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February 2003

 Reproduced with permission from Nature @2003 Splits at Clay The recent administrative changes at the Clay Mathematics Institute were the subject of a piece ("Resignations rock mathematics institute") by Geoff Brumfiel in the January 23 2003 Nature. It had been widely known in the mathematics community that former AMS President Arthur Jaffe had resigned under pressure from his position as head of the CMI, and that Fields medalists Alain Connes and Edward Witten had left the scientific advisory board in protest. Brumfiel reports on the reactions to the news. Hyman Bass, AMS president: "The Clay has raised our visibility and provided a significant amount of resources to mathematicians. It has become a major part of the field." In fact, according to Brumfiel, the Clay Institute provided about $3 million in grants and awards supporting mathematical research last year. Peter Sarnak: "For pure mathematics, the CMI has been a godsend. If it disappeared, it would be very serious." But it does not seem to be disappearing. Brumfiel quotes Eric Woodbury, the CMI's chief administrator: "Our level of programming and research support will be continued." New board members, including two Fields medalists, are in place and a new president will soon be named. "And the$7 million in prizes won't go away." Ah yes, the prizes. The Clay gained media fame in May 2000 with the offer of million-dollar prizes for the solution of any one of seven mathematical problems.

Gödel in Science. Keith Devlin contributed "Kurt Gödel -- Separating Truth from Proof in Mathematics" to the December 6, 2002 Science. His article, in the "Essays on Science and Society" series, centers on Gödel's Incompleteness Theorem, which Devlin paraphrases: "in any axiomatic mathematical system that is sufficiently rich to do elementary arithmetic, there will be some statements that are true but cannot be proved (from the axioms)." He presents the historical and scientific background and then sketches the proof: "In essence, Gödel took the familiar Liar Paradox and showed how to reproduce it within any axiom system that supported arithmetic." He "took a similar statement, 'This statement is not provable,' and showed how it could be formulated as a mathematical formula within arithmetic." Devlin gives an eagles-eye view of the technical problems involved in this process: "This required, first of all, coding statements as numbers ... . Gödel's next step was to show how the concept of provability could be captured within arithmetic." Then: "If one assumed that the axiom systems were consistent (that is, they did not lead to any internal contradictions), the statement clearly could not be provable (since it declared its own unprovability). Hence it was true--but unprovable." A brisk but satisfying explanation in this context! Devlin concludes by explaining why the Incompleteness Theorem did not bring mathematics research to a halt: "Today ... mathematicians regard it simply as confirming the limitations of what can be achieved with axiom systems."

"Math's Wild and Crazy Guy" is how the January 6 2003 Washington Post describes Maryland's James A. Yorke, the recipient, two weeks before, of the Japan Prize for his work in chaos theory. He shared the prize with Benoit Mandelbrot, "another major mojo in the chaos biz." Peter Carlson, the author of the piece, takes us on a visit to Yorke's lab where we him talk about current projects. The Rat Genome: "We're not the official guys doing it, but we hope our results are better than theirs." An improved computer model for weather forcasting, in collaboration with the National Weather Service. An epidemiological study of AIDS. Yorke shows us a double pendulum: "You see -- the motion gets pretty complicated. ... This is what chaos is. It's predictable in the short run but not in the long run. Chaos is about lack of predictability, basically. Obviously, the spin of the pendulum is determined by physical laws, but it's very hard to predict because very small changes in the spin cause very big changes in the output." And then, of course, chaos intrudes in the lab. There's asbestos work going on, so they have to keep moving the computers from room to room. And Yorke's graduate student's motherboard just fried. The article is available online.

Figure Skating. "Most mathematicians would prefer a double integral to a triple axel" is how Charles Seife starts his piece in the January 31, 2003 Science about the mathematical analysis of the new rules developed for judging international figure-skating competitions. The rules were in response to the brouhaha at last winter's Olympics Pairs Final. "One key change is that the traditional panel of nine judges would be expanded to 14; five of those judges' votes would be randomly discarded. In theory, this would reduce the effectiveness of a corrupt judge or group of judges by raising the specter of their votes' not counting." Elyn Rykken (Muhlenberg College, Allentown PA) and her collaborators reported at the January AMS-MAA Meetings that the method is flawed. "It's especially unfair and capricious for the competition," Seife quotes her as saying. Her team ran a simulation of the Women's Final at those same Games. "Sarah Hughes comes in first about one-quarter of the time, while Elena Slutskaya comes in first three-quarters of the time." An ideal judging method, according to Rykken, would yield an identical outcome for identical sets of judges' scores. In fact the International Skating Union seems to have kept that in mind. Their spokesman Roland Jack told Seife "that the same judges are eliminated throughout the whole skating program to make it as consistent as possible."

Mathematical oncology. "Clinical oncologists and tumor biologists posess virtually no comprehensive model to serve as a framework for understanding, organizing and applying their data." This statement is featured in a box at the top of Robert A. Gatenby and Philip K. Maini's "Concepts" piece in the January 23 2003 Nature. They point out that despite the glut of publication (over 21000 articles on cancer in 2001) oncology has not been pursuing "quantitative methods to consolidate its vast body of data and integrate the rapidly accumulating new information." The explanations they suggest are mostly cultural For example: "... medical schools have generally eliminated mathematics from admission prerequisites ..." They also blame "those of us who apply quantitative methods to cancer" for not having "clearly demonstrated to our biologist friends a dominant theme of modern applied mathematics: that simple underlying mechanisms may yield highly complex observable behaviors." An illustration from Wolframscience.com drives home the point. They end with an apology for mathematical modeling, showing how a verbal schema may be be enriched and strengthened by incorporation into a mechanistic and quantitative model which can handle, through computation, properties such as stochasticity and nonlinearity which cannot be handled by verbal reasoning alone. "As in physics, understanding the complex, nonlinear systems in cancer biolgy will require ongoing, interdisciplinary, interactive research in which mathematical models, informed by extant data and continually revised by new information, guide experimental design and interpretation."

Democracy and despotism are contrasted mathematically, at least for animal societies, in Larissa Conradt and Tim Roper's "Group decision-making in animals" (Nature, January 9 2003). The decisions they analyze are when to stop an activity. For example, the halting problem for deer: how does a herd of deer "decide" when to stop traveling? Conradt and Roper construct an elementary mathematical model and calculate the total inconvenience in two modes: democracy, (the herd stops when the majority of the deer are ready to stop) and despotism (the herd stops when the deer leader stops). Their finding: "We show that under most conditions, the costs to subordinate group members, and to the group as a whole, are considerably higher for despotic than for democratic decisions." There are only two hypotheses: (1) inconvenience increases "linearly with the difference between a member's optimal and the group's realized activity duration." (2) costs of stopping too late or too early are symmetrical. The authors also analyze the unsymmetrical situation: "For example, if stopping too early is twice as costly as stopping too late, the group should stop its activity when two-thirds of its members want to stop." (This is a "modified democratic" mode). They only claim to have quantitative, testable predictions about group decision-making in non-humans, although they do mention that "in many human societies a two-thirds majority rather than a 50% majority is required for decisions that are potentially more costly if taken than if not taken."

-Tony Phillips
Stony Brook