Zeit und Ort

Montag, 29.6.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

An irreducible \(G_2\)-manifold is a Riemannian 7-manifold \(M\) with holonomy group equal to the exceptional Lie group \(G_2\). When \(M\) is closed, the Teichmüller space \(T(M)\) of \(G_2\) metrics on \(M\) divided by diffeomorphisms isotopic to the identity is a smooth, finite-dimensional manifold by a result of Joyce. Yet its topology, and that of its quotient by the smooth mapping class group, remains elusive. Using ideas of Crowley, Goette, and Hertl, we exhibit the first known example of a \(G_2\)-manifold \(M\) together with infinitely many diffeomorphisms that both act freely on \(T(M)\) and preserve a connected component. The diffeomorphisms are 7-dimensional analogs of diffeomorphisms of K3 surfaces constructed recently by Farb and Looijenga, and much like the Farb-Looijenga examples, these diffeomorphisms minimize topological entropy among their isotopy class.

Zeit und Ort

Montag, 15.6.26, 16:15–17:45, Seminarraum 404

Zeit und Ort

Montag, 18.5.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

Llarull proved that the round sphere is extremal, meaning that one cannot simultaneously increase both its scalar curvature and its metric. Goette and Semmelmann generalized this result and proved that scalar-rigid maps are Riemannian submersions. In this talk, I present a recent generalization showing that a scalar-rigid f: M → N is not only a Riemannian submersion, but that M is essentially a Riemannian products of the base manifold with a Ricci-flat fiber. The proof is based on spin geometry for Dirac operators and an analysis connecting Clifford multiplication with the representation theory of the curvature operator. This is joint work with Oskar Riedler.

Zeit und Ort

Montag, 11.5.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

The existence of closed geodesics on compact Riemannian manifolds has been studied for a long time. In this talk, I present the ideas of Asselle and Mazzucchelli to generalize results for the compact case to the larger class of manifolds without close conjugate points at infinity, i.e. manifolds such that every sufficiently short geodesic outside a given compact set does not contain any conjugate points. Furthermore, I compare their approach with a similar one by Benci and Giannoni involving the sectional curvature of the manifold.