Alexander Engel – Page 3

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

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

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

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

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

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

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.

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