April 2004
New result on arithmetic progressions of prime numbers: These stories report on recent research shedding new light on an outstanding old problem. The research concerns arithmetic progressions of prime numbers. Arithmetic progressions are sequences of numbers in which each number differs by a fixed amount from its predecessor; one example is the sequence 3, 5, 7, in which each number in the sequence is 2 more than its predecessor. This particular sequence is a prime arithmetic sequence because 3, 5, and 7 are all prime numbers. The question examined in the new research is whether there are infinitely many prime arithmetic progressions, and how long such progressions can be. The best result previously known was proved in 1939 by a mathematician named van der Corput, who showed that there are infinitely many prime arithmetic progressions of length 3. Ben Green and Terence Tao now claim they can show that there are infinitely many prime arithmetic progressions of every finite length. Their work, posted on the web in the preprint "The primes contain arbitrarily long arithmetic progressions", has yet to be fully checked before being accepted by mathematicians as correct. Still, the result has generated a good deal of interest and enthusiasm.  Allyn Jackson "Computer Experiments Are Transforming Mathematics," by Erica Klarreich. Science News, 24 April 2004, pages 266268. Besides using logic and formal proof, mathematicians have often experimented with numbers as a means for determining new results. For example, Gauss made many discoveries empirically, although at least one of his conjectures waited over a century for formal proof. Now, mathematicians have a new tool for running experiments and obtaining answers: computers. Klarreich describes two recent examples: a simple formula for the number π found using a computer program that searched for relationships between π and numbers with known formulas, and certain exciting discoveries made in the field of hyperbolic geometry using software known as "SnapPea." Some mathematicians believe that experimental results should be considered as another means, beside formal proof, of determining mathematical truth. But the more mainstream view seems to remain that formal proof is necessary, in part because things that may look true based upon experimentation could actually be false. And, in AMS President David Eisenbud's words, "Proof is the path to understanding."  Claudia Clark "Mathematics With a Moral," by Robert Osserman. The Chronicle of Higher Education, 23 April 2004, page B10. Two major mathematical announcements mark the beginning and the end of the last 10 years: Andrew Wiles' proof of Fermat's Last Theorem in 1993 and Grigori Perelman's assertion last year that he had solved the Poincaré Conjecture. But, as Osserman points out, achievements such as Perelman's do not exist in a vacuum: besides Henri Poincaré, major players included Bernhard Riemann "whose radical rethinking of the foundations of geometry in 1854 would eventually lead to the solution of the Poincaré Conjecture" and William Thurston, "whose work provided the essential ingredients for connecting the ideas of Riemann with the conjecture of Poincaré." And this doesn't take into account other mathematicians who contributedand are still contributingto this endeavor. What moral can be drawn from this story? Osserman sees two. First, if attempts to solve a problem fail, formulating a more difficult problem may lead to new insights and ultimately a solution to the original problem. Second, contrary to being a lonely enterprise, mathematics is more typically a social endeavor involving fruitful collaboration. And even when working alone, the individual cannot help but depend upon the results generated by other mathematicians.  Claudia Clark "Is space rolled up like a funnel?" by Stephen Battersby. New Scientist, 17 April 2004, page 12. This article reports on a recent paper proposing that the universe is shaped like an infinitely long funnel, skinny at one end and widening at the other. The model, called the Picard topology, is based on data from the Wilkinson Microwave Anisotropy Probe, which has provided a detailed look at fluctuations in the microwave background radiation. Over the next year or so, the probe and other experiments should provide further clues as to whether the funnel idea is correct.  Allyn Jackson "Sixteen steps to Isaac": Review of From Newton to Hawing, edited by Kevin Knox and Richard Noakes. Reviewed by Roy Herbert. New Scientist, 17 April 2004, page 53. This book is about the work of those who have held the Lucasian professorship in mathematics at the University of Cambridge. The first was Isaac Barrow in 1663; currently it is Stephen Hawking. Others on the list are Newton, Babbage, and Dirac. Reading this "staggering book" is a major enterprise, the reviewer writes. But the reader's reward is a journey "through a magnificent history of mathematics and physics, nothing less than humanity's constant effort to understand the universe."  Allyn Jackson "No Half Wits," by Marc Lallanilla. ABCNEWS.com, 12 April 2004. Researchers at the U.S. Army Research Institute for the Behavioral and Social Sciences at Fort Benning, GA, and the University of Melbourne, Australia, compared the responses of mathgifted students to average students on a computerized visual test. The results, published in Neuropsychology, received attention as the study showed that mathematicallygifted teens showed no difference between the two halves of their brains while matching various patterns. The researchers concluded that giftedness in math, music, or art "may be the byproduct of a brain that is functionally organized itself in a different way." However, they caution parents and educators that many other factors are in play. The following news sources covered the research results: "It adds up ... mathematicians are better at using their heads." The Scotsman, 12 April 2004;  Annette Emerson "Strung out across the universe": Review of The Fabric of the Cosmos, by Brian Greene. Reviewed by Michael Duff. New Scientist, 10 April 2004, page 52. With his bestselling book The Elegant Universe, Brian Greene showed a rare talent for explaining complex theories of modern physics in a way the general public could understand. His new book, The Fabric of the Cosmos, "does not disappoint," the reviewer writes. "Topics that have been treated ad nauseam by other popularisers, such as special relativity, black holes and the big bang, are given a new lease of life with Greene's userfriendly analogies and skilful turn of phrase." The book presents a "convincing case" that the mysterious Mtheory may be the ultimate allencompassing theory that physicists have been searching for.  