MATH NEWS ARCHIVE

Christianity sanctifies Sunday as a day of rest and worship. In the early 19th century, some Protestant communities interpreted the Sabbath sobriety as an injunction against dancing, games and other entertainments. But in Massachusetts a loophole was found.
Nowhere in the Bible could the church leaders of Salem find a prohibition against puzzles, and in the absence of a "no," they filled the gap with a resounding "yes." At the time, Salem was a center of brisk trade with China, and ship captains would often deliver a wooden chest filled with ivory puzzles as a gift for merchants, what came to be known as "Sunday boxes." A particularly fine example of a Sunday box is one of the centerpieces of a major new exhibition of mechanical puzzles to open next week at the Lilly Library at Indiana University.
The exhibition, which contains many world-class specimens of mathematical and physics-based puzzles, is the first taste of a collection of more than 30,000 puzzles being donated to the library by Jerry Slocum, a retired engineer and a former vice president of Hughes Aircraft, who has been collecting puzzles and researching their history for more than half a century.
Mr. Slocum is the author of 10 books on the history of puzzles, including a recently published account of the sudoku-like 15 Puzzle, which precipitated a puzzle mania across America in the 1880's.
Sitting in his private puzzle museum at his home in Beverly Hills, Calif., Mr. Slocum spoke about the exhibition and the convoluted, puzzlelike stories behind some of the pieces. Around him on shelves stacked from floor to ceiling sat thousands of puzzles of almost every conceivable shape and form: wooden, metal, wire, porcelain, plastic, glass and cardboard. There were Russian puzzle rattles, ancient Chinese puzzle mirrors and a rare example of an American Indian puzzle purse used to carry a version of gaming chips.
Of all the puzzles in the Sunday boxes, Mr. Slocum said, one of the most challenging was a deceptively simple-looking toy called Chinese rings. Its solution requires what mathematicians call a recursive sequence of moves.
The Chinese rings example in the library exhibition is particularly finely made. A set of rings are threaded over a long, thin loop with wires attached to each ring, tethering it below. Each ring can be taken off the loop or put back on only if the one next to it is on but the others farther down the chain are off. The goal is to get all the rings off.
According to legend, the puzzle was invented in the second century by a Chinese general who gave it to his wife to keep her busy while he was away at war.
Logically, Mr. Slocum said, the puzzle is closely related to the Towers of Hanoi problem, which requires one to move a tower of increasingly smaller blocks from one peg to another.
In recursive puzzles like these, as the number of rings (or blocks) increases, the number of moves required to solve the puzzle increases exponentially. Recursive problems are well known to computer scientists, but it is harder for most of us to get a grip on this elusive concept.
Chinese rings make the problem tangible, Mr. Slocum noted, and reveal in a hands-on fashion the exponential growth entailed. There are typically nine rings in a classic set of Chinese rings; if a player makes no mistakes, the puzzle requires 341 moves to solve. Mr. Slocum can solve it in three to four minutes.
But on a table next to the Sunday box sat a version with 65 rings. A perfect solution in that case would take 18,446,744,073,709,551,616 moves, Mr. Slocum said. "Assuming one move every second, that would be 56 billion years, or four times the age of the universe," he said.
This enigmatic object called to mind the White Queen's advice to Alice about how the more one practices, the better one gets at believing in impossible things. Though solvable in principle, in practice this puzzle can never be completed. Mr. Slocum's collection is a mind-boggling compendium of seemingly impossible, wildly improbable and sometimes breathtakingly difficult puzzles.
Breon Mitchell, director of the Lilly Library, said in an interview that the library was attracted to the collection "because we believe puzzles are important in the history of thought, in the history of mathematics and philosophy, and also the history of science."
Scott Kim, who writes a puzzle column for Discover magazine, said this was the first time a major collection of puzzles would be available in an academic setting. "Puzzles have always interested scientists and engineers," Mr. Kim said. "Many popular things, such as comic books, eventually become subjects for scholarly and academic study. Puzzles are on that cusp right now."
Among the star pieces in the Lilly Library show is an original Rubik's Cube signed by the Hungarian mathematician Erno Rubik, and a prototype of the first Rubik's Cube with six rows of six blocks on each side. That is an object long believed impossible to make, Mr. Slocum said. Finally, last year, the Greek inventor Panayotis Verdes managed to build one.
The Rubik's Cube is an example of a sequential movement puzzle, one of 10 basic categories in what Mr. Slocum called his "puzzle taxonomy." Other categories include disentanglement puzzles (Chinese rings), interlocking solid puzzles (three-dimensional jigsaws) and take-apart puzzles, which include among their subcategories trick locks, trick knives and secret-compartment puzzles.
A beautiful example of the compartment puzzle is another standout of the Lilly Library show. Made by the Japanese puzzle master Akio Kamei, it appears to be a simple, albeit finely crafted wooden cube. But on the top is a hint of how to gain access to its secret compartment. Inlaid in the dark wood is a series of tiny circles of paler wood. The pattern matches the arrangement of stars in the constellation Cassiopeia, and the box will open only when the constellation is correctly aligned. A mechanism inside the box includes a compass that triggers a hidden lock.
Mr. Kamei has made a specialty of such science-based puzzles, and the library exhibition includes several striking examples.
Perhaps the most famous class of physics-based puzzles is one of the most ancient: puzzle vessels. Usually built in the form of a cup or a jug, these vessels offer the challenge that one must drink from them, or fill them up, without spilling any liquid. They have strategically placed holes, so it is immediately clear that a trick is entailed. Early precursors to the form date to at least the 10th century B.C., and Mr. Slocum's collection includes examples from China, Peru, Germany, France and the Middle East.
Not all puzzles are complex. Mr. Slocum said that many of his favorites were the simplest, and that just because a puzzle was simple to look at did not mean it was easy to solve. He particularly likes one that consists of just two three-dimensional pieces that have to be arranged to form a tetrahedron. Another consists of four flat pieces that have to be arranged in the shape of the letter T.
"Both require geometrical reasoning that is counterintuitive," he said. "Good puzzles always go against the grain of our thinking."
Visitors to the Lilly Library will be able to play with a range of puzzles and view animations of various geometric puzzles. This involvement is a critical feature of the exhibition, said Dr. Mitchell, the library director.
"Generally," he said, "you can only study puzzles from books, but when you have three-dimensional puzzles, it's hard to get a sense of them from books alone." In keeping with the spirit of the show, the drawers and cupboards that hold the puzzles will themselves be puzzles.
"The first person who tries to open one each day will have to solve it," Mr. Slocum said, his eyebrows arching slyly. "With puzzles, there is really no substitute for trying them out yourself."
Correction: July 28, 2006
An article in Science Times on Tuesday about a new exhibition of mechanical puzzles at Indiana University included an incorrect estimate from a collector for the time it would take to solve his 65-piece Chinese ring puzzle. At a rate of one move per second, the 18,446,744,073,709,551,616 moves would take nearly 585 billion years, not 56 billion.
Celebrating Puzzles, in 18,446,744,073,709,551,616 Moves (or So)

