Independence and dichotomies in dynamics and operator algebras

该项目将包括一系列研究和教育活动，主要围绕动力系统中的随机性、对称性、悖论性和维度进行研究。我们的目标是理解组合、几何和拓扑现象在一个新的动力学范围内相互联系的方式，这个范围是由熵理论、无限群理论和核C*-代数的分类程序的最新进展所开辟的。调查将植根于独立性的组合和概率概念，以及它们作为来源或对位关系的各种二分法。该计划的研究和教育组成部分将与其他几个数学分支建立联系，包括几何和组合群论、代数拓扑、凸几何和描述集合论。动力学的数学理论试图建立一个严格的框架，以理解人们在物理系统随时间演变时可能观察到的各种行为。在一个混沌系统中，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精确，例如通过熵的概念。这些概念都与独立的概率概念有关，而独立的概率概念最基本的表现形式就是反复抛硬币。一般系统通常会以复杂的方式结合确定性和不确定性方面。该项目旨在开发新的工具，以识别各种动态形式的独立性及其对相关数学结构（如算子代数）的影响。在实践方面，这种努力的意义在于它对研究信息论中涉及编码传输和纠错的问题具有重要意义。

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.