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

9322498瓦戈纳 本计画将研究位移空间与流动的动力学，它们的对称群，以及与代数K理论的关联。 通常有许多方法来描述或构造具有基本相同动力学行为的移位系统。 即使在一维的情况下，有效地识别两个具体的模型是等效的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或对称性。 一维移位系统的分类和对称群问题的一种方法，近年来在该领域的一些主要问题上取得了进展，涉及代数K理论的类比。 反过来，离散时间和连续时间动力系统及其对称群的几何学很自然地产生了一种新的有序环的正代数K-理论，它使用了矩阵上通常的行和列运算，但施加了某些不等式。 在这里调查的主要数学结构，转移动力系统和马尔可夫链，是有用的模型，从微分方程，统计力学，信息和编码理论等领域的现象。 然而，他们的有用性是受损的令人惊讶的困难，决定当两个转移系统，产生了不同的方式享受基本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被称为可逆细胞自动机。 现在，研究者已经证明，一个非常抽象的代数理论，称为代数K理论，在解决这些问题时是非常宝贵的。 代数K-理论的类比在符号动力学的一些主要问题上出人意料地取得了令人兴奋的进展。 此外，从另一个角度来看待问题已经导致了一种新型的K理论，它的问题在本质上是令人感兴趣的，因为它们将使人们重新认识动力系统及其对称性。 ***

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. ***