Allyn Jackson "In Math, Computers Don't Lie. Or Do They?" by Kenneth Chang. The New York Times, 6 April 2004. The Annals of Mathematics will soon publish Thomas Hales's proof of the Kepler Conjecture, a statement about the most efficient way to pack spheres. The Annals is not publishing the proof's computer sections, however; they will be published in Discrete and Computational Geometry. Referees have been checking the entire proof, hundreds of pages long, for years but although they found no mistakes, they can not be completely sure that all the computer calculations in the proof are correctthus, the proof is being split. In Chang's article, Robert MacPherson, an Annals of Mathematics editor, says the part of Hales's proof being published in the Annals has been verified and "We feel that he [Hales] has made a serious contribution to mathematics." Several mathematicians are quoted in the article, which gives background on Kepler's Conjecture and attempts at proving it. The article also contains a discussion of the role of computers in proofs. Chang writes, "Dr. Hales said that final publication, after a review process originally expected to last a few months, would be almost anticlimactic. 'For me, the big moment was when I completed the proof... and I don't think anything will change when I see it in print.'" Devlin, in an interview with National Public Radio's Linda Wertheimer, also discusses the conjecture and its proofpointing out how spherepacking is connected to datapackingand reveals that he received "a bag of hate mail" after writing an article on the changing nature of proof.  Mike Breen "The Height of Vanity," by Don Troop. The Chronicle of Higher Education, 9 April 2004, page A8. Scientists at the Institute of Physics in London have found a formula to determine the maximum high heel height a person can wear before he or she tips over. The formula is based on the Pythagorean Theorem, but also involves quantities such as the number of years' experience the wearer has with high heels and the size of the shoe (in British measurement). Although the article is brief, it does give the complete formula.  Mike Breen "Drawing to a Close," by Peter Monaghan. The Chronicle of Higher Education, 9 April 2004, pages A14A16. This article is about New Zealand architect Mark Burry's efforts to fnish Antoni Gaudí i Carnet's Sagrada Família in Barcelona. The Sagrada Família is an enormous church with huge spires, which has remained incomplete since Gaudí's death in 1926. The work is so complicated that conventional architectural software is not sophisticated enough for Burry's work. There isn't much mathematics in the article, but there are occasional references to geometry and the article does mention some of the quadric surfaces.  Mike Breen "Dashing a dream," by Tom Siegfried. Dallas Morning News, 5 April 2004. This article explores the significance of the number Omega, discovered by mathematician Gregory Chaitin in 1974. Earlier, Chaitin had proposed a new definition of randomness, thereby rediscovering an idea of the 19th century mathematician Gottfried Wilhelm von Leibniz. Chaitin suggested that something is random if the information it contains cannot be compressed into something smaller than itself. More precisely, a number is random if the shortest computer program that can be used to calculate the number is as long as the number itself. (Any program or number can be translated into a string of 0s and 1s, and it is the length of those strings that are being compared.) Omega is a "maximally random" number, in the sense that it can be shown that there is no program that will produce Omega as output and that is shorter than Omega itself. "And the moral, [Chaitin] says, is that no single system of axioms will suffice for understanding (or compressing) mathematics," Siegfried writes. "So mathematicians need intuition. And mathematics must embody creative thought."  Allyn Jackson "Autistic Genius?": Review of two books on autism. Reviewed by Allan Snyder. Nature, 1 April 2004. Snyder summarizes and critiques two recent books: Autism and Creativity: Is There a Link between Autism in Men and Exceptional Ability?, by Michael Fitzgerald (BrunnerRoutledge, 2003) and Autism: Mind and Brain, edited by Uta Frith and Elisabeth Hill (Oxford University press, 2003). Both books examine the spectrum of autistic behaviors and individuals and the idea of "genius". The review includes a photograph of mathematician Ramanujan with the caption, "Did autism help shape the thinking of Indian mathematician Srinivasa Ramanujan?"  Annette Emerson "Mathematicians Honored for 'Index Theorem' Concept," by David Perlman. San Francisco Chronicle, 26 March 2004. On March 25 the Norwegian Academy of Science and Letters announced that it awarded the 2004 Abel Prize jointly to Sir Michael Francis Atiyah, University of Edinburgh, and Isadore M. Singer, Massachusetts Institute of Technology. Atiyah and Singer will receive the Prize for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics. This article quotes David Eisenbud (President of the American Mathematical Society and Director of the Mathematical Sciences Research Institute in Berkeley, California): "One of the most remarkable developments in my scientific lifetime has been the new fields of interaction between physics and mathematics, and these two men have been responsible for a major part of that development." Various media worldwide posted the news release and/or covered the news: Some of the articles include a brief explanation of the index theorem. "MIT professor wins major international math prize," by Scott Allen, The Boston Globe, 30 March 2004  Annette Emerson "A Regal Bearing," by JR Minkel. Scientific American, April 2004, page 34. This short piece summarizes work by physicist Hans J. Herrmann of the University of Stuttgart on a spacefilling arrangement of bearings. In Herrmann's arrangement, larger bearings are placed at the six vertices of a regular octahedron, with smaller spheres arranged in a fractal pattern. The research is published in "SpaceFilling Bearings in Three Dimensions" in the 30 January issue of Physical Review Letters.  Mike Breen Reviews of Count Down: Six Kids vie for Glory at the World's Toughest Math Competition "Summing up Achievement: Teens plus a math competition equal a fascinating story," by Marta Salij. Detroit Free Press, 28 March 2004. Steve Olson has written a book about the U.S. team in the 2001 Intermational Mathematical Olympiad, which was held in Washington, D.C. The book profiles the team members, the way they approach problems, and their interests outside mathematics. Salij, Fuson, and Arnesen were very enthusiastic about the book, while Helman wishes Olson delved more into the personalities of the team members.  Mike Breen

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