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A monthly survey of math news
15 milliseconds of fame. Four (count 'em) mathematicians make exceedingly brief appearances in the July 28 2003 New Yorker. Bernard Morin and Jacques Hadamard surface in Oliver Sachs' meditation on visual thinking in the blind: "The Mind's Eye." Morin because he is blind and a master of spatial imagination. "With the reallocation of the visual cortex to touch and other senses, these can take on a hyperacuity that perhaps no sighted person can imagine." Hadamard (elsewhere in the article) because in "The Mathematician's Mind" he published Einstein's mental self-analysis: "The physical entities which seem to serve as elements in thought are ... more or less clear images which can be 'voluntarily' reproduced and combined." The names of Eugenio Calabi and Shing-Tung Yau flicker momentarily in Woody Allen's "Strung Out," a meditation on string theory and its repercussions in everyday life. "'My pleasure,' she said, smiling coquettishly and curling up into a Calabi-Yau shape. I could feel my coupling costant invade her weak field ..."
Women at the EDGE. "Safety in Numbers" is a piece by Robin Wilson in the July 18 2003 Chronicle of Higher Education. Wilson does the numbers: "In 2002, 42 percent of the undergraduate mathematics majors in the country were women ... in 2000 only 17 percent of those tenured in math at four-year institutions were women." What happens in between? Wilson surveys the programs set up to better prepare women to face the rigors of life as a math graduate student, focusing mainly on EDGE (Enhancing Diversity in Graduate Education), a four-week summer "boot camp" sponsored by the NSF and the Mellon Foundation. According to Rhonda J. Hughes (Bryn Mawr) one of the EDGE co-founders, it is special in its concentration on fundamental material and on the construction of proofs that "hold together and will stand up to careful scrutiny at the graduate level." The basic principles of all the programs, according to Lenore Blum: "Getting a critical mass of girls or women together to do math, making math a positive experience, and having networking and mentorship." Blum points to the success of Title IX, in sports, and suggests that the same approach will work for math: "Nothing works like getting them out there ..." Wilson quotes one small questioning voice: EDGE participant Naomi Utgoff, on her way from Brandeis to grad school at Penn, who wonders whether programs like EDGE are "kind of unfair" to men.
Death and the heptagon. On the front page of the July 29 2003 Wall Street Journal is a story by Peter Landers with the headline: "Dying Mathematician Spends Last Days on Area of Polygon." The mathematician is David Robbins (CCR, Princeton), recently diagnosed with pancreatic cancer. His doctors tell him he has two years to live at the most, and maybe much less. How does he choose to spend the short time he has left? Robbins wants to find the formula that gives the area of a cyclic heptagon (all vertices on a circle) in terms of the lengths of its sides. Landers leads us through the history of the problem. For a triangle, automatically cyclic, the area is given in terms of the sides a, b, c and the semi-perimeter s = (a+b+c)/2 by the somewhat mysterious Heron's Formula: the area is the square root of the product s(s-a)(s-b)(s-c). The cyclic quadrilateral was worked out by Bhramagupta around 650 A.D. (formula not given). Möbius himself took a stab at the pentagon in 1828, but did not get the answer. That was Robbins' achievement: he found the formula, and that for the hexagon, in 1994. But Robbins is much better known for his work on the "alternating-sign matrix conjecture." Landers takes a stab at explaining the difference between "recreational" mathematics (like the heptagon) and more serious stuff like the matrix conjecture, but he does not get it. (The difference is not in the math, but what we make of it. See the next item.)
A heap of trouble. The July 3 2003 issue of Nature has a news feature by George Szpiro entitled "Does the proof stack up?" The topic is the fate of the research paper describing Thomas Hales' five year old proof of the Kepler Conjecture: the optimal way to pack equal spheres is the face-centered cubic arrangement used by grocers to stack oranges. The proof was unusual in that, after "reducing the infinite number of possible stacking arrangements to 5,000 contenders," it relied on a compter program to calculate the density of each arrangement; thereby verifying that face-centered cubic was the densest. Nevertheless Robert MacPherson (IAS, Princeton) asked Hales and his graduate student collaborator Sam Ferguson to submit their manuscript to the Annals of Mathematics. Understanding the complexity of the project, he named a team of twelve referees. But the referees have given up. Checking all three gigabites of code, inputs and outputs turned out to be more than twelve humans could handle. So the paper is to be published with "a cautionary note ... stating that proofs of this type ... may be impossible to review in full." Hales is unhappy and has started a project to use computers to check every line of his proof; he estimates 20 person-years of work to carry it through.
"Please Pile on the Problems? This Has to Be Math Camp" is the title of Michael Winerip's On Education column in the July 30 2003 New York Times. Winerip tells about Max Warshauer and the summer camp he runs for top high school students each summer at Southwest Texas State University. Warshauer recuits "math majors from M.I.T., Stanford and Harvard" as counselors, and keeps an eye out for students from small towns like Brady, Tex. (population 6,000) and Sulphur Springs ("my last two high school science teachers have made it a point to let the class know that they do not believe in the theory of evolution"), and from homes like Shamika's where "Money was always tight. ... So I'd shy away from expensive things like ballet lessons. Math was something I could do in my room." Shamika is now on her way to Stanford.