Strong Shift Equivalence Theory

从离散时间光滑动力学到统计力学，再到信息和编码理论，瓦格纳转移动力系统和马尔可夫链都是研究各种现象的有用模型。在实践中，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移位系统。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。另一个自然而相关的问题是研究两个系统之间的不同等价性或对称性。给定系统的对称性通常被称为可逆元胞自动机。一维移位系统的分类和对称群问题的一种方法已经取得了丰硕成果，其中包括强移位等价理论。近年来，这导致了在位移等价问题上的进展。这在符号动力学中是一个诱人的问题，可以用矩阵和图形用初等术语提出，自1974年以来一直悬而未决。它通过对矩阵进行通常的行和列运算，得到了一种新型的有序环的正代数K-理论，但施加了某些不等式。强移位等价理论是代数和拓扑学的结合，它起源于被称为移位系统及其对称群的系统的动力学。事实证明，它与数学外部动力学领域有意想不到的有趣联系，如代数K理论、循环同调和拓扑量子场论。瓦格纳将继续探索这些新发展。*

9971501Wagoner Shift dynamical systems and Markov chains are useful models forphenomena in areas ranging from discrete time smooth dynamics, tostatistical mechanics, to information and coding theory. In practice,there are generally many ways of describing or constructing shiftsystems that have essentially the same dynamical behaviour. Even inthe one-dimensional case, effectively recognizing when two concretemodels are equivalent is a fundamental open problem. Another naturaland related problem is to study the different equivalencies orsymmetries between two systems. Symmetries of a given system areoften called reversible cellular automata. One approach to theclassification and symmetry group problems for one-dimensional shiftsystems that has been fruitful involves strong shift equivalencetheory. In recent years, this has led to progress on the ShiftEquivalence Problem. This is a tantalizing question in symbolicdynamics that can be posed in elementary terms using matrices andgraphs and that has been open since since 1974. It has led to a newtype of positive algebraic K-theory for ordered rings by way of theusual row and column operations on matrices but with certaininequalities imposed. Strong shift equivalence theory is a combination of algebra andtopology that grew out of the dynamics of systems known as shiftsystems and their symmetry groups. It turns out to have unexpectedand interesting connections with areas of mathematics outsidedynamics, such as algebraic K-theory, cyclic homology, and topologicalquantum field theory. Wagoner will continue to explore these newdevelopments.***