Mathematical Sciences: Continuous Complexity and Dynamical Systems

舒布9616920，研究者继续研究连续问题和动力系统的复杂性理论。主要问题是：1)建立一个涉及科学计算和数值分析的计算和复杂性理论，以及2)作为动力系统，特别是混沌动力系统的一种属性的统计稳健性的有效性程度。这项研究涉及到一大批数学家，并对抽象数学和计算机科学之间的关系以及抽象数学和物理与工程之间的关系产生了影响。复杂性理论发展了一种方法需要多少工作才能为一类问题中的典型问题提供解决方案的界限。常见且重要的例子包括求解线性或非线性方程组的方法。该项目的一部分实际上就是针对这个问题。方程求解是许多数学的核心，与它的计算方面一起，是工程、物理、经济学和许多其他学科使用数学的主要方式。该项目的另一部分研究关于某些类型的混沌系统可能非常相似的问题。预测混沌系统的具体行为是困难的，因为系统的任何测量中的小误差都会被放大。但从统计意义上讲，混沌系统可能是相对较好的；如果是这样，那么样本测量集可能会提供有关系统行为的有用信息，即使任何单一测量都容易出错。研究者研究了动力系统中的这些性质，特别是在混沌系统中。这对混沌系统的统计分析有重要的影响-因此对天气和气候研究、农业和工程学的实际影响。

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.