Mathematical Sciences: Symbolic Dynamics and Related Topics

Boyle Boyle将研究涉及符号动力学的各种主题，包括满射细胞自动机的周期点的密度；实的整数和其他子环上的非负矩阵的逆谱问题；该问题的代数推广；适合于这些系统的一般代数分析的函数型可约移位的分类及其与C*代数的关系；拓扑轨道等价；一般符号动力学框架和剩余熵。要对符号动力学有一个粗略的初始概念，可以考虑研究涉及非常长的0和1磁带的问题。一个适当的理想化是考虑无限磁带。当人们考虑数据传输和存储问题时，这样的研究是有意义的。它也是研究物理系统如何随时间演化的关键之一：想象一下将一个系统分成(比方说)两个部分，并根据这两个部分的行程将一个由0和1组成的序列与一个点关联起来。通过相关的行程来研究一个系统实际上是很有用的。正如在数学中所发生的那样，为此而开发的理论被证明与其他领域表面上无关的问题有关。这一建议的研究对象是两个例子：了解非负矩阵的代数结构，这在许多数学应用中是基本的；以及改进一个框架，以理解与C*-代数这一数学主题的某些关系。这项提议的另一个主题涉及理解细胞自动机(生命的计算机游戏就是一个例子)，它可以被视为物理学和进化的模型。最后一个主题，轨道等价性，涉及到研究不依赖于路线排序方式的动力系统的性质。

Abstract Boyle Boyle will study various topics involving symbolic dynamics, including the density of periodic points for surjective cellular automata; the inverse spectral problem for nonnegative matrices over the integers and other subrings of the reals; algebraic generalizations of this problem; the classification of reducible shifts of finite type in a functorial setting suitable to the general algebraic analysis of these systems and their relations with C* algebras; topological orbit equivalence; general symbolic dynamical frameworks and residual entropy. To have a crude initial idea of symbolic dynamics, think of studying problems involving very long tapes of zeros and ones. An appropriate idealization is to consider infinite tapes. Such a study is significant when one considers problems of data transmission and storage. It also is one key to the study of how physical systems evolve over time: imagine cutting up a system into (say) two pieces, and associating to a point a sequence of zeros and ones according to its itinerary through the two pieces. It is actually useful to study a system by way of the associated itineraries ... As happens in math, the theory developed for this has turned out to be relevant to superficially unrelated problems in other areas. Two examples are objects of study of this proposal: understanding the algebraic structure of nonnegative matrices, which are basic in many applications of mathematics, and improving a framework for understanding certain relations with the mathematical subject of C*-algebras. Another topic of this proposal involves understanding cellular automata (the computer game of Life is an example) which can be viewed as models for physics and evolution. The final topic, orbit equivalence, involves the study of properties of a dynamical system which do not depend on the way itineraries are ordered.