Algebraic and Analytic Structures in Dynamics

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Abstract Rychlik Rychlik will investigate the following problems: The existence of special classes of solutions in non-linear difference equations and ordinary differential equations which correspond to homoclinic and heteroclinic connections. The existence of algebraic integrals for a class of symplectic maps. The measure of periodic trajectories in billiard systems. The existence of traveling wave and breather solutions to PDEs. Development and implementation of algorithms for dealing with systems of multi-variate polynomial equations with many parameters that often arise in applications. The problems selected for this project generally invite the use of techniques from algebraic geometry and complex analysis. The main objective of the current project is to show that advanced techniques from these fields are useful in solving problems of dynamics. A result of this research will be a suite of algebraic and complex-analytic methods general enough to be applied to other problems.