Random Matrices, Inverse Spectral Methods and Asymptotics (08w5017)


Estelle Basor (American Institute of Mathematics)

(Concordia University)

(IPhT CEA Saclay)

(Centre de recherches mathematiques, Universite de Montreal, and Concordia University)

Alexander Its (Indiana University - Purdue University Indianapolis)

(University of Arizona)


The Banff International Research Station will host top researchers in its workshop on "Random Matrices, Inverse Spectral Methods and Asymptotics" next week, October 5 - October 10, 2008. The study of random matrix theory in physics dates back at least to the middle of the last century, and originates in studies of the statistical properties of high lying energy levels of large atomic nuclei. Matrices are finite dimensional analogs of the linear operators that appear in the quantum theory of such complex systems, and this approach was based on the idea that "universal" behaviour appears at a statistical level, independently of the detailed quantum dynamics involved, which were too complex in any case to allow for computation of individual energy levels. The mathematical theory has been considerably developed since that time, and newer applications were found in a stunning variety of seemingly unrelated areas of mathematics and physics. These include, on the mathematical side: combinatorial problems relating to enumeration of inequivalent graphs on surfaces; the computation of fundamental topological quantities characterizing inequivalent surfaces and sets of marked points on them, computation of the statistical properties of certain types of random sets, such as the lengths of increasing subsequences in arbitrary sequences of random numbers, or the number of ways an integer can be partitioned into a sum of other integers, and the intriguing problem of distribution of the points where a certain complex function, the Riemann zeta function vanishes, which implies very detailed information regarding the statistical distribution of the prime numbers. On the physics side, the spectral theory of random matrices has turned out to be of direct importance in computations of the features characterizing the growth e.g., of crystals, the boundaries between fluids having different viscosities, some of the fundamentals of quantum gravity and the basic forces governing interactions of the fundamental particles of matter at the most microscopic scales.

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 and with 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 Tecnologí­a (CONACYT).