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.
