Mathematical Sciences: Coding and Convergence in Ergodic Theory

遍历理论是现代分析的一个活跃的核心领域。它的起源在于十九世纪末经典统计力学的理论表述。现代观点源于庞加莱、伯克霍夫、冯·诺伊曼和其他巨人在本世纪初发展起来的关于递归和遍历的深刻数学理论。随着学科的发展和复杂程度的不断提高，遍历理论与其他数学分支，如动力系统、概率论、泛函分析、数论、微分拓扑学和微分几何，以及与数学物理、信息论和计算机设计的应用都建立了密切的关系。彼得森教授致力于遍历理论的几个方面的工作，特别是那些与调和分析、概率和最近的信息论有关的方面。他的主要工作涉及极大函数和几乎所有地方的收敛；例如，最大遍历定理到等式的精化，遍历定理中的收敛速度，遍历希尔伯特变换，以及与谱理论和调和分析的关系。他还研究了遍历理论的应用，特别是符号动力学，以在嘈杂的信道上有效地对信息进行编码。在这里，彼得森教授建议继续他对符号动力学和几乎处处收敛的研究。在前一门课中，他将考虑无限字母表上符号动力系统的一般性质，以及一类与磁记录信号控制有关的例子。在后一个领域，他的研究将强调遍历希尔伯特变换和保测变换的谱测量之间的联系。

Ergodic theory is an active, central area of Modern Analysis. Its origins lie in the theoretical formulation for classical statistical mechanics at the end of the nineteenth century. The modern point of view derives from the profound mathematical theory of recurrence and ergodicity, as developed by Poincare, Birkhoff, von Neumann and other giants earlier in this century. As the subject has evolved and sophistication continually increased, ergodic theory has acquired close relationships with other branches of mathematics, such as dynamical systems, probability theory, functional analysis, number theory, differential topology, and differential geometry, and with applications as far ranging as mathematical physics, information theory, and computer design. Professor Petersen has worked on several aspects of ergodic theory, especially those connected with harmonic anlaysis, probability, and more recently information theory. A major part of his work has concerned maximal functions and almost everywhere convergence; for example, the refinement of the maximal ergodic theorem to an equality, speed of convergence in the ergodic theorem, the ergodic Hilbert transform, and the relationship with spectral theory and harmonic analysis. He has also conducted research in the applications of ergodic theory, specifically symbolic dynamics, to efficient coding of information across noisy channels. Here, Professor Petersen proposes to continue his research into both symbolic dynamics and almost everywhere convergence. In the former subject, he will consider general properties of symbolic dynamical systems on an infinite alphabet as well as a class of examples that relate to the control of signals for magnetic recording. In the latter area, his research will emphasize connections between the ergodic Hilbert transform and the spectral measure of a measure-preserving transformation.