Linkage

Matt Blaze reminds all Ph.D. hopefuls that the way to get into a Ph.D. program is to apply to it. Emailing individual professors will not help: “No one gets in to a program that way. Just apply.”

New Wikipedia article on quasi-polynomial growth (\(\mathbb{M}\)): it’s not just about time complexity. Here “quasi-polynomial” means upper-bounded by \(\exp O\bigl((\log n)^c\bigr)\) for some constant \(c\); polynomial would be \(c=1\). Unfortunately, the same word is used in closely related areas of mathematics to mean something totally unrelated, functions that are sort of like polynomials but with some periodicity in their coefficients.

Atlantic on the consequences of West Virginia University’s shutdown of programs in humanities, foreign languages, environmental-health sciences, education administration, and mathematics (\(\mathbb{M}\), archived), under the guise of turning it into a trade school that will focus only on programs directly related to future employment. Via YC but I’m not even going to link to them because the comments are (as usual) so bad.

I’m particularly pleased by the numerical coincidence of having a post included in the \(11011110_2\) Carnival of Mathematics (\(\mathbb{M}\)). It’s the one on pyramidology.

Progress Around Borsuk’s Problem (\(\mathbb{M}\)). There exist bounded subsets of \(d\)-dimensional Euclidean space that cannot be covered by fewer than exponentially many sets of smaller diameter. Here, exponentially many means, as a function of the dimension. Previously only exponential in the square root of the dimension was known. For background see Wikipedia on Borsuk’s conjecture and for the preprint announcing the new result see “Convex bodies of constant width with exponential illumination number”, by Arman, Bondarenko, and Prymak. The preprint is formulated in terms of illumination rather than diameter, but it’s the same thing; see footnote 2 of Gil Kalai’s blog post.

Erdős–Anning theorem (\(\mathbb{M}\)): every point set in the Euclidean plane for which all distances are integers must be collinear or finite. Newly promoted Wikipedia Good Article.

Cut-away Henneberg’s surface, Dan Piker. This algebraic minimal surface (the first known non-orientable minimal surface) is self-intersecting, but its structure can be made more obvious by cutting it on two trefoil knots and a figure-eight to eliminate all the self-intersections.

A side note to some recent progress in additive combinatorics (\(\mathbb{M}\)), on “how women are written out of stories of mathematical discovery”: the recent discoveries prove a conjecture formulated by Katalin Marton, after work by Gregory Freiman. Progress on her conjecture had been made by Imre Ruzsa, giving proper credit to Marton. So of course her conjecture became known as the Freiman–Ruzsa conjecture. The new research preprint by Gowers, Green, Manners, and Tao tries to correct the record by its title, “On a conjecture of Marton”. See also Tao’s blog posts on the solution of the conjecture and its formalization.

AI-generated visual anagrams (\(\mathbb{M}\)), described by Robin Houston as “cool, but not going to put Scott Kim out of a job”.

Nim and other oriented graph games (\(\mathbb{M}\), via). A one-hour video from 1966 in which Andrew Gleason lectures on combinatorial game theory: impartial finite games, their equivalence and nim-sums, and the Sprague-Grundy theorem according to which all such games are equivalent to nim positions. If you’ve read On Numbers and Games or Winning Ways you probably know all this, but it’s a nice clear presentation and interesting to see how well-understood this was already at this time, long before those books popularized the subject.

The next biennial Conference on Fun With Algorithms, to be held in Sardinia in June 2024 (\(\mathbb{M}\)), is also hosting a new “Computer Science Salon des Refusés”. The salon talk proposals should concern work that is at least eight years old, was submitted to a major venue, rejected, published elsewhere, and became highly influential. Get your submissions ready for both tracks, with submission deadline February 20.

The Borsuk-Ulam explorer (\(\mathbb{M}\)) finds pairs of antipodal points on the Earth with the same temperature and pressure. By the Borsuk–Ulam theorem, at least one such pair always exists.

I gave up on Dropbox a year or two back when they tried to take over my desktop, but I have plenty of contacts who still use it. If you do, and you keep anything on it that you don’t want OpenAI (and hence the world) to see, there are some steps you need to take to opt out of OpenAI sharing (\(\mathbb{M}\)).

Imaginary cubes and their puzzles (\(\mathbb{M}\)), Hideki Tsuiki. These are subsets of a cube that, like a cube, have square orthogonal projections in each axis-aligned plane. Tsuiki markets a physical puzzle in Japan involving packing nine of these shapes (not all the same) into \(2\times 2\times 2\) cube; it is a nice puzzle, but not very difficult and with many solutions.