# Special issue of Geombinatorics on recent progress on the chromatic number of the plane

The recent breakthrough work by de Grey that showed the chromatic number of the plane could not be equal to 4 (and so must be 5, 6 or 7) has been published, along with a few other papers in a special issue of the journal Geombinatorics. There are free copies of all the articles in this subscription journal around the place, so I thought I’d gather links to all of them here.

Apparently, Exoo and Ismailescu managed to rule out the case that the chromatic number is 5 independently and at about the same time as de Grey, but wanted to improve the construction and shrink the graph they used, and so were scooped while they kept working in secret.

# Question on HSM.SE: which one is Benacceraf?

I’m trying to track down a picture of Benacceraf in his younger days, to illustrate a seminar on the history of numbers for undergraduates. Ideally close in time to his famous essay What numbers could not be. I found two photos from Princeton University Philosophy department from the late 60s and 70s respectively, but the people in them are not personally identified. The photos can be seen at this History of Science and Mathematics Stackexchange question, where I invite answers, or let me know in the comments.

# Profile on Geordie Williamson

Maths prodigy comes home to establish \$5 million world-class maths centre

A sample quote:

I’m standing with Williamson in his office at the University of Sydney, on the seventh floor of the Carslaw Building, looking down at the campus’s historic sandstone quadrangle. The Carslaw is renowned by staff and students as being the university’s ugliest building, an antipodean approximation of a KGB regional headquarters. “The standard joke is that the best thing about working in Carslaw is that you don’t have to look at it,” Williamson says. His office is similarly unappealing – essentially a concrete bunker with bad carpet – and yet it has everything a mathematician might need; a whiteboard, marker pens, a computer, and most important of all, a couch. “When I moved in here, I told them ‘I need a big couch!’ ” he says. “As opposed to things like medicine and science, which require specialised equipment – microscopes, X-rays – mathematicians can do most of their work with a pencil and paper. Really, you spend most of your time sitting around talking.”

Fortunately, Williamson is a good talker – jaunty and light, his sentences tripping along before ending with an upward inflection, like a little trampoline kick-out off the final syllable. He’s a little goofy. He smiles a lot; his eyes go wide. You get the sense that inside his head is a banging dinner party where all these brilliant ideas are elbowing one another to get out and roam around. Turning on his computer, he talks me through a slide display about representation theory – his area of expertise – and how it can, via spectral analysis of fundamental frequencies, explain why a whistle sounds different to a violin, and why, consequently, you’d rather listen to a concerto played by violins than a concerto played by whistles. An intriguing-looking textbook lies open on his desk, the pages crammed with cryptic glyphs and a photo of a Mayan pyramid. There’s also a stack of shiny new books. “Our latest publication,” he says, handing me one. I turn it over and read the back cover. “In this book,” it says, “we conjecture that translation functors give an action of the (diagrammatic) Hecke category of the affine Weyl group on the principal block.”

I want to ask: what is a “functor”? Who is Hecke? And why is the word diagrammatic in brackets? But instead, I ask: “Where can we get a sandwich around here?”

The book is on the arXiv, and published in Astérisque.

# Narrow fjords, Whitney jets and smooth functions

I don’t usually like ResearchGate, but someone has recently used it in a way that saved me and my co-author Alexander Schmeding a headache: just over a week ago they used the (subscriber-only) commenting feature to point out that we had a gap in a proof in v3 of our paper Extending Whitney’s extension theorem: nonlinear function spaces. The replacement (v4) has now just gone up on the arXiv, and contains a property of regular closed sets of metric spaces that I have dubbed “no narrow fjords”, whose definition is due to Bierstone.

Thankfully, no results needed to be changed, except the addition of this hypothesis in the general theorems; the sets mentioned as examples and in the application all satisfy the condition.

# Trying to compute Brunerie’s number

In Yet Another Cartesian Cubical Type Theory yacctt Anders Mörtberg gave, as a side comment, an update on progress to compute Brunerie’s number from the term of type nat produced by the constructive proof, in Homotopy Type Theory, that $\pi_4(S^3)= \mathbb{Z}/n\mathbb{Z}$, for some $n$.

Right now Favonia is running the computation with a machine with 1TB of RAM, and it’s been running for … 90 hours or something, and we don’t know yet what will happen. (6:53)

This has to be the most serious computation whose answer we already know should be 2! Forget multi-TB proof certificates, this is the real deal…

# Completing Saks spaces and recovering the Banach space structure of M(A)

In the previous post in the series, I introduced Cooper‘s notion of Saks spaces, which are Banach spaces with an additional, coarser, locally convex topology satisfying some mild conditions on the unit ball. We also saw that any $C^*$-algebra gives rise to a canonical Saks space (here ignoring the algebra structure, which turn up again soon), using the strict topology as the second topology. In this post I will describe the completion of a general Saks space, and show that the underlying Banach space of the algebra $M(A)$ of multipliers (and the strict topology thereon) can be recovered from the Saks space completion.