proudly presents  Carnival of Mathematics Number 234

Welcome to the 234th edition of Carnival of Mathematics!

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Welcome to the 234th edition of Carnival of Mathematics, meta-hosted by the remarkable aperiodical.com.

Custom dictates that I should offer a little celebration of the number 234. Last time I was honoured with a Carnival hosting gig I cheated and looked for an occurrence of my number in the Online Encyclopedia of Integer Sequences. This time I will meta-cheat and look up the 234th entry at OEIS.

A000234 is a sequence that appeared in the original book (N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973). "This sequence gives the number of solutions to the inequality \(\sum_{i=1,2,...} x_i^{2/3} \leq n\) with the constraint that \(1 \leq x_1\leq x_2 \leq x_3\leq \ldots\) is a list of at least \(1\) and no more than \(n\) integers."

There's an example, luckily! The 3rd entry in the sequence is 8 because there are eight ways you can take up to three \(2/3\) powers of integers and get a positive summation \(\leq 3\). One of these eight is \(1^{2/3}+2^{2/3}\approx 2.587\). Just missing out is \(1^{2/3}+3^{2/3}\approx 3.080\).

As social media crisply has it: WTF!? Well, OEIS helpfully reproduces the original source: "Statisical mechanics and partitions into non-integral powers of integers", in which B. K. Agarwala and F. C. Auluck remark that "The thermodynamical approach to the partition theory, apart from its intrinsic interest, draws attention to aspects and generalisations of the partition problem that would, otherwise, perhaps go unnoticed." So there you are: you are innocently playing with a little puzzle that throws up the sequence \(1,3,8,18,37,72,\). You look it up in OEIS, and BOOM! Your puzzle now has to explain what it's doing moonlighting with the energy levels of heavy nuclei.

It's a big scary world out there!

But talking of mathematical physics, first up in November's Carnival postbag is Not Even Wrong's post on "The Impossible Man", the new biography of Roger Penrose by Patchen Barss. Peter Woit's blog has a huge reach and there are already 32 comments on this post which include personal anecdotes, links to Penrose-related physics things and much well-informed debate on the merits of Barss' book.

Book launches aren't usually news stories but the publication of the Penrose book was. More traditionally, Cambridge University announced the news that Hannah Fry was to be its first Professor of the Public Understanding of Mathematics. Since 2007 Cambridge already had a Winton Professor of the Public Understanding of Risk (which was David Spiegelhalter until 2020). The name changed a few years ago and it is now the Harding Professor of Statistics in Public Life. What can you read into such re-badging? Maybe Hannah Fry is destined to occupy a chair of Mathematics in Public Life. She would be equally well-suited, having very much been one of the public faces of mathematics during the last ten years.

Fry's advocacy is badly needed: in other news, the London Mathematical Society's president Jens Marklof made a statement on the website of the Campaign for Mathematical Sciences deploring the axing of mathematics at Birkbeck, University of London. It was during his ten years as an applied mathematician at Birkbeck that Penrose discovered his famous singularity theorem (not refuted by Roy Kerr in 2023 — see the aforementioned comments at Not Even Wrong). More serious for mathematics in public life, Birkbeck is the only university in London dedicated solely to evening teaching. Stuck in a day job unable to nurture a real talent for mathematics? There are many such; I've taught some, at a Birkbeck outpost in Stratford (excellent students, execrable blackboard!) Now their talents look like being stifled.

For actual, forget-the-real-world, mathematical news, I often hear it first at Gil Kalai's blog Combinatorics and More. This month he reported on the resolution of a problem open for 60 years, and for which the solution has been known, but not confirmed, for more than 30. Wait a minute, it's not forget-the-real-world after all, it's gritty reality! Wikipedia has a picture to put you in the picture:
        
It moves at the Wiki page! It has six curve sections but the optimal, as reported by Gil Kalai, has eighteen. And Jineon Baek proves it's optimal. It takes him 120 pages divided into eight chapters with a helpful two-and-a-half page table of symbols. It's on the arxiv (linked from Gil Kalai) which means it is a purported proof. But it's Baek's Phd thesis, basically, and will have been under quite a few microscopes already; I won't personally be hiring a sofa shifter who promises to do better.

Of course the mathematical future increasingly looks mechanised and JOLT ML give us an article which "...sums up some recent progress in machine learning for mathematics, and gives an introduction to formalisation. It also gives an overview of some recently formalised bits of maths research." There have been some formal (computer generated from axioms) proofs of some very deep mathematics in the last couple of years and it's easy to imagine Jineon Baek's work getting this ultimate-microscope once-over. Whether some kind of AlphaProof thing would have beaten Baek to it, if he'd been doing his PhD, say, ten years from now, who knows? But JOLT ML's report will get you up to speed.

Terence Tao's blog is another place to hear the latest on AI in mathematics (his is some of the 'very deep mathematics' referred to above). In fact, his What's New is a mixture of

1. Very deep reflections on how the mathematical world is evolving
2. Very deep new things which I can't even understand the statements of, and sometimes
3. Things whose statements are easy to grasp, even if I'm probably not going to have the stamina to really profit from what follows (not so much TLDR as TMLHWDR, if you get me).

Peter Cameron's blog fits the same three categories as Terence Tao's actually, so maybe it's just a top mathematician thing. His post on Location parameters and twin vertices was a no. 3 and very representative of his hallmark bigger-picture genius for building bridges between on-the-face-of-it unrelated topics. Which strikes me, going back to AI and mathematics, as a kind of genius very far beyond what current machines exhibit.

Of course category theory says you don't have to build bridges between different things if you opt for a level of abstraction where they aren't even different. I am old enough to remember this being called 'abstract nonsense' by pragmatic colleagues, but I know perfectly well that it is the lingua franca of modern algebra. And that those who boast that they don't know anything about aren't so different from those who boast that they are ignorant of mathematics full stop (with the implication that they've got better things to be good at). So offerings like Abuse of Notation's Category Theory Illustrated should definitely not be treated as TMLHWDR, especially when the hard work is so lavishly illustrated. We are told "Note that the chapter itself is old, it's only the last section, which describes associativity using bikes that is new" which I find (those are my italics!) intriguing and enticing.

And lastly but by no means leastly Tim Richardt's blog gives us some amazing animations of stereographic projections of the The Barning-Hall Tree (all Pythagorean triples) and the Stern–Brocot Tree (all rational numbers). Angle geometry builds a bridge here which seems tremendously fruitful (for good measure Richardt links back to a previous post giving an excellent introduction to Stern–Brocot and building yet another bridge — to information entropy). Actually, this wasn't submitted to Carnival using the official dead-simple form, nor did I find it myself; I owe it to Colin Beveridge's excellent weekly round-up Double Maths First Thing (I link here to its residency at Aperiodical but you can sign up from there to get Double Maths delivered to your doorstep with the milk, first thing on Wednesday mornings).

Thanks for reading, clicking through, submitting to future Carnivals! Indeed, submit right here to Carnival 235, to be hosted by John D Cook.