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+

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.

The twin primes conjecture states that there are infinitely numbers $n$ such that $n$ and $n+2$ are prime. David Roberts on Google+ refers to this as “to put it mildly, EXTREMELY HARD to prove”. An equivalent statement is that there are infinitely many primes $p$ and $q$ such that $|p-q|<3$, and this, says David, allows the production of weaker conjectures:

Conjecture($N$): there are infinitely many primes $p$ and $q$ such that $|p-q|<N$.

Anyway, according to a blog post by Peter Woit of Columbia University, who apparently got an email announcing it, there is a seminar today at Harvard at 3pm local time, in which “Yitang Zhang will present new results on ‘Bounded gaps between primes’”. Peter says that Zhang claims a proof of Conjecture($70,\!000,\!000$), that is: there are infinitely many primes $p$ and $q$ such that $|p-q|<70,\!000,\!000$.

We await further news.

Source: Peter Woit’s post announcing the seminar: Number Theory News.

via David Roberts on Google+

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.

Travelling waves

The idea of travelling waves came to the fore in the 1950s – Russian scientist Boris Beloüsov discovered an oscillating reaction (warning, may hurt eyes) and reputedly submitted an article on it to a journal, only for it to be rejected as thermodynamically impossible. Beloüsov did what any of us would have done and threw his toys out of the pram, completely withdrawing from science.

Luckily, the discovery was picked up by Zhabotinsky, who persevered with it, or else we’d have neither that brilliant technicolour nor a good model for how crime spreads.

Reaction-diffusion-advection

Rodríguez’s model, a reaction-diffusion-advection system, is made of three elements. Anyone care to guess what they are? Well done. A reaction term, roughly meaning that once crime happens somewhere it tends to carry on happening there; a diffusion term, which damps down crime, and an advection term, which tends to spread crime around. More specifically:

\[ s_t = \Delta s - s + s_0(x) + (\alpha(x,t))u(x,y), \]
where $s(x,t)$ is the propensity towards crime, $(\alpha(x,y))$ is the total payoff (ill-gotten gains minus costs) of committing crime, $s_0$ is the base propensity and $u(x,y)$ is the moving average of crime.

One of the model’s innovations is to allow for the community attitude towards crime to vary. When everyone is perfectly happy to commit profitable crime ($s_0 > 0$ and $\alpha > 0$), it turns out (to everyone’s surprise) that crime tends to spread. When the population is neutral, things aren’t much better – when $s_0 = 0$ and crime is profitable enough, a hot-spot can still form; when the local populace is ready with the perfectly-legal pitchforks ($s_0 < 0$), though, you have two distinct spatial equilibria: you still get occasional waves of crime – but they’re mixed with waves of no crime at all.

Defined: the gap

Moreover, if you can surround the crime zone with a big enough ‘gap’ where no crime takes place – for example, by increasing police presence there, or bulldozing large sections of the city¹ – you can prevent the wave spreading. It’s a similar idea to containing forest fires: if you surround a fire with fireproof stuff, eventually it burns out.

Rodríguez says that simply increasing police numbers isn’t the solution to stopping crime: “It’s not enough to crack down on crime without cracking down on the attitude of the community.”

Further reading

• Stanford News summary
• Dr Rodríguez’s slides for a talk on the topic
• Link to submitted paper.

Thanks to George Stagg for bringing this to our attention.

1. I don’t think Dr Rodríguez is advocating this, although it would probably work.

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.¹ 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.

Geoffrey B. West is a physicist. Geoffrey West, like any self-respecting physicist, thought he would take a look outside his specialisation and find out what governs how living things work. How about size? Perhaps some animal traits follow a nice simple power law function of size. Big things like redwoods and elephants seem to live for a long time and move slowly, while little things like hamsters and algae flash in and out of existence like living’s for chumps. Previously, Geoffrey West had thought about cities, and concluded that, funny coincidence, they too are influenced by power laws².

So he did what a scientist has to do and came up with a theory and tested it against some data and wrote a paper and then a book and soon it was time to go to the media and chat about³ how remarkable it is that this really simple mathematical model fits so many things, apart from all the exceptions.

The problem with power laws is, drawing a log-log plot is a terrible way to spot one, and spotting one is otherwise quite tricky. As Cosma Shalizi explained in his post, ”So you think you have a power law – well isn’t that special?“, there’s a lot to bear in mind when looking for power laws, and we’re pretty terrible as a species at thinking about exponential curves. So Geoffrey West’s ideas are not particularly widely accepted, and many criticisms focus on the lack of appropriate statistical tests of their power.

An NPR blogger wrote a blog post titled “Nature Has A Formula That Tells Us When It’s Time To Die“, joining Geoffrey West’s ideas with Shanghai artist Yunfan Tan’s work and a paper by Marba, Duarte and Agusti from 2007 in a very ooo deep science kinda TED-y/New Scientisty way. This blogger, Robert Krulwich, had previously written another post on basically the same subject which took great pleasure in anthropomorphising things which shouldn’t really be anthropomorphised. The second article repeated lots of claims about power laws but with even less detail than before, while anthropomorphising ‘Nature’ more strongly.

Scientists immediately jammed up the comment section switchboard pointing out all the exceptions to the rule and the lack of any kind of statistic describing the model’s explanatory power. Also, how this log-log graph really doesn’t make a convincing case for anything, especially not the claimed line of best fit:

The relationship between plant mortality rate (D) and the individual mass (M) of plants, from Allometric scaling of plant life history by Marba, Duarte and Agusti (2007)

The NPR journalist did the appropriate thing and deferred blame to Dr West, saying his theories are “controversial”, before pointing out that the paper in Science presenting the idea has been cited “more than 1500 times”⁴, so it “has weight”. As I write, comments seem to have been closed and removed from the NPR piece.