Alain Connes

If there is a mathematician eagerly waiting for the Large Hadron Collider near Geneva to start up next year, it is Alain Connes of the Collège de France in Paris. Like many physicists, Connes hopes that the Higgs particle will show up in detectors. The Higgs is the still missing crowning piece of the so-called Standard Model--the theoretical framework that describes subatomic particles and their interactions. For Connes, the discovery of the Higgs, which supposedly endows the other particles with mass, is crucial: its existence, and even its mass, emerges from the arcane equations of a new form of mathematics called noncommutative geometry, of which he is the chief inventor.
Connes's idea was to extend the relation between geometric space and its commutative algebra of Cartesian coordinates, such as latitude and longitude, to a geometry based on noncommutative algebras. In commutative algebra, the product is independent of the order of the factors: 3 x 5 = 5 x 3. But some operations are noncommutative. Take, for example, a stunt plane that can aggressively roll (rotate over the longitudinal axis) and pitch (rotate over an axis parallel to the wings). Assume a pilot receives radio instructions to roll over 90 degrees and then to pitch over 90 degrees toward the underside of the plane. Everything will be fine if the pilot follows the commands in that order. But if the order is inverted, the plane will take a nosedive. Operations with Cartesian coordinates in space are commutative, but rotations over three dimensions are not.
To gain a clearer vision of what goes on in nature, physicists sometimes resort to "phase space." Such a space is an alternative to Cartesian coordinates--a researcher can plot the position of an electron against its momentum, rather than simply its x and y locations. Because of the Heisenberg uncertainty principle, one cannot measure both quantities simultaneously. As a consequence, position times momentum does not equal momentum times position. Hence, the quantum phase space is noncommutative. Moreover, introducing such noncommutativity into an ordinary space--say, by making the x and the y coordinates noncommutative--produces a space that has noncommutative geometry.
Through such analyses, Connes discovered the peculiar properties of his new geometry, properties that corresponded to the principles of quantum theory. He has spent three decades refining his thinking, and even though he laid down the basics in a 1994 book, researchers beat a path to listen to him. On a day plagued by typical March showers and wind, about 60 of the crème de la crème of French mathematicians fill Salle 5 at the Collège de France. Like a caged lion, the 59-year-old Connes walks quickly back and forth between two overhead projectors, talking rapidly, continually replacing transparencies filled with equations. Outside, police sirens scream amid student protestors trying to occupy the Sorbonne next door in response to the French government's proposed new employment law.
To Connes, physics calculations not only reflect reality but hide mathematical jewels.
Connes seems oblivious to the commotion--even afterward, while crossing the rue Saint-Jacques past blue police vans and officers in riot gear, he keeps talking about how his research has led him to new insights into physics. As an example, Connes refers to the way particle physics has grown: The concept of spacetime was derived from electrodynamics, but electrodynamics is only a small part of the Standard Model. New particles were added when required, and confirmation came when these predicted particles emerged in accelerators.
But the spacetime used in general relativity, also based on electrodynamics, was left unchanged. Connes proposed something quite different: "Instead of having new particles, we have a geometry that is more subtle, and the refinements of this geometry generate these new particles." In fact, he succeeded in creating a noncommutative space that contains all the abstract algebras (known as symmetry groups) that describe the properties of elementary particles in the Standard Model. The picture that emerges from the Standard Model, then, is that of spacetime as a noncommutative space that can be viewed as consisting of two layers of a continuum, like the two sides of a piece of paper. The space between the two sides of the paper is an extra discrete (noncontinuous), noncommutative space. The discrete part creates the Higgs, whereas the continuum parts generate the gauge bosons, such as the W and Z particles, which mediate the weak force.
Connes has become convinced that physics calculations not only reflect reality but hide mathematical jewels behind their apparent complexity. All that is needed is a tool to peer into the complexity, the way the electron microscope reveals molecular structure, remarks Connes, whose "electron microscope" is noncommutative geometry. "What I'm really interested in are the complicated computations performed by physicists and tested by experiment," he declares. "These calculations are tested at up to nine decimals, so one is certain to have come across a jewel, something to elucidate."
One jewel held infinities. Although the Standard Model proved phenomenally successful, it quickly hit an obstacle: infinite values appeared in many computations. Physicists, including Gerard 't Hooft and Martinus Veltman of the University of Utrecht in the Netherlands, solved this problem by introducing a mathematical technique called renormalization. By tweaking certain values in the models, physicists could avoid these infinities and calculate properties of particles that corresponded to reality.
Although some researchers viewed this technique as a bit like cheating, for Connes renormalization became another opportunity to explore the space in which physics lives. But it wasn't easy. "I spent 20 years trying to understand renormalization. Not that I didn't understand what the physicists were doing, but I didn't understand what the meaning of the mathematics was that was behind it," Connes says. He and physicist Dirk Kreimer of the Institut des Hautes Études Scientifiques near Paris soon realized that the recipe to eliminate infinities is in fact linked to one of the 23 great problems in mathematics formulated by David Hilbert in 1900--one that had been solved. The linkage gave renormalization a mathematically rigorous underpinning--no longer was it "cheating."
The relation between renormalization and noncommutative geometry serves as a starting point to unite relativity and quantum mechanics and thereby fully describe gravity. "We now have to make a next step--we have to try to understand how space with fractional dimensions," which occurs in noncommutative geometry, "couples with gravitation," Connes asserts. He has already shown, with physicist Carlo Rovelli of the University of Marseille, that time can emerge naturally from the noncommutativity of the observable quantities of gravity. Time can be compared with a property such as temperature, which needs atoms to exist, Rovelli explains. What about string theory? Doesn't that unify gravitation and the quantum world? Connes contends that his approach, looking for the mathematics behind the physical phenomena, is fundamentally different from that of string theorists. Whereas string theory cannot be tested directly--it deals with energies that cannot be created in the laboratory--Connes points out that noncommutative geometry makes testable predictions, such as the Higgs mass (160 billion electron volts), and he argues that even renormalization can be verified.
The Large Hadron Collider will not only test Connes's math but will also give him data to extend his work to smaller scales. "Noncommutative geometry now supplies us with a model of spacetime that reaches down to 10-16 centimeter, which still is a long way to go to reach the Planck scale, which is 10-33 centimeter," Connes says. That is not quite halfway. But to Connes, the glass undoubtedly appears half full.
The Geometer of Particle Physics

Professor Tanya Monro

SCIENCE is just as creative as playing music, according to one of the world's youngest professors.
Professor Tanya Monro, 33, always wanted to be a musician, either a pianist or a cellist, but her year 9 science teacher opened her eyes to the "beauty and elegance" of science and she decided, aged 14, to become a physicist.
She still plays both the piano and the cello, and it continues to teach her to be persistent and patient in order to improve. Success is not all about being the best and brightest. "I've learnt (that) mainly you need perseverance and enthusiasm. While being bright helps, it takes a lot of stubbornness and doggedness to be a good researcher."
Author Richard Feynman was an inspiration when Professor Monro was first interested in physics. "He made me realise that physics doesn't have to be complicated to be profound and important. That simplicity of approach attracted me and I live by that even now."
She fell in love with photonics - the science and technology of the generation and control of light using glass optical fibres - during a summer job with a photonics research project in her first year at Sydney University.
"It's exciting. You can have a completely new idea, do the research, test the idea by making a new kind of optical fibre and putting light through it, and then take it forward to develop a solution to a real-world practical problem."
In 1998 she won the Bragg Gold Medal for the most outstanding PhD thesis in physics at an Australian institution, given by the Australian Institute of Physics. Professor Monro then left Australia for a research job at the University of Southampton in England, where she ran a project to develop a new class of optical fibres for patterning large plasma displays.
"I learnt to bring together the strengths of different specialities: chemists to develop new types of glass, engineers to make the optical fibre, mathematicians to predict how glass flows during the fibre fabrication process, physicists to predict how light travels through new classes of optical fibres and to test them experimentally, and people from industry to let us know what new type of fibres were of interest for their applications. Working with scientists from such diverse backgrounds allows for really creative and technologically important research to be done."
A position as the inaugural professor in photonics at the University of Adelaide lured her back to Australia last year. She is now also the director of the new Centre of Expertise in Photonics, which opened a few weeks ago at the University of Adelaide in partnership with the Defence Science and Technology Organisation. The organisation is giving $400,000 a year for up to five years, plus additional support for research so far exceeding $1 million. The South Australian Government has contributed more than $400,000.
The centre is working on the design, fabrication, development and applications of a new class of optical fibres - soft glass fibres - which have significant benefits over the conventional silica-based optical fibres.
"We heat the glass to soften it and push it at high pressure through extrusion dyes to get a wide range of different shapes and to introduce holes into the optical fibre cross-section," Professor Monro says. "The holes give you a great deal of freedom to control the way in which light travels along the fibre. This allows you to develop fibres that are very good at sending high power - a lot of light - down the fibre without melting. This is useful for optical computing, using optical fibres instead of electronic systems for data processing."
Professor Monro has learnt that industry works differently to the academic world. "Industry has a more closed and more competitive approach to science research. You don't collaborate as much as we should as academics. It's about finding the fine line (between commercial sensitivities and collaboration)."
Giving birth to twins recently has enforced Professor Monro's work philosophy. "I love the work I do, but there's more to life than work - even when it is as involving as research can sometimes be," she says. "It's important not to let work take over. Family forces you to strike a balance, because you never finish work - there is always something more to do. After a break, you come back to work fresh and with new perspectives. I learnt after my first child that if I didn't want to be at work, I was not effective."
NEXT LESSONS
· Be persistent and patient to improve your work.
· Working with scientists from diverse backgrounds allows for creative and technologically important research to be done.
· Be aware of the fine line between collaboration and competition.
· Success is not always about being the best and brightest - you need to be single-minded and enthusiastic.
· There is always more work to do - don't let it take over your life.
Let's get physical

" Noeud borroméen de trois tétraèdres", estimé 10-12000 euros(Photo DR)

