Lehmer’s conjecture and the Fuglede-Kadison determinant

Lehmer’s conjecture is one of the most striking open problems in number theory. It roughly postulates that the complex roots \(a\) with \(|a| > 1\) of a polynomial with integral coefficients cannot simultaneously be close to the unit circle (unless all non-zero roots lie on the unit circle). More precisely, the Mahler measure of a … Continue reading "Lehmer’s conjecture and the Fuglede-Kadison determinant"

Wolf Prize 2022

The Wolf Prize in Mathematics 2022 is awarded to George Lusztig for “groundbreaking contributions to representation theory and related areas.” The Wolf Prize is an international award granted in Israel for “achievements in the interest of mankind and friendly relations among people …”. Wikipedia writes further: “Until the establishment of the Abel Prize, the Wolf Prize was probably the closest equivalent of … Continue reading "Wolf Prize 2022"

Positive scalar curvature and the conjugate radius

A classical result in Riemannian geometry is the theorem of P. O. Bonnet and S. B. Myers stating that a complete Riemannian \(n\)-manifold \(M\) with Ricci curvature bounded from below by \(n-1\) has diameter at most \(\pi\). In the introduction of Bo Zhu’s recent preprint arXiv:2201.12668 the following ‘analogue’ of the Bonnet-Myers Theorem for scalar … Continue reading "Positive scalar curvature and the conjugate radius"

Properly positive scalar curvature

An interesting (at least to me) research theme in the geometry of manifolds is the question about the existence of positive scalar curvature metrics on closed manifolds. Since I also like to do coarse geometry, I therefore also consider the corresponding question on non-compact manifolds. But what is the ‘corresponding’ question on non-compact manifolds? Currently, … Continue reading "Properly positive scalar curvature"

An implication of the Farrell-Jones conjecture

A ‘well-known’ implication of the Farrell-Jones conjecture (for a given group G) is that the map \[\widetilde{K_0(\mathbb{Z}G)} \to \widetilde{K_0(\mathbb{Q}G)}\] in reduced algebraic K-theory is rationally trivial. What at first might seem as a technical statement about algebraic K-theory turns out to have an interesting geometric consequence. It implies the Bass conjecture, which is equivalent to … Continue reading "An implication of the Farrell-Jones conjecture"