Mathematical Sciences: Continuous Complexity and Dynamical Systems

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)计算和复杂性理论的构建，这与科学计算和数值分析有关；2)统计鲁棒性作为动力系统，特别是混沌动力系统的一种特性的有效性程度。所涉及的研究引起了大批数学家的兴趣，并对抽象数学和计算机科学与抽象数学、物理和工程之间的关系产生了影响。复杂性理论规定了一种方法需要做多少工作才能解决一类问题中的一个典型问题。常见和重要的例子包括寻找线性或非线性方程组的解的方法。这个项目的一部分确实是针对这个问题的。方程求解是许多数学的核心，它与计算方面一起，是工程学、物理学、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的问题与某些类型的混沌系统非常相似。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。但混沌系统在统计意义上可能相对较好；如果是这样，那么即使任何单一的测量都容易出错，采样测量集也可能提供有关系统行为的有用信息。研究者在动力系统特别是混沌系统中研究这些性质。这对混沌系统的统计分析具有重要意义，因此对天气和气候研究、农业和工程具有实际意义。

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.