Le mathématicien et psychanalyste Jean-Michel Vappereau a rencontré Jacques Lacan en 1969 et cherché avec lui à réaliser « le noeud à quatre ». Sa collection d'oeuvres graphiques et de manuscrits du maître sera mise demain aux enchères.
«DANS TRÈS PEU de temps, tout le monde s'en foutra de la psychanalyse», disait Jacques Lacan en 1977, quatre ans avant sa mort. On ne se fout pas, en tout cas, de la pensée lacanienne, comme devrait le montrer la vente aux enchères qui aura lieu demain à Paris (Artcurial-hôtel Dassault). En 1991, la vente de son cabinet de la rue de Lille avait fait salle comble et son précieux divan, mis à prix 2 000 F, avait été attribué pour 98 000 F. Pour la vente du 30 juin, qui n'est pas approuvée par la famille Lacan, le catalogue s'est déjà arraché. Il est vrai que c'est la première fois que des documents d'archives relatifs aux travaux du maître sont mis aux enchères. Près de 120 documents et dessins datant de 1972 à 1979, rassemblés par Jean-Michel Vappereau.
Ce mathématicien rencontre pour la première fois Jacques Lacan en 1969 par l'intermédiaire de Roland Dumas, qui fut son avocat. Il fera partie du cercle avec lequel le psychanalyste se penche sur des énigmes topologiques ; ensemble, ils chercheront à réaliser le « noeud à quatre ».
La collection qu'il met en vente «correspond aux années décisives de l'aventure topologique, de l'énigme des noeuds», commente Jacques Roubaud, mathématicien, écrivain, membre de l'Oulipo (Ouvroir de littérature potentielle), ami de Queneau et de Perec. «Ces documents témoignent d'une période décisive dans la recherche du DrLacan, dit Roubaud. Ils donnent accès au chantier d'une pensée de travail, avec ses tâtonnements, ses intuitions brusques, ses découvertes.»
Voici, avec des estimations allant de 1 000 à 10-12 000 euros, des chaînes, des tresses, des cercles, des tétraèdres, des quatresses, un triangle de Pascal, des noeuds borroméens. Ce noeud qui a pris tant d'importance dans ses théories, qui était devenu, comme le souligne Roland Dumas, «à la fois un symbole, un instrument de recherche et une éternelle interrogation».
Les manuscrits, écrits à l'encre bleue, sont pour la plupart de courts documents. Cela va de l'ordonnance de Valium aux lettres en passant par des textes humoristiques ou poétiques (un poème signé « là quand ») pleins de ces jeux de mots qui faisaient sa marque de fabrique («Le comble du comble, c'est que je suis comblé») ou une variation sur le syllogisme « Tous les hommes sont mortels ».
Lacan dans ses oeuvres

