Rigidity Results in von Neumann Algebras and Orbit Equivalence

拟议的项目将扩大PI？在von Neumann代数和轨道等价遍历理论等领域的研究。PI将利用新的思想和技术，以获得额外的刚性结果冯诺依曼代数所产生的群体行动的概率空间。PI正在寻求产生W*E超刚性作用的新例子的一般方法，这是理论中的一个核心问题，因为它统一了轨道等价和冯诺依曼代数的极端刚性形式。此外，PI将继续他之前对某些冯诺依曼代数的结构性质的研究，例如获得合并自由积的额外Bass-Serre刚性类型结果。在过去的十年里，波帕？的变形/刚性理论导致了冯诺依曼代数，轨道等价和描述集理论的许多长期存在的问题的解决方案。它还引发了新的令人兴奋的平行发展，如彼得森？的推导理论冯诺依曼代数，一种技术，旨在使更多的洞察力领域的上同调方面的团体表示理论。此外，它产生了一系列新的公开问题，涉及冯诺依曼代数的分类和等价关系的研究，其中一些是当前提案的对象。PI预期与无限群表示理论和几何群论有更多的联系。最后，他将寻求吸引新的年轻研究人员在该领域，并利用与其他领域的数学共同组织会议和非正式研讨会在冯诺依曼代数的连接。

The proposed project will expand PI?s prior research in the fields of von Neumann algebras and orbit equivalence ergodic theory. PI will exploit new ideas and techniques to obtain additional rigidity results for von Neumann algebras arising from group actions on probability spaces. The PI is seeking general methods that produce new examples of W*E superrigid actions, a problem of central interest in the theory as it unifies extreme forms of rigidity from both orbit equivalence and von Neumann algebras. In addition, the PI will continue his prior investigation of structural properties of certain von Neumann algebras, such as obtaining additional Bass-Serre rigidity type results for amalgamated free products. Over the last decade, Popa?s deformation/rigidity theory led to the solutions of many long-standing problems in von Neumann algebras, orbit equivalence and descriptive set theory. It also triggered new exciting parallel development, such as Peterson?s derivation theory for von Neumann algebras, a technique designed to bring more insight to the field from cohomological aspects of groups representation theory. Moreover, it generated a new array of open problems concerning the classification of von Neumann algebras and the study of equivalence relations, some of which are the object of the current proposal. The PI anticipates additional connections with infinite group representation theory and geometric group theory. Finally, he will seek to attract new young researchers in the field and to exploit connections with other areas of mathematics by co-organizing conferences and informal seminars in von Neumann algebras.