Mathematical Sciences: Aspects of Integrability

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The principal investigator will study nonlinear phenomena in which chaotic and solitonic behavior occur. Topics to be studied include Hamiltonian theory over non-commutative rings, algebraic and analytical aspects of the D-bar method, solutions of algebraic equations using algebraic geometric methods, Painleve equations and their appearance in the matrix models of quantum gravity, and generalizations of the hypergeometric functions. Integrable systems and inverse scattering problems play a fundamental role in establishing links between fundamental physics and the way in which mathematical models are used to describe it.