Deformation/rigidity theory in von Neumann algebras and ergodic theory

形变/刚性理论是由索林·波帕于2000年初由S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的一些长期存在的问题方面取得了极大的成功。可变形性质(如Haagerup性质、自由积或无界上循环)与刚性性质(如性质(T)或谱间隙)之间的并置使人们能够发现von Neumann代数中同时出现这两种现象的隐藏结构。这使人们对这些von Neumann代数的结构性质有了新的认识，并反过来又在其他领域找到了应用，如测度群理论，或L2不变量理论。首席调查员将继续调查和发展这些想法，重点是它们与遍历理论的各个方面的密切联系。冯·诺依曼代数在1930年的S和40年的S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·诺依曼代数已经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一直与可测动力学的研究密切相关，最近随着新发展的刚性现象的出现，这些联系开始重新出现。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。进一步发展刚性将反过来导致这些不同领域之间的新见解和联系。

Deformation/rigidity theory, initiated by Sorin Popa in the early 2000's, 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 property (T) or 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 into 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 L2-invariants. The Principal Investigator will continue to investigate and develop these ideas, focusing on their close connection to aspects of ergodic theory. Von Neumann algebras were introduced in the 1930's and 40's 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' 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 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 will in turn lead to new insights and connections among these various fields.