Linkage

Gil Kalai on recent developments in Roth’s theorem (\(\mathbb{M}\), see also). Salem and Spencer and later Behrend proved in the 1940s that subsets of \([1,n]\) with no triple in arithmetic progression can have nearly linear size, and Klaus Roth proved in 1953 that they must be sublinear. The upper bounds have slowly come down, to \(n/\log^{1+c} n\) in this new result, but they’re still far from Behrend’s \(n/e^{O(\sqrt{\log n})}\) lower bound.

Peter Cameron describes Peter Sarnak’s Hardy Lecture (\(\mathbb{M}\)). It’s on the spectral theory of graphs. If you know about this you probably already know that regular graphs with a big gap between the largest eigenvalue (degree) and the second largest are very good expander graphs. It turns out that 3-regular graphs with gaps elsewhere in their spectrum are also important in the theories of waveguides and fullerenes, and some tight bounds on where those gaps can be are now known.

Optimal angle bounds for quadrilateral meshes (\(\mathbb{M}\)). Christopher J. Bishop meshes any simple polygon (why simple?) with max angle 120° and min angle max(60°, min of the polygon). Nice techniques involving conformal mapping, hyperbolic tessellation, and thick/thin decompositions of hyperbolic convex hulls of ideal sets. Also amusing to see him have to disambiguate my name from David B. A. Epstein’s within a single paragraph.

One-dimensional diagonal cellular automata generate Sierpinski carpets and intricate branching structures (\(\mathbb{M}\), see also). Via the June 27 update to mathpuzzle.com which also has plenty of other neat stuff involving tilings, drawings of symmetric graphs, graceful labeling, rectangle dissection into similar rectangles, etc.

Terry Tao on mathematical notation (\(\mathbb{M}\)), in response to a MathOverflow question about why there’s more than one way to write inner products.

Carmesin’s 3d version of Whitney’s planarity criterion (\(\mathbb{M}\)): a simply-connected 2-dimensional simplicial complex (meeting a technical condition, “locality”) can be topologically embedded into Euclidean space if and only if a certain ternary matroid on its faces has a graphic dual. The proof relies on Perelman’s proof of the Poincaré conjecture! Simply-connected complexes are pretty restrictive but they include e.g. the cone over a graph, which embeds if and only if the graph is planar.

Fast-growing functions revisited (\(\mathbb{M}\)). News of recent developments relating the busy beaver function with Graham’s number, and proofs of some older claims.

Wikipedia, the free online medical encyclopedia anyone can plagiarize: Time to address wiki‑plagiarism (\(\mathbb{M}\), via). In this editorial in Publishing Research Quarterly, Michaël R. Laurent identifies five PubMed-indexed papers that copied content from Wikipedia without crediting it (noting that this is much more prevalent in predatory book and journal publishing), and argues that doing this should be treated as a form of academic misconduct.

Facebook temporarily blocks posts of links to dreamwidth (\(\mathbb{M}\), via). Maybe it was just a mistake? And I guess the decentralization of Mastodon would make doing this to Mastodon posts somewhat harder. But this continued walling-off of the open web is not a good thing.

How much we don’t know about Fibonacci (\(\mathbb{M}\)). Entry F in Joseph Nebus’s 2020 mathematics A-to-Z.

Rational dodecahedron inscribed in unit sphere (\(\mathbb{M}\)). It’s easy to inscribe a dodecahedron in the unit sphere: just use a regular one of the appropriate size. And it’s not hard to construct a dodecahedron combinatorially equivalent to the regular dodecahedron but with integer coordinates. Now Adam Goucher shows how to do both at once, in answer to an old MathOverflow question.

Descartes on Polyhedra (\(\mathbb{M}\), see also). This book is mainly on whether Descartes (circa 1630) knew Euler’s formula \(E-V+F=2\) (before Euler in 1752, but after Maurolico in 1537). It also covers Descartes’ invention of polyhedral figurate numbers beyond the cubes and pyramidal ones known to the Greeks. Descartes’ manuscript has an interesting history: found after his death in a desk, sunk in the Seine, copied by Leibniz, both copies lost, and Leibniz’s copy finally rediscovered in 1860.

Spherical geometry is stranger than hyperbolic (in how it looks from an in-universe viewpoint) (\(\mathbb{M}\), via).