Linkage

Congratulations to Robert Hickingbotham (\(\mathbb{M}\)), a student of David Wood at Monash University who just finished his PhD with an impressive portfolio of research in graph structure theory. Next step: a post-doc at ENS Lyon.

Not a context where I would have expected to see self-overlapping curves (boundaries of immersed disks): models of 2d quantum gravity (\(\mathbb{M}\)). I don’t understand much of this preprint but the connection makes me want to know more.

From Zero to Hero: a Euclidean proof (\(\mathbb{M}\)) of Heron’s formula on triangle areas, from which you can also get Descartes’ theorem on the radii of tangent circles.

Boaz Barak posts the sad news of the death of Luca Trevisan (\(\mathbb{M}\), see also), researcher on pseudorandomness and complexity, and blogger at “in theory”

Tumbling downhill along a given curve (\(\mathbb{M}\)), Eckmann, Sobolev, and Tlusty, in Notices of the AMS. Almost every repeating path can be traced by a carefully weighted and shaped roller, rolling down an inclined plane, so that it returns to its original orientation after two repetitions through the path. The authors conjecture but do not prove that only one repetition suffices. Robin Houston suggests that this would make an interesting test case for 3d printing.

US academic freedom under fire: Florida government lawyer asserts that the government, and not professors, has the right to control what the professors present in university classes (\(\mathbb{M}\), archived).

Three of Scopus’s top-ten philosophy journals turn out to be fake journals filled with fake papers (\(\mathbb{M}\)). Although this problem was discovered and publicized by a group of philosophers, I’m hearing at third hand that many academics in this area are unconcerned with the ongoing crisis in academic publishing; they’d rather focus on the philosophical questions such as on what basis we derive our beliefs on academic integrity or on what criteria a journal ranking system should try to reflect than on the practical question of how to prevent their discipline from becoming overwhelmed by AI-generated slop. See also a related piece at Daily Nous on how to detect AI generation in philosophy journal submissions and what to do when it is detected.

A uniform tiling that fold-transforms into another uniform tiling, Gerard Westendorp. It’s just the snub hexagonal tiling; I guess the trick is in the choice of mountain and valley folds to use.

Bill Gasarch collects open problems honoring Luca Trevisan (\(\mathbb{M}\)). The problems should either be by or inspired by Trevisan.

Generative Escher meshes (preprint and code; \(\mathbb{M}\)): Escher-like tilings from text prompts using stable diffusion, by Noam Aigerman and Thibault Groueix. It’s no Escher yet, but it’s interesting to see what more can be done with these systems when they’re hooked up to something more structured (in this case, code for handling periodic tilings) rather than being expected to handle everything themselves.

The Pullulating Polyps of OMICS (\(\mathbb{M}\)). Someone with the likely name of Smut Clyde enlightens us on the unending search of a predatory-publishing empire for fresh but old journals that are not named OMICS, to take over and sell to authors who want to buy publications but have some taste in where they publish. One of the victims: the briefly-respectable and now-ironically-named Mathematica Æterna. Or, if you see the non-word “mobileular” in a paper, run. (It is machine-mangled-English for “cellular”.)

A question I would be quite interested in, , in connection with my recent paper on Dehn invariants and a guide on making polyhedra with one non-right angle by Robin Houston (\(\mathbb{M}\)): for the angles \(\theta\) that can be realized as the single non-right angle of a polyhedron, is there a fixed upper bound (not depending on \(\theta\)) for the number of edges that you need to use?

More generally what I really want to know is whether there is a fixed bound on the number of edges of a polyhedron that realizes a given Dehn invariant, depending only on the rank of the invariant. The single-non-right-angle version of the question is for rank-one invariants and for a restricted subset of polyhedral realizations. The Dehn invariant version of the question isn’t interesting for angles that are rational multiples of \(\pi\), because their Dehn invariant is zero and can be realized by a cube. But the single-non-right-angle version of the question might still be interesting in that case, too.

In follow-up discussion, Houston suggests that each combinatorial type of single-non-right-angle polyhedron should uniquely determine the remaining angle. Infinitely many angles are realizable (by adding known irrational angle constructions) so, if true, the number of edges needed for single-non-right-angle realizations would be unbounded. It follows from known results on realizability of Dehn invariants that one cannot smoothly interpolate between two realizations with the same combinatorial type but different angles, but it’s not obvious why realizations with two different angles might lead to this kind of forbidden interpolation.

Microsoft CEO claims that, for content on the open web, copyright is meaningless (\(\mathbb{M}\), via): “Anyone can copy it, recreate with it, reproduce with it. That has been freeware, if you like.”

Today in stuff I don’t understand but looks interesting, a seven-dimensional counterexample to an old conjecture of Milnor (\(\mathbb{M}\)): “Fundamental Groups and the Milnor Conjecture”, Elia Bruè, Aaron Naber, and Daniele Semola, arXiv:303.15347, via “Strangely Curved Shapes Break 50-Year-Old Geometry Conjecture”, Quanta. More precisely the construction gives a seven-dimensional manifold with non-negative (Ricci) curvature, whose fundamental group is \(\mathbb{Q}/\mathbb{Z}\). Milnor conjectured that the fundamental group of non-negatively-curved spaces would always be finitely generated, but this one isn’t.

Two videos about the infinite wild slipknot (\(\mathbb{M}\)): a 3d print by Henry Segerman and an animation of it being undone by Hsin-Po Wang.