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，部分原因是作为一种工具，为量子物理奠定了数学基础。自那以后，冯·诺依曼代数已成为一个独立的研究领域，并进一步应用于数学领域，如遍历理论、沃库列斯库的自由概率理论、琼斯的子因子和平面代数理论、纽结理论等。Von Neumann代数的发展历史上也与可测动力学的研究密切相关，最近随着一种新发展的刚性现象的出现，这些联系又开始重新出现。从那时起，对这种刚性现象的研究导致了冯·诺依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带来这些不同领域之间的新的洞察和联系，并将为在其他领域探索这一现象提供机会。这个项目将研究von Neumann代数中新出现的形变/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/刚性理论由Popa在本世纪初提出，在过去的十年里，它在回答von Neumann代数和遍历理论中的一些长期存在的问题方面取得了极大的成功。形变性质(如Haagerup性质、自由积或无界余圈)与刚性性质(如所谓的性质T或谱间隙)之间的并置使人们能够发现von Neumann代数中出现这两种现象的隐藏结构。这导致了对这些von Neumann代数的结构性质的新的认识，并反过来又在其他领域找到了应用，例如测量群论，或不变量理论。变形/刚性理论是应用于群作用遍历理论的相应技术。这已经带来了一些令人惊叹的结果，但这里只触及了皮毛。该项目还将更全面地调查这些相互作用。

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.