The d-bar method: Inverse scattering, nonlinear waves, and random matrices (12frg176)


(University of Arizona)

(University of Kentucky)


The "The d-bar method: Inverse scattering, nonlinear waves, and random matrices" workshop will be hosted at The Banff International Research Station.

International exprerts will gather at the Banff International Research Station for a two-week intensive research focus on the "∂-method" in inverse scattering theory. Examples of inverse scattering include medical imaging, in which images of a patient are reconstructed from the patient's response to electromagnetic radiation, and geophysical prospection, in which underground reservoirs of oil are located by surface measurements of acoustic waves. In each case, the physical process (propagation of electromagnetic or acoustic waves) is governed by a partial differential equation (PDE) which depends on the quantity to be reconstructed. The "inverse" problem is to use measurements of the solution of the PDE to find the physical quantity--that is, to "invert" the solution to find the equation.rn In a fascinating mathematical twist, the inverse scattering method can be used to solve so-called completely integrable differential equations, including nonlinear equations which model the propagation of nonlinear waves in fluids, optical media, and plasmas. Moreover, the same "completely integrable method" can be applied to solve problems in the theory of orthogonal polynomials, with applications to approximation theory; random matrix models, which are used to model the statistics of nuclear energy levels, to study quantum chaotic scattering, to understand certain two-dimensional quantum field theories, and many other areas of applied science. The purpose of this intensive workshop is to make progress in the study of 2+1-dimensional (two space, one time) differential equations, and related problems in the study of normal matrices and orthogonal polynomials. The analog of the Riemann-Hilbert method in this setting is the ∂-method, which the participants will refine and develop collaboratively by focussing, at first, on very specific problems which should admit a complete solution and provide valuable insight into the structure of an anticipated general method. The methods that are developed will provide new mathematical tools to study nonlinear wave propagation as well as mathematically related problems in probability theory, approximation theory, and quantum physics.

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 Tecnologí255a (CONACYT).