Corrections appended.
Professor of Mathematics Jeremy Kahn won the 2012 Clay Research Award — one of the most prestigious awards in mathematics — for his work on hyperbolic geometry. Kahn and his collaborator Vladimir Markovic published two papers in 2009 and 2011 containing "major breakthroughs in mathematical research," according to the award's website.
Hyperbolic geometry studies the properties of a hyperbolically curved space. Think the surface of a saddle, where there is more distance between any two points than on a flat plane. In hyperbolic space, the sum of angles in a triangle can be less than 180 degrees.
In the 2009 paper, Kahn and Markovic established that if you take a two-dimensional hyperbolic space — the saddle surface — and you put it in a three-dimensional hyperbolic space, it is impossible to completely flatten it. But it is possible for it to sit in the space so that it is only very slightly curved. Properties of the space correlate with properties of the surface, allowing them to apply ideas from one space to the other.
"It's an unusually simple theorem," said Kahn. "That's part of the attraction."
Kahn, who came to Brown this past fall, first became interested in exploring these spaces when he was working at the State University of New York at Stony Brook in 2007. Though Kahn had not been working on hyperbolic space at the time, his adviser encouraged him to attend a talk by another student — Vladimir Markovic. After the lecture, he presented Markovic with a possible approach to solving a problem, and thus, the partnership was born.
"I loved working with (Kahn)," said Markovic. "We're good friends of a similar age and have other things in common besides math." But he added that when they get together, "it's mainly just hard work."
Last year, Kahn and Markovic explored the patterns that could "tile" a hyperbolic surface. In this second award-winning paper, the duo showed that it is possible to approximate the same cover using differently shaped tiles, proving what is known as the Ehrenpreis Conjecture.
"You can take a pattern that's repeating and find a larger repeating unit and think of that larger thing as what's repeating," Kahn said. "And then you could replace it with a different repeating pattern so it would only repeat on this larger scale."
This form of discovery was nothing new to Kahn, who said mathematics was "a significant part of (his) identity, even at age three or four." Back then, he played with differently-sized rods, forming them into squares and cubes. Years later, he went to Harvard as an undergraduate and the University of California at Berkeley for graduate school.
But then he hit a snag. "I could imagine everything," he said. "But I couldn't write it down on a piece of paper."
It took meeting collaborators like Markovic to jump-start his publishing career. He would visit Markovic, who was in England at the time, for chunks of two to four weeks' length.
"When we proved the first theorem, I visited him for 10 days in June, and all we did was talk about things," Kahn said. "Maybe write a little bit on the board or something, but we had essentially no record of what we had done."
He came home knowing he had solved a big problem, but he had no proof. It took another month of correspondence to hammer out the Ehrenpreis Conjecture paper.
Kahn and Markovic wrote their other paper just as quickly.
"It's very unusual to prove a major theorem, of course, and it's extraordinarily unusual to work it out essentially in two weeks over the summer," Kahn said. "And it's practically unheard-of to do it twice in a row."
But Kahn and Markovic are currently shooting for a hat trick and have already set out to prove another major theorem.
An earlier version of this article incorrectly referred to Kahn and Markovic's 2009 paper as a proof of the Ehrenpreis Conjecture. In fact, the Ehrenrpreis Conjecture relates to the tiling of a hyperbolic space, and Kahn and Markovic proved the conjecture in their 2011 paper. The article also incorrectly stated that Kahn's adviser at the State University of New York at Stony Brook encouraged him to attend Markovic's lecture. In fact, Kahn's collaborator Mikhail Lyubich encouraged him. The Herald regrets the errors.
