Mathematical Sciences: Symbolic Dynamics and Related Topics

Boyle将研究涉及符号动力学的各种主题，包括满射元胞自动机的周期点密度；非负矩阵在整数及其它实数子矩阵上的逆谱问题这个问题的代数推广；适合于这些系统的一般代数分析的泛函集合中有限型可约移位的分类及其与C*代数的关系；拓扑轨道等效；通用符号动态框架与残差熵。为了对符号动力学有一个粗略的初步认识，想想研究涉及很长一长串0和1的问题。一个适当的理想是考虑无限的磁带。当考虑到数据传输和存储问题时，这样的研究是有意义的。这也是研究物理系统如何随时间演变的关键之一：想象一下，把一个系统切成（比如说）两块，并根据它在两块中的行程将一个0和1序列与一个点联系起来。通过相关的行程来研究一个系统实际上是很有用的。正如在数学中发生的那样，为此而发展的理论已被证明与其他领域表面上无关的问题有关。本课题的研究对象有两个：一是理解非负矩阵的代数结构，这是许多数学应用的基础；二是改进理解与C*-代数的数学主题之间某些关系的框架。这个提议的另一个主题涉及到理解细胞自动机（电脑游戏《生命》就是一个例子），它可以被视为物理和进化的模型。最后一个主题，轨道等价，涉及到一个动力系统的性质的研究，它不依赖于路径的排序方式。

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.