Independence and dichotomies in dynamics and operator algebras

该项目将包括一些围绕动力系统中的随机性、对称性、矛盾性和维度的研究和教育活动。本课程的目的是了解组合、几何和拓扑现象在动力学中新拓宽的范围内相互联系的方式，这一范围是由最近在熵理论、无限群理论和核C*-代数的分类程序中开辟的。调查将植根于独立的组合和概率概念，以及它们作为来源或对立点所涉及的各种二分法。该计划的研究和教育部分将与其他几个数学分支联系起来，包括几何和组合群论、代数拓扑学、凸几何和描述集合论。动力学的数学理论寻求建立一个严格的框架，以理解一个物理系统随着时间的演化可能观察到的行为类型。在一个混乱的系统中，我们对系统未来状态的预测能力是有限的或为零的，有不同的方法可以使这种不可预测性在数学上变得精确，例如通过熵的概念。这些概念都与独立的概率概念联系在一起，这种概念最基本的形式就是反复抛硬币。一个一般的系统通常会以一种错综复杂的方式将确定性和不确定性两个方面结合在一起。该项目旨在开发新的工具，用于识别各种动力学伪装下的独立性的存在及其对相关数学结构(如算子代数)的影响。在实践方面，这种努力的兴趣在于它对研究涉及码传输和纠错的信息论问题具有重要意义。

The project will consist of a number of research and educational activities centered around the study of randomness, symmetry, paradoxicality, and dimensionality in dynamical systems. The goal is to understand the way that combinatorial, geometric, and topological phenomena interrelate within a newly broadened scope in dynamics that has been opened up by recent advances in entropy theory, the theory of infinite groups, and the classification program for nuclear C*-algebras. The investigations will be rooted in the combinatorial and probabilistic notions of independence and the various dichotomies to which they relate either as a source or as a counterpoint. Both the research and educational components of the program will bridge to several other branches of mathematics including geometric and combinatorial group theory, algebraic topology, convex geometry, and descriptive set theory.The mathematical theory of dynamics seeks to establish a rigorous framework for understanding the kinds of behavior that one might observe in a physical system as it evolves in time. In a chaotic system our ability to make predictions about the future state of the system is limited or null, and there are different ways in which this unpredictability can be made mathematically precise, for example via the notion of entropy. These concepts are all tied to the probabilistic notion of independence, which is exhibited in its most basic form by the repeated tossing of a coin. A general system will typically combine both deterministic and indeterministic aspects, often in an intricate way. This project aims to develop new tools for identifying the presence of independence in its various dynamical guises and its effect on associated mathematical structures such as operator algebras. On the practical side, the interest in such an endeavor lies in its significance for the study of problems in information theory involving code transmission and error correction.