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.

Coherence of one-relator groups

The recent series of results about one-relator groups (see, e.g. the previous two blog posts) culminates now in the final breakthrough: The conjecture of Baumslag from the 70s (that all one-relator groups are coherent) is resolved. The corresponding preprint by Marco Linton was put today on the arXiv:2303.05976.

Coherent groups

Recall that a group G is called coherent if every finitely generated subgroup of G is finitely presented. Main examples are 3-manifold groups and (virtually) polycyclic groups. Instead of trying to motivate the study of coherent groups in my own words, let me instead just refer to Section 2 of the article `An Invitation to … Continue reading "Coherent groups"

Regularity of minimizing hypersurfaces

Let us consider the following classical problem from geometry (the case \(n=3\) is basically Plateau’s problem): Let \(\Gamma\) be a smooth, closed, oriented, \((n−1\))-dimensional submanifold of \(\mathbb{R}^{n+1}\). If we consider all the smooth, compact, oriented hypersurfaces \(M \subset \mathbb{R}^{n+1}\) with \(\partial M = \Gamma\), does there exist one with least area among them? In the … Continue reading "Regularity of minimizing hypersurfaces"

Illustrating the Impact of the Mathematical Sciences

A series of posters and some other related media were produced by the National Academy of Sciences of the USA to showcase mathematics of the twenty-first century and its applications in the real world: link. If you still don’t know what to put on your office walls, have a look at those posters!