Making waves

For the stadium system yet another interesting thing can happen, known as a "bouncing ball mode." Bouncing ball modes were observed experimentally and only recently proven to exist by Andrew Hassell.

In their QUE conjecture, Rudnick and Sarnak hypothesized that for a large class of systems, unlike the stadium there are no scars or bouncing ball states and in fact all states become evenly distributed. Holowinsky and Soundararajan's work shows that the conjecture is true in the number theoretic setting.

Highly excited states

The conjecture of Rudnick and Sarnak deals with certain kinds of shapes called manifolds, or more technically, manifolds of negative curvature, some of which come from problems in higher arithmetic. The corresponding waves are analogous to highly excited states in quantum mechanics.

Soundararajan and Holowinsky each developed new techniques to solve a particular case of QUE. The "waves" in this setting are known as Hecke eigenforms. The approaches of both researchers work individually most of the time and miraculously when combined they completely solve the problem. "Their work is a lovely blend of the ideas of physics and abstract mathematics," said Brian Conrey, Director of the American Institute of Mathematics.

According to Lev Kaplan, a physicist at Tulane University, "This is a good example of mathematical work inspired by an interesting physical problem, and it has relevance to our understanding of quantum behavior in classically chaotic dynamical systems."

Soundararajan's talk will take place at noon on Friday, October 10, at the Stanford Math Department.

About the American Institute of Mathematics

The American Institute of Mathematics, a nonprofit organization, was founded in 1994 by Silicon Valley businessmen John Fry and Steve Sorenson, longtime supporters of mathematical research. AIM is one of the seven mathematics institutes in the U.S. funded by the National Science Foundation. The mission of AIM is to expand the frontiers of mathematical knowledge through sponsoring focused research projects and workshops and encouraging collaboration among mathematicians at all levels. AIM currently resides in Palo Alto, California, while awaiting the completion of its permanent headquarters in Morgan Hill, California. For more information, visit www.aimath.org.

Contacts:
Brian Conrey, Executive Director of AIM, 650.845.2071,
Peter Sarnak, Eugene Higgins Professor of Mathematics at Princeton University and Chair of AIM's Scientific Board, 609.258.4200 or 609.258.4229,