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.

该项目将动力学和算子代数这两个数学领域联系起来。动力系统的一个基本例子是这样的系统，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体的瞬时位置和速度。给定任意时刻的状态，物理定律决定随后任意时刻的状态。举一个离散的例子，考虑由程序控制的计算机的状态。该程序是一种规则，用于使用当前状态来确定一个时间步骤后的状态，并通过重复确定未来任意数量的步骤。对于合适的动力系统，我们可以将算子代数关联起来，算子代数是一种对动力系统的某些结构进行编码的代数结构。离散动力系统具有数值不变量，即其平均维数，它描述了其复杂性的一个方面。算子代数有一个数值不变量，即比较半径，它的定义没有考虑动力学。该项目的中心目标是寻求新的方法将这些联系起来，从而以意想不到的方式连接两个不同的数学领域。该项目还将通过培训研究生和本科生来促进美国劳动力的发展。更准确地说，所考虑的动力系统是服从群体在紧凑度量空间上的自由最小行为。平均维数是在动力学中发明的，没有考虑算子代数。粗略地说，它测量了每次动态迭代时所看到的空间部分尺寸的增加。关联算子代数是代数中斜群环的泛函分析版本，使用空间上连续函数代数的群作用。比较半径是对艾略特分类程序失败时所需条件的严重程度的数值测量。它最初应用的代数与动力学无关。推测比较半径是平均尺寸的一半。该项目旨在解决鲜为人知的不平等：比较半径至少是平均维度的一半。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查进行评估，被认为值得支持标准。