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Issue: 04/30/03
Mathematician's research 'knot' for the impatient
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04/30/03
Paul Francuch
When Ian Agol decided he wanted to make the study and teaching of mathematics his career, he knew the job came with strings attached.
Agol, assistant professor of mathematics, statistics and computer science, specializes in knot theory, the study of closed loops of string.
“You can encode a knot so that a computer can understand it, in terms of writing down a set of symbols that represent a particular projection of the knot,” he says.
In March, Agol learned that he had won a research fellowship from the New York-based Alfred P. Sloan Foundation. He’ll get $40,000, which he plans to spend on computer software and travel for professional collaborations and study.
Math department chair Jerry Bona says Agol’s work is well known and appreciated internationally.
“Ian works on three-manifold topology, one of the central areas of mathematics research in the last three decades,” says Bona.
“He has made several fundamental contributions and developed new construction that cast light where none was expected.”
“I study three-dimensional spaces,” Agol says. “Topologists consider two spaces to be equivalent if one can be deformed into the other without tearing or gluing. We’re not worried about the geometry as much as the shape of spaces up to deformation.”
Agol majored in math as an undergrad at the California Institute of Technology.
“I was interested in math and physics but I didn’t do so well in some physics classes, so I went into math,” he laughs.
“I like mathematics. Many academic fields are imprecise, but mathematics is probably the most precise of any field.”
While Agol jokes about his less than stellar performance in physics, his research into three-dimensional hyperbolic spaces has been used by astrophysicists in considering the possible shapes of the universe.
And there are those knotty problems that come with untangling all that string.
“One of the things I try to understand is the unknotting problem. Given a very tangled-up loop of string, how hard is it for a computer program to analyze the loop of string and distinguish whether it is round?
“It may be hard to show that a knot cannot be untangled to be round, but once I’ve figured out how knotted it is, I’ve proven a theorem that implies that I can show someone else fairly quickly whether or not the knot can be untangled — much more quickly than it took me to figure out whether it could be untangled.
“The notion of quickness here is relative, but has a mathematical meaning that is used by computer scientists to distinguish the hardness of various algorithms.”
While abstract to many people, knot theory has interested mathematicians for more than a century. It is also used in genetics.
Agol is the 15th UIC faculty member to receive a Sloan Research Fellowship, which is awarded to young scientists judged likely to make fundamental contributions to their fields.
The award is seen as a good predictor of future accomplishments. Twenty-six Sloan recipients have gone on to win Nobel prizes.
Above: Ian Agol studies knot theory.
Photo: Troy Heinzeroth
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