New Tools for Old Problems

AIM - October 2007

circle When mathematicians notice connections between two distinct areas of their discipline, you can be sure something interesting will develop. That is exactly what happened at a recent workshop at the American Institute of Mathematics (AIM).

The scientific advisory board of AIM noticed that new methods in ergodic theory, a subject arising from mathematical physics, may have applications in other areas. So AIM approached Anatole Katok (Penn State), Elon Lindenstrauss (Princeton), and Ralf Spatzier (University of Michigan) about organizing a workshop designed to explore these new connections. This led to the the workshop, “Emerging Applications of Measure Rigidity.” The workshop attracted researchers from several areas of mathematics, all with the goal of determining whether these new techniques in ergodic theory could have applications in their own specialty.

AIM has a tradition of sponsoring workshops that bring together a diverse group of researchers with the goal of finding some common ground. “It can be a challenge for mathematicians in different areas to work together,” said AIM Director Brian Conrey. “But there is a big payoff when you get to apply a new set of tools to old problems.”

Two of the workshop participants were Manfred Einsiedler (University of Washington) and Elon Lindenstrauss (Princeton) who, together with Anatole Katok, had done some of the work in ergodic theory that motivated the workshop. During a discussion session, number theorists Akshay Venkatesh (MIT) and Philippe Michel (Universite de Montpellier II) noted a possible connection between these recent ideas and some all but forgotten, 50-year-old work of the Russian mathematician Yu. Linnik.

After some quick study of the old literature, they found that Linnik’s work did indeed point to a remarkable connection between the disciplines. “Michel’s observation proved to be a gold mine, which we have successfully probed for two years already,” says Venkatesh. “Work of W. Duke (UCLA) completely settled Linnik’s problem in the quadratic field case. One of our main accomplishments to date is the resolution of the cubic field case — a problem studied by Linnik himself, but which has taken the intervening fifty years of technological development — both in number theory and ergodic theory — to address. More importantly, this work has uncovered rich mathematical structures, which, I hope, will provide much interesting material for future work.”

Venkatesh summarized the new work in the following way: “Number theory has its roots in pure mathematics in the classical sense, while ergodic theory stems from the study of long-term evolution of complicated physical systems, such as planetary motion. In technical terms, the key fact which makes it all work is that closed orbits of tori on homogeneous spaces are related both to the torus actions, and to class groups of number fields.”