Mathematical Sciences: Continuous Complexity and Dynamical Systems

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Shub 9616920 The investigator continues studies of the complexity theory of continuous problems and dynamical systems. The main issues are: 1) the construction of a theory of computation and complexity which speaks to scientific computation and numerical analysis, and 2) the extent of validity of statistical robustness as a property of dynamical systems, especially chaotic dynamical systems. The research involved is of interest to a large class of mathematicians and has implications for the relations between abstract mathematics and computer science on the one hand and abstract mathematics and physics and engineering on the other. Complexity theory develops bounds on how much work a method requires to produce the solution to a typical problem in a class of problems. Common and important examples include methods to find the solutions of a system of linear or nonlinear equations. Part of the project indeed at aims at just this issue. Equation solving is at the heart of much of mathematics and, together with its computational aspects, is a main way that mathematics is used by engineering, physics, economics, and many other disciplines. The other part of the project studies questions about much alike certain kinds of chaotic systems may be. Predicting the specific behavior of a chaotic system is difficult, because small errors in any measurement of the system are amplified. But chaotic systems may be relatively nice in a statistical sense; if so, then sets of samples measurements may provide useful information about the system's behavior even though any single measurement is error-prone. The investigator studies these properties in dynamical systems and particularly in chaotic systems. There are important implications for the statistical analysis of chaotic systems --- hence practical consequences for weather and climate studies, agriculture, and engineering.