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This month's topics:

Math project wins Intel

According to Science (March 20, 2009) the 2009 Intel Science Talent Search competition was won by Eric Larson of Eugene, Oregon "for his classification of new fusion categories, a type of algebraic structure with applications in string theory and quantum computation." Definitions can be found in the paper Larson and David Jordan, an MIT graduate student, have posted here on the arXiv. More information about the winner is available in an online article by Megan Crepeau of The Oregonian (March 10, 2009). Among other achievements, "Larson has built a reputation as an accomplished pianist, winning a gold medal four times at the Oregon Junior Bach Festival." [Larson's advisors for this project were Pavel Etingof and David Jordan (MIT), Arkady Vaintrob and Viktor Ostrik (University of Oregon).-TP]

Big earmark for mathematical talent detection

Also in the March 20 2009 Science was a news report by Jeffrey Mervis: "Senate Majority Leader Hands NSF a Gift to Serve the Exceptionally Gifted." Mervis tells us that "buried in the $410 billion federal spending measure enacted last week is a provision that gives NSF$3 million this year 'to establish a mathematical institute devoted to the identification and development of mathematical talent.' The directive, which is backed by Senate Majority Leader Harry Reid (D-NV), is aimed at serving supersmart children whose needs aren't being met in school." The context, as Mervis explains, is an NSF competition "to choose a new batch of university-based mathematical research institutes". One of the contenders (which would be located in Reno) has the non-standard mission of focussing "on prospective mathematicians ... rather than the academic mathematicians--faculty members, postdocs, and graduate students-- who populate the existing centers;" the PIs are experts in working with "profoundly gifted" children. Mervis ends by remarking that, whatever happens to the proposal, "it's a good bet that the needs of this special student population have moved up on the list of NSF priorities."

"Explosive Percolation"

sounds like a terrible way to start the day, but detecting it in random networks answers an outstanding question in probability theory. The work, reported in Science (March 13, 2009) is due to Dimitris Achlioptas (Santa Cruz), Raissa D'Souza (Davis) and Joel Spencer (NYU). The domain of inquiry here is the way the size C of the largest connected set evolves as more and more edges are added at random to an initial set of disjoint vertices. With no other conditions, this is "the classic Erdös-Rényi (ER) model." The "percolation transition" is the emergence of a giant component.

The proportion C/n of vertices in the largest component as a function of the ratio r of edges to vertices in the Erdös-Rényi (ER) model, in a bounded size model (BF = Bohman-Frieze) and in the product model (PR). Image courtesy of Raissa D'Souza.

In ER networks with n vertices it is known that when rn edges are added, with r < 1/2, "the largest component remains miniscule." In contrast, just after r > 1/2, C starts growing as roughly (4r-2)n. This is the black curve in the figure, and is an example of a continuous transition. A more general model takes two random links e1, e2 at each step and uses a rule to select which one of them to add. The rule is bounded-size "if its decision depends only on the sizes of the components containing the four end points of {e1, e2} and, moreover, it treats all sizes greater than some (rule-specific) constant K identically." The Bohman-Frieze rule (blue curve) is one of these; it is believed that all such rules have continuous transitions, although "a fully rigorous argument has remained elusive."

The authors "provide conclusive numerical evidence that, in contrast, unbounded-size rules can give rise to discontinuous percolation transitions." The example they study is the product rule (PR; red curve) where the edge selected maximizes the product of the numbers of vertices in the components it joins. Notice that the critical point is displaced but that, more significantly, "the fraction of vertices in the largest component jumps from being a vanishing fraction of all vertices to a majority of them 'instantaneously.'"

Mirrors that don't reverse, and others

The Misner-Thorne-Wheeler "Gravitation" in the flesh and reflected in a non-reversing mirror. Photograph courtesy of Andrew Hicks.

"Mathematician creates impossible, rule-bending mirrors" is the headline in the gadget blog Gizmodo (February 25, 2009); "At Drexel, he designs amazing mirrors" is the more subdued title from the January 12 Philadelphia Inquirer. The New Scientist published a gallery of his mirror images on February 23. Andrew Hicks has captured the media's attention with his unusual mirrors, and he claims it's all done with mathematics. In his research article on the subject ("Designing a mirror to realize a given projection," J. Opt. Soc. Am. A 22 323-330) he describes how the desired relation between object and image determines a vector field in space; if that vector field is the gradient of a function F, then a locus F = constant gives a suitable reflecting surface. Otherwise he gives a method for a best gradient approximation.

"The genius who came in from the cold"

 Misha Gromov. Photo: Gérard Uferas. That's how Stéphane Foucart, writing in Le Monde (online edition, March 26, 2009), characterizes Mikhail Gromov. His winning of the 2009 Abel Prize had just been announced. Foucart interviewed some of Misha's peers for his profile of the renowned geometer. Jean-Pierre Bourguignon describes him as "a conceptualizer, someone who in just one article will string out a hundred theorems which the mathematical community will take ten or fifteen years to absorb." Marcel Berger attributes to Einstein the statement that one only gets to have one or two ideas in one's life, and goes on to say "Misha Gromov has had five or six." Foucart fills in some background by quoting from an interview Misha gave to the physicist Georges Ripka, published in "Vivre savant sous le communisme" by Editions Belin in 2002. "I had wanted to emigrate [from the Soviet Union] ever since I was 14." Bourguignon explains how he managed it, leaving the university in the early 1970s, cutting off all contact with the mathematical community and stopping all written mathematical work. The idea was to become obscure enough to be allowed, as a Jew, to emigrate to Israel. This happened in 1974, but Gromov changed route in Rome and came to Stony Brook instead. Bourguignon: "Those were extraordinary years. He was so to speak 'debriefed' by the Long Island mathematicians, and was able to spill out all that he had kept in his head and had not been able to write down during his years of 'mathematical clandestinity.' We were astounded."

In a companion article ("From Poincaré to Gromov, a French tradition") Foucart explains how the French educational system, and especially the Écoles normales supérieures, funnel the best students towards universities and research. And Gromov himself gives an analysis of the difference between French and American mathematics (quoted from Ripka's book): "In America, many mathematicians quickly become limited because they are constrained by having to publish and to get grants. If they are not expert in a field, they don't get them. That makes them intellectually very narrow, both in what they know and in their attitude towards other fields. (...) In France things go much better because of the education inherited from Bourbaki, who developed an inter-mathematical curriculum. A French mathematician is much better educated in mathematics than an American one." [My translations -TP.]

Tony Phillips
Stony Brook University
tony at math.sunysb.edu