It seems that big mathematical advances are like buses – you wait ages for one, and then two come along at once. Also revealed yesterday was a proof of the odd Goldbach conjecture: that all odd numbers greater than 7 can be written as the sum of exactly three odd primes. The proof is contained in Major arcs for Goldbach’s theorem, a paper submitted to the arXiv by Harald Helfgott, who’s a mathematician at the École Normale Supérieure in Paris. This new paper completes the work started in Helfgott’s previous paper, Minor arcs for Golbach’s problem, published last year.
The strong Goldbach conjecture states that every even number can be written as the sum of two primes. This is still unproven, and remains one of the long-standing unproven results in number theory. Sadly, it’s the opinion of Terence Tao, among others, that the method used to prove the weak conjecture probably won’t work on the strong conjecture.
The paper: Major arcs for Goldbach’s theorem by Harald Helfgott
via Terry Tao on Google+
Update 14/05/2013: The seminar was successful: Zhang announced that his proof has already been refereed for the Annals, and everyone seems happy with it.
Hard Maths news now: there’s a rumour going round that Yitang (Tom) Zhang of the University of New Hampshire reckons he can prove that there are infinitely many different pairs of primes at most 70,000,000 apart.
Nothing puts your home insurance premium up like having been burgled in the past – because it means you’re more likely to be burgled again. Stanford researcher Nancy Rodríguez, with colleagues Henri Berestycki (who is first author, for the record) and Lenya Ryzhik, has developed a travelling waves model to explain this phenomenon – and, more importantly, how to stop it.
Crime, according to past research, tends to cluster in particular neighbourhoods – and even individual houses. Once a crime epicentre has been established, criminal activity tends to spread out in a wave pattern, gradually engulfing larger and larger areas.
This is a story about a particularly awful bit of science reporting. It’s taken me a while to sort out what’s going on. First I thought it was bad maths, then I thought it was bad science, and now I think it might be bad journalism. But it might be none, all, some, or a superset of those. I didn’t really want to write this post because crowing about other people’s apparent failures isn’t great, but we took a team decision to cover it because of entertainment reasons. I’m going to make an attempt to tell you what I’ve found, a little bit of what I think, and then we can rejoin Team Positive Mental Attitude when it’s all over.
A paper by Lorenzo Pérez-Rodríguez, Roger Jovani and François Mougeot in Proceedings B, “Fractal geometry of a complex plumage trait reveals bird’s quality“, claims that the measurement of the fractal dimension of a red-legged partridge’s chest plumage is a good indicator of its health.
I know what you’re thinking: another ‘non-mathematicians pick trendy term to describe something rather different’ story, but actually the authors do quite a good job of explaining and justifying their method. I’m convinced!
Plumage patterns are the product of reaction-diffusion systems which probably don’t really produce fractal dimension, but the researchers needed a fairly easy and consistent way of measuring the complexity of patterns. A healthy bird can produce more melanin, which can produce more complicated patterns. For the level of detail needed, the researchers say that the box-counting method of computing fractal dimension is a quick way of measuring the effect they’re looking for.
Paper: Fractal geometry of a complex plumage trait reveals bird’s quality, by Lorenzo Pérez-Rodríguez, Roger Jovani and François Mougeot, in Proceedings of the Royal Society B.
Every year, Princeton University Press gathers together a small anthology of the best writing on mathematics from the past 12 months.
The Best Writing on Mathematics 2012 was released last week. Now that Princeton’s web servers have been dried out after Hurricane Sandy’s visit, I can give you its blurb:
This annual anthology brings together the year’s finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2012 makes available to a wide audience many articles not easily found anywhere else–and you don’t need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today’s hottest mathematical debates. Here Robert Lang explains mathematical aspects of origami foldings; Terence Tao discusses the frequency and distribution of the prime numbers; Timothy Gowers and Mario Livio ponder whether mathematics is invented or discovered; Brian Hayes describes what is special about a ball in five dimensions; Mark Colyvan glosses on the mathematics of dating; and much, much more.
The “much, much more” alluded to above includes our very own Peter Rowlett’s collection of essays “The unplanned impact of mathematics”, which was published in Nature last year. And at only £13.95, just £1.95 more than what Nature is asking for Peter’s article alone, The Best Writing on Mathematics 2012 is a steal.
The Best Writing on Mathematics 2012 at Princeton University Press. $19.95/£13.95 in paperback or ebook.
Would you be interested in taking part in a sort of online video-chat seminar about recreational maths? Then read on!