I wanted to share the results of this funny survey with you: 100 Questions About Mathematical Conventions. But beware, if you click on the link you will loose an hour of your life time. 😉
Do you know when two groups \(G\) and \(H\) are called isoclinic to each other? I have never heard of this notion before, but there was a paper on the arXiv listing today using it in its abstract; so I looked in this paper, not because I was interested in its content, but just because … Continue reading "Isoclinism of groups"
Heute kam die Ankündigung, dass die Stiftung Innovation in der Hochschullehre im Februar wieder eine Freiraum-Förderung ausschreiben wird (hier ein Link zu dem Blogbeitrag über die vorhergehende solche Förderung). Hierbei wird wieder, wie auch beim letzten Mal, zuerst eine Lotterie stattfinden, welche rein auf Glück eine gewisse Anzahl von Antragsskizzen durchlässt zum eigentlichen Verfahren. Die … Continue reading "Lotterien in Antragsverfahren"
One of the landmark results about the topology of 4-manifolds is Freedman’s proof of the 4-dimensional Poincaré conjecture. The problem with this result was that after some time people claiming to understand its proof disappeared, so that in the last decade the validity of this result was openly questioned (e.g., link). In the end, a … Continue reading "Pseudo-isotopies of 4-manifolds"
Letztens wurden die Projekt verkündet, welche bei Freiraum 2023 gefördert werden. Im Rahmen von Freiraum können Mittel für die Verwirklichung von Projekten in der Lehre beantragt werden, und ich hatte mich diesmal gefragt, ob es etwas spannendes aus der (reinen) Mathematik darunter gibt. Da es keine Auflistung nach Fächern gibt, arbeitete ich mich durch die … Continue reading "Freiraum 2023"
Think about your favourite non-trivial group. Now add a generator to it and then add any relation. Is the resulting group still non-trivial? The above question is known as the Kervaire conjecture. Phrased more concretely, if \(G\) is any non-trivial group and \(r \in G \ast \mathbb{Z}\), is \((G \ast \mathbb{Z})/\langle\!\langle r\rangle\!\rangle\) again non-trivial? This … Continue reading "The Kervaire Conjecture"
Two months ago I blogged about the resolution of the famous problem whether a single shape can tile the entire plane, but only aperiodically (link). There was just one drawback to the solution: It required not only translating and rotating the shape, but also reflecting it. In a recent preprint (arXiv:2305.17743) this drawback was overcome … Continue reading "A chiral aperiodic monotile"
Today a paper was posted on the arXiv by Blackadar, Farah and Karagila (arXiv:2304.09602) that examines Hilbert spaces and C*-algebras in ZF set theory without the assumption of any Axiom of Choice (not even with Countable Choice). Since many standard tools from functional analysis like the Hahn-Banach theorem or the Baire category theorem fail without … Continue reading "Hilbert spaces and C*-algebras without Choice"