Some good discussion is happening elsewhere, in a MetaFilter thread about the article, where several commenters have raised good points both about criticisms of the power-law model, and about why this kind of thing is worth thinking about and might have some value somewhere in it.

Now let’s do something else and forget this ever happened!

Via Slashdot.

PS: shout-out to Ronan Mehigan, who informed me that the way to get back from the page containing the graph above to its parent paper was to google the paper’s title. Journal sites: a fractal of bad design.

1. Unlike the BBC, we are not funded in a unique way, so we have no public service remit. I think that’s right.
2. Also in his oeuvre is a paper whose abstract claims that “Fractal-like networks effectively endow life with an additional fourth spatial dimension.”
3. Eagle-eyed readers will note that edge.org thinks ‘ten¹⁴‘ is an acceptable way to write numbers. We do not share this opinion.
4. Google makes it just over 2000

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.

via Slashdot

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.

Way back when, before we even launched this site, Katie and I made a video about a paper I’d found called Cryptographic and Physical Zero-Knowledge Proof Systems for Solutions of Sudoku Puzzles. The paper described a scheme for proving you can solve a sudoku puzzle without revealing your solution, using only paper and scissors. We tried it out, and it worked!

I’ve been wanting to do more videos like that, but papers describing new mathematical things to do with household objects are quite rare. I also have the problem that most of my Interesting Esoterica collection goes unread, even though there are quite a few things I’d really like to look at in more detail. So I’ve connected those two thoughts and come up with the ‘recreational maths seminar’.

The idea is this: I pick a paper from my collection that looks interesting, and post it along with a date to get together and discuss it. We all pile in to a Google+ On Air hangout and work through it for an hour or two, and leave as better-educated people, with a YouTube recording of the session for people who couldn’t make it.

I had a small test-run a couple of days ago with Colin Beveridge, where we looked at a very interesting paper titled Twin Towers of Hanoi. Here’s Google+’s recording of the session (I wasn’t particularly paying attention to the fact it was being recorded, so it starts and ends quite abruptly):

I need to pick a time that will suit the most people, and that I can make, so if you like the sound of my idea please put a comment below saying which of the following are convenient for you. I expect each seminar will last between one and two hours.

• Weekdays between 1800 and 2300 GMT
  (1300-1800 EST, 1900-0000 CET, 0300-0800 EDT)
• Weekdays between 0700 and 0900 GMT
  (0200-0400 EST, 0800-1000 CET, 1800-2000 EDT)
• Weekends between 0700 and 1100 GMT
  (0200-0600 EST, 0800-1200 CET, 1800-2200 EDT)
• Weekends between 1600 and 2300 GMT
  (1100-1800 EST, 1700-0000 CET, 0100-0800 EDT)

Here’s a snippet:

Actually, uncongratulations to Prof Rathke, who doesn’t exist, and congratulations instead to Nate Eldredge whose Mathgen program created the paper, along with severe disapprobations to Scientific Research Publishing of “P. O. BOX 54821, Irvine CA”, for apparently not checking that even the title of the paper they received makes sense.

Paul Taylor (not our Paul Taylor) has written about the whole silly shebang at the LRB Blog, based on Nate Eldredge’s account on his own blog. The comments from the “anonymous referee” are hilarious!

At the end of August, Shin Mochizuki released what he claims is a proof of the abc conjecture (link goes to a PDF). Barring someone spotting a huge error, it’s going to take a long time to verify. It’s mainly quiet at the moment, apart from a claimed set of counterexamples to one of Mochizuki’s intermediate theorems posted by Vesselin Dimitrov on MathOverflow, which was quickly shut down because the community there didn’t approve of MO being used to debate the validity of the proof. No doubt there are other niggles being worked out in private as well.

At the start of September, Justin Moore uploaded to the arXiv what he claimed was a proof that Thompson’s group F is amenable. Like Mochizuki’s abc proof, experts thought Moore’s proof was highly credible. We were waiting for my chum Nathan to write about it, since his PhD was all about Thompson’s groups F and V, but it turns out we don’t need to: at the start of this week, Justin retracted his paper because of an error which “appears to be both serious and irreparable”. The amenability of Thompson’s group F has been proven and disproven many times, so I still want Nathan to tell me (and you) all about it.

In lighter news, via Richard Green on Google+, recent uploads to the arXiv show that Goldbach’s conjecture and the Riemann hypothesis are true. I’d love to know how it feels to upload a six-page paper which you know proves something like the Riemann hypothesis. It must be a lovely state of mind. Certainly much better than what people like Moore and Mochizuki must go through, waiting for the first email to arrive telling them they’ve made a terrible mistake and their work is not yet complete.

If I’ve inspired you to have a go yourself, look at Wikipedia’s list of unsolved problems in mathematics and take a crack at one this weekend. Can’t hurt¹ to try!

1. Disclaimer: depending on levels of ability, perseverance and agreement with consensus reality, attempts to solve these problems may well ruin your life

This diagram was constructed by Khalegh Mamakani and Frank Ruskey of the University of Victoria in Canada. Their page about the diagram contains plenty of pretty graphics to accompany the intro to the paper, which explains what they’ve achieved quite clearly.

Paper: A New Rose : The First Simple Symmetric 11-Venn Diagram (arXiv preprint)

Pretty pictures: Ruskey’s page about the construction