This month's topics:

Andrew Wiles' Abel Prize


The announcement from the Norwegian Academy of Science and Letters went out on March 15, 2016. Among American media, The New York Times reprinted the short Associated Press bulletin without comment. NPR did slightly better on their website, repeating this nice detail from the Academy's text:

  • In 1963, when he was a ten-year-old boy growing up in Cambridge, England, Wiles found a copy of a book on Fermat's Last Theorem in his local library. Wiles recalls that he was intrigued by the problem that he as a young boy could understand, and yet it had remained unsolved for three hundred years. "I knew from that moment that I would never let it go," he said. "I had to solve it."

On Time's website Rishi Iyengar linked to Oxford's release, which features a video of Wiles acknowledging the prize.

Coverage in the British press was considerably more extensive (Sir Andrew is British, developed his proof at Princeton, and is now back at Oxford). Several of their reporters were able to speak with the awardee himself and to relay characteristic remarks.

  • (quoted by Davide Castelvecchi, Nature, March 15) "It was very, very intense. ... Unfortunately as human beings we succeed by trial and error. It's the people who overcome the setbacks who succeed." (Castelvecchi also sketches out the history of the solution, and the link to the Shimura-Taniyama conjecture).
  • (by Ian Sample, The Guardian, March 15) "This problem captivated me. It was the most famous popular problem in mathematics, although I didn't know that at the time. What amazed me was that there were some unsolved problems that someone who was 10 years old could understand and even try. And I tried it throughout my teenage years. When I first went to college I thought I had a proof, but it turned out to be wrong." "There were two or three moments, and one particular moment right at the end when suddenly I understood how to think about the whole thing, and because you've put in all those years of slog, those moments show you the whole vista at once." "The proof didn't just solve the problem, which wouldn't have been so good for mathematics. The methods that solved it opened up a new way of attacking one of the big webs of conjectures of contemporary mathematics called the Langlands Program, which as a grand vision tries to unify different branches of mathematics. Its given us a new way to look at that."
  • (by Jacob Aron, The New Scientist, March 15) "In the years since then I have encountered so many people who told me they have entered mathematics because of the publicity surrounding that, and the idea that you could spend your life on these exciting problems, that I've realised how valuable it actually it is." And on subsequent developments in the field: "I think it has gone better than I could have hoped. There are still lots and lots of challenges, but it has come to be an ever-expanding part of number theory."
  • (by Simon Singh, The Telegraph, March 20) "I had this very rare privilege of being able to pursue in my adult life what had been my childhood dream. I know it's a rare privilege, but if you can tackle something in adult life that means that much to you, then it's more rewarding than anything imaginable."

(Photo by Alain Goriely.)

New patterns in the primes

Robert J. Lemke Oliver and Kannan Soundararajan posted their "Unexpected Biases in the Distribution of Consecutive Primes" on arXiv, March 15, 2016. Even though this is an amazing result that almost anyone can comprehend, it was only picked up in the press (as far as I know) by Jacob Aron in The New Scientist and by David Larousserie in Le Monde. Aron's piece (the print version was titled "'Random' primes pair up on the sly") states the new findings on these terms: "Soundararajan and Lemke Oliver noticed that primes ending in 1 are less likely to be followed by another ending in 1 than primes ending in 3, 7 or 9. That shouldn't happen if primes are truly random -- consecutive primes shouldn't care about their neighbour's final digit. The pair found that in the first hundred million primes, a prime ending in 1 is followed by another ending in 1 just 18.5 per cent of the time. If they were random, you'd expect to see two primes ending in 1 next to each other 25 per cent of the time." He quotes Soundrarajan: "It was very weird. It's like some painting you are very familiar with, and then suddenly you realise there is a figure in the painting you've never seen before." Here are some more statistics from the arXiv posting; these survey the first million prime numbers, and record the number of times a prime congruent to 1 or 2 modulo 3 was followed (as the next prime in the list) by one or the other congruence (aside from 3 itself, which was not counted, these are the only possibilities)

  first prime $\equiv 1$ first prime $\equiv 2$
next prime $\equiv 1$ 215,873 283,957
next prime $\equiv 2$ 283,957 216,213

showing "substantial deviations from the expectation that all four quantities should be roughly 250,000," as the authors put it. They go on: "The purpose of this paper is to develop a heuristic, based on the Hardy-Littlewood prime $k$-tuples conjecture, which explains the biases seen above."

Larousserie spoke with Lemke Oliver, who told him "When I first saw this difference I was in shock, but I was confident because the simulation is easy to do." And that the phenomenon may not have practical applications, even in coding, but "it shows ways to seek other biases at the heart of the prime numbers or in other sets."

Who needs Math? (Cont.)

