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"

The Alon-Jaeger-Tarsi conjecture states the following: For any (finite) field F with at least four elements and any non-singular matrix M over F there is a vector x such that both x and Mx have only non-zero entries. The problem is easy to state – one can understand it after attending any Linear Algebra I … Continue reading "Alon-Jaeger-Tarsi conjecture"

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"

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"

Not too long ago I blogged about the first counter-example to Kaplansky’s unit conjecture (link) stating that there are no non-trivial units in the group ring K[G] for K a field and G a torsion-free group. A related conjecture of Kaplansky (one that I was not aware of until recently) is that K[G] is directly … Continue reading "Kaplansky’s direct finiteness conjecture"

It is an interesting and important fact that a contractible manifold (without boundary) is not necessarily homeomorphic to Euclidean space. This makes the classical Cartan-Hadamard theorem, stating that a contractible manifold equipped with a Riemannian metric of non-positive sectional curvature is diffeomorphic to Euclidean space, even more powerful. One can ask now whether one can … Continue reading "Topological CAT(0)-manifolds"