Cosmic geometries The cover art from *Science News*, November 17, 2007. Design by Anders Sandberg (Future of Humanity Institute, Oxford), used with permission. This elegant image means to illustrate "the link between laws of physics as they are perceived in universes with different geometries, even different numbers of dimensions" (from the caption in *Science News Online*). The accompanying article, by David Castelvecchi, sketches some recent developments related to Juan Maldacena's 1997 ideas about string-particle duality: "Just as a hologram creates the illusion of the third dimension by scattering light off a 2-D surface, gravity and the however many dimensions of space could be a higher-dimensional projection of a drama playing out in a flatter world." Castelvecchi quotes Maldacena to the effect that recently "very strong evidence" has been found that the conjecture is true. But then we read: "Unfortunately, the equations ... seem a good match only for the mathematics of strings living in a contracting universe." So what about this universe here? A semi-theological argument has it that "It would be too much of a coincidence ... if such a seemingly miraculous mathematical duality were to apply to a particular kind of abstract universe but not to our own." On the other hand Abhay Ashketar (Penn State) reminds us, as Castelvecchi puts it, that "In the 1860s, Kelvin pointed out that many of the known properties of chemical elements could arise naturally if atoms were knotted vortices in the fabric of the ether. The uncanny coincidence went away once physicists demonstrated that the ether probably didn't exist." Euclid in China, in 1607 400 years ago, the first six volumes of Euclid's *Elements* were published in China, in Chinese. Last October the Partner Institute for Computational Biology (Shanghai) marked the anniversary with a meeting, reported on by Richard Stone under the title "Scientists Fete China's Supreme Polymath" (*Science*, November 2, 2007). Stone is referring to Xu Guangqi, a prominent Ming-dynasty scholar/administrator, who along with the Jesuit missionary Matteo Ricci carried out the translation. Matteo Ricci and Xu Guangqi, from Kircher's *China Illustrata* (1667). Athanasius Kircher was a Jesuit colleague of Ricci's; the image evokes Xu's conversion to Catholicism. Xu's long career spanned agriculture ("His experiments in Shanghai with yams, then a new import from South America, led to the widespread adoption of the high-energy crop."), weaponry ("Xu also trained imperial soldiers to use a newfangled device from Europe, the cannon.") and calendar reform. His most lasting contribution may have been the vocabulary he and Ricci developed for their translation. They chose the characters *ji he * for "geometry," as well as the Chinese terms for "point," "line," "parallel," etc. which remain in use today. First encounters in strange places Candamin *et al.* give the Sierpinski gasket as an example of the kind of fractal for which they can compute the mean first-passage time from one point S to another T. A typical random path is shown. "First-passage times in complex scale-invariant media" by a team (S. Candamin, O. Bénichou, V. Tejedor, R. Voituriez, J. Klafter) at Paris-VI and Tel-Aviv University appears in the November 1 2007 *Nature*. It leads off with the definition of first-passage time (FPT): "How long does it take a random walker to reach a given target point?" and continues: "Our analytical approach provides a universal scaling dependence of the mean FPT on both the volume of the confining domain and the source-target distance." In all cases the mean FPT `<T>` from point S to point T scales linearly with the volume `N` of the medium; and scales with a power of the distance `r` from S to T, according to the relative size of the "walk dimension" `d`_{w} and the fractal dimension `d`_{f} . The walk dimension is defined so that the first time a random walk reaches a point at distance `r` from its start scales as `r`^{dw}; the fractal dimension so that the number of sites within a sphere of radius `r` scales as `r`^{df}. For the Sierpinski gasket illustrated above, `d`_{f} = ln 3/ln 2 and `d`_{w} = ln 5/ln 2 so we are in the `d`_{f} < d_{w} regime for which their general result gives `<T>` scaling as `r`^{dw - df}; a prediction the authors buttress with numerical simulations. In an interview with *Nature* on the "Authors" page, Bénichou explains how his team worked around the problem of boundary conditions: "... we use a mathematical trick to isolate and replace the confinement effect. Then, we relate the mean FPT in confined conditions to properties of random walks in infinite space, which are easier to estimate." *Nature* also published a detailed appraisal of this work, by Michael Shlesinger of the ONR, in their "News and Views" section. Math for the birds, or maybe not More on random walks in a "NewsFocus" piece in the November 2, 2007 *Science*: a popular mathematical model of how animals search for food has recently come into question. In "Do Wandering Albatrosses Care About Math?" John Travis tells us how a 1996 report, "One of the first studies in which recording devices tracked animal movements, ... brought a little-known mathematical tool to bear on the study of animal foraging. It showed ecologists that a model of random motion called a Lévy flight (named for Paul Lévy) described the way albatrosses searched for food." Lévy flights are "characterized by many short hops, with much longer jumps on rare occasions." They are useful in physics, and they seemed to be just the thing for albatrosses, who fly tremendous distances across the ocean in search of food. Except that it turned out that the flight-recorder data analysis (which assumed that when a bird was dry it was in the air) was misinterpreting long periods before and after foraging when birds were sitting on their nests. Travis quotes Stephen Buckland, a statistician working in ecology: "It's mathematicians taking a simplistic tool and pretending it is relevant to the real world." Others disagree, and are "equally adamant that Lévy flights are a useful tool for ecologists." Tony Phillips Stony Brook University tony at math.sunysb.edu |