What is the curl of a vector field, really?

The impetus for this post came from David Butler (blog, Twitter, also found on #mtbos, the “math-twitter-blog-o-sphere”), who wanted an explanation of what the curl of a vector field is without recourse to explanations involving physical phenomena, like the electromagnetic field, or fluid flow and little rotating paddles and so on. What he wanted was a description of how the curl is a derivative, that is, the linearisation of something, much like the gradient of a vector field (or indeed an ordinary derivative) tells us about the linearisation of a real-valued function. This led me on an interesting journey in how to turn how I think of the curl (as the 2-form that is the exterior derivative of a 1-form), into something palatable for the students learning it for the first time, without implicitly teaching them new material in order to grasp the explanation. In the end, it didn’t quite turn out like I expected, and the answer was kind-of hidden in existing theory of continuum mechanics; I grew to suspect that what I was trying to do had been done before, but didn’t know where to look. Click through to get what I feel is a comprehensive answer, complete with pretty pictures! Continue reading “What is the curl of a vector field, really?”

Nalini Joshi nominated for VP of the IMU!

 

Edit (31 July): And she is in! Together with Loyiso Nongxa of South Africa, Nalini will be Vice-President for 2019-2022 (The IMU has a tradition of having more than one VP, in case that looks odd). The President will be Carlos Kenig.

Elsevier finally pull the pin on Germany

Elsevier last week stopped thousands of scientists in Germany from reading its recent journal articles, as a row escalates over the cost of a nationwide open-access agreement.

The move comes just two weeks after researchers in Sweden lost access to the most recent Elsevier research papers, when negotiations on its contract broke down over the same issue.

Dutch publishing giant cuts off researchers in Germany and Sweden, Nature News, 19 July 2018, doi:10.1038/d41586-018-05754-1

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 4 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.

Added 23 August: Another interesting profile/interview, in the Campus Morning Mail.

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. 

Screen Shot 2018-07-05 at 2.17.04 pm

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.

Continue reading “Narrow fjords, Whitney jets and smooth functions”