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A Mathematician's Pursuit of Symmetry

Through his award-winning work, Robert Moody has opened doors to the past and the future

Robert Moody wouldn't stand out in a crowd. On a street corner, you wouldn't pick him out as someone with a branch of algebra named after him, a person whose work changed the course of much of mathematics and mathematical physics in the second half of this century.

Moody you would pick out as the guy down the block who likes to tinker in his garage in the evening and on weekends.

And, in a sense, Moody, who is a professor of mathematical sciences at the U of A, is a tinkerer: he likes to find out how things work. Not toasters, though, or carburetors. Things like fivefold symmetry and the subtleties of superstring theory.

During 30-plus years as a mathematician, Moody's contributions have set him apart from all hut a handful of colleagues in the world — a fact that will be underlined this summer when he will receive the prestigious Eugene Wigner Medal of the Group Theory and Fundamental Physics Foundation.

The presentation, which will take place in August at the Foundation's congress held in Goslar, Germany, will end a bit of a wait for Moody. He is a co-winner of the medal for 1994-95, hut since the Foundation meets only every second year, the presentation ceremonies had to wait. (Proof, perhaps, that to devotees of group theory and fundamental physics, time is relative.)

The U of A mathematician is being honored jointly with Victor Kac (that's pronounced "kats"), formerly of the University of Moscow and now of MIT, for the development of the Kac Moody algebras. In the mid 1960s, beginning from completely different points of view, Kac and Moody independently enlarged the paradigm of the algebras invented in the 19th century Sophus Lie (pronounced "lee"), whose work provided the mathematical objects — Lie groups — used to study continuous symmetry.

"Symmetry is a very big part of modern in mathematics," explains Moody, "because it is one of the key ways in which we can describe order and repetitive structure in nature."

Exciting times

Before the breakthroughs made by Moody and Kac, mathematicians worked extensively with "discrete" symmetry — the simple sort of symmetry seen in figures such as squares and hexagons and in crystals — and "continuous" symmetry — the symmetry of a sphere, for instance. Using Lie groups as their tools, Moody and Kac expanded this type of mathematical pursuit to encompass "infinite dimensional" symmetry.

"All the mathematics we did deals with infinite-dimensional objects," says Moody, admitting that this "sounds totally bizarre to normal ways of thinking — but all sorts of things in mathematics and physics involve infinite-dimensional spaces."

Symmetry is not just of abstract mathematical interest. Continuous symmetries play an important role in quantum mechanics, relativity theory, and most of theoretical physics. And by the 1980s, physicists working in the areas of elementary particle theory, gravity, and two-dimensional phase transitions had discovered that the Kac Moody symmetries could also he used to formulate descriptions of nature.

Moody, who taught at the University of Saskatchewan then, recalls the excitement of the time: "For a few very exciting years mathematicians and physicists were closely following each other's works. The physicists had, to some extent, actually come across [the Kac Moody algebras] without knowing it, really. At this point, which is around 1980, a lot of things clicked in people's minds, and it opened up a tremendous amount of involvement of the mathematical physicists, who had a whole bunch of new tools to show us."

As the Kac Moody algebras have found their place in Lie theory — "they've been subsumed into the whole fabric of what people think about now," says Moody — the excitement that came with their freshness has naturally waned, but the Wigner Medal is proof that the importance of Moody's contribution to the expansion of Lie theory is not forgotten.


And Moody, who came to the U of A in 1989, continues to draw attention with his work. George Seligman, the James E. English professor of mathematics at Yale, says that Moody's work has opened doors to the past and to the future, Commenting about Moody's new treatise, Lie Algebras with a Triangular Decomposition, English says that with this work (written jointly with the U of A's Arturo Pianzola) Moody has "consolidated with a very broad scientific community the position of leadership that we in his general field have long accorded him."

Seligman's colleague at Yale, the distinguished mathematician Efim Zelmanov says Moody's work has changed the course of much of mathematics and mathematical physics in the second half of this century. He likens Moody not to a tinkerer hut to a composer. "Leo Landau once said that there are 'physicist-composers' and 'physicist-performers.' In my opinion Dr Moody is undoubtedly a 'mathematician-composer'."

Tinkerer or composer — and mathematician, for certain — Robert Moody has shown he stands apart.

Published Summer 1996.

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