Prof. Alex Eskin is among the 16 scholars of mathematics, theoretical physics, theoretical computer science and the mathematical modeling of living systems who have been selected as 2014 Simons Investigators.
Eskin is the Arthur Holly Compton Distinguished Service Professor in Mathematics.
The Simons Foundation will provide each investigator with $500,000 of support over the next five years, enabling them to undertake long-term study of fundamental questions.
The foundation’s announcement of this year’s investigators said, “Alex Eskin is a leading geometer with important contributions to geometric group theory, ergotic theory and number theory. He has applied ideas from dynamical systems to solve counting problems in the theory of Diphantine equations, the theory of mapping class group and mathematical billiards on rational polygons.”
Eskin’s research deals with a longstanding problem related to ergodicity or randomness. An example would be the dynamics of a billiard ball on a frictionless polygonal table. If a ball becomes set in motion, it would travel forever, all the while colliding with the walls of the table.
“If the table is a square or an equilateral triangle, one can easily show that there are only two possible behaviors,” Eskin said. “Either the ball repeats the same path forever, or it travels randomly in the entire polygon, eventually visiting the neighborhood of any point.”
He directs his work toward the basic mathematical problem of understanding the behavior of a billiard ball in more general polygons. “This becomes much more difficult because most polygons do not tile the plane,” he said, meaning that they do not fit together to cover a given space without gaps or overlaps.
Such problems are motivated by problems in statistical mechanics, which deals with the motions of large numbers of interacting particles. “Given any physical system, one may ask whether the motion is ‘ergoic,’ namely, provably random in a certain sense,” Eskin said. This concept, which goes back to Austrian physicist Ludwig Boltzmann (1844-1906), “has been highly fruitful for mathematics,” he said.
In situations in which the angles of the polygon are rational (can be written as a simple fraction or ratio), the problem of interest to Eskin connects to other fields of mathematics: Lie groups and Teichmuller theory.
Eskin is the third UChicago faculty member to become a Simons Investigator, joining Ngô Bao Châu, the Francis and Rose Yuen Distinguished Service Professor in Mathematics; and Dam Thanh Son, the University Professor of Physics, who both were selected in 2013.
A member of the American Academy of Arts and Sciences and a fellow of the American Mathematical Society, Eskin’s honors also include the Clay Research Prize.