### A chiral aperiodic monotile

Two months ago I blogged about the resolution of the famous problem whether a single shape can tile the entire plane, but only aperiodically (link). There was just one drawback to the solution: It required not only translating and rotating the shape, but also reflecting it. In a recent preprint (arXiv:2305.17743) this drawback was overcome … Continue reading "A chiral aperiodic monotile"
Posted on

### 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.
Posted on

### 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 "The invariant subspace problem in Hilbert spaces"
Posted on

### 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 "Metric spaces and C*-algebras"
Posted on

### 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"
Posted on

### 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"
Posted on

### 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 "Uniform and coarse embeddings of Banach spaces"
Posted on

### 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"
Posted on