Dissertation Enhancement: Noncommutative Dynamical Systems

9802696达达拉特该奖项为日本东京大学马里乌斯·达达拉特博士颁发的为期12个月的国家科学基金会学位论文增强奖提供支持。达达拉博士的学生纳撒尼尔·P·布朗先生将与数学系教授川桥康之博士合作开展一项名为《非自变量动力系统》的研究项目。动力系统是一种数学模型，人们试图通过它来理解物理现象的行为。经典的或可交换的动力学系统的研究与现代物理学的许多方面密切相关，包括混沌和量子理论。这项研究将尝试将经典动力系统理论应用于非对易环境，并将研究与交叉积C*-代数有关的一系列几何和代数问题。研究的第一部分将考察某些交叉积C*-代数何时被嵌入到AF代数中，从而允许关于交叉积C*-代数的特定几何和代数性质的推导。研究的第二部分将建立在交叉积C*-代数的一些已知分类结果的基础上；具体地说，它将试图确定某些交叉积何时本质上相同。最后，本研究将尝试为这些非对易系统发展拓扑熵的创新概念。熵在对易系统的研究中起