Le lieu du rendez-vous était simple : Centre Pompidou, galerie Sud, niveau 1. Jean-Luc Godard avait décidé de parcourir ce Voyage(s) en utopie, JLG, 1946-2006, À la recherche d'un théorème perdu en compagnie d'un journaliste. Une exposition (voir critique p. 20) dont les prémices ont fait couler beaucoup d'encre. Parce qu'il y a, à l'origine de Voyage(s) en utopie... le projet d'une autre exposition, dont le titre devait être Collage(s) de France, qui n'a pu se faire. La presse a écrit sur les responsabilités de cet échec, et Dominique Païni, qui devait être le commissaire de l'exposition et qui a dû finalement se retirer après trois années de collaboration avec le cinéaste, est intervenu dans les médias à ce sujet.
Mais, aujourd'hui, Jean-Luc Godard n'est pas à Beaubourg pour revenir sur ces épisodes. Il est là, le cigare souvent à la bouche, pour accompagner son visiteur, dans la plus grande décontraction, à travers les trois pièces - nommées « Avant-hier », « Hier » et « Aujourd'hui » - qui constituent son exposition. Pendant plus de deux heures, celui qu'on a appelé « l'ermite de Rolle » déambule entre les maquettes ou les objets, les téléviseurs diffusant des extraits de films, les nombreuses citations sans auteur mentionné, en forme et en verve. Sa parole multiplie les changements de direction avec la même vélocité dont fait preuve l'équipe du Brésil avec un ballon : talonnades, déviations, passes de la poitrine. Et cela, aussi bien quand les mots brillent au soleil ou se camouflent dans la brume. Si Godard ne rechigne pas à s'arrêter devant tel ou tel élément exposé, il ne cache pas sa préférence en faveur d'une visite en mouvement, pour ne pas réduire la perception, fixer la sensation, figer le sens. Cet entretien ne doit pas se concevoir autrement.
L'exposition « Voyage(s) en utopie... » est née d'un premier projet d'exposition, dont le titre aurait été « Collage(s) de France », mais qui n'a pu être menée à bout. Comment les avez-vous articulées ?
Jean-Luc Godard : La première exposition a été empêchée, barrée ; la seconde a été pensée comme une introduction à cette impossibilité. Il y a donc des allers et retours entre les deux.
J'ai essayé d'être le plus clair possible, en disant : « Une exposition n'a pas pu avoir lieu. Il en reste quelques briques, quelques ruines, sous forme de maquettes. » Ce devait être le principe de la première exposition : édifier de grandes ruines. Les gens n'auraient rien compris, mais ils auraient senti. Dans l'exposition Voyage(s) en utopie..., on ne peut pas vraiment sentir, parce que j'ai essayé d'établir le scénario suivant : en soixante ans de cinéma, je suis finalement arrivé à ces ruines dont on n'a pas voulu. On peut toujours essayer de l'expliquer, comme un guide le ferait. Mais je n'ai rien à expliquer. Il s'agit d'une postface, sauf qu'on la met au début. Ce n'est donc pas du tout clair.
Vous vous refusez à toute explication, comme si vous vous méfiez du dire, de la parole sur, comme si vous en dénonciez même l'inanité...
Je ne peux plus parler sans images. Ce qui me revient immédiatement en mémoire, ce ne sont pas forcément des mots, mais des actions, des souvenirs... Je ne nie pas l'importance du texte. Personne n'aime autant les livres que moi, et personne ne l'a autant montré que moi. Mais il m'est difficile de tenir une discussion « de texte sur du texte », comme disait Péguy.
Si on se lance dans une discussion de mots au lieu de chercher à établir des rapports entre les choses, on reste en vase clos. C'est exactement ce qui se passe dans le domaine politique ou culturel. Dans Notre Musique (sorti sur les écrans en 2004, NDLR), l'écrivain Jean-Paul Curnier prononce cette phrase de Claude Lefort : « En faisant de la politique un domaine de pensée séparé, les démocraties modernes prédisposent au totalitarisme. » La science est plus forte parce qu'elle accepte un antagonisme entre l'expérience et la théorie. Dans l'art, il y a longtemps que cela n'existe plus. Il n'y a qu'à voir « l'exposition Duchamp » qui se déroule à côté (une exposition Claude Closky, artiste qui a reçu en 2005 le prix Marcel-Duchamp, se tient en effet actuellement au Centre Pompidou, juste à côté de l'exposition Voyage(s) en utopie..., NDLR).
Cette réticence à produire du « texte sur du texte » éclaire aussi les relations singulières que vous entretenez désormais avec les médias...
Nous - la Nouvelle Vague - avons cru qu'il fallait intervenir dans la presse pour nous faire une place, et nous avons pensé qu'on nous accepterait. J'ai cru, pris dans l'engrenage, que je pourrais convaincre qu'il était possible de faire de la télévision autrement. J'y suis donc apparu, un peu en clown, mais c'était très sincère. Aujourd'hui, je regarde cela avec un sentiment de honte, parce qu'il en reste un caractère prétentieux, autoritaire. Mais c'était bon enfant. Et puis, à un moment donné, la télévision n'a plus voulu de moi. Ce qui est bien, du reste. Aujourd'hui, je n'ai plus envie d'expliquer. Je peux, à la rigueur, déplier.
Dans l'exposition, les rapports que vous établissez entre les choses ne sont pas sans évoquer, selon vous, des formules mathématiques. Il est même question d'« un théorème perdu ». La présence des mathématiques y est forte, où l'on rencontre par exemple le nom d'un mathématicien comme Georg Friedrich Riemann...
Je fais des relations sous forme imagée. Si jamais un mathématicien vient voir l'exposition, je pourrais ainsi avoir avec lui un petit dialogue. Je lui dirais : « Vous êtes un littérateur, un écrivain. Il n'y a presque que des lettres dans vos théorèmes. Il y a très peu de chiffres. Et si vous êtes un écrivain, vous êtes un poète. » Jeune, je pensais que je me destinerais aux mathématiques. Je croyais être très doué. Ce ne fut pas le cas. Mais j'aimais bien ça.
Riemann est très difficile à lire. Certains mathématiciens sont plus linéaires dans leur démarche, plus explicatifs. Lui, il avance en faisant des sauts. Dans le même ordre d'idée, Fermat, qui était un contemporain de Pascal, a écrit dans la marge de son théorème : « Je n'ai pas beaucoup de place ici pour le démontrer, mais la solution de ce problème est extrêmement simple. » On a mis trois cents ans à le résoudre. C'étaient les débuts de l'algèbre géométrique et de la géométrie analytique, avec Descartes. Quand Fermat se plaignait de n'avoir pas assez de place, je crois qu'il voulait dire qu'il n'avait pas suffisamment de place pour exécuter des figures, contrairement aux mathématiciens arabes.
Les mathématiciens montrent mais ne disent rien. Je me suis longtemps identifié à des mathématiciens malheureux. Du début du XIXe siècle en particulier. À Évariste Galois, par exemple, qui est mort à 22 ans, et qui a fondé la théorie des ensembles. À Niels Abel aussi, un Norvégien très pauvre. Il avait démontré, à l'âge de 19 ans, l'impossibilité de résoudre les équations algébriques du 5e degré. Il a souhaité rencontrer, à Paris, un mathématicien très célèbre, Cauchy, de l'Académie des sciences, mais celui-ci n'a pas voulu le recevoir. Abel est reparti à pied en Norvège. Où il est mort aussi très jeune. Aujourd'hui, il existe un prix Abel en Norvège. Pour moi, ceux-là sont des amis.
Pour cette exposition, vous avez réalisé un film de montage, Vrai/faux passeport, où vous attribuez des bonus et des malus à des images de cinéma ou de télévision. En procédant ainsi, ne participez-vous pas à la faillite généralisée du jugement critique ?
Ce sont des citations à comparaître pour avoir la possibilité de juger des films. On m'a beaucoup reproché d'accorder des bonus et des malus. Or, c'est ce que pratiquent aujourd'hui tous les journaux : ils accordent des étoiles. Je ne fais rien d'autre que de partir de ce qui se pratique. Je mets un bonus ou un malus à telle ou telle séquence. Soit. Mais comme je procède par opposition dialectique - voici comment untel parle de la Palestine, voici comment tel autre en parle - le spectateur a la possibilité de juger du bien-fondé de ce bonus ou de ce malus. Les extraits de film sont comme des pièces à conviction.
Si je devais répondre explicitement à la question : « Pourquoi est-ce que je mets un malus ou un bonus ? », il me faudrait faire un autre film, ou un livre, qui me prendrait vingt ans. Ce ne serait pas la Critique de la raison pure, mais la Critique du cinéma pur... Mais dans Vrai/faux passeport, je ne m'intéresse pas au pourquoi, je pose la question du comment. Le spectateur, ensuite, peut voir ou revoir le film, même en DVD, et repasser le film en jugement. Et cela l'aidera peut-être ensuite pour d'autres films. Aujourd'hui, le spectateur a tous les moyens techniques pour regarder des films où et quand il le veut. Mais il préfère une autorité qui décrète, il a besoin d'être guidé, il a besoin de savoir combien on a mis d'étoiles...
Dans la pièce de l'exposition intitulée « Hier », et qui a pour sous-titre « Avoir », sont diffusés un certain nombre d'extraits de films du patrimoine, et des extraits de quelques-uns de vos films. C'est sans aucun doute la salle la plus sereine de l'exposition. Ressentez-vous de la nostalgie pour cet « Hier » ?
J'ai simplement voulu dire ceci : en soixante ans de cinéma, j'ai été influencé par ces films allemands, russes, américains..., et par quelques films français. Ça m'a amené à faire les films que j'ai réalisés, certains avec Anne-Marie Miéville, et de là je suis passé à l'archéologie du cinéma. Est-ce qu'il y a de la nostalgie ou de la mélancolie ? Non. Ma seule mélancolie personnelle vient du fait qu'on aurait pu m'aider mieux qu'on ne l'a fait. Je suis connu, mais pas reconnu. Mais aujourd'hui, c'est un peu tard.
Dans cette salle, les films dialoguent entre eux comme si vous vous effaciez derrière eux, ou comme si le visiteur se trouvait là où ils dialoguent en vous...
Imaginez que Scorcese ait eu à réaliser cette salle. Il n'aurait pas pu. Dans son histoire du cinéma italien, il raconte qu'il a vu tel film en telle année. Il a besoin de s'affirmer lui-même en s'appuyant sur l'existence de l'autre. C'est comme les livres sur moi. Ceux qui les écrivent touchent des droits d'auteur mais il ne connaissent rien. Tout est faux.
Dans la dernière salle, « Aujourd'hui », dont le sous-titre est « Etre », vous êtes en revanche, beaucoup plus sévère. La critique que vous faites de notre époque est implacable : nous sommes dans l'ère télévisuelle, c'est-à-dire celle du vide...
Il n'y a en effet que du rien : je ne le dis pas du monde entier, je le dis de la télé et de l'appartement où on regarde la télé. Si j'avais été plus explicite, j'aurais écrit en gros : « Là, à Pompidou, surnommé le garage, où on donne le moins d'argent possible, il n'y a rien ». Les gens se seraient dit : « Ah oui, d'accord, il est un peu vache avec le Centre Pompidou... »
Au début, je voulais garder ce qui restait de l'ancienne exposition. Comme ça, il y aurait eu au moins quelque chose. Et puis, petit à petit, j'ai fait tout enlever, jusqu'à ce qu'on arrive à l'idée d'un presque rien, à l'instar de ce qu'on dit par exemple du journal télévisé : ce n'est pas grand-chose... Beaucoup de gens vont se dire en entrant dans cette salle « Aujourd'hui » : ce n'est pas vrai, ce rien ne correspond pas à ce qui se passe aujourd'hui. Mais il s'agira d'une mauvaise interprétation. Il faut simplement voir ce qui est montré : des téléviseurs à plat dans un appartement. C'est décomposé. C'est un peu du Derrida. Mais, pour moi, c'est plus compréhensible que du Derrida...
Pourquoi ces écrans posés à plat ?
Puisqu'on les appelle des écrans plats, je ne vois pas pourquoi on ne les mettrait pas à plat. Dans cette salle, on est effectivement dans la platitude. On a la cuisine, la chambre à coucher, la cuisine, le bureau, c'est plat. Il n'y a pas de profondeur.
On y trouve aussi trois enveloppes vides, à côté d'une balance, sur lesquelles sont inscrits ces quelques mots : « Plus jamais ça », « Les lendemains qui chantent », et « L'Appel de Stockholm »...
Ces enveloppes introduisent un peu de profondeur, un peu de temps. Mais c'est mon temps à moi. Ce sont les premières phrases dont on m'a dit qu'elles étaient importantes. J'ai entendu : « Plus jamais ça ». On m'a parlé des « lendemains qui chantent ». Et j'ai signé l'appel de Stockholm à dix-sept ou dix-huit ans, sans savoir d'ailleurs ce dont il s'agissait exactement. Aujourd'hui, je constate que ces phrases ne pèsent plus grand chose. Le temps s'est effacé. Mais cette disparition du temps, elle, pèse lourd.
Et aujourd'hui, où en est le cinéma ?
Aujourd'hui, tout ce que je souhaite, c'est qu'un producteur me donne son avis sur le projet de film que je lui présente. Mais il ne me le donne plus. Il n'y a plus de producteur, il n'y a plus que des distributeurs. Le cinéma avait trois stades : la production (la caméra), la distribution (le projecteur), l'exploitation (la salle). Aujourd'hui, n'existe plus que la distribution qui produit pour distribuer. Du cinéma, il reste une métaphore. Mais à partir de cette métaphore, on peut encore expliquer le monde. Les mathématiciens croient que c'est à partir des mathématiques. Moi je crois que c'est à partir du cinéma. Je ne me dis pas cela depuis très longtemps, depuis sept ou huit ans seulement. Mais je crois que ce sera encore valable pour ce qu'il me reste de vie.

