Deformation/rigidity theory in von Neumann algebras and ergodic theory

冯·诺伊曼代数在20世纪30年代和40年代被引入，部分原因是作为发展量子物理学数学基础的工具。冯·诺伊曼代数从此成为一个独立的兴趣领域，并进一步应用于数学领域，如遍历理论、Voiculescu的自由概率论、Jones的子因子理论和平面代数、结理论等。冯·诺伊曼代数的发展在历史上也与可测量动力学的研究密切相关，这些联系最近在一种新发展的刚性现象中开始重新出现。对这种刚性现象的研究导致了冯·诺伊曼代数和其他数学领域之间的新联系。刚性理论的进一步发展将反过来导致这些不同领域之间的新见解和联系，并将为在其他领域利用这一现象提供机会。本项目将研究von Neumann代数中新出现的变形/刚性理论，并继续探索该理论与遍历理论之间的深层联系。形变/刚性理论是由Popa在21世纪初提出的，在过去的十年里，它在回答冯·诺依曼代数和遍历理论中一些长期存在的问题方面取得了极大的成功。可变形性（如Haagerup性质、自由积或无界环）与刚性（如所谓的性质T或谱隙）之间的并并，使人们能够在冯·诺伊曼代数中发现两种现象发生的隐藏结构。这导致了对这些冯·诺伊曼代数的结构特性的新见解，并反过来在其他领域如测量群论或不变量理论中找到了应用。与变形/刚性理论同时发展的是相应的应用于群体作用遍历理论的技术。这已经导致了一些惊人的结果，但这里只是表面的触及。这个项目也将更全面地研究这些相互作用。

Von Neumann algebras were introduced in the 1930s and 1940s in part as a tool for developing a mathematical foundation for quantum physics. Von Neumann algebras have since become a field of independent interest with further applications to areas of mathematics such as ergodic theory, Voiculescu's free probability theory, Jones's theory of subfactors and planar algebras, knot theory, and many others. The development of von Neumann algebras has also historically been closely connected to the study of measurable dynamics, and these connections have recently begun to reemerge in the presence of a newly developed rigidity phenomenon. The investigation of this rigidity phenomenon has since led to new connections between von Neumann algebras and other areas of mathematics. Furthering the development of rigidity theory will in turn lead to new insights and connections among these various fields, and it will provide opportunities to exploit this phenomenon in other areas. This project will investigate the newly emerging deformation/rigidity theory in von Neumann algebras, as well as continue to explore the deep connection between this theory and ergodic theory. Deformation/rigidity theory, initiated by Popa in the early 2000s, has been extremely successful over the last decade in answering a number of longstanding problems in von Neumann algebras and ergodic theory. The juxtaposition between deformability properties such as Haagerup's property, free products, or unbounded cocycles, with rigidity properties such as so-called property T or the spectral gap allows one to discover hidden structure in a von Neumann algebra where both types of phenomena occur. This has led to new insight in the structural properties of these von Neumann algebras, and in turn has found applications to other areas such as measured group theory, or the theory of invariants. Developing alongside deformation/rigidity theory is the corresponding techniques applied to ergodic theory of group actions. This has already led to a number of striking results, and yet here the surface has only been scratched. This project will also investigate more fully these interactions.