Dynamics and C*-algebras

动力学和 C* 代数

• 批准号: RGPIN-2016-04104
• 负责人: Putnam, Ian
• 金额: $ 2.4万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=662733
• 关键词: Dynamics; algebras

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.***********

算子代数最初被研究为量子力学系统的模型，因为算子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被应用于许多其他数学领域。对于动力系统，即物理系统时空演化的数学模型来说，这种方法尤其成功。这种算子代数和动力系统之间的相互作用是我研究的主题。******在动力系统中有相当一般的算子代数的构造。在Alain Connes的非交换几何规划中，它们作为系统轨道空间的替代。这首先提供了一系列工具，通过它们的相关算子代数来理解动力学。在过去的30年里，这种方法尤其有效，因为Connes的计划提供了强有力的新想法。作为一个例子，Krieger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子代数实例来源，理解它们的结构一直是该领域的主要目标。在过去的25年里，乔治·埃利奥特（George Elliott）的可适应C*-代数分类计划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（Toms）和温特（Winter）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一些想法和技术工具，这些想法和技术工具至今仍在使用。******我的建议是继续调查这些相互作用。特别强调混沌系统，我最近将Krieger的不变量扩展到更广泛的混沌动力系统。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，同时也在C*-代数理论中开发工具来阐明动态结构。******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。***********