Noncommutative Symmetries and Renormalization

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AbstractPopaPopa 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.