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

• 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

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.

这个项目连接两个数学领域，动力学和算子代数。动力系统的一个基本例子是这样一个系统，它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。给定任何时刻的状态，物理定律决定了以后任何时刻的状态。对于一个离散的例子，考虑由程序控制的计算机的状态。该程序是一个规则，用于使用当前状态来确定一个时间步后的状态，并通过重复来确定未来任何数量的步骤。对于合适的动力系统，我们可以将算子代数联系起来，算子代数是一种代数结构，它编码了动力系统的一些结构。离散动力系统有一个数值不变量，即平均维数，它描述了系统复杂性的一个方面。一个算子代数有一个数值不变量，它的比较半径，这是在没有考虑动力学的情况下定义的。该项目的一个中心目标是寻求新的方法来联系这些，从而以一种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。更确切地说，所考虑的动力系统是紧致度量空间上顺从群的自由极小作用。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，它测量了随着动力学的每次迭代而看到的空间部分的维度的增加。伴随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的代数上的群作用。比较半径是一个数值度量，用来衡量如果艾略特分类程序失败，需要的条件有多严重。代数，它是第一次适用于没有相关的动力学。猜想是比较半径是平均维数的一半。该项目旨在解决人们所知较少的不平等问题：比较半径至少是平均尺寸的一半。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

K-Theory and the Universal Coefficient Theorem for Simple Separable Exact C*-Algebras Not Isomorphic to Their Opposites

K-理论和简单可分精确C*-代数与逆元不同构的通用系数定理

• DOI: 10.1093/imrn/rnac358
• 发表时间: 2023
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Phillips, N Christopher;Viola, Maria Grazia
• 通讯作者: Viola, Maria Grazia

A simple nuclear C∗-algebra with an internalasymmetry

具有内部不对称性的简单核 C-代数

• DOI: 10.2140/apde.2023.16.711
• 发表时间: 2023
• 期刊: Analysis & PDE
• 影响因子: 2.2
• 作者: Hirshberg, Ilan;Phillips, N. Christopher
• 通讯作者: Phillips, N. Christopher