Noncommutative Symmetries and Renormalization

AbstractPopaPopa将继续他的调查强刚性和超刚性的背景下，运营商代数，非交换遍历理论和描述集理论。他以前的结果为算子代数的各种关键问题提供了强有力的新方法，如Connes的刚性猜想和Jones的“千禧年问题”。另一方面，他预计的角度来看，算子代数理论将使他作出额外的重要贡献，非交换遍历理论。与他的合作者和学生，他希望找到这些不同领域之间重要的新联系。Effros计划继续他的调查的组合技术中使用的越来越广泛的非交换分析。其中包括尼卡和斯派克的自由概率论方法，以及费恩曼图的康纳斯-克雷默理论。他目前与Aguiar，Anshelevich，Nica和他的学生Mihai Popa合作。量子力学的发现是二十世纪物理学最引人注目的进步。这一主题的矛盾概念现在已经得到很好的理解，它们在当前的技术中发挥着越来越重要的作用。量子理论需要全新的数学工具，这是冯·诺依曼首先研究的。由此产生的“算子代数”理论已成为现代数学中最令人兴奋和最有影响力的领域之一。波帕打算调查的一些中心问题的主题，使用的结果，他已经发现，提供之间的联系算子代数和不同的领域，如遍历理论，群论，和描述集理论。Effros将进一步探索在费曼图和量子概率论理论中发挥越来越重要作用的“组合”概念。

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.