Confirmed Participants Schedule and Abstracts (PDF) Final Report (PDF)

L-functions, ranks of elliptic curves, and random matrix theory (07w5114)


(American Institute of Mathematics)

Michael Rubinstein (University of Waterloo)

(University of Bristol)


Number Theorists, analysts, computational specialists and physicists came together at BIRS this week, July 8-13, 2007 to try to understand some very exotic mathematical phenomena. The search is for certain special functions in number theory, called L-functions, which can have very useful properties.

"One of the best ways to create a new mathematical tool is to discover a function with fantastic properties," said Michael Rubinstein of the University of Waterloo, one of the conference's organizers. "We are tapping into Random Matrix Theory to point the direction."

Random Matrix Theory (RMT) was largely developed by physicists studying statistical properties of the energy levels of excited nuclei. "The fact that Random Matrix Theory now serves a role in Number Theory - whose chief studies are prime numbers and whole number solutions to equations - is somewhat mind boggling," admits Nina Snaith of the University of Bristol,a co-organizer and one of the pioneers in applying RMT to Number Theory questions. But the vast amount of computational evidence amassed in the months leading up to this workshop shows that new breakthroughs will come from combining RMT and Number Theory. The organizers and participants are hoping that by the end of the week they will have constructed several new examples of useful L-functions.

"This workshop involves many people from different disciplines but focused on a common goal," said David Farmer of the American Institute of Mathematics, who is planning his own workshop on L-functions later this summer.

"The workshop talks are giving everyone a solid foundation,but what is really important is that the schedule allows plenty of time for interaction. I am getting lots of good ideas here."

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines within the industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the US National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnologa (CONACYT).