### Spaces of positively curved Riemannian metrics

It is by now a classical topic in index theory to study on a (closed) Riemannian (spin) manifold the space of all Riemannian metrics of positive scalar curvature. We have several results showing that this space is usually highly complicated from a homotopy theoretic point of view (provided it is non-empty). Instead of studying positivity … Continue reading "Spaces of positively curved Riemannian metrics"

### Quasi-local operators

In the first post of this series we asked at the end two questions – in this post we start working towards the answers in the general setup of the third post of this series. Our setup from the third post is the following: We have a metric space $$(X,d)$$ and we consider a bounded, … Continue reading "Quasi-local operators"

### The unplanned impact of mathematics

In 2011 Nature published a short article about The unplanned impact of mathematics which I stumbled upon just now. Since it is just 4 pages long, I can recommend reading it just for fun – or for one’s ease of mind. 😉

### Computer assistance and pure mathematics

Today I want to tell you a story of a preprint in pure mathematics that came into existence only by crucial help of precise computer computations. To explain the results, let us first define for a set $$A \subset \mathbb{N}_{>1}$$ of natural numbers $f(A) := \sum_{n \in A} \frac{1}{n \log(n)}\,.$ For $$k \ge 1$$ let … Continue reading "Computer assistance and pure mathematics"

### Validity of results

Can you vouch for the validity of the results in papers that you cite and use in your own articles? In April this year Blagojević, Cohen, Crabb, Lück and Ziegler posted a preprint on the arXiv (arXiv:2004.12350) writing in the abstract “This invalidates a paper by three of the present authors […] who used a … Continue reading "Validity of results"

### Conjugation Curvature

Recently I saw some papers on the arXiv on conjugation curvature of finitely generated groups (also called medium-scale curvature, transportation curvature, metric Ricci curvature or comparison curvature for Cayley graphs). I got a bit interested in it and so decided to write up a short post about it. Let $$G$$ be a finitely generated group … Continue reading "Conjugation Curvature"

### Collatz conjecture

Given a natural number n, the Collatz sequence it generates is the following: if n is even, then divide it by 2, if n is odd, then multiply it by 3 and add 1; and now iterate this procedure. The Collatz conjecture states that you will always end up with the number 1 after finitely … Continue reading "Collatz conjecture"

### Large scale properties of 3-manifold groups

A week ago there was a preprint posted on the arXiv by Peter Haïssinsky and Cyril Lecuire about Quasi-isometric rigidity of three manifold groups (arXiv:2005.06813). Building on work by many other people, they complete the proof that the class of 3-manifold groups is quasi-isometrically rigid, meaning the following: if a finitely generated group G is quasi-isometric … Continue reading "Large scale properties of 3-manifold groups"

### Banach conjecture

There was a paper today in the arXiv mailing list (arXiv:2006.00336) proving yet another case of the Banach conjecture. I never heard of this conjecture before, but it is easy to state and seems to me to be a foundational recognition principle for those Banach spaces that are actually Hilbert spaces. The conjecture was stated … Continue reading "Banach conjecture"

### Shaw Prize 2020

The Shaw Prize 2020 in the Mathematical Sciences goes to Alexander Beilinson and David Kazhdan “for their huge influence on and profound contributions to representation theory, as well as many other areas of mathematics.” More information can be found here: https://www.shawprize.org/laureates/mathematical-sciences/2020.