Insights into symmetry

Abel Prize awarded for advances in group theory

The 2008 Abel Prize was awarded to John Griggs Thompson of the University of Florida and Jacques Tits of the Collège de France for their contributions to group theory, the mathematical field that analyzes symmetry. The Abel Prize is widely considered mathematics’ equivalent of the Nobel Prize.

Thompson, a professor at the University of Florida, proved fundamental results about symmetry that led to the discovery of the basic building blocks of all finite symmetries, one of the greatest findings in 20th century mathematics. Tits showed how to create geometrical objects with the symmetry of almost all of these building blocks.

The study of symmetry has influenced much of mathematics and portions of science during the last 150 years. Physicists use symmetry to understand the deepest patterns of the universe, computers use it to transmit information across noisy channels without errors and chemists use it in crystallography.

Mathematicians consider an object to be symmetric if you can pick it up, move it around in some way, put it back down and have it look just the same as when you started. For example, a heart shape has mirror symmetry: look at the right half in a mirror and it looks exactly like the left half. A snowflake has rotational symmetry because you can rotate it a one-sixth turn and it looks unchanged. A repeating wallpaper frieze has translational symmetry: pick it up and move it over, and it looks just the same.

Draw any pattern you please on a piece of paper, and any symmetries it has will be combinations of these three types of movements, mathematicians have shown. But what kinds of symmetry (also called symmetry groups) might patterns or objects in three dimensions have, or even, if our minds can imagine them, might they have in higher dimensions still? This is the question that Thompson’s work set the stage to answer.

The first step toward answering this question came late one night in 1832 as Évariste Galois hastily scribbled down notes on his work before a morning duel. All finite symmetry groups, he wrote, are combinations of a few basic, indivisible symmetry groups, just as every molecule in chemistry is a combination of a few different types of indivisible atoms.

The symmetry group of a triangle, Galois realized, couldn’t be broken down any further, nor could that of a pentagon. But a 15-sided polygon is another story. Galois observed that a single one-fifteenth rotation can be accomplished by rotating forward two-fifth of a turn (using the symmetries of a pentagon) and then back one-third of a turn (using the symmetries of a triangle). So the symmetry group of a 15-gon is actually a combination of the symmetries of a triangle and a pentagon. (To view an animation of this, click here and scroll down to pink and purple figure.)

The problem was that Galois didn’t know all the elements in the periodic table of symmetry, called “finite simple groups.” What were these atoms of symmetry? He knew of a few, like the triangle and the pentagon, but he didn’t know whether there were more, how to find any additional ones or how to know when he had them all. Unfortunately, he never got to engage in the search: He was killed in his morning duel.

Progress on the problem crawled for the next 130 years. Mathematicians knew of several families of simple groups, and they knew of five peculiar simple groups that weren’t in any of the families. From time to time, a new one of these strange groups was discovered, but mathematicians were making little progress on the problem as a whole. “Nobody really had a clue of what to do,” says Ronald Solomon of Ohio State University in Columbus.

“Then Thompson arrived on the scene in the 1950s,” Solomon says, “and completely revolutionized the field.” Thompson introduced a powerful set of techniques to analyze the structure of groups like the “signalizer method” and “Thompson factorization.”

In 1962, he and the late Walter Feit used these techniques to make a breakthrough that suddenly made the full classification of finite simple groups seem possible. They proved a 60-year-old conjecture, a task that had been thought to be next to impossible. Except for the most elementary, well-known examples, they showed, all of the finite simple groups had to have an even number of symmetries. The proof was 255 pages long, perhaps the longest proof developed to that time. Thompson used the result to classify a special class of groups, known as the N-groups.

Dozens of group theorists jumped to the task of proving the full classification theorem. The job was so huge that it took more than two decades to finish. In the process, theorists discovered 21 more of the peculiar, isolated simple groups, until the number reached 26.

The complete theorem is tens of thousands of pages long, spread over journal articles with more than 100 authors.
The mind-boggling complexity of the proof has led some to doubt whether it’s complete and correct. Solomon quotes Michael Aschbacher of the California Institute of Technology, one of the leaders in the effort, as saying that although he thinks it’s complete and correct, he wouldn’t bet his house on it. “But his house is in California and mine is in Ohio, so his is more expensive,” Solomon says. “Maybe I would bet my house on it. But I wouldn’t bet my child.”

Even with these slight doubts, “the solution is considered by many to be one of the absolute high points of 20th century mathematics,” says Richard Lyons of Rutgers University, the State University of New Jersey, at New Brunswick. “And the idea’s that Thompson had in the ’60s and early ’70s formed the foundation and for the most part the entire structure of the proof.”

Not all groups were originally derived from the symmetries of a geometric object. Groups are used in many different areas of mathematics, illuminating connections between fields that seem very far apart. Nonetheless, Jacques Tits found that nearly any group can indeed be seen as symmetries.

He constructed geometric objects he called “buildings” with symmetries that correspond precisely to the elements of nearly any group. They are a kind of generalization of the Platonic solids like the dodecahedron or icosahedron, creating a sort of very high-dimensional crystal. The geometry of these crystals reveals the structure of the groups they came from, giving mathematicians a new way to understand them.

The award will be given May 20 in a ceremony in Oslo, Norway.

SYMMETRICAL SNOWFLAKE Snowflakes show nearly perfect six-fold rotational symmetry. Kenneth Libbrecht,
SYMMETRY ON THE WALL This wallpaper frieze has translational symmetry.
IN EIGHT-DIMENSIONS The Gossett Polytope, an eight-dimensional crystal depicted here, is the skeleton of the Tits building of the most complex of Tits’ geometries. It is closely related to The Monster, the largest of the “sporadic groups” — isolated simple groups that don’t belong to a family of groups. J. Stembridge

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