Popa will continue his investigation of strong rigidity and superrigidity in the context of operator algebras, non-commutative ergodic theory, and descriptive set theory. His previous results provide powerful new approaches to various key problems for operator algebras, such as the Connes' Rigidity Conjecture and Jones' "Millenium Problems". On the other hand, he anticipates that the perspective of operator algebra theory will enable him to make additional important contributions to non-commutative ergodic theory. With his collaborators and students, he expects to find important new links between these different areas. Effros plans to continue his investigation of the combinatorial techniques used in an increasingly broad range of non-commutative analysis. These include the Nica and Speicher approach to free probability theory, as well as the Connes-Kreimer theory of Feynman diagrams. He is currently collaborating with Aguiar, Anshelevich, Nica, and his student Mihai Popa.
The discovery of quantum mechanics provided the most dramatic advance in physics during the Twentieth century. The paradoxical notions of this subject are now well understood, and they are playing an increasingly important role in current technology. Quantum theory requires completely new mathematical tools, which were first investigated by von Neumann. The resulting theory of "operator algebras" has become one of the most exciting and influential areas of modern mathematics. Popa intends to investigate some of the central questions of the subject by using results that he has discovered which provide links between operator algebras and such disparate areas as ergodic theory, group theory, and descriptive set theory. Effros will further explore the "combinatorial" notions that are playing an increasingly important role in the theory of Feynman diagrams and quantum probability theory.
" The PI's work during 2006-2011 focused on the study of symmetries of von Neumann algebras and the investigation of rigidity phenomena appearing in this context. This type of algebras were introduced by John von Neumann in the 1920s, in his effort to give a rigorous approach to quantum mechanics. His work was motivated by the discovery that observables in particle physics behave like in nite matrices, rather than functions on the ground space, with consecutive measurements corresponding to matrix multiplication, where the order in the product matters, i.e. the product in these algebras is non-commutative. Thus, von Neumann algebras and their symmetries appear naturally whenever one studies quantum systems. Von Neumann and his collaborator F. Murray have related these algebras with group theory and ergodic theory by noticing that actions of groups on probability measure spaces give rise to a remarkable class of von Neumann algebras, called factors. The calculation of the symmetry groups of such factors is an important and diffcult problem. Rigidity in this framework occurs whenever the group action can be recognized by merely knowing its associated algebra. Another manifestation of rigidity is when all the symmetries of the algebra come from the symmetries of the group. During his NSF Grant 2006-2011, the PI continued to develop his deformation/rigidity theory, an intricate, effcient set of techniques aimed at studying such phenomena, that he initiated in 2001-2006. He obtained a number of important results over the last five years, which have surprised the mathematical community. His results are closely interconnected to rigidity phenomena in other areas of mathematics and led to deep interdisciplinary activity. The PIs work in rigidity theory already had considerable impact in several areas, such as group theory, ergodic theory, logic. A large number of research articles and PhD thesis have resulted from his work. He expects this work to have direct and indirect impact in applied mathematics and computer science.
