Mathematical Sciences: Dynamical Systems and Complexity

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The main objectives of this project are to enhance the understanding of dynamical systems, to study the relation of these dynamical systems to topology and to algorithms in numerical analysis, and to study the global geometric properties of these algorithms and their computational process. The principal investigator proposes a theory of computation and a model for computation over the real numbers. This theory includes recursive function theory, universal machines, and NP completeness. For example, degenerate fixed points occur in the analysis of the iterative processes for polynomial root finding and interior point methods for linear programming. Blowing up techniques for the linear programming problem involve the dynamics of the n-sphere. An understanding of this problem has implications for the asymptotic rate of attraction to the optima for discrete versions of the linear and the projective rescaling algorithms of linear programming.