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"

An aperiodic monotile

It is finally done – the long standing question whether there exists a single tile that can cover the entire plane, but only aperiodically, is answered! Here is the answer (arXiv:2303.10798): Many other people have already written blog posts, etc. about this, hence I will refrain from repeating everything and just refer to those other … Continue reading "An aperiodic monotile"

Abel Prize 2023

The Able Prize 2023 is awarded to Luis A. Caffarelli for his seminal contributions to regularity theory for nonlinear partial differential equations including free-boundary problems and the Monge–Ampère equation.

Coherence of one-relator groups

The recent series of results about one-relator groups (see, e.g. the previous two blog posts) culminates now in the final breakthrough: The conjecture of Baumslag from the 70s (that all one-relator groups are coherent) is resolved. The corresponding preprint by Marco Linton was put today on the arXiv:2303.05976.