NSF-BSF: Dynamics and Operator Algebras beyond the Elliott Classification Program

NSF-BSF：艾略特分类计划之外的动力学和算子代数

基本信息

批准号：
2055771
负责人：
Norman Phillips
金额：
$ 27.92万
依托单位：
University of Oregon Eugene
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055771&HistoricalAwards=false
关键词：
NSF BSF Dynamics Operator Algebras

项目摘要

This project links two mathematical fields, dynamics and operator algebras. An elementary example of a dynamical system is the system whose state at any given time is the instantaneous positions and velocities of the sun, planets, moons, asteroids, and other astronomical bodies in the solar system. Given the state at any time, physical laws determine the state at any later moment. For a discrete example, consider the states of a computer governed by a program. The program is a rule for using the current state to determine the state one time step later, and, by repetition, any number of steps in the future. To suitable dynamical systems, one can associate an operator algebra, a kind of algebraic structure which encodes some of the structure of the dynamical system. A discrete dynamical system has a numerical invariant, its mean dimension, which describes one aspect of its complexity. An operator algebra has a numerical invariant, its radius of comparison, which was defined without consideration of dynamics. A central aim of the project is to seek new ways to relate these, thus connecting two different areas of mathematics in an unexpected way. The project will also contribute US workforce development through the training of graduate and undergraduate students.More precisely, the dynamical systems under consideration are free minimal actions of amenable groups on compact metric spaces. Mean dimension was invented in dynamics, with no thought of operator algebras. Roughly, it measures the increase of the dimension of the part of the space one sees with each iteration of the dynamics. The associated operator algebra is a functional analyst's version of the skew group ring in algebra, using the group action on the algebra of continuous functions on the space. The radius of comparison is a numerical measure of how badly a condition needed if the Elliott classification program fails. The algebras to which it was first applied were not related to dynamics. The conjecture is that the radius of comparison is half the mean dimension. The project is aimed at the inequality for which less is known: that the radius of comparison is at least half the mean dimension.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目链接了两个数学字段，动力学和操作员代数。动态系统的一个基本示例是该系统，其在任何给定时间的状态是太阳，行星，卫星，小行星和太阳系中其他天文机构的瞬时位置和速度。鉴于国家的任何时候，物理法在以后的任何时刻决定了国家。在一个离散的示例中，请考虑由程序管辖的计算机状态。该程序是使用当前状态稍后确定状态的规则，并通过重复来确定未来数量的任何步骤。对于合适的动态系统，可以将操作员代数（一种代数结构）关联，该代数结构编码了动态系统的某些结构。离散的动态系统具有数值不变的，其平均维度，它描述了其复杂性的一个方面。运算符代数具有数值不变的，它的比较半径，该数字是不考虑动力学的而定义的。该项目的一个主要目的是寻求新的方法来关联这些方法，从而以出乎意料的方式将数学的两个不同领域连接起来。该项目还将通过培训研究生和本科生来为美国的劳动力发展做出贡献。更确切地说，所考虑的动态系统是对紧凑型公制空间的自由行动。平均维度是在动力学中发明的，没有考虑操作员代数。粗略地，它测量了一个动力学的每次迭代，都可以衡量一个空间部分的尺寸的增加。相关的操作员代数是功能分析师在代数中的偏斜组环的版本，使用在空间上连续函数的代数上的组动作。比较半径是一个数值度量，即如果Elliott分类程序失败，则需要如何严重条件。最初应用的代数与动态无关。猜想是比较半径是平均维度的一半。该项目的目的是对鲜为人知的不平等：比较半径至少是平均维度的一半。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，认为值得通过评估来获得支持。