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伯格不确定原理。从那时起，算子代数的这种同样的非交换性质被应用于数学的许多其他领域。这对于动力系统来说尤其成功，动力系统是物理系统的空间和时间演化的数学模型。算子代数和动力系统之间的相互作用是我研究的主题。从动力系统出发，有非常一般的算子代数的构造。在阿兰·康尼斯的非对易几何方案中，这些起到了替代系统轨道空间的作用。这首先提供了一系列工具，用于通过相关的算子代数来理解动力学。这在过去的三十年里尤其有效，因为康奈斯的项目提供了强大的新想法。作为例子，Krieger通过C*-代数的K-理论给出了符号系统的新的不变量。与我的合著者的结果表明，类似的结果可以为其他系统的轨道结构提供完全不变量。动力系统的构造也提供了丰富的算符代数的例子，了解它们的结构一直是该领域的一个主要目标。George Elliott关于顺从C*-代数的分类程序在过去25年中一直是算子代数的最大领域之一，许多最令人印象深刻的结果，如汤姆和温特的结果，都是来自动力系统的例子。我自己早期在这个领域的工作产生了今天仍然在使用的想法和技术工具。我的建议是继续我对这些相互作用的调查。其中特别强调混沌系统，我最近将Krieger不变量推广到更广泛的混沌动力系统。这为研究分形学提供了创新的工具。目标是更好地理解不变量，同时也是C*-代数理论中阐明动态结构的工具的发展。在另一个方向，我的目标是开发工具，为一对C*-代数提供定量度量，尽管每个C*-代数都是从复杂的动力学构造的，但以一种相当简单的方式彼此相关。

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.