This month's topics:
John Allen Paulos' review of a new book about Grisha Perelman (New York Review of Books, April 29, 2010) gives him the opportunity to teach the Review's readers some topology. The book is "Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century" by Masha Gessen (Houghton-Mifflin-Harcourt); and according to Paulos it is "a fascinating biography of ... the fearsomely brilliant and notoriously antisocial Russian mathematician." Paulos starts with a lightning summary of the principal episodes in Perelman's life, and then spends an awkward couple of paragraphs mulling Gessen's suggestion that Perelman may have Asperger's syndrome; Paulos refers to the British psychologist Simon Baron-Cohen who "maintains that there is some neurological reason for the strong correlation between mathematical talent and Aspberger's syndrome. Whether true or not, mathematicians do score consistently higher on what he calls his AQ (autism-spectrum quotient) test ..." And he tells us the old joke about the definition of an extroverted mathematician: the one who looks at your feet while he's talking.
The second half of the review focuses on the actual mathematics in unusual, and unusually correct, detail. First a definition of topology ("the branch of geometry concerned with the basic properties of geometric figures that remain unchanged when they are stretched and shrunk, deformed and distorted, or subjected to any 'smooshing,' as long as they're not ripped or punctured.") Then some examples of topological properties: knotting, separation, dimension, boundaries, genus (the doughnut and the coffee cup, of course) and simple connectedness: "imagine stretching a rubber band around the surface of a ball", etc., with reference to the picture reproduced above. "Poincaré was aware of the fact that a two-dimensional sphere --the topological term for the two-dimensional surface of a three-dimensional ball-- could be defined by this property of simple connectivity. That is, any simply connected two-dimensional closed surface ... is topologically equivalent to the surface of a ball. He wondered if simple connectedness might characterize three-dimensional spheres as well. The statement that it does so is the Poincaré Conjecture." After working at giving the NYRB readers some clue as to what a 3-dimensional sphere might be ("... can't be visualized except in cross-sections--or, it is said, by a very few mathematicians like William Thurston of Cornell University ...") Paulos sketches the history of the problem including the Ricci flow ("Think of blowing hot air into a crinkled-up balloon."), the problem of singularities ("places where the process breaks down and part of the shape starts to stretch on and on, beyond bound--a little like dividing by zero"), how to correct them ("a repair must be made using a controlled process of grafting on pieces of other shapes that topologists call 'surgery'"), the fundamental problem ("there was no guarantee that repairs could be made for every type of singularity and for every recurrence of the same type"), Perelman's solution ("...dazzlingly showed that all possible singularities were reparable, and ... demonstrated how to do the requisite surgeries and put all the stringy and lumpy pieces of the blob together") and the subsequent turmoil in the mathematical world. A tremendous amount of good information in a very small space.Abstract mathematician = psycho?
"'Nyet' to $1 million? Math genius may reject award" came over the AP wires on March 29, 2010, and appeared in The Oregon Herald on March 30; authors Malcolm Ritter and Irina Titova, dateline St. Petersburg. The article speculates on the reasons Gregory Perelman might have for turning down his Clay Institute prize. Saying "nyet" to a megabuck is clearly a sign of pathology; the authors consulted a couple of experts, and reported:
"Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry" appeared in the January 8 2010 Science. The authors, a 9-person team based in Great Britain and in Germany and led by Radu Coldea (Oxford), report the experimental realization of a 20-year-old prediction by A. B. Zamolodchikov (Int. Jour. Mod. Phys. A 4, 4235 (1989)). "The simplest of systems, the Ising chain, promises a very complex symmetry, described mathematically by the E8 Lie group." They add: "Lie groups describe continuous symmetries and are important in many areas of physics. They range in complexity from the U(1) group, which appears in the low-energy description of superfluidity, superconductivity, and Bose-Einstein condensation, to E8, the highest-order symmetry group discovered in mathematics, which has not yet been experimentally realized in physics."
The authors worked with cobalt niobate CoNb2O6, in large crystals where the magnetic Co2+ ions had arranged themselves into nearly isolated zigzag chains. These are the Ising chains. At temperatures below 2.95 K "The chains order ferromagnetically along their length with magnetic moments pointing along the local Ising direction." A strong external transverse magnetic field will undo the ordering, placing the system in what they describe as a "quantum paramagnetic phase;" a continuous transition between the two phases occurs at a specific critical field strength. Zamolodchikov had predicted that at that critical point the system would manifest excitations behaving like the particles of a particular integrable quantum-field theory, with energies in specific ratios given by a representation of E8. In particular the lightest two particles would have masses m1 = m, m2 = 2mcos(π/5), i.e. in the "golden ratio" one to the other. This is the phenomenon that Coldea and his collaborators recorded: "Just below the critical field, the spin dynamics shows a fine structure with two sharp modes at low energies, in a ratio that approaches the golden mean predicted for the first two meson particles of the E8 spectrum." They conclude: "Our results demonstrate the power of symmetry to describe complex quantum behaviors."
This report was picked up as a Nature "News and Views" item by Ian Afflek (March 18, 2010).
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