Department of Mathematics
University of Fribourg

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

Organizers: Anand Dessai, Philipp Reiser, Sam Hagh Shenas Noshari

Fall 2022

Monday 3 October 2022 17:00 0.05 (PER 23) Philipp Reiser (University of Fribourg)
Surgery on Riemannian manifolds of positive Ricci curvature

The surgery theorem of Gromov and Lawson is a powerful tool to construct Riemannian manifolds of positive scalar curvature. For the stricter condition of positive Ricci curvature, however, it is not known if a surgery theorem in the same generality holds. If one imposes additional restrictions on the metric and the dimensions involved, then there exist surgery results for positive Ricci curvature by Sha-Yang and Wraith. In this talk we review these results and present a generalization of the surgery theorem of Wraith.

Monday 10 October 2022 No seminar Plancherel Lecture
Monday 17 October 2022 17:00 0.05 (PER 23) Philipp Reiser (University of Fribourg)
Metrics of positive Ricci curvature on simply-connected 6-manifolds

In this talk we use the results of the previous talk to construct metrics of positive Ricci curvature on manifolds obtained by plumbing. As application we show that one obtains infinite families of new examples of simply-connected 6-manifolds with a metric of positive Ricci curvature.

Monday 24 October 2022 17:00 0.05 (PER 23) Sam Hagh Shenas Noshari (University of Fribourg)
GKM actions: A short survey

Equivariant cohomology as introduced by A. Borel is an algebraic invariant associated to continuous Lie group actions that, roughly speaking, captures information about the fixed point set of the action, very much like ordinary cohomology encodes the presence of higher-dimensional "holes" on a topological space. Determining the equivariant cohomology of an action is, in general, a challenging task. The purpose of this talk is to introduce a class of actions for which it actually is possible to compute - and to visualize - their equivariant cohomology, the so-called GKM actions. We will briefly review some necessary background on equivariant cohomology, then explain how to associate a graph to GKM actions and how to recover their equivariant cohomology from this graph. Afterwards, we will discuss recent results about or involving GKM actions.

Monday 31 October 2022 No seminar
Monday 7 November 2022 No seminar
Monday 14 November 2022 17:00 0.05 (PER 23) Jan Nienhaus (University of Münster)
Four-periodicity and a Conjecture of Hopf with Symmetry

An almost 100 year old conjecture of Hopf states that even-dimensional positively curved manifolds have positive Euler characteristic. While this seems out of reach in full generality, some progress has recently been made assuming in addition that the positively curved metric has some symmetries. If a torus acts on a closed manifold, the Euler characteristic of the manifold is the same as that of the fixed point set, and the problem is reduced to understanding which fixed point sets may occur. We prove that if the metric admits an isometric T4-action, then all fixed point components have the rational cohomology ring of spheres or complex or quaternionic projective spaces, in particular have positive Euler characteristic, proving the Hopf conjecture for manifolds with T4-symmetry. This builds upon work of Kennard, Wiemeler and Wilking, who proved the same for manifolds admitting a T5-action in 2021.

Monday 21 November 2022 No seminar
Monday 28 November 2022 No seminar
Monday 5 December 2022 17:00 0.05 (PER 23)
Monday 12 December 2022 17:00 0.05 (PER 23) Christian Ketterer (University of Freiburg (D))
Monday 19 December 2022 No seminar

Spring 2023

Monday 20 February 2023 17:00 0.05 (PER 23) Uwe Semmelmann (University of Stuttgart) tba