Mathematical Sciences: Shift Dynamics, Symmetry and and Algebriac K-Theory

数学科学：位移动力学、对称性和代数 K 理论

• 批准号: 9322498
• 负责人: John Wagoner
• 金额: $ 15.6万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-08-01 至 1998-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9322498&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Shift; Dynamics; Symmetry

9322498 Wagoner This project will investigate the dynamics of shift spaces and flows, their symmetry groups, and connections with algebraic K-theory. There are generally many ways of describing or constructing shift systems which have essentially the same dynamical behaviour. Even in the one-dimensional case, effectively recognizing when two concrete models are equivalent is a fundamental open problem. Another natural and related problem is to understand the different equivalences or symmetries between two systems. One approach to the classification and symmetry group problems for one-dimensional shift systems which has led to progress on some of the main problems in the field in recent years involves analogies from algebraic K-theory. In turn, the geometry of both discrete-time and continuous-time dynamical systems and their symmetry groups has very naturally given rise to a new type of positive algebraic K-theory for ordered rings, using the usual row and column operations on matrices, but with certain inequalities imposed. The main mathematical structures under investigation here, shift dynamical systems and Markov chains, are useful models for phenomena in areas ranging from differential equations, to statistical mechanics, to information and coding theory. However, their usefulness is impaired by the surprising difficulty of deciding when two shift systems which arose in different ways enjoy essentially the same dynamical behaviour. This is related to understanding the different equivalences or symmetries between two systems. (Symmetries of a given system are often called reversible cellular automata.) Now the investigator has shown that a very abstract algebraic theory known as algebraic K-theory can be invaluable in addressing some of these problems. Analogies from algebraic K-theory have led unexpectedly to exciting progress on some of the main problems in symbolic dynamics. Moreover, looking at matters the other way round has led to a new type of K-theory, whose problems are of interest both intrinsically and for the insight they will bring back to dynamical systems and their symmetries. ***

9322498 Wagoner 该项目将研究位移空间和流动的动力学、它们的对称群以及与代数 K 理论的联系。 通常有多种描述或构建具有基本相同的动态行为的换档系统的方法。 即使在一维情况下，有效识别两个具体模型何时等效也是一个基本的开放问题。 另一个自然且相关的问题是理解两个系统之间的不同等价性或对称性。 一维移位系统的分类和对称群问题的一种方法涉及代数 K 理论的类比，该方法近年来在该领域的一些主要问题上取得了进展。 反过来，离散时间和连续时间动力系统及其对称群的几何结构很自然地产生了一种新型有序环正代数 K 理论，使用矩阵上常见的行和列运算，但施加了某些不等式。 这里研究的主要数学结构，位移动力系统和马尔可夫链，是从微分方程到统计力学，再到信息和编码理论等领域的有用模型。 然而，它们的实用性因难以确定以不同方式出现的两个轮班系统何时具有基本相同的动态行为而受到损害。 这与理解两个系统之间的不同等价性或对称性有关。 （给定系统的对称性通常称为可逆元胞自动机。）现在，研究人员已经证明，一种非常抽象的代数理论（称为代数 K 理论）对于解决其中一些问题具有无价的价值。 代数 K 理论的类比意外地在符号动力学的一些主要问题上取得了令人兴奋的进展。 此外，以相反的方式看待问题催生了一种新型的 K 理论，其问题在本质上以及它们将带回动力系统及其对称性的见解方面都令人感兴趣。 ***