Algebraic and analytic aspects of integrable systems and painleve equations

Description:

This volume contains the proceedings of the AMS Special Session on Algebraic and Analytic Aspects of Integrable Systems and Painlevé Equations, held on January 18, 2014, at the Joint Mathematics Meetings in Baltimore, MD. The theory of integrable systems has been at the forefront of some of the most important developments in mathematical physics in the last 50 years. The techniques to study such systems have solid foundations in algebraic geometry, differential geometry, and group representation theory. Many important special solutions of continuous and discrete integrable systems can be written in terms of special functions such as hypergeometric and basic hypergeometric functions. The analytic tools developed to study integrable systems have numerous applications in random matrix theory, statistical mechanics and quantum gravity. One of the most exciting recent developments has been the emergence of good and interesting discrete and quantum analogues of classical integrable differential equations, such as the Painlevé equations and soliton equations. Many algebraic and analytic ideas developed in the continuous case generalize in a beautifully natural manner to discrete integrable systems. The editors have sought to bring together a collection of expository and research articles that represent a good cross section of ideas and methods in these active areas of research within integrable systems and their applications.

subjects: Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Applications to integrable systems,  Dynamical systems and ergodic theory -- Infinite-dimensional Hamiltonian systems -- Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.).,  Algebra,  Difference and functional equations -- Difference equations -- Difference equations, additive,  Partial differential equations -- Equations of mathematical physics and other areas of application -- NLS-like equations (nonlinear Schrödinger),  Special functions (33-XX deals with the properties of functions as functions) -- Hypergeometric functions -- Generalized hypergeometric series, $.,  Quantum theory -- Groups and algebras in quantum theory -- Relations with integrable systems,  Painlevé equations,  Congresses,  Ordinary differential equations -- Differential equations in the complex domain -- Painlevé and other special equations; classification, hierarchies;,  Ordinary differential equations -- Differential equations in the complex domain -- Isomonodromic deformations,  Algebraic geometry -- Birational geometry -- Birational automorphisms, Cremona group and generalizations,  Approximations and expansions -- Approximations and expansions -- Interpolation,  Differential equations,  Dynamical systems and ergodic theory -- Infinite-dimensional Hamiltonian systems -- Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)