Linkage

How fascination with a knight’s tour puzzle eventually led June Huh to a Fields Medal (\(\mathbb{M}\)).

Why natural logarithms are abbreviated ln and not nl (\(\mathbb{M}\)). I found an early example of this notation in an 1813 book by Mauricius de Prasse, predating the 1893 examples of Cajory and the MacTutor History of Mathematics Archive, but I suspect it may be even older.

Timothy Chow on the NF consistency proof and Lean (\(\mathbb{M}\)), relating to the recent verification in Lean of a proof by Randall Holmes that NF is consistent (assuming “some fragment of ZFC”). Chow argues out that one use of proof assistants seen here, to break a logjam where a claimed proof cannot be published because no willing referees can be found, “convincingly demonstrates the value of proof assistants in the process of producing new mathematical knowledge”.

Fulkerson 100: A Workshop in Celebration of Ray Fulkerson’s 100th birthday, July 17–19 at the University of Waterloo (\(\mathbb{M}\)). And if that conflicts with something else (as it does with 8OSME for me and with CCCG for the Canadian computational geometers), there’s an alternative: The D.R. Fulkerson Centennial Celebration, September 20–21 at Cornell University.

Hexagonal Tiling Honeycomb (\(\mathbb{M}\)). A covering of hyperbolic space by horoballs, intersecting in a pattern that forms a hexagonal tiling of each horoball, and its number-theoretic description as a subset of Minkowski space-time.

Japan’s “Wasan” Mathematical Tradition: Surprising Discoveries in an Age of Seclusion (\(\mathbb{M}\)). Includes a description of the discovery by Seki Takakazu of determinants, predating their discovery by Leibniz, and of Bernoulli numbers, predating their discovery by Bernoulli.

My PhD advisor, Zvi Galil, is featured by the Association for Computing Machinery in its “people of ACM” column (\(\mathbb{M}\)).

When faced with the question of “Do you feel this article is of broad interest to mathematicians from diverse fields?”, often used to excuse bias against combinatorics papers, David Wood has a good answer: “at least the same level as all the papers in press at the journal website”.

After seeing several recent student presentations on quantum algorithms (\(\mathbb{M}\)), it occurs to me (as I’m sure it has occurred to others) that we desperately need some way of communicating these concepts at a higher and more intuitive level than slide after slide of tensor algebra equations. If we tried to describe classical algorithms but our language was limited to circuit diagrams, how far would we get?

Gasarch on what counts as “closed form” and what does not (\(\mathbb{M}\)), using the Ordered Bell numbers as an example. He asks, in the form of a Platonic dialogue: is \(\tbinom{n}{i}\) or \(\tfrac{n!}{i!(n-i)!}\) closed form? What, if anything, makes that sort of formula different from the formula \(H(n)\), denoting the \(n\)th ordered Bell number?

Packings of smoothed polygons (\(\mathbb{M}\)). Tom Hales and Koundinya Vajjha prove Mahler’s first conjecture, on the centrally symmetric shape with the lowest possible packing density. This shape is conjectured to be the smoothed octagon, formed by rounding the corners of a regular octagon, but the first conjecture is simply that it is piecewise linear and hyperbolic, like the smoothed octagon, with finitely many pieces.

Reviewer 2 is a LLM (\(\mathbb{M}\)). AI/ML researcher Cynthia Rudin writes from her personal experience that the pervasive use of low-quality reviewers in AI/ML conference peer review has become worse, with some reviewers pawning off their work to the machines, no appeal against rejection caused by a bad review (as has long been usual for conference reviewing), and the possibility that one’s work, supposedly confidential while under review, has gone into the machine’s training set. The use of LLMs by reviewers is forbidden but largely consequence-free.

Graph theorist Reinhard Diestel has a new book (\(\mathbb{M}\), see also): Tangles, A structural approach to artificial intelligence in the empirical sciences. If I understand it correctly, his setup involves a population (of people, states of a physical system, data points in a machine learning data set) and a collection of binary questions that each might answer (opinion poll answers, experiments, classifiers). The “tangles” of the title can be thought of as hypothetical members of the population whose answers to triples of questions are consistent, in a certain sense (Why triples? It’s technical.) Consistent might mean, for instance, that there is a real population member with the same answers to that triple (but not to all the other questions). From this setup one can find a small number of questions that distinguish all tangles from each other (they will answer differently to at least one of the questions) or, if there are no tangles, an explanation for why not in the form of a small number of triples of questions that cannot all be consistently answered. This analysis can then be used both to cluster the population and to provide structure in the system of questions. This is all a broad generalization of methods originally used in a highly technical way as part of the theory of graph minors and their extensions to matroid minors. The task Diestel has set himself is an ambitious one: describe all this readably in a book aimed at non-mathematicians. I’m not really the right reader to tell whether he is successful in this goal, but it’s an interesting attempt.