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

9322498瓦格纳这个项目将研究移位空间和流的动力学，它们的对称群，以及与代数K理论的联系。通常有许多方法来描述或构造基本上具有相同动力学行为的换挡系统。即使在一维的情况下，有效地识别两个具体模型何时等价也是一个基本的开放问题。另一个自然和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到代数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又非常自然地产生了一种新的有序环的正代数K-理论，它使用通常的矩阵行和列运算，但施加了某些不等式。这里研究的主要数学结构，换挡动力系统和马尔可夫链，是从微分方程组到统计力学，再到信息和编码理论等领域的现象的有用模型。然而，它们的有用性受到了令人惊讶的困难的影响，即确定以不同方式出现的两个移位系统何时享受本质上相同的动力学行为。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，这位研究人员已经证明，一种非常抽象的代数理论，即代数K-理论，在解决其中一些问题方面可以发挥无价的作用。代数K理论的类比出人意料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，反过来看问题导致了一种新型的K理论，它的问题既从本质上讲是有意义的，也是因为它们将带回动力系统及其对称性的洞察力。***

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. ***