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A monthly survey of math news
Unknotting the unknot. The February 9, 2001 issue of Science has a nice piece by Charles Seife entitled ``Loopy Solution Brings Infinite Relief.'' The subject is the recent discovery, by Jeff Lagarias (AT&T) and Joel Haas (U.C. Davis) of an upper bound on the number of Reidemeister moves required to remove all the crossings from the projection of a topologically unknotted curve. The three Reidemeister moves are elementary local changes in the projection:
\ / \ / \ / \ / \ / \ / \ Move I \ / \ <----> \ / / \ \ / \ / \ / \/ \/
\ / \ / \ \ / \ \ / / \ Move II \ / \ / <----> / \ / / \ / / \ / \ / \
\ / \ / -- \ -------- \ / \ / \ / \ Move III \ \ <----> \ / \ / \ / \ -------- \ -- / \ / \
They correspond to moves you might actually make trying to unsnarl a tangle. So there is finally an upper bound on just how long it might take to do it. Seife: ``Finite numbers, however, can still be ridiculously large. All Lagarias and Hass guarantee is that if a knot crosses itself n times, you can untangle it in no more than 2100,000,000,000n Reidemeister moves. In other words, if every atom in the universe were performing a googol googol googol Reidemeister moves a second from the beginning of the universe to the end of the universe, that wouldn't even approach the number you need to guarantee unknotting a single twist in a rubber band. ... Still, [Lagarias] says, just showing that a limit exists may inspire future researchers to whittle it down to a reasonable size. (Macedonian swordsmen need not apply.)''
Knots on the air. National Public Radio's Weekend Edition for February 24, 2001 featured an interview with Math Guy Keith Devlin, ``our white knight in the world of mathematics'' according to NPR regular Scott Simon. The occasion was the appearance in Science of the item mentioned above, on the calculation by Jeff Lagarias and Joel Haas of an upper bound on the number of moves required to disentangle an unknotted loop, in terms of the number of crossings in the tangle. Scott spoke for all of us slightly befuddled citizens: ``What's this knot theory?" Keith rose to the occasion: ``Mathematics is not just about numbers, it's the study of patterns. In arithmetic we study patterns of numbers, in geometry we study patterns of shape, calculus studies patterns of motion, and so forth. In the 19th century some mathematicians ... said: `Can we use mathematics to study the patterns of knots?'" After Keith had explained the theorem and its importance, Scott set him up: ``I do not want to suggest for a moment that an entire theory of mathematical thought ... is useless, but ... I'm trying to figure out if there's another practical application that some of us would recognize.'' This allowed Keith to sketch in vivid terms (somewhat fanciful, but probably useful) how knots turn up in modern physics and molecular biology. The segment is available online.
``Iannis Xenakis, the Greek-French composer who often used highly sophisticated scientific and mathematical theories to arrive at music of primitive power, died yesterday at his home in Paris.'' Thus starts his obituary, written by Paul Griffiths, the New York Times music critic, in the February 5, 2001 issue of that newspaper. Xenakis spent his career using mathematical models to replace conventional techniques of composition. He used calculations, for example, ``to determine pitches of notes and their placement in time.'' Metastasis, his first published composition, was ``regulated by Poisson's Law of Large Numbers.'' The premiere, in 1955, caused a sensation and put Xenakis in the first rank of successful and influential composers. Balanchine used Metastasis for a ballet. Later Xenakis founded the Équipe de Mathématique et Automatique Musicales, in Paris, and produced works like the Polytope de Cluny for electronic sound with laser projections.
Jock Math I. Mathematics does continue to enrich our lives. In a piece in the February 3, 2001 New York Times, (``What good is Math? An Answer for Jocks'') Patricia Cohen explains how the new XFL football league has harnessed the power of combinatorics to optimize the bringing of mayhem to the masses. Two mathematicians, Jeff Dinitz (football fan) and Dalibor Froncek, volunteered their scheduling services to the XFL last year and got the job. ``You can do one thing relatively well and the other relatively well, but when you want to put those things together you can't do that without a deep knowledge of some mathematical methods," according to Froncek. The article does not say much about what these methods are, except that they involve graphs, and that Dinitz says: ``The cool thing is it's mathematics, but there isn't an equation in it.'' The XFL were ``very, very happy'' with the work, says Rich Rose, senior league consultant, ``We fully plan on using them in 2002.''
Jock Math II. You may wonder, halfway through the season, if your team has a mathematical chance of winning the league. If your sport is soccer (``football''), then well may you wonder. It turns out that this is an NP-complete problem, equivalent to the notorious traveling salesman problem, and therefore computationally as hard as a problem can get. This information comes from a piece by Justin Mullins in the January 27 2001 New Scientist, entitled ``Impossible Goal'' and explaining this recent discovery, due to Walter Kern and Daniël Paulusma of the University of Twente, and also, independently, to Thorsten Bernholt, Alexander Gülich, Thomas Hofmeister and Niels Schmitt of the Dortmund University Computer Science department. ``Fans had a much easier time in the days when teams got 2 points for a win and 1 for a draw. Kern and Paulusma have shown that this is mathematically simpler than a travelling salesman problem, and the time to solve it increases more slowly as it gets bigger. The switch a few years ago to 3 points for a win turned it into an NP-hard problem. ''
Chaos? Complexity? The currect issue of Skeptic is devoted to Chaos Theory and features an article ``Chaos and Complexity: Should We Be Skeptical?'' by Massimo Pigliucci of the U.T. Knoxville Department of Botany. This piece was picked up in the February 9, 2001 edition of the on-line journal ScienceWeek, from which the following quotations are taken. Pigliucci elucidates the distinction between the common use of ``chaos'' (confusion, tohu-bohu) and the mathematical one, which refers to ``a deterministic (i.e., non-random) phenomenon characterized by special properties that make the predictability of outcomes very difficult." He surveys complexity theory. ``Essentially, ... an attempt to study systems that satisfy two conditions: a) the system is made of many interacting parts; b) the interactions result in emergent properties that are not immediately reducible to a simple sum of the properties of the individual components." The ScienceWeek report explains: `` `emergent behaviors' apparently occur in many complex systems involving living organisms. One example is the idea that human consciousness is an emergent property of a complex network of neurons in the brain. The major problem of complexity theory is how to model such emergent behavior: how to devise mathematical laws that allow emergent behavior to be explained and predicted.''