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

这个项目连接了两个数学领域，动力学和算子代数。动力系统的一个基本例子是，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的系统。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意数量的步骤。对于合适的动力系统，可以将算子代数联系起来，算子代数是一种编码动力系统某些结构的代数结构。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，它的比较半径是在不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为美国的劳动力发展做出贡献。更确切地说，所考虑的动力系统是紧化度量空间上可服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，它测量的是人们在每次动态迭代中看到的空间部分维度的增加。关联算子代数是代数中斜群环的泛函分析版本，利用空间上连续函数代数上的群作用。比较半径是一个数值度量，如果Elliott分类程序失败，需要的条件有多严重。它最初应用的代数与动力学无关。假设比较半径是平均维数的一半。该项目针对的是鲜为人知的不平等：比较半径至少是平均维度的一半。该奖项反映了美国国家科学基金会的法定使命，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。

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.