This month's topics:
The Emperor of Math
"There's no such thing as bad publicity," they say, and it certainly seems to be true in the case of Shing-Tung Yau. Yau is one of the foremost American mathematicians, but his eminence was unknown to the general American public until Sylvia Nasar's New Yorker piece (August 28, 2006; on Perelman's proof of the Poincaré conjecture and subsequent events) put his name on the national stage. Her portrayal of Yau was not flattering, but without it it is safe to say he would never have had the full-scale royal treatment he was accorded in the October 17 2006 New York Times. Picture this: the entire top half of the Science section front page is given over to his full-length portrait with, superimposed, "Shing-Ting Yau" and, in huge type, "The Emperor of Math." No question mark, no quotation marks: he has been anointed. Has any mathematician ever been so celebrated in the press? The article itself, by Dennis Overbye, gives a full and balanced portrayal of Yau's remarkable life and career, with memorable quotes from Brian Greene ("He corners equations like a lion after its prey, then he seals all the exits") and Yau himself, on his relation with the Chinese leadership: "If I didn't have the Fields medal, I would be dead to them.""Numbers are Male, said Pythagoras,
and the Idea Persists." That's the title of Margaret Wertheim's piece in the October 3 2006 New York Times Science section, her reaction to the National Academy of Sciences' report Beyond Bias and Barriers: Fulfilling the Potential of Women in Academic Science and Engineering (2006) , released on September 18. The report confirms that many female scientists continue to experience both overt and covert discrimination. Wertheim remarks that "when it comes to the mathematically intensive sciences like physics and astronomy, it is not just bureaucracies that stand in the way. Female physicists, astronomers and mathematicians are up against more than 2,000 years of convention that has long portrayed these fields as inherently male."
Wertheim traces the tradition back to the Pythagoreans, who "associated the mind/spirit side of reality with maleness and the body/matter side with femaleness" so that "thinking about numbers, or doing mathematics, was an inherently masculine task. Mathematics was associated with the gods, and with transcendence from the material world; women, by their nature, were supposedly rooted in this latter, baser realm." She traces how "... this godly-mathematical connection also sat easily with the Catholic tradition of a male-only priesthood" and concludes: "It is not just bureaucratic will that needs to shift; it is the cultural zeitgeist."Next year in Marienbad: chaos
The first 3 moves in a 5 x 6 game of Chomp. A: the initial configuration; the object is not to be forced to select the green cookie. B: after Player 1's first bite. C: after Player 2's first bite. D: after Player 1's second bite. Each bite takes a cookie and all the cookies north and east of it.
Chomp is a 2-dimensional version of Nim, the game popularized in L'année dernière à Marienbad. But while a simple strategy exists for Nim, Chomp is much harder. It is known that there is always a winning strategy for Player 1 but the strategy itself is unknown in general, except for a few special cases like n x n, 2 x n, and n x 2. In "Chaotic Chomp" (Science News Online, July 22, 2006) Ivars Peterson reports on developments in the analysis of the 3 x n case. Chomp dates back to 1974 (in fact, it is equivalent to a game discovered in 1950) but was taken up a few years ago by Doron Zeilberger, a mathematician at Rutgers, who decided it would be "an ideal problem for illustrating the role that computers can play in mathematical research." Zeilberger introduced the notation (x, y, z) to describe the position in 3 x n Chomp which has x columns of 3 cookies, y columns of 2, and z columns of 1, and published in 2000 an algorithm generating for each x an algorithm for playing the game with an arbitrary y and z. He returned to the problem in 2003 with faster algorithms and on the basis of the results speculated "It seems that we have 'chaotic' behavior, but in a vague, yet-to-be-made-precise sense." Peterson focuses on the recent work of Eric Friedman (Computer Science, Cornell) and Adam Landsberg (Physics, Claremont colleges), who have fleshed out this intuition: "By using mathematical tools originally developed for calculating properties of physical systems, Friedman and Landsberg show that the exact location of winning and losing cookies in Chomp varies unpredictably with small changes in the size of the initial array." The figure below uses Zeilberger's notation and shows in blue, for x = 300, the "instant winner" positions (positions from which you can leave your opponent in a losing position with smaller x). The chaotic region is clearly visible. Furthermore, they made the remarkable discovery that Chomp is renormalizable. As Peterson explains it, "the geometry of winning positions for small values of x and winning positions for large values of x is roughly the same, after a suitable change in scale." Specifically, the W600 figure, scaled down by a factor of 2 in each direction, is essentially indistinguishable from the W300 shown here.
Winning positions (blue) for a 3-row Chomp game with 300 height-3 columns. The y and z coordinates refer to the number of height-2 and height-1 columns, respectively. Image courtesy Adam Landsberg.
Zeilberger's papers (excellent reading) are available online1, online2. Friedman and Landsberg's paper is also available online, as a PDF file. For a history of the problem, see Andries Brouwer's page on the game."Less happy + less confidence = good math student"
That was one web wag's reaction to an October 18 2006 story by Ben Feller, the Associated Press education writer, picked up by CNN.com with the headline "Confident students do worse in math; bad news for US." Feller: "The nations with the best scores have the least happy, least confident math students, says a study by the Brookings Institution's Brown Center on Education Policy." Tom Loveless, the author of the Brookings study, analyzed "the happiness factor" using data from an earlier study, the 2003 SS. As Feller explains it, the SS was "a test of fourth-graders and eighth-graders across the globe. Along with answering math questions, students were asked whether they enjoyed math and whether they usually did well in it." The result: "The eighth-grade results reflected a common pattern: The 10 nations whose students enjoyed math the most all scored below average. The bottom 10 nations on the enjoyment scale all excelled." The "bad news" is "The results for the United States hover around the middle of the pack, both in terms of enjoyment and in test scores." But Feller also reports: "Within a given nation, the high-confidence kids did better than their peers." So what is detected here is a pattern in national intellectual traditions and attitudes, rather than a paradoxical quirk in adolescent psychology. Loveless' report is available as a PDF file online.
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