Operator Algebras and Dynamics

本课题将研究L^2-不变量理论与遍历理论之间的关系。前者通过将群von Neumann代数技巧应用于泛覆盖空间的代数拓扑，为CW-复形提供了不变量，并在几何和K-理论中有应用。遍历理论提供了不变量，如熵和平均维度。它们最初是为整数组动作定义的，随后是为服从组动作定义的，最近是为SOFIC组动作定义的。主要研究者还打算将SOFIC群作用的熵理论推广到算子代数上的作用和超线性群的作用。经典拓扑学研究几何对象的形状。这个项目中涉及的不变量理论通过使用所谓的冯·诺伊曼代数实现了这一目标，冯·诺伊曼代数最初是作为一种数学工具引入物理学中研究量子力学的。遍历理论研究系统在进化中的长期行为(例如，物理学中的“沙堆模型”)。在过去的几年里，在这两个领域之间发现了一些令人惊讶的联系，这种联系导致了这两个领域的应用。首席调查员计划深化和扩大这种联系。

This project will study relationships between L^2-invariant theory and ergodic theory. The former provides invariants for CW-complexes through the process of applying group von Neumann algebra techniques to the algebraic topology of universal covering spaces, and it has applications to geometry and K-theory. Ergodic theory provides invariants such as entropy and mean dimension. These were originally defined for integer group actions, subsequently for amenable group actions, and most recently for sofic group actions. The Principal Investigator also intends to extend the entropy theory of sofic group actions to actions on operator algebras and actions of hyperlinear groups. Classical topology studies the shape of geometric objects. The invariant theory involved in this project achieves such a goal by using so-called von Neumann algebras, which were first introduced as a mathematical tool to study quantum mechanics in physics. Ergodic theory studies the long-time behavior of systems under evolution (e.g., the "sandpile model" in physics). In the last few years, some surprising connections between these two fields have been discovered, connections that lead to applications in both areas. The Principal Investigator plans to deepen and broaden such connections.