Un entretien avec Jean-Luc Godard à propos de son exposition au Centre Pompidou : « Je n'ai plus envie d'expliquer »

Las sonrisas de satisfacción y orgullo son inevitables y no es para menos. Después de varios meses de preparación, la noche del pasado martes, los chicos de la delegación peruana de matemática retornaron triunfantes de su participación en la edición 47 de la Olimpiada Internacional de Matemática celebrada en Eslovenia del 8 al 18 de julio último.
Daniel Chen Soncco con tan solo 15 años ganó una de las medallas de plata: "Estoy recontracontento. No pensé que en esta prueba tan difícil iba a lograr esto". Jossy Alva, también de 15 años, se hizo merecedora de una de bronce. Franco Vargas, María Teresa Huánuco, Paolo Aguilar y Aldo Quillas recibieron mención honrosa. Con estos resultados, de un total de 90 países, estos chicos dejaron al Perú en el puesto 40 del ránking general de la olimpiada con un total de 85 puntos acumulados. El país ganador fue China con 214 puntos, seguido por Rusia y Corea del Sur.
Además de la competencia, nuestros representantes tuvieron la oportunidad de recorrer la ciudad de Luvenia y conocer un poco sobre su cultura. "Este año la organización estuvo mejor que el año anterior, pero los problemas estuvieron más difíciles", cuenta Daniel Chen Soncco, sin que sus manos dejen de jugar con la medalla cuadrada que le cuelga del cuello.
RESOLVIENDO PROBLEMAS
Durante dos días los concursantes se volcaron en la resolución de seis problemas. El que Jossy disfrutó más fue una inecuación por la manera bastante creativa de resolverlo. Sin embargo, nos cuenta que el más difícil fue uno en el que le daban un polígono que había que dividirlo en varios triángulos. "Te pedían demostrar que la suma de algunos de ellos era mayor que el doble del área del polígono", explica.
Otro problema, no de tipo matemático, que la delegación tuvo que resolver fue el financiamiento del viaje. A diferencia de años anteriores, este año no contaron con el auspicio de alguna entidad. Solo gracias a la unión de esfuerzos de los padres de familia y de los colegios en los que estudian, se pudo cubrir el costo de los pasajes. El sector privado y público bien gracias.
Jóvenes peruanos regresan victoriosos de Olimpiada Internacional de Matemática

El equipo argentino, entrenado por profesoras del CBC, obtuvo en la competencia internacional de Eslovenia dos medallas de plata, dos de bronce y una mención de honor, informó la UBA.
La 47. Olimpíada Internacional de Matemática, realizada el domingo en la República de Eslovenia, fue como las anteriores favorable para la actuación del equipo argentino, que obtuvo dos medallas de plata, dos de bronce y una mención especial.
Los estudiantes fueron preparados, como es habitual, por las profesoras del CBC Patricia Fauring y Flora Gutiérrez.
Los alumnos distinguidos fueron Julián Eisenschlos, de la Ciudad de Buenos Aires, con medalla de plata; Ignacio Rossi, de Escobar, provincia de Buenos Aires, también con medalla de plata; Roberto Morales, de la ciudad de La Plata, con medalla de bronce, premio que también obtuvo Pablo Nicolás Zimmermann, de Rosario.
Por último, Fernando Martín Vidal, de Villa Cañás, Santa Fe, fue premiado con Mención de Honor.
La Olimpíada Internacional de Matemática es la más exigente en su género.
Un platense premiado en Olimpiada Matemática de Eslovenia

The raison d'etre of doing science is aptly summed up in the statement that 'if life was not beautiful, it won't be worth knowing and if life was not worth knowing, it won't be worth living'.
Before deliberating upon the subliminal text of the above statements in the context of Mathematics, I should begin by trying to dispel certain myths about Mathematics that have been in currency for a long time now. To begin with, mathematicians have very little to do with numbers in the same way as a research scientist has hardly anything to do with technology which is merely a welcome outcome of the hard work that he is supposed to put in with a view to understanding the mysteries of nature. In fact, the definition of mathematics as 'working with numbers' has been out of date for nearly 2500 years now. One can no more expect a mathematician to be able to add a column of numbers rapidly and correctly than one can expect a painter to draw a line or a surgeon to carve a turkey- popular legend attributes such skills to these professions- but popular legend is wrong. There is, to be sure, a part of mathematics called number theory, but even that doesn't deal with numbers in the legendary sense. A number theorist and an adding machine would find very little to talk about. A machine might enjoy proving that the sum of the first five cubes is 225 but many mathematicians would enjoy the theorem that sum of the cubes of the first n integers is always a perfect square, whereas the 'infinity' involving the word 'every' in that statement would frighten and paralyze any computing machine. Plainly, mathematics is not numbers or machines. It is also not determination of heights of mountains by trigonometry or compound interest by algebra. A more altruistic but an equally important view of mathematics consists basically in the idea of 'doing things for their own sake'. Thus, one may encounter this attitude in music as much as in Art or even in medicine. To quote a famous mathematician P.R.Halmos " I never quite trust a doctor who says he chose his profession out of a desire to benefit humanity. I am uncomfortable and skeptical when I hear such things. I must prefer the doctor to say that he became one because he liked the idea, because he thought he would be good at it, or even because he got good grades in biology. I like the subject for its own sake in medicine as much as in music and I like it in Maths." What is it then that mathematics is all about?
To answer this question, let me begin by saying that mathematics is an important human endeavour, essentially a science of patterns- real or imagined- visual or mental, arising from the natural world or from within the human mind. Far from being too abstract to matter and besides being a uniquely human endeavour , mathematics helps us understand the universe and ourselves. It informs our perceptions of reality-both the physical, biological and the social worlds without, and the realms of ideas and thought within. Like a scientist who does not study nature because it is useful to do so, but because he takes pleasure in his subject, so does a mathematician take pleasure in mathematics because it is beautiful. What I mean is that intimate beauty of mathematics that comes from the harmonising order of its parts and which only pure intelligence can grasp. Here I quote the famous French mathematician Henri Poincare who said" Intellectual beauty is self-sufficing and it is for it, more perhaps than for the future good of humanity, that the scientist condemns himself to long and painful labour". In that sense, mathematics has a lot in common with poetry where it is the search for beauty that inspires the activity of the poet as it does that of the mathematician. Pushkin had captured this analogy succinctly when he had said" Inspiration is needed as much in geometry as it is needed in poetry". This puts into perspective Bertrand Russel's quote stated in the beginning of this article, not only in the context of mathematics but equally pertinently from the viewpoint of natural sciences where it is invariably the search for beauty that drives a scientist's work in understanding the myteries of nature. And that entails not merely a comprehension of the processes involved in how do the things work but, perhaps more importantly, why do the things behave the way they do. This is precisely what constitutes the scientific temper which, however, should not be deemed to be at variance with other forms of intellectual activity. It is a pity that there are those swearing fealty to the cause of education whose vision of a 'reformed and modernized' system of education is to do away with the 'traditional' in favour of the more contemporary curriculum, as if that would act as a panacea for all our educational ills. They would do well to remember that no worthwhile purpose of education can be hoped to be served if 'modernising' the curriculum is achieved at the cost of 'traditional' subjects like literature, philosophy or history etc. which are at least as vital for human development as are the more contemporary ones that are fancied to be in accord with the 'zeitgeist'.
On the other hand - and at a more utilitarian level - mathematics surpasses every other human endeavour in terms of the value it has brought to human life through science and technology where the role of mathematics has been immense. It is this unexpected role of mathematics it has played in the development of science over the centuries that it is the topic of a famous and widely-read classic by the Physics Nobel Laureate Eigene Wigner 'The unreasonable effectiveness of mathematics in the natural sciences", a theme that is echoed by the fact that mathematics has come to be recognised as the language of Science, where its role is not limited to the physical sciences alone but more recently to biological sciences as well. Contemporary research pertaining to the 'topology' of the DNA molecules is a case in point where the ideas and techniques from 'Knot Theory'- a branch of modern differential topology- have been employed to gain a deeper understanding of the 'geometry' of the DNA molecule. There is also this ubiquitous individual phenomenon of mathematicians being motivated primarily by a subtle mixture of ambition and intellectual curiosity, and not by a wish to benefit society. According to Professor Timothy Gowers, the Field Medalist ( equivalent of Nobel Laureate in Mathematics), this is because mathematics is cheap and occasionally produces breakthroughs of enormous economic benefit, either directly as in the case of public-key cryptography, or indirectly as a result of providing the necessary theoretical underpinnings of science. Here it is pertinent to briefly talk about the oft-quoted division of mathematics into 'pure' and 'applied' mathematics. As things stand, the main difference between these parts of mathematics is the intellectual curiosity that motivates the work or, perhaps more accurately, the kind of intellectual curiosity that is relevant. How in the first place does one choose one's research problem and what is it about the problem that attracts a mathematician? Do you want to know about Nature or about Logic" Do you prefer concrete facts to abstract relations? If it is nature you want to study, in other words, if the concrete has the greater appeal, then you are an applied mathematician. In this case, the question always comes from outside, from the 'real' world and the satisfaction the scientist/mathematician gets from the solution comes, to a large extent, from the light it throws on facts. The conclusion from these speculations is inescapable: between the two-pure and applied- there is identity with small differences instead of diversity with important connections! We have a case for saying that mathematics is a glorious creation of the human spirit and deserves to live even in the absence of any practical applications which, in any case, are aplenty.
A study of the growth and development of mathematics reveals that care for the beautiful invariably leads to the same selection as care for the useful. This is borne out by developments, say in 'Group theory' and 'Hilbert spaces'- important areas of mathematics- that have been witnessed over a period of time, resulting in epoch-making discoveries in the theoretical underpinnings of the processes unfolding in the physical world. After all, today's research is tomorrow's technology! All the same, it may be of no practical use that ? ( read as pi)- an important number in mathematics-is irrational, but if we can know, it would surely be intolerable not to know! The so-called pure mathematicians do mathematics because for them it is fun in the same way perhaps that people climb mountains for fun. It may be an extremely arduous and even fatal pursuit, but is fun nevertheless. Mathematicians enjoy themselves because they do sometimes get to the top of their mountain and, anyhow trying to get up there does seem to be worthwhile. In essence, mathematicians are engaged in discovering and mapping out a real world- a world of thought, which is of a kind on the basis of which the physical world is, to a certain extent, also constructed. And the most inscrutable object in this world of thought is surely the concept of infinity and the ways to tackle it. Thus for the most part, mathematics ought to be looked upon as a philosophy to deal with the ways to tackle and tame the 'infinite'.
While hoping that the idea of 'what mathematics is not about' has been reasonably hammered home, there still is a need for identifying ways in which it is possible to tell a good piece of mathematics from one which is not. The bottomline is that it has to be new without being silly. Mere accuracy in performance does not make good music nor does resemblance in appearance qualify for good painting. In the same way, mere logical correctness doesn't make good mathematics. The fact is that one of the greatest joys of the subject has to do with discovering surprising connections between various branches of mathematics, which are-on the face of it- as different as the chalk is from the cheese. The interplay of ideas from 'Knot Theory' which belongs to Differential Topology and 'Operator Algebras'( a branch of Functional Analysis) within mathematics on the one hand and 'Knot Theory' and the study of the DNA molecule (outside of mathematics) on the other , is a case in point. It is sad that at least in this part of the country, Science is merely perceived as a profession where you learn the skill say, to play the computer keyboard or transacting business with your credit card. This is symptomatic of an affliction where scientific temper has not been allowed to flourish. In a similar vein, mathematics has been touted purely as a number game which is supposed to be learnt-and mastered!- through rote learning which, sadly, is the flavour of the month in most of the educational institutions, both public and private. This prurient thirst for trivia involving encouragement of rote learning and similar other tips for 'passing the exam' is not going to enhance the cause of education unless the entire educational system is overhauled at all levels involving the school, college and the university level where comprehension and not memorizing the material is emphasized. The incorporation of negative marks in the recently held entrance tests in the university of Kashmir is no doubt a small but an important beginning in this direction.
Mathematics is by no means a spectator's game where one can hope to have fun without participating in the game in the first place. In order to enjoy the thrill of discovery in mathematics, one has to 'get one's hands dirty' as it were, by 'doing' mathematics, which entails a willingness for 'slogging it out', for total commitment and a self-effacing obsession with the problem under study. Among other factors responsible for an unwillingness, even among bright students, to pursue a career in science is the abysmally low quality of teaching of science in general and of mathematics in particular in our educational institutions. Unless the system of education is drastically reformed where such values are cultivated and nurtured as part of a system, we are condemned to be second-raters in the field of science where nothing substantial can be achieved in the absence of a culture in which scientific ethos tempered by pure thought and reason would flourish. I conclude with the following quote of Professor Steven Weinberg which sums up the essence of the scientific activity as discussed in the above lines in no uncertain terms: The effort to understand the universe is one of those few things that lifts human life a little above the level of farce and gives it some grace of tragedy.
(The author is Professor of Mathematics and Dean Academics, Kashmir University, Srinagar. He can be mailed at aminsofi@rediffmail.com)
Revisiting the Relevance of the Queen of Sciences

