Linkage

Hilbert’s 13th, unsolved (\(\mathbb{M}\)). You can solve polynomials of degree at most four using one-argument algebraic functions like \(\sqrt x\). If \(RD(n)\) denotes the number of arguments needed for degree-\(n\) polynomials, then \(RD(4)=1\). Hilbert asked whether \(RD(7)=2\). Vladimir Arnold showed in the 1950s that you can solve all polynomials with two-variable continuous (but not algebraic) functions, but mathematicians are only now catching on that the algebraic problem is still open. See also some recent bounds on \(RD\).

Fogleworms (\(\mathbb{M}\)), partitions of \(n\times n\) grids into \(n\)-vertex grid paths, their enumeration, and a crafty project to visualize them.

A figure with Heesch number 6: Pushing a two-decade-old boundary (\(\mathbb{M}\)), Bojan Bašić in the Mathematical Intelligencer. When a shape cannot tile the plane, its Heesch number measures how far you can tile before getting stuck: if you surround the shape by layers of the same shape, how many layers can you make? Casey Mann’s previous record of five looked like a row of five hexagons with extra crenellations. This one uses six hexagons, with simpler crenellations.

Exponentiation by squaring, in somewhat cryptic form, in a work by Pingala from India from over 2000 years ago (\(\mathbb{M}\)).

I’ve been trying to understand the structure of the Kurilpa Bridge in Brisbane, supposedly “the world’s largest tensegrity bridge” (\(\mathbb{M}\)). The clearest description I’ve found is from Tadashi Design. Wikipedia is more cagy, calling it a “hybrid tensegrity bridge”, as it also includes features of cable-stayed bridges where the deck hangs from cables attached to tower piers. But the Tadashi Design site shows long sections far from the piers, so it seems the tensegrity is not just for show.

How origami folding patterns might help in the design of better face masks (\(\mathbb{M}\)).

More US universities using covid as an excuse to treat faculty badly (\(\mathbb{M}\)): the Kansas state university system guts tenure, and the University of Florida asks students to snitch on faculty who refuse to endanger themselves by teaching in person.

Proposed New York waterfront tower is a handlebody of high genus (\(\mathbb{M}\), via, via2).

Although it also has other names, the graph \(K_{1,1,n}\) has been called the “thagomizer graph”, and its associated graphic matroid has been called the “thagomizer matroid” (\(\mathbb{M}\)). The term appears to have been introduced by Katie Gedeon in arXiv:1610.05349 in honor of the famous Far Side cartoon, whose terminology has also been adopted by some paleontologists.

This new preprint looks interesting: Point-hyperplane incidence geometry and the log-rank conjecture, Noah Singer and Madhu Sudan, arXiv:2101.09592 (\(\mathbb{M}\)). In the plane, \(n\) points and \(m\) lines can only touch \(\Theta\bigl((mn)^{2/3}+m+n\bigr)\) times. In 3d, points and planes can have mn incidences but only by sharing a common line. This paper connects similar problems in high dimensions to the log-rank conjecture, a famous unsolved problem in communication complexity.

Unknot recognition in quasi-polynomial time (\(\mathbb{M}\), via). Title of talk announcement by Marc Lackenby. No details or preprint yet but judging solely from the title and non-fringe status of the author this sounds like big news.

Frederic Green has published another review of my book “Forbidden Configurations in Discrete Geometry” in the latest SIGACT News (\(\mathbb{M}\), official but paywalled url). Thanks to Joe O’Rourke for the heads-up: I last checked my mail at the office, where my physical copies of SIGACT News would go if they went anywhere, months ago, and even then it looked like magazines weren’t getting through.

Mathematics on the cutting block at Leicester again: Gowers, nCat, petition (\(\mathbb{M}\)). The plan is to eliminate research in pure mathematics at the University of Leicester, fire eight professors, and hire three back in purely teaching positions. I’m not sure who the eight are – the staff list includes some other disciplines – but Leicester mathematicians in Wikipedia include Katrin Leschke and Sergei Petrovskii.

Two newly-listed Good Articles on Wikipedia: Curve of constant width and Ronald Graham (\(\mathbb{M}\)).