| The Poincaré Conjecture. The New York Times, in their Science section for April 15, 2003, ran a piece by Sara Robinson entitled "Celebrated Math Problem Solved, Russian Reports." The problem is the 100-year-old Poincaré Conjecture; the Russian is Grigory Perelman of the Steklov Institute in St. Petersburg. As Robinson describes it, Perelman is claiming even more: a proof of a conjecture due to William Thurston, that "three-dimensional manifolds are composed of ... homogeneous pieces that can be put together only in prescribed ways." The Poincaré Conjecture, about the possible topology of a three-dimensional manifold in which every loop can be shrunk to a point, follows because now it would be known what possible geometric structure such a manifold could have. Robinson comments briefly on the method of proof. There is a natural way for the geometry of a manifold to evolve in time: this is the Ricci flow, "an averaging process used to smooth out the bumps of a manifold and make it look more uniform." Its application to Thurston's geometrization conjecture was pioneered by Richard Hamilton (now at Columbia) and carried out in full, we hope, by Perelman. Robinson remarks on the interesting parallels between Perelman's odyssey and that of Andrew Wiles (who recently proved Fermat's Theorem) and also on Perelman's eligibility, if his proof sustains scrutiny, for one of the Clay Mathematical Institute's million-dollar prizes.|
The Times picked up the story again in the "Week in Review" section on Sunday, April 20: "A Mathematician's World of Doughnuts and Spheres," by George Johnson.
"Poincaré proof adds up to potential payday" is the tack Nature chose to follow in a News in Brief item (April 24, 2003). The math got mangled: "Closed two-dimensional surfaces without holes can be transformed onto the surface of a sphere, and Henri Poincaré suggested that similar surfaces with higher dimensions should also transform back to spheres." But they did give a link to one of Perelman's preprints.
Coot Math. Nature for April 3, 2003 ran an article by Bruce Lyon (UC Santa Cruz) with the title "Egg recognition and counting reduce costs of avian conspecific brood parasitism." It turns out that "brood parasitism within species is ... widespread in birds." One Blue Wren will lay her eggs in the nest of another Blue Wren. A defense against this parasitism only exists in some species, one of which, the American Coot Fulica americana, is featured in this article. Coots exploit the difference in color and spotting patterns between eggs of one female and another. The odd eggs are banished to the periphery of the nest. Another more subtle adjustment is in overriding the upper bound on clutch size. Coots use "an external cue, such as the number or surface area of eggs in the nest" to tell them when to stop laying. But some female coots whose nests had been parasitized counted only their eggs and ignored parasitic eggs when making their clutch-size decisions, providing "a convincing, rare example of counting in a wild animal." Lyon's article is highlighted in a "News and Views" piece, by Malte Andersson, in the same issue of Nature.
Wolfram at the Cooper-Hewitt. The National Design Triennial is on show (until January 25) at the Cooper-Hewitt National Design Museum (a branch of the Smithsonian) in New York. And part of the exhibit is a piece ("Totalistic Cellular Automata") by Stephen Wolfram, the man who brought you Mathematica and A New Kind of Science. "TCA" is a 4 by 8 grid of rectangles, 19 of which are occupied by the intricate, triangular patterns that trace the first hundred or so steps in the life of a cellular automaton with inital seed 1. The color schemes vary from rectangle to rectangle, and accentuate the resemblance between these images and the surfaces of sumptuous oriental carpets. Herbert Muschamp, reviewing the show in the April 25 2003 New York Times gave Wolfram his "BEST GRAPHIC DESIGN BY AN EMINENT SCIENTIST" citation, with the comment "A picture must be worth more than 1,000 pages of brain-teasing prose."
"The Superformula" Nature Science Update ran a piece on April 3, 2002 by John Whitfield: "Maths gets into shape." Whitfield was reporting on an article by Johan Gielis (Nijmegen) in the March 2003 American Journal of Botany in which Gielis proposes his superformula ("A generic geometric transformation that unifies a wide range of natural and abstract shapes"). The superformula, in slightly different notation, is the following polar equation:
which, for various values of the parameters A, B, M, p, q, n and various choices of the function f(φ) does in fact give a wide variety of interesting shapes. Whether this mathematical unity is of any botanical significance is harder to see. Whitfield quotes Ian Stewart (Warwick): "I'm not convinced ... , but it might turn out to be profound if it could be related to how things grow" as is the case, for example, with D'Arcy Thompson's explanation of the logarithmic spiral in mollusk shells. Gielis' position, as quoted by Whitfield: "Description always precedes ideas about the real connection between maths and nature." A botanical Kepler awaiting his Newton. Meanwhile, Gielis has applied for a patent on his discovery: Methods and devices for synthesizing and analyzing patterns using a novel mathematical operator, USPTO patent application No. 60/133,279 (1999).
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