### Pseudo-isotopies of 4-manifolds

One of the landmark results about the topology of 4-manifolds is Freedman’s proof of the 4-dimensional Poincaré conjecture. The problem with this result was that after some time people claiming to understand its proof disappeared, so that in the last decade the validity of this result was openly questioned (e.g., link). In the end, a … Continue reading "Pseudo-isotopies of 4-manifolds"

### Formalizations in LEAN

I already blogged a few times about computer assisted verification of proofs, for example here: link. It seems to me that this topic is accelerating fast right now, especially formalizations in LEAN. In addition to the example linked above (a verification of a recent result of Clausen-Scholze), an important new result in combinatorics was recently … Continue reading "Formalizations in LEAN"

### Freiraum 2023

Letztens wurden die Projekt verkündet, welche bei Freiraum 2023 gefördert werden. Im Rahmen von Freiraum können Mittel für die Verwirklichung von Projekten in der Lehre beantragt werden, und ich hatte mich diesmal gefragt, ob es etwas spannendes aus der (reinen) Mathematik darunter gibt. Da es keine Auflistung nach Fächern gibt, arbeitete ich mich durch die … Continue reading "Freiraum 2023"

### Uncomplemented subspaces in Banach spaces

In any Hilbert space $$H$$, every closed subspace $$M$$ therein is complemented, i.e. there exists a closed subspace $$N$$ with $$M \oplus N \cong H$$. One possible choice for $$N$$ is always the orthogonal complement $$N := M^\perp$$. If we consider Banach spaces instead, then the situation changes. For example, it is `well known’ that … Continue reading "Uncomplemented subspaces in Banach spaces"

### 2024 Breakthrough Prize in Mathematics

The 2024 Breakthrough Prize in Mathematics goes to Simon Brendle for transformative contributions to differential geometry, including sharp geometric inequalities, many results on Ricci flow and mean curvature flow and the Lawson conjecture on minimal tori in the 3-sphere. The 2024 New Horizons in Mathematics Prize goes to Roland Bauerschmidt for outstanding contributions to probability … Continue reading "2024 Breakthrough Prize in Mathematics"

### Metric spaces and C*-algebras II

In the previous post about metric spaces and C*-algebras we saw the definition of the uniform Roe algebra $$C^*_u(X)$$ of a metric space $$X$$ and discussed its remarkable property that two metric spaces (of bounded geometry) $$X$$ and $$Y$$ are coarsely equivalent if and only if their uniform Roe algebras $$C^*_u(X)$$ and $$C^*_u(Y)$$ are Morita … Continue reading "Metric spaces and C*-algebras II"

### 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"

### Can you hear your location on a manifold?

Can one hear the shape of a drum? is a famous paper by Mark Kac from 1966. The precise mathematical question stated therein is the following: If the Laplace operators of the two Riemannian manifolds with boundary (M,g) and (N,h) have the same eigenvalues (for Dirichlet boundary conditions), must (M,g) and (N,h) be isometric? I … Continue reading "Can you hear your location on a manifold?"