John Adams HS junior Fareed Mohammed
with the silver medal he won at the
National High School Math Championship in Okla.

Queens math whiz Fareed Mohammed had never seen a calculus problem before staring at four of the mind-melting equations at the National High School Mathematics Championship.
But that didn't stop the John Adams High School junior from solving the problems - and snagging a silver medal at the prestigious annual math competition July 1 and 2.
"It was the hardest thing I've ever seen, I never did calculus in my whole life," Mohammed, 17, told the News. "I had to use my past knowledge to figure it out."
Mohammed was one of just 24 other finalists selected to participate in the annual two-day math melee held by the American Society for Mathematics in Oklahoma City.
Other New York finalists included Matt Zee, 18, of West Nyack; Sulinya Ramanan, 17, of Poughkeepsie and Kenneth Mandel, 16, of Sea Cliff, L.I. Outreach programs that help find the math whiz kids in New York schools are sponsored by the Daily News Educational Dept.
Despite the grueling mathematical challenges, Mohammed said he made friends with many of the other finalists - fellow bright kids with whom he will stay in touch via e-mail.
"I'm a normal kid, right; I like to kid around and have fun," he said. "I made a lot of friends from [places] like Texas and Georgia, all over the United States."
American Society for Mathematics president Richard Neal called Mohammed the "most liked" person of the competition, which tests students on topics unfamiliar to them unless they've taken college courses.
"The areas are pretty expansive; logic, modular arithmetic and other things they wouldn't normally encounter in their curriculum," Neal said.
The finalists have bright futures, he said, adding that "about half of them are going to MIT."
Mohammed - who moved from his native Republic of Trinidad and Tobago to Richmond Hill in Queens two years ago - got a boost from Councilman Joseph Addabbo (D-Howard Beach) and State Sen. Ada Smith (D-Jamaica), who helped collect $800 for his airfare and expenses.
"It was money well spent," Addabbo said. "He did very well and represented not only the school, not only the borough, but the whole state."
Mohammed said that after attending college, he wants to be a medical engineer and "as a sideline, also find a cure for cancer."
Before curing cancer, however, he said he wants to return to the math competition. "Next year, I plan on getting the gold medal."
Originally published on July 13, 2006
Teen adds silver at math meet

Irving Kaplansky

Irving Kaplansky, a mathematician who broke ground in exploring concepts central to algebra and multiplication, died on June 25 at his home in the Los Angeles community of Sherman Oaks. He was 89.
The cause was respiratory failure, his family said.
From 1945 to 1984, Dr. Kaplansky taught at the University of Chicago, where he joined his famous former teacher, Saunders Mac Lane, who worked on topology and category theory, an abstract branch of algebra with applications in computer science. Dr. Mac Lane died in 2005.
Dr. Kaplansky's interests were similarly broad, and he explored the properties of groups of numbers called commutative groups, also known as Abelian groups, in which the order that a group's members are multiplied does not affect their outcome.
He published "Infinite Abelian Groups" (1954, 1969) and "took a big step in showing how far you could go with infinite elements" that are commutative, said David Eisenbud, director of the Mathematics Sciences Research Institute in Berkeley, Calif. From 1984 to 1992, Dr. Kaplansky directed the institute.
J. Peter May, a former student of Dr. Kaplansky and a professor of mathematics at the University of Chicago, praised his "exceedingly incisive mind that saw through to the essentials in mathematical arguments with precision and clarity."
Dr. Kaplansky went on to write "Commutative Rings" (1970), a work that Dr. Eisenbud said remained in use and was "narrowly focused on its subject, a subject that, partly because of this book, has since gone much further." Dr. Kaplansky later wrote about an area bridging algebra and topology, a field that involves the study of real or abstract spaces, in "Lie Algebras and Locally Compact Groups" (1971).
A noted pianist, he also composed music, often on mathematical themes, and contributed to performances of Gilbert and Sullivan productions in Chicago.
Irving Kaplansky was born in Toronto. He received his bachelor's and master's degrees from the University of Toronto before earning a doctorate in mathematics from Harvard in 1941.
After early work at Columbia, Dr. Kaplansky moved to Chicago in 1945. He was named a professor of mathematics there in 1955, and a professor emeritus in 1984. He became an American citizen in the 1950's.
Dr. Kaplansky was a member of the Institute for Advanced Study in Princeton, N.J., and was elected to the National Academy of Sciences in 1966. He also was president of the American Mathematical Society.
In 1989, the society awarded him its Leroy P. Steele Prize for Lifetime Achievement.
Dr. Kaplansky is survived by his wife of 55 years, the former Chellie Brenner.
He is also survived by a daughter, Lucy, a singer-songwriter, of Manhattan; two sons, Alex, of Hillsborough, N.J., and Steven, of Sherman Oaks; and two grandchildren.
As a musician entranced with the mathematical possibilities of music, Dr. Kaplansky once wrote a melody based on assigning notes to the first 14 decimal places of pi. Called "A Song About Pi,'' it received lyrics in 1971 from a Chicago colleague, Enid Rieser, and has been sung by Dr. Kaplansky's daughter in her act.
Irving Kaplansky, 89, a Pioneer in Mathematical Exploration, Is Dead

George Gadanidis

Western Associate Education professor George Gadanidis has been honoured by the Fields Institute for his continuing work in the field of mathematics.
Gadanidis was one of six Canadians recently inducted as a Fields Institute Fellow, a lifetime appointment conferred on individuals who have made outstanding contributions to the Fields Institute, its programs, and to the Canadian mathematical community. a The Fields Institute, located in Toronto, is recognized as one of the world's leading independent mathematical research institutions. With a wide array of pure, applied, industrial, financial and educational programs, the Institute attracts over 1,000 visitors annually from every corner of the globe, to collaborate on leading-edge research programs in the mathematical sciences.
Gadanidis has given significant support to the Fields program in Mathematics Education and continues to serve on the steering committee of the Fields Mathematics Education Forum.
Gadanidis honoured for mathematical contributions

Two male Erigone spiders on a grass seed head.
The lower one is in a pre-ballooning posture
ready to disperse, known as the "tip-toe" position.

