SPP 2026 – Geometry at Infinity

The invariant subspace problem in Hilbert spaces

I was a bit surprised today to see a preprint by Per Enflo on the arXiv (arXiv:2305.15442). He is already 79 years old (Wikipedia)! Per Enflo is famous for several fundamental results in functional analysis (see the English Wikipedia article linked above for an overview). The one related to this post is his counter-example (the … Continue reading

Metric spaces and C*-algebras

Let \(X\) be a metric space of bounded geometry. The latter means that for all \(r > 0\) exists an \(n_r \in \mathbb{N}\) such that every ball in \(X\) of radius \(r\) has at most \(n_r\) elements. This especially implies that \(X\) is discrete. It might seem a bit strange for some to consider discrete … Continue reading

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

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

Uniform and coarse embeddings of Banach spaces

Let \(X\) and \(E\) be Banach spaces. The metrics on these spaces induce both uniform and coarse structures, and we can ask whether the following two statements are equivalent to each other: \(X\) uniformly embeds into \(E\). \(X\) coarsely embeds into \(E\). Let us first write down what it means for a map \(\phi \colon … Continue reading

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

Ars legendi-Fakultätenpreis Mathematik 2023

Der Ars legendi-Fakultätenpreis Mathematik ist dieses Jahr an Prof. Dr. Claudia Kirch verliehen worden. Der Preis wird für herausragende, innovative und beispielgebende Leistungen in Lehre, Beratung und Betreuung vergeben.

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.