Derivations, quantum Dirichlet forms, and deformation/rigidity theory in von Neumann algebras

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。Von Neumann代数理论是由F.Murray和J.von Neumann最先发展起来的一种非对易积分理论，用于理解群的表示理论以及为量子力学提供一个数学框架。最近，von Neumann代数中新的刚性现象的出现，加深了人们对von Neumann代数以及轨道等价、遍历理论和可测等价关系等领域中许多长期存在的问题的理解和解决。最近也出现了von Neumann代数上的形变/刚性理论、自由概率、群上同调和导子/量子Dirichlet形式之间的联系。本文的目的是探讨这些新的联系，以便更多地了解正在发展的关系，并利用这种理解从新的角度来探讨一些深层次的问题。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The theory of von Neumann algebras is a non-commutative integration theory which was first developed by F. Murray and J. von Neumann as a tool for understanding representation theory for groups as well as providing a mathematical framework for quantum mechanics. Recently the appearance of new rigidity phenomena in von Neumann algebras has led to deeper understanding and to the solutions to many longstanding problems both in von Neumann algebras and in other areas such as orbit equivalence ergodic theory and measurable equivalence relations. There has also emerged recently a connection between deformation/rigidity theory, free probability, group cohomology, and derivations/quantum Dirichlet forms on von Neumann algebras. It is the intent of this author to investigate these new connections in order to gain more understanding of the relationship which is developing, and to use this understanding in order to approach some deep problems from new perspectives.