Sphere packing in the New Scientist
Jacob Aron's report, published online on August 12, 2014, bears the title "Proof confirmed of 400-year-old fruit-stacking problem." The problem: Johannes Kepler's conjecture that "no packing of congruent balls in Euclidean 3-space has density greater than the face-centered cubic packing" (this quote from the Flyspeck Project completion report, released on August 10). As Aron puts it, "The problem is a puzzle familiar to greengrocers everywhere: what is the best way to stack a collection of spherical objects, such as a display of oranges for sale? In 1611 Kepler suggested that a pyramid arrangement was the most efficient, but couldn't prove it." Thomas Hales (U. of Pittsburgh) "broke the problem down into ... thousands of possible sphere arrangements ... and used software to check them all." He submitted the proof to the Annals of Mathematics in 1998. "But the proof was a 300-page monster that took 12 reviewers four years to check for errors. Even when it was published ... in 2005, the reviewers could say only that they were '99 per cent certain' the proof was correct." This was not enough for Hales. His Flyspeck Project, started in 2003, "used two formal proof software assistants ... both of which are built on a small kernel of logic that has been intensely scrutinised for any errors -- this provides a foundation which ensures the computer can check any series of logical statements to confirm they are true." And now: "On Sunday, the Flyspeck team announced they had finally translated the dense mathematics of Hale's proof into computerised form, and verified that it is indeed correct."
The Flyspeck completion report gives a sample of mathematical text and how it appears translated into machine-digestible logical notation:
`(!V. packing V
==> (?c. !r. &1 <= r
==> &(CARD(V INTER ball(vec 0,r))) <=
pi * r pow 3 / sqrt(&18) + c * r pow 2))`
"In standard mathematical language, this states that for every packing V (which is identified with the set of centers of balls of radius 1), there exists a constant c controlling the error term, such that for every radius r that is at least 1, the number of ball centers inside a spherical container of radius r is at most pi * r^3 / sqrt(18), plus an error term of smaller order. As r tends to infinity, this gives the density bound pi / sqrt(18) = 0.74+, which is the density of the face-centered-cubic packing."
Aron widens the context: "A computer-verified proof of a 400-year-old problem could pave the way for a new era of mathematics, in which machines do the grunt work and leave humans free for deeper thinking."
2014 Fields Medals in the media
The 2014 Fields Medals, awarded in August at the ICM in Seoul, received unusually prominent international coverage principally because, for the first time, one of the four prizes went to a woman: the Iranian mathematician Maryam Mirzakhani (Stanford). The other medalists were Artur Avila (IMPA, CNRS-Jussieu), Manjul Bhargava (Princeton) and Martin Hairer (Warwick).
"Top Math Prize Has Its First Female Winner" ran in the New York Times on August 12, 2014 under Kennth Chang's byline. "An Iranian mathematician is the first woman ever to receive a Fields Medal, often considered to be mathematics' equivalent of the Nobel Prize." Chang quotes an email from Ingrid Daubechies, the ICM President: "All researchers in mathematics will tell you that there is no difference between the math done by a woman or a man, and of course the decision of the Fields Medal committee is based only on the results of each candidate. That said, I bet the vast majority of the mathematicians in the world will be happy that it will no longer be possible to say that 'the Fields Medal has always been awarded only to men.' " Chang briefly describes the research area of each of the four medalists, and then goes on with details about women's status in mathematics. " ... mathematics is still dominated by men, who earn about 70 percent of the doctoral degrees." He tallies the prizes. Abel Prize: "all 14 recipients so far are men", Wolf Prize: "No woman has won the Wolf Prize in Mathematics." He asks Curt McMullen, Mirzakhani's Ph.D. advisor at Harvard, "why it had taken so long for a woman to be recognized." McMullen: "I would prefer to look forward and celebrate this occasion, and see it as a sign of positive trends in society and in science."
"Fields Medal in Mathematics: a woman elected for the first time", by Daniel Larousserie, ran in Le Monde the same day. "A great first on the mathematical planet: a woman earns her place among the four recipients of one of the most prestigeous scientific awards: the Fields Medal, considered as the Nobel Prize in the area." "Maryam Mirzakhani, 37, Iranian and professor at Stanford University in the United States ... shares the honor with the Franco-Brazilian Artur Avila, the Canado-American Manjul Bhargava the Austrian Martin Hairer." Larousserie also quotes the same text from Ingrid Daubechies, but adds: "A sign of the times, this Belgian researcher, professor at Duke University, became in 2010 the first woman to direct the international community of mathematicians." He describes Mirzakhani's work: "She is a specialist in geometry and in the dynamics of slightly bizarre surfaces, like hyperbolic, saddle-shaped, surfaces. One of her major results bears on deformations of surfaces with "handles" (somewhat like soft pretzels) and on the enumeration of the closed curves one can draw on those surfaces. Abstract to say the least, these objects correspond nevertheless to real physical situations, in contemporary theories trying to describe the infinitely small by unifying gravity and quantum mechanics." The article contains information about Avila in the context of French mathematics, and the news that when his medal and Grothendieck's are counted as French, "France becomes the most often rewarded nation with twelve medals, tied with the United States." [My translations -TP].
Indian-origin wizard wins 'Nobel Prize' of Mathematics was the Times of India's take on the news, contributed by Chidanand Rajghatta from Washington, on August 13. "Mathematicians of Indian- and Iranian-origin are among the four winners of the 2014 Fields Medal, widely considered the Nobel Prize for maths that has been broadly dominated by white males since it was instituted in 1936. The award going to Princeton University's Manjul Bhargava, a Canadian-American maths wizard was no surprise; although he is the first person of Indian origin, he was the hot favorite in pre-award polls among peers." There is more material on the other medalists ("The sensational co-winner is Maryam Mirzakhani ...", with another quote from Ingrid Daubechies) and on the institution of the Fields Medals themselves; then some nice details about Bhargava. "Apparently, his peers pretty much expected it. Which is not surprising for someone who became a tenured full professor within two years of finishing graduate school, an Ivy League record, and the second youngest full professor in Princeton's history. That's not all. Before you think all he does is crunch numbers, Bhargava is also an accomplished tabla player (tutored by Zakir Hussain) and has the number on Sanskrit ... . He sees close links between his three loves noting how beats of tabla and rhythms of Sanskrit poetry are highly mathematical. Such recognition came to him early. In past interviews, he has often recounted how in Grade 3, he became curious about how many oranges it takes to make a pyramid. Just as well his mathematician mother and chemist father were well-to-do: they indulged him with oranges till he figured out the answer, which was not long coming. Now he's at the pinnacle of his calling."
The Wiener Zeitung (August 13, 2014) ran side-by-side images of Mirzakhani and Hairer with the caption: "The first woman and the first Austrian with Fields Medals." The accompanying Austria Presse Agentur piece is mostly about Hairer. "If you ask Martin Hairer about his profession, he immediately corrects a misapprehension: 'Mathematicians don't calculate' - it's more about proof. And now this Austrian mathematician, who works at Warwick University in Britain, has proved that he belongs at the top of the world." We learn that his degree from the University of Geneva was in physics, but that he switched to mathematics in search of more permanent results. "If I prove a mathematical theorem, then it stays true. A physical theory, on the other hand, can be completely false after ten years." [my translations -TP]
A sobering note from NPR. How Does Winning Math's Fields Medal Affect Productivity?, aired in the "Morning Edition" on August 18, 2014. Shankar Vedantam is interviewed the topic by the host, David Greene. Vedantam: "The [new] research asks what's the effect of winning a big prize at an early age? How does it affect your future productivity? A couple of economists, Kirk Doran at Notre Dame and George Borjas at Harvard decided to look at winners of the Fields Medal to answer this question. ... What Doran and Borjas realized is that they could compare the productivity of people who won the Fields Medal against the productivity of those 37-year-old people who didn't win the Fields Medal. And Doran told me that when he did this he found something really striking." [Archived recording] Doran: "People who win the Fields Medal produce many fewer papers after winning than one would have guessed from the previous trends in their output. Or from the trends of a very similar control group."
"Doran and Borjas analyzed why this was happening and what they find is that mathematicians who win this big prize start trying their hand at unrelated fields in mathematics. It as though before the prize these mathematicians are really single-minded about their specific area of study but after they win this huge accolade they say, let me go try do something else. And when you get into a new field there's this learning curve, so it's not surprising that your productivity in this new discipline would start to fall." [Archived recording] Doran: " I think it's not what John Charles Fields expected. He expected that in giving people a prize to honor and extol their previous work, that would encourage them to do more of the same work. What we find is that the opposite takes place." Vedantam unwinds the paradox: "The question in my mind really is how you think about genius. Are they people who are so smart that you can just take them from one field and plunk them in another field and they're going to continue to do spectacular work? In which case, what's happening with the Fields Medal is just great because you're taking these smart minds and putting them in new disciplines. On the other hand if you think of genius as really being domain specific, people are really good at doing this one narrow thing, then taking them away from that narrow thing into other fields, you would have to argue, is counterproductive."
Putting more math into common discourse
Last May 24, the Slate web magazine posted an installment of Mike Pesca's daily news and culture podcast "The Gist," that includes an interview with Jordan Ellenberg (starts at about 8:35). Ellenberg recently published "How Not To Be Wrong." After some initial badinage about people's seeming pride in their mathematical ineptitude, about the Laffer curve ("you can explain it to a Congressman in six minutes and he can talk about it for six months") and about some faulty applications of linear regression, Pesca asks Ellenberg what he hopes to achieve with his book. "I'd like to see some of the deep ideas of math, ideas which are part of our cultural heritage, that people have hammered out over many centuries of hard work, I'd like to make a world where it is safe and accepted and customary to talk in that language. Not in a book that's declared as a math book, not in a magazine that's declared as a science magazine, but in a regular editorial in the New York Times, or a regular article in a magazine, or a regular piece on Slate. In the same way that it's no longer considered weird or technical to talk about the incentives people face: that language has moved out of the academy and people have seen how useful it is. I think there's space for more of mathematics to get out there as well."
Stony Brook University
tony at math.sunysb.edu