Rigidity Results in von Neumann Algebras and Orbit Equivalence

拟议的项目将扩展 PI 先前在冯诺依曼代数和轨道等效遍历理论领域的研究。 PI 将利用新的想法和技术来获得由概率空间上的群作用产生的冯诺依曼代数的额外刚性结果。 PI 正在寻找产生 W*E 超刚性作用新例子的通用方法，这是该理论的核心兴趣问题，因为它统一了轨道等效和冯·诺依曼代数的极端形式的刚性。此外，PI 将继续之前对某些冯诺依曼代数的结构特性的研究，例如获得合并自由积的附加 Bass-Serre 刚性类型结果。在过去的十年中，波帕的变形/刚性理论解决了冯诺依曼代数、轨道等效和描述集合论中许多长期存在的问题。它还引发了新的令人兴奋的并行发展，例如彼得森的冯诺依曼代数推导理论，该技术旨在从群表示理论的上同调方面为该领域带来更多见解。此外，它还产生了一系列新的开放问题，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目标。 PI 预计与无限群表示理论和几何群论有更多联系。最后，他将寻求吸引该领域新的年轻研究人员，并通过共同组织冯诺依曼代数会议和非正式研讨会来探索与其他数学领域的联系。

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.