SPP 2026 – Page 3 – Geometry at Infinity

Virtually free-by-cyclic groups

Last month I blogged about coherent groups (i.e. groups whose finitely generated subgroups are finitely presented). There I also referred to an article of Daniel Wise about coherent groups that contains many open problems at the end. One of these problems is whether every one-relator group is coherent (a question posed by Baumslag in the … Continue reading

Differentials in the Adams spectral sequence

In this post we consider the Adams spectral sequence which computes the stable homotopy groups of spheres at the prime 2. The classes \(\{h_j\}_{j\ge 0}\) on the 1-line of the second page are called Hopf classes since they are related to the Hopf invariant: Adams proved that Hopf invariant one classes can only exist in … Continue reading

Coherent groups

Recall that a group G is called coherent if every finitely generated subgroup of G is finitely presented. Main examples are 3-manifold groups and (virtually) polycyclic groups. Instead of trying to motivate the study of coherent groups in my own words, let me instead just refer to Section 2 of the article `An Invitation to … Continue reading

Regularity of minimizing hypersurfaces

Let us consider the following classical problem from geometry (the case \(n=3\) is basically Plateau’s problem): Let \(\Gamma\) be a smooth, closed, oriented, \((n−1\))-dimensional submanifold of \(\mathbb{R}^{n+1}\). If we consider all the smooth, compact, oriented hypersurfaces \(M \subset \mathbb{R}^{n+1}\) with \(\partial M = \Gamma\), does there exist one with least area among them? In the … Continue reading

Illustrating the Impact of the Mathematical Sciences

A series of posters and some other related media were produced by the National Academy of Sciences of the USA to showcase mathematics of the twenty-first century and its applications in the real world: link. If you still don’t know what to put on your office walls, have a look at those posters!

Double Soul Conjecture

Recall the Soul Theorem of Cheeger and Gromoll: If \((M,g)\) is a complete and connected Riemannian manifold of non-negative sectional curvature, then there exists a closed, totally convex and totally geodesic embedded submanifold whose normal bundle is diffeomorphic to \(M\). Such a submanifold is called a soul of \((M,g)\) and the Riemannian metric induced on … Continue reading

Progress on the union-closed sets conjecture

The union-closed sets conjecture is the following extremely easy to state conjecture about subsets of finite sets: Assume that \(\mathcal{F}\) is a family of subsets of \(\{1, 2, \ldots, n\}\) which is union-closed; this means that for any two sets \(A,B\) in \(\mathcal{F}\) their union \(A \cup B\) is also a member of \(\mathcal{F}\). Then … Continue reading

Status-Bias im Peer-Review

In der aktuellen Forschung & Lehre ist ein Beitrag mit dem Titel Status-Bias im Peer-Review-Verfahren, den ich sehr interessant fand. Der Beitrag beginnt mit folgenden Worten: Forschungsarbeiten von renommierten Wissenschaftlerinnen und Wissenschaftlern werden trotz gleicher Qualität im Peer-Review-Verfahren deutlich besser bewertet als Arbeiten weniger bekannter Forschender. Zu diesem Ergebnis kommt ein Wissenschaftlerteam um Professor Jürgen … Continue reading