Researchers have developed a new model that explains how spiders are able to 'fly' or 'parachute' into new territory on single strands of silk – sometimes covering distances of hundreds of miles over open ocean. By casting a thread of silk into the breeze spiders are able to ride wind currents away from danger or to parachute into new areas. Often they travel a few metres but some spiders have been discovered hundreds of miles out to sea. Researchers have now found that in turbulent air the spiders' silk moulds to the eddies of the airflow to carry them further.
The team at Rothamsted Research, a sponsored institute of the Biotechnology and Biological Sciences Research Council (BBSRC), realised that the existing 20 year old models to explain this phenomenon – known as 'ballooning' – failed to adequately deal with anything other than perfectly still air. Called Humphrey's model it made assumptions that the spider silk was rigid and straight and the spiders were just blobs hanging on the bottom. It could not explain why spiders were able to travel long distances over water, to colonise new volcanic islands or why they were found on ships. The new Rothamsted mathematical model allows for elasticity and flexibility of a ballooning spider's dragline – and when a dragline is caught in turbulent air the model shows how it can become highly contorted, preventing the spider from controlling the distance it travels and propelling it over potentially epic distances.
Dr Andy Reynolds, one of the scientists at Rothamsted Research, explained: "Researchers knew that spiders could use ballooning to cover long distances but no previous model has adequately explained how this worked. By factoring in the flexibility of the dragline that the spiders cast into the breeze have shown how it can contort and twist with turbulence, affecting its aerodynamic properties and carrying its rider unpredictable distances. Spiders are key predators of insects and can alleviate the need for farmers to spray large quantities of pesticide. But they can only perform this function in the ecosystem if they arrive at the right time. With our mathematical model we can start to examine how human activity, such as farming, affects the dispersal of spider populations."
Dr Dave Bohan, a member of the research team, commented on how mathematical models and traditional bioscience observation come together: "To really understand the factors at play on ballooning spiders we need to watch them in action. We have already observed spiders ballooning through still air and we are now planning to take them into a wind tunnel to watch how they handle turbulent flows. Once we have done that we can refine the model further." Professor Julia Goodfellow, Chief Executive of BBSRC, the organisation which funded the project, said: "The exciting thing about this research is that it not only explains a long-standing question but also shows how ecologists, mathematicians and physical scientists can draw on each others strengths. The future face of bioscience is highly interdisciplinary and will require more collaboration between, for example, mathematicians and ecologists working together to answer biological questions."

Images available at: http://www.bbsrc.ac.uk/media/pressreleases/06_07_12_spiders.html
Image available is: Two male Erigone spiders on a grass seed head. The lower one is in a pre-ballooning posture ready to disperse, known as the 'tip-toe' position.
Contact
Dr David Bohan, Rothamsted Research, Tel: 01582 763133 ext. 2454, email david.bohan@bbsrc.ac.uk
Dr Andy Reynolds, Rothamsted Research, email: andy.reynolds@bbsrc.ac.uk
Dr James Bell, Rothamsted Research, email: james.bell@bbsrc.ac.uk
Rothamsted Research
Elspeth Bartlet, Tel: 01582 763133 ext 2260, email: elspeth.bartlet@bbsrc.ac.uk
BBSRC Media Office
Matt Goode, Tel: 01793 413299, email: matt.goode@bbsrc.ac.uk
Tracey Jewitt, Tel: 01793 414694, email: tracey.jewitt@bbsrc.ac.uk
Notes to Editors
An article on this research appears in the July 2006 issue of BBSRC Business available soon. Business is the quarterly research highlights magazine of the UK's Biotechnology and Biological Sciences Research Council.
Further information about the Ecosystem Dynamics and Biodiversity Group at Rothamsted Research is available at: http://www.rothamsted.bbsrc.ac.uk/pie/EDG/ Rothamsted Research in Harpenden, Hertfordshire is one of the largest agricultural research institutes in the country and is sponsored by the BBSRC. For more information please visit: http://www.rothamsted.ac.uk/
About BBSRC
The Biotechnology and Biological Sciences Research Council (BBSRC) is the UK funding agency for research in the life sciences. Sponsored by Government, BBSRC annually invests around £350 million in a wide range of research that makes a significant contribution to the quality of life for UK citizens and supports a number of important industrial stakeholders including the agriculture, food, chemical, healthcare and pharmaceutical sectors. http://www.bbsrc.ac.uk
How parachute spiders invade new territory

Van Gogh - The Starry Nigth

Vincent van Gogh is known for his chaotic paintings and similarly tumultuous state of mind. Now a mathematical analysis of his works reveals that the stormy patterns in many of his paintings are uncannily like real turbulence, as seen in swirling water or the air from a jet engine.
Physicist Jose Luis Aragon of the National Autonomous University of Mexico in Queretaro and his co-workers have found that the Dutch artist's works have a pattern of light and dark that closely follows the deep mathematical structure of turbulent flow.
The swirling skies of The Starry Night, painted in 1889, Road with Cypress and Star (1890) and Wheat Field with Crows (1890) — one of the van Gogh's last pictures before he shot himself at the age of 37 — all contain the characteristic statistical imprint of turbulence, say the researchers.
These works were created when van Gogh was mentally unstable: the artist is known to have experienced psychotic episodes in which he had hallucinations, minor fits and lapses of consciousness, perhaps indicating epilepsy.
"We think that van Gogh had a unique ability to depict turbulence in periods of prolonged psychotic agitation," says Aragon.
In contrast, the Self-portrait with Pipe and Bandaged Ear (1888) shows no such signs of turbulence. Van Gogh said that he painted this image in a state of "absolute calm", having been prescribed the drug potassium bromide following his famous self-mutilation.
Measured chaos
Scientists have struggled for centuries to describe turbulent flow — some are said to have considered the problem harder than quantum mechanics. It is still unsolved, but one of the foundations of the modern theory of turbulence was laid by the Soviet scientist Andrei Kolmogorov in the 1940s.
He predicted a particular mathematical relationship between the fluctuations in a flow's speed and the rate at which it dissipates energy as friction. Kolmogorov's work led to equations describing the probability of finding a particular velocity difference between any two points in the fluid. These relationships are called Kolmogorov scaling.
Aragón and colleagues looked at van Gogh's paintings to see whether they bear the fingerprint of turbulence that Kolmogorov identified. "'Turbulent' is the main adjective used to describe van Gogh's work," says Aragn. "We tried to quantify this."
Darkness and light
The researchers took digital images of the paintings and calculated the probability that two pixels a certain distance apart would have the same brightness, or luminance. "The eye is more sensitive to luminance changes than to colour changes," they say, "and most of the information in a scene is contained in its luminance."
Several of van Gogh's works show Kolmogorov scaling in their luminance probability distributions. To the eye, this pattern can be seen as eddies of different sizes, including both large swirls and tiny eddies created by the brushwork.
Van Gogh seems to be the only painter able to render turbulence with such mathematical precision. "We have examined other apparently turbulent paintings of several artists and find no evidence of Kolmogorov scaling," says Aragon.
Edvard Munch's The Scream, for example, looks to be superficially full of van Gogh-like swirls, and was painted by a similarly tumultuous artist, but the luminance probability distribution doesn't fit Kolmogorov's theory.
The distinctive styles of other artists can be described by mathematical formulae. Jackson Pollock's drip paintings, for example, bear distinct fractal patterns.
Van Gogh painted perfect turbulence

di: Costantino D'Onorio De Meo

Il problema della fondazione dell'aritmetica come scienza sintetica a priori.

