Dynamics and C*-algebras

算符代数最初是作为量子力学系统的模型来研究的，因为算符乘法的非交换性可以用来编码海森堡不确定性原理。从那时起，算子代数的这种非交换性质已经应用于许多其他数学领域。这在动力系统中尤其成功，动力系统是物理系统时空演化的数学模型。算子代数和动力系统之间的这种相互作用是我研究的主题。从动力系统出发构造算子代数有相当普遍的方法。在阿兰·康纳斯的非对易几何程序看来，这些作为系统轨道空间的替代。这首先提供了一系列的工具，通过其相关的运营商代数的动力学理解。这在过去的三十年里特别有效，因为Connes的计划提供了强大的新思想。作为一个例子，克里格通过C*-代数的K-理论给出了符号系统的新不变量。与我的合著者的结果显示类似的结果可以提供完整的不变量的轨道结构的其他系统。* 动力系统的构造也提供了丰富的算子代数的例子，理解它们的结构一直是该领域的主要目标。乔治埃利奥特的分类程序顺从C*-代数一直是一个最大的领域运营商代数在过去的二十五年和许多最令人印象深刻的结果，如那些汤姆斯和冬季，一直为例所产生的动力系统。我自己在这一领域的早期工作产生了今天仍在使用的想法和技术工具。我的建议是继续调查这些相互作用。有特别强调混沌系统，我最近扩展克里格的不变量为更广泛的一类混沌动力系统。这为分形几何的研究提供了创新的工具。我们的目标是更好地理解不变量，同时也是C*-代数理论中阐明动力学结构的工具的发展。在另一个方向上，我的目标是开发工具，为一对C*-代数提供定量测量，尽管每个C *-代数都是由复杂的动力学构造的，但它们以相当简单的方式相互关联。

Operator algebras were initially studied as models for quantum mechanical systems, because the noncommutativity of multiplication of operators can be used to encode the Heisenberg uncertaintly principle. Since then, this same noncommutative nature of operator algebras has been applied in many other areas of mathematics. This has been particularly successful for dynamical systems, the mathematical models for the spatial and temporal evolution of physical systems. This interaction between operator algebras and dynamical systems is the subject of my research.******There are quite general constructions of operator algebras from dynamical systems. In the view of Alain Connes' program of noncommutative geometry, these act as a replacement for the space of orbits of the system. This first provides a range of tools for understanding the dynamics through their associated operator algebras. This has been particularly effective over the past thirty years, as Connes' program has provided powerful new ideas. As an example, Krieger gave new invariants for symbolic systems through K-theory of the C*-algebras. Results with my co-authors show how analogous results can provide complete invariants for the orbit structure of other systems. ***The construction from dynamical systems also provides a rich source of examples of operator algebras and understanding their structure has been a major goal for the field. George Elliott's classification program for amenable C*-algebras has been one of the largest areas of operator algebras over the past twenty-five years and many of the most impressive results, such as those of Toms and Winter, have been for examples arising from dynamical systems. My own work in this area in the early days produced ideas and technical tools which are still in use today.******My proposal is to continue my investigations into these interactions. There is special emphasis on chaotic systems for which I have recently extended Krieger's invariant for to a much broader class of chaotic dynamical systems. This gives innovative tools for the study of the geometry of fractals. The goal is a better understanding of the invariant but also the development of tools within C*-algebra theory which elucidate the dynamical structure.******In another direction, my goal is to develop tools that provide quantitative measures for a pair of C*-algebras which, although each is constructed from complicated dynamics, are related to each other in a fairly simple fashion.***********