Category: Formalization and AI

One of most important mathematical results from 2021 was Gardam’s refutation of the unit conjecture (blog post): He constructed the first non-trivial unit in a group ring. The coefficients of the group ring of the first counter-example were \(\mathbb{F}_2\), and for three years the question whether one can generalize his idea to get counter-examples in … Continue reading "Non-trivial units of complex group rings"

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 "Unit conjecture disproved!"

Today I want to tell you a story of a preprint in pure mathematics that came into existence only by crucial help of precise computer computations. To explain the results, let us first define for a set \(A \subset \mathbb{N}_{>1}\) of natural numbers \[f(A) := \sum_{n \in A} \frac{1}{n \log(n)}\,.\] For \(k \ge 1\) let … Continue reading "Computer assistance and pure mathematics"