Vladislav Goldberg

Vladislav Goldberg, PhD, a distinguished professor in the department of mathematical sciences at New Jersey Institute of Technology (NJIT), and an expert in web geometry, was honored last month for a lifetime of scholarship. The International Geometry in Odessa Conference in the Ukraine lauded the 70-year-old mathematician during a multi-day conference. Goldberg, born and schooled in Moscow, emigrated to the US in 1979 during the immigration wave of the 1970s that brought into the US a number of highly educated Jewish scientists. Today, Goldberg has retained a network of scholarly friends in Russia, the US, Israel and many other countries.
Goldberg is renowned for his understanding of a little-known branch of geometry: web geometry. Only a small number of scholars study this field, although their expertise is frequently tapped by economists and physicists, especially those scientists studying thermodynamics. The late S.S. Chern, PhD, of the University of California, Berkeley, numbered among the past century's noted mathematicians who worked in web geometry.
Web geometry focuses on the non-changing, or invariant, properties of a series of curved lines laid over a grid of horizontal and vertical lines.
Goldberg and a small international group of other web mathematicians create the rules for studying or understanding web geometry. If there is only one family of curves to overlay the grid, a curvilinear three-web is produced," said Goldberg. "Two families of curves produce a curvilinear four-web. We can use an infinite number of curves in each web family, but only a finite number of families produce a web. The grid, which is the foundation, or starting point, always counts as the first two families."
Economists and physicists use web geometry as a tool to prove their theories. Through the years, Goldberg has served many colleagues in other disciplines. His most memorable collaboration was in 2003 with Nobel Prize economists Paul A. Samuelson of MIT, Thomas Russell of University of Santa Clara and James B. Cooper of Johannes Kepler Universität Linz, Austria. The trio challenged him to answer an unsolved problem. "They asked to me find the conditions under which a curvilinear web can be mapped into a web with all transversal families being families of straight lines," said Goldberg.
Goldberg accepted the challenge and in 2004 and again in 2006, he co-authored two breakthrough works refuting a 1938 assertion by Wilhelm Blaschke, a noted German mathematician credited with founding the field of web geometry. Blaschke (1862-1955), a colorful character in his own right, ended up snubbed by nations and academics later in life for his public Nazi sympathies during World War II.
In Einfuehrung in die Geometrie der Waben (1955) Blaschke wrote that it was "hopeless" for mathematicians to find the conditions under which a curvilinear web can be mapped into a new web with transversal, nonintersecting straight lines.
"By hopeless, he meant that the problem had impossible calculations to carry out by hand," explained Goldberg. Of course, Blaschke's words were penned prior to the arrival of personal computing. Some 60 years later, Goldberg and coauthors, with the aid of advanced computer software programs, proved Blaschke wrong. In 2004, Maks A. Akivis, of the Jerusalem Institute of Technology, Goldberg and Valentin V. Lychagin, of the University of Tromso, Norway, each a renowned mathematical author and scholar, solved the problem for all webs, except three-webs. Selecta Mathematica published "Linearizability of d-webs, d> 4, on two-dimensional manifolds" in December of 2004. Then, in 2005, Goldberg and Lychagin solved the more difficult variant of the problem, finding the conditions under which a curvilinear three-web can be mapped into a linear three-web with transversal straight lines. The Journal of Geometric Analysis published "On the Blaschke conjecture for 3-webs" in March of 2006. Comptes Rendus Mathematique, published a six-page version of the same work in August of 2005.
Since 1958, Goldberg has published four monographs, eight textbooks, three book chapters and more than 120 scientific papers. Goldberg received his master's and doctoral degrees in mathematics from Moscow State University. From 1964 to 1978, Goldberg was a professor in the department of mathematics at Moscow Institute of Steel and Alloys. In 1979, after immigrating to the US, Goldberg worked at Lehigh University for two years as a visiting professor. In 1981, he joined NJIT as a full professor. In 1985, NJIT made Goldberg one of the university's earliest distinguished professors. At the ceremony this past May in Ukraine, Goldberg's colleagues from different countries warmly lauded his contribution to science and education. They presented him with a formal bound document which was read aloud to him at the ceremony. The document said: "We wish that you continuously move ahead and successfully complete all your plans and intentions. Let the welfare and success and health of all your family members assure your peace of mind and good mood. Let your loyal friends and your highly professional colleagues, united by joint goals, support your professional success which will bring progress and success to our common cause."

Ron Eglash, associate professor of science and technology studies at Rensselaer, has created a suite of 11 computer software programs that focus on individual facets of African American, Native American, or Latin American culture where math plays a role in design. Called "culturally situated design tools" (CSDTs), the programs educate students about the mathematics principles used to design cornrow hairstyles, Mangbetu art, Navajo rugs, Yupik parka patterns, Pre-Columbian pyramids, and Latin music, among others. New research reported in the June 2006 issue of American Anthropologist suggests that use of CSDTs can raise math achievement and may improve technological career aspirations for ethnic minority students.

About Rensselaer
Rensselaer Polytechnic Institute, founded in 1824, is the nation's oldest technological university. The university offers bachelor's, master's, and doctoral degrees in engineering, the sciences, information technology, architecture, management, and the humanities and social sciences. Institute programs serve undergraduates, graduate students, and working professionals around the world. Rensselaer faculty are known for pre-eminence in research conducted in a wide range of fields, with particular emphasis in biotechnology, nanotechnology, information technology, and the media arts and technology. The Institute is well known for its success in the transfer of technology from the laboratory to the marketplace so that new discoveries and inventions benefit human life, protect the environment, and strengthen economic development.
Math Lessons Get a Makeover: New Tools Spark Student Interest, Achievement in Mathematics

Eberhard Zeidler (66) was awarded the prestigious Alfried Krupp Prize at a ceremony in Essen (Germany) on 13 June 2006. The prize is awarded to leading scientists for their life's work by the Alfried Krupp von Bohlen und Halbach Foundation. The prize was given in recognition of Zeidler's achievements in the field of non-linear functional analysis and for his scientific contribution to issues concerning the application of mathematical research in the natural sciences. The award was set up in 1998 and carries prize money of €52,000.

Springer author wins the Alfried Krupp Science Prize

Citation

Lumsden, C. (2006). Of Biocultural Mathematics and Mind. Evolutionary Psychology, 4:57-74.

Date of article 30 June 2006 Email Charles Lumsden Links

Of Biocultural Mathematics and Mind

Da quasi due mesi a Palazzo Ducale a Genova la gente paga (2 euro) per fare matematica. Sono quasi 2000 le persone che hanno partecipato finora a MateFitness, la Palestra della Matematica. Con l'aiuto di studenti universitari-animatori persone di tutti i generi si sono divertite col «puzzle di Pitagora», con le «fette di torta» di legno per capire le frazioni, con i sudoku e con un altro centinaio di attrazioni matematiche.
Nonostante la sua pessima fama di materia barbosa e complessa, se presentata al pubblico in maniera accessibile, la matematica diventa interessante e autenticamente divertente.
Secondo uno studio condotto da Irving Biederman della Southern California University, apparso su American Scientist, il cervello si gratifica per aver afferrato un concetto difficile o interessante producendo sostanze simili agli oppiacei. E allora forse, una volta vinta la resistenza iniziale, si può diventare "dipendenti" dalla matematica!
Un meccanismo gratificante di questo tipo è il mezzo che l'evoluzione ha scelto per sviluppare l'intelligenza, e spiega in maniera plausibile la curiosità cognitiva in generale.
Sempre a proposito di matematica un altro recentissimo studio condotto da Yijun Liu, Yiyuan Tang e colleghi della Dalian University of Technology, di Dalian, in Cina, ha ipotizzato che essa sia strettamente legata al linguaggio, e che la lingua madre abbia un'influenza sul modo in cui il cervello affronta diversi compiti numerici. Sottoponendo a risonanza magnetica funzionale volontari sottoposti a test di aritmetica è stato riscontrato che per risolvere un problema di addizione, i soggetti di madrelingua inglese mostrano la presenza di attività nell'area che sovrintende al linguaggio, mentre i soggetti di madrelingua cinese utilizzano regioni cerebrali più legate all'elaborazione delle informazioni visive, in accordo con la tipologia ideografica della loro lingua.
Link | Una bella risorsa on line è il sito Matematica dell'Università Bocconi

La matematica: fra oppio e linguaggio

Sam Vaknin, Ph.D. is the author of Malignant Self Love - Narcissism Revisited and After the Rain - How the West Lost the East. He served as a columnist for Central Europe Review, PopMatters, Bellaonline, and eBookWeb, a United Press International (UPI) Senior Business Correspondent, and the editor of mental health and Central East Europe categories in The Open Directory and Suite101. Until recently, he served as the Economic Advisor to the Government of Macedonia. Sam Vaknin's Web site is at http://samvak.tripod.com
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What is Intuition?

—Jennifer Allen
PORTRAIT OF THE PSYCHOANALYST AS AN ARTIST