Mathematical Sciences: The Complex Geometry of Integrable Hamiltonian Systems

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The two principal investigators will study the isolation of a geometric aspect of integrability, singularity analysis of Toda equations and stratification of Bruhat cells, solutions to the sine-Gordon equations involving constant mean curvature tori immersed in spheres, and the universal enveloping algebra associated with properties of Poisson brackets. This project concerns deep properties of Hamiltonian systems. These are systems of differential equations which describe in particular the motion of mechanical devices. Such devices must conserve both energy and momentum. Recently, various algebraic methods as well as Painleve analysis of integrability has been used to make considerable progress toward the understanding of such systems. This project will contribute to these developments.