Andrew Hacker is back on the New York Times Sunday Opinion pages with "The Wrong Way to Teach Math" (February 27, 2016), adapted from his The Math Myth and Other STEM Delusions (published on March 1). He revisits the ideas from his 2012 "Is Algebra Necessary?" on the same pages, as well as his Who Needs Advanced Math? Not Everybody interview in "Education Life" (also in the Times, February 5) and his February 26 interview, "The Case Against Mandating Math for Students" in the Chronicle of Higer Education. Here are some of his points:

  • "Most Americans have taken high school mathematics, including geometry and algebra, yet a national survey found that 82 percent of adults could not compute the cost of a carpet when told its dimensions and square-yard price." "The Organization for Economic Cooperation and Development recently tested adults in 24 countries on basic 'numeracy' skills. ... The United States ended an embarrassing 22nd, behind Estonia and Cyprus."
  • "Is more mathematics the answer? In fact, what's needed is a different kind of proficiency, one that is hardly taught at all."
  • "What citizens do need is to be comfortable reading graphs and charts and adept at calculating simple figures in their heads. Ours has become a quantitative century, and we must master its language. Decimals and ratios are now as crucial as nouns and verbs."
  • "We teach arithmetic quite well in early grades, so that most people can do addition through division. We then send students straight to geometry and algebra, on a sequence ending with calculus. Some thrive throughout this progression, but too many are left behind."

Keith Devlin responds with "Andrew Hacker and the Case for and Against Algebra" in the Huffington Post, March 1, 2016. He gives several examples in the articles and in the book (which he has reviewed elsewhere) that document Hacker's lack of familiarity with what is actually being taught, and a certain lack of basic understanding of mathematics. But this is his main point: "For a variety of reasons, the subject now taught in schools under the name of algebra is a travesty of the powerful way of thinking and problem solving developed in the Muslim world in the 8th and 9th Centuries, called 'algebra' today after the Arabic term al-Jabr. If Hacker had instead used his NYT connection to argue for a major make-over of 'school algebra' (as I think we should call the object of his criticisms), he would have garnered massive support from the pros, including me. As it is, his support came exclusively from those who, like Hacker himself, have no idea what algebra is or how significant it is in today's world."

So Devlin is telling us that Hacker is onto something, but is just calling it by the wrong name: "it is a pity that, because he is so far removed from mathematics as it is actually practiced in today's world, Hacker misses the large target that I am pretty certain he is trying to hit--a target that deserves to be hit. Namely, the degree to which the mathematics taught in many of the nation's schools has drifted away from the real thing used every day by large numbers of people, to the point where much of what is taught is not only of little use, but can do real harm. Kids who are put off math in school will find their life choices significantly narrowed."

Christopher Zeeman, 1925-2016

The great British topologist died last month. Excerpts from two of his obituaries:

By Ian Stewart in The Guardian. "Sir Christopher Zeeman, who has died aged 91, was one of the true greats of British mathematics." "Christopher had a charismatic personality enhanced by an impressive beard. He could be a little intimidating the first time you met him, but he was friendly, generous and tireless in his drive to encourage the growth of new mathematics. He was also unusually active in public engagement: talking to schools, giving public lectures, appearing on the radio."

"In the early 1970s, Christopher came across some provocative ideas of the French mathematician René Thom, on a new way to think about mathematical models of reality. Among Thom's suggestions was a list of 'catastrophes élémentaires'-- a topological classification of sudden changes. This so appealed to Christopher's mathematical inclinations that he changed his research field from topology to what soon became known as catastrophe theory. He invented a 'catastrophe machine', [a model here] in which a circular disc on a pivot was attached to two elastic bands, one with a free end. As that end moved around, the disc would suddenly flip to a drastically different position, a vivid example of discontinuous behaviour resulting from a continuously changing cause. Catastrophe theory, renamed singularity theory, has now established itself as an important technique in many applied areas. One of Christopher's early proposals, a 'clock and wavefront' model demonstrating a key stage in the development of a vertebrate, was recently shown to be correct."

By Andrew Ranicki in The Telegraph. "Sir Christopher Zeeman, the mathematician, who has died aged 91, had a unique combination of mathematical and administrative abilities, allied to an overwhelming personal charm." "Zeeman was a pioneer of communicating the excitement of mathematics to the general public. In 1978 he delivered the Royal Institution Christmas Lectures on 'Mathematics into Pictures', which made him into the first mathematical television star." [Those lectures are all available online and give an excellent idea of the range of Zeeman's interests and of that charm]. Ranicki also mentions catastrophe theory: "Even if he sometimes oversold them, Zeeman's applications of catastrophe theory to the social sciences (prison riots, marital strife, economics, animal behaviour, and so on) caught the public imagination."

Tony Phillips
Stony Brook University
tony at