riemannian and metric geometry, together with some of the main achievements and current lines of research. We will put special focus on manifolds of positive sectional curvature and those of positive Ricci curvature, and also on certain curvature conditions which interpolate between them.