SPP 2026 – Geometry at Infinity

Unit conjecture disproved!

There are three conjectures about group rings of torsion-free groups that are attributed to Kaplansky. To state them, let \(K\) be a field, \(G\) be a torsion-free group and denote by \(K[G]\) the corresponding group ring. The unit conjecture states that every unit in \(K[G]\) is of the form \(kg\) for \(k \in K\setminus\{0\}\) and … Continue reading

Database of online seminar talks

Due to the current situation there are numerous online seminar talks organized all over the world. I was quite happy to discover this week that the website researchseminars.org maintains a useful database of online seminars and conferences, focusing on mathematics and related fields. The website is supported by the American Mathematical Society, the MIT and … Continue reading

Tetrahedra

Consider the following three basic questions about tetrahedra: Does a given tetrahedron tile space? Which tetrahedra are scissors-congruent to a cube? Can one describe the tetrahedra all of whose six dihedral angles are a rational number of degrees? The first question goes back to Aristotle, the second is from Hilbert’s list of problems, and the … Continue reading

The commutator subgroups of surface groups

Every subgroup of infinite index in a surface group is a free group. There are many ways to see this, for instance, using a bit of topology. An infinite index subgroup corresponds to a covering space with infinitely many sheets and this covering space is a non-compact surface. By a result of Whitehead it deformation … Continue reading

Condensed mathematics

This post is inspired by Alex Engel’s post “Validity of results II” on incorrect results and computer proof checking. Before I get there, let me start with some background first and reveal the connection later. Dustin Clausen and Peter Scholze have put forward the idea of “condensed mathematics” which aims at replacing topological spaces by … Continue reading

Gegenseitige Korrektur in Lehrveranstaltungen

Dieses Semester halte ich eine Vorlesung für Studierende im dritten Semester (ich habe hier über den mathematischen Inhalt gebloggt). Zu dieser Vorlesung gibt es offiziell keinen Übungsbetrieb, aber ich gebe den Studierenden trotzdem hin und wieder (freiwillige) Übungsaufgaben auf, denn wie wir alle wissen lernt man Mathematik am besten wenn man sich selbst an ihr … Continue reading

Validity of results, II

In July I wrote a post about Validity of results. It was about a recent preprint pointing out an erroneous proof in an influential paper. By chance I stumbled yesterday upon this question on MathOverflow: Extent of unscientific, and of wrong, papers in research mathematics. The question starts with a link to Kevin Buzzard’s slides … Continue reading

How to hear the corners of a drum

A short while ago I was blogging about the classical isospectrality problem which goes back to Mark Kac’ famous question: Can one hear the shape of a drum? Some basic information about it can be found in that blog post of mine: Can one hear orientability? The original question of Mark Kac was answered negatively by … Continue reading