The Kervaire Conjecture

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"

Shaw Prize 2023 in Mathematical Sciences

The Shaw Prize this year is awarded to Vladimir Drinfeld and Shing-Tung Yau for their contributions related to mathematical physics, to arithmetic geometry, to differential geometry and to Kähler geometry. More information can be found here: link.

Positive vs nonnegative sectional curvature

The sectional curvature is one of the main invariants of Riemannian manifolds. But despite its importance, there is actually little known about questions like: What is the distinction (for closed manifolds) between admitting a metric of positive sectional curvature vs admitting a metric of nonnegative sectional curvature? Another problem is also to distinguish positive sectional … Continue reading "Positive vs nonnegative sectional curvature"

Hilbert spaces and C*-algebras without Choice

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"