From bispectrality to integrable systems, orthogonal polynomials, heat equations and W-algebras

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The bispectral problem appeared first in the pioneering work of Duistermaat and Grunbaum. It asks to classify spectral problems possessing a hidden symmetry between the space and the spectral variables. The goals of this proposal is to establish and explore new connections between integrable systems, orthogonal polynomials, heat equations and Lie algebras, closely related with each other and mysteriously linked to the bispectral property. The first part of the proposal aims at the construction and the classification of multivariable classical orthogonal polynomials; their duality properties (between the continuous and the discrete variables) and relations to integrable systems and algebraic geometry. Another part of the proposal explores the connection between rank one bispectral operators and dynamical systems of particles on the line (such as Calogero-Moser and Ruijsenaars-Schneider). The bispectral property in this case leads to the construction of new interesting W-algebras and their representations. The PI has also proposed a new approach to study heat equations based on soliton equations (such as KdV and Toda lattice) and Sato?s Grassmannian, and to use this approach to characterize the bispectral operators by a finiteness property of the heat kernel. The classical (one dimensional) orthogonal polynomials have played a crucial role in mathematics and physics in the last centuries. The construction and the classification of such polynomials in more than one variable is an old and important problem and its solution will have numerous applications in both pure and applied mathematics. The connection with algebraic geometry and integrable systems serves as a useful bridge which translates powerful techniques, explains the bispectrality from duality and leads to various extensions. Heat equations, arising in many physical applications, can be studied in a uniform way, by connecting them to soliton equations. The characterization of bispectrality in terms of representation theory of Lie algebras brings the purely algebraic side of the proposal. The interdisciplinary nature of this project will establish new links between the different branches of mathematics and physics and will stimulate communications and collaborations between specialists in the various areas involved. Research projects are also designated for students, whose involvement in the project will advance discovery and understanding while promoting teaching and training.