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A monthly survey of math news
This month's topics:
Nine lawyers and a mathematician
The Iraq Study Group, which delivered its report on October 6, 2006, was characterized four days later on the New York Times Op-Ed page as "nine lawyers and a mathematician." In fact former (1994-1997) Secretary of Defense William Perry, a member of the Group, earned a BS and MS from Stanford and a PhD from Penn State, all in mathematics. The Op-Ed piece author, Andrew Exum, points out how a few of the Group's military suggestions "could spell disaster," and ends by saying: "Given that the panel consulted with five times as many politicians as military officers, such oversights shouldn't come as a surprise. But the mathematician on the panel ... should have known better." Exum led a platoon of Army Rangers in Iraq in 2003.The math of swarms
School of "silversides," Bonaire, N.A., March 2000. Image courtesy Kent Wenger.
"Math explains how group behavior is more than the sum of its parts" is the subtitle to Erica Klarreich's report "The Mind of the Swarm" in the November 25 2006 Science News. Examples of the behavior in question: "a flock of birds swooping through the evening sky, ... a school of fish making a hairpin turn, an ant colony building giant highways, or locusts marching across the plains." One of Klarreich's sources is Iain Couzin (Oxford, Princeton) whose 2002 article (with several co-authors) "Collective Memory and Spatial Sorting in Animal Groups" (J. theor. Biol. 218, 1-11) gave a simple mathematical Ising-type model (the "alignment zone" model) which duplicates some of the exotic behavior of schools of fish. Specifically, for a certain range of parameter values the simulated school would look like a torus, with all the fish swimming around a common axis. Klarreich quotes Couzin: "When we first saw [the doughnut] pattern in the simulations, I thought 'That's really weird!' But then we found in the literature that it really does appear in nature. ... There's nothing in the individual rules that says, 'Go in a circle,' but it happens spontaneously." The key to a general understanding of these collective phenomena, Klarreich tells us, seems to be "a trio of physics and engineering principles-- nonlinearity, positive feedback, and phase transitions."Math propaganda on Bay Area TV
On December 5, 2006, KGO-7, San Francisco's ABC station, aired a 4-minute segment (available online) with the title "Top Mathematicians meet in Berkeley." It's not actually a news story; it's a piece about "the world's largest research center for pure mathematics," the Mathematical Sciences Research Institute up on Grizzly Peak across the Bay. "Top mathematicians meet there to scratch out formulas on a chalk board that nobody understands. We take a look at how some of these complex equations are fueling the drive to discover advancements in everything from science to social issues." Narrated by ABC7 reporter Alan Wang, that "look" is a smoothly edited series of four vignettes. In each one Wang, off-camera, identifies the mathematician we are watching, who then says one or two sentences about a practical application of his work. The camera cuts away to illustrative video, probably from stock: a computer-chip assembly engine, a voting machine, a football touchdown (Cal over UCLA, naturellement), an eye examination. Wang wraps it up: "[MSRI] Director David Eisenbud says these thinkers are the unrecognized heroes who forge the tools that make the inventions. And this is where much of it happens." But there is more. Over footage of i-Pods and cell-phones we hear: "most of these innovators say they have no desire to pursue intellectual rights for their work." Cut to Eisenbud: "It's like discovering a waterfall. It's like discovering something beautiful in nature, and it's just there. It's real. All we're doing is uncovering the truth."
[Scores. Platonic realism: 100. Industrial funding for "pure math," and for MSRI in particular: 100. Intellectual excitement about mathematics itself: 0. Women, minorities and young people in mathematics: 0. With respect to this last point, note that the 5 mathematicians who appear are all white, all male, and all with PhDs before 1987, most of them significantly earlier. -TP]"I was a teenage angle trisector."
So starts Brian Hayes' "Foolproof," a meditation on the nature of mathematical proof, in the January/February 2007 American Scientist. The trisection problem serves as a thread running through the story, which ranges through Hobbes' reading of Euclid (backwards, according to legend, starting at Book I Proposition 47), Socrates' geometric arguments in the Meno (paraphrased "very loosely"), Oliver Byrnes' 1847 color-coded version of Euclid's angle bisection construction (see below), quick glances at the proofs of the four-color theorem and the Kepler conjecture (too electronic), the classification of finite simple groups (too long), Perelman's proof of the Poincaré conjecture (too difficult) and at a couple of problems where one can easily be misled, just to show how hard it is to sort the false proofs from the real. (Hayes' subtitle: "Mathematical proof is foolproof, it seems, only in the absence of fools.") There is a nice reference to David Bressoud, who "emphasizes that the most important function of proof is not to establish that a proposition is true but to explain why it's true." Finally we sew up the trisection argument with Pierre Laurent Wantzel's 1837 proof (separately translated for us by Hayes) that the problem is, in general, impossible.
Euclid's constructive proof that an angle can be bisected, from Oliver Byrne's color-coded 1847 edition. Image courtesy the University of British Columbia, where the entire text is available online.Mathematician becomes "Genome Sleuth"
On December 12, 2006, the New York Times "Scientist at Work" series featured Nick Patterson, a mathematician. His PhD, from Cambridge, was in finite group theory. Patterson told the Times' Ingfei Chen: "I'm a data guy. What I know about is how to analyze big, complicated data sets." He honed this skill on code-breaking, first for the British, then for the U.S. Department of Defense. After some 20 years as a cryptographer, applying the Hidden Markov Model to "predict the next letter in a sequence of ... text" he turned this skill to predicting the next data point is a series of stock prices, working for the hedge fund managed by mathematician/financier Jim Simons. When he started, according to Chen, the fund was worth $200 million; seven years later, it was up to $4 billion. "Their methods apparently worked." But now the data guy is on to a third career: "Genome Sleuth Nick Patterson" was the caption for his photograph in the Times. And apparently the methods are still working. An article by him and four of his colleagues at the Broad Institute (Cambridge MA) ran in the June 29 2006 Nature. The title: "Genetic evidence for complex speciation of humans and chimpanzees." The team ran a comparison of the human, chimpanzee and related genomes on a much larger scale (by a factor of 800) than had ever been attempted. Chen: "Two strange patterns emerged. Some human DNA regions trace back to a much older common ancestor of humans and chimps than other regions do, with the ages varying by up to four million years. But on the X chromosome, people and chimps share a far younger common ancestor than on other chromosomes. ... the data appeared best explained if the human and chimp lineages split but later began mating again, producing a hybrid that could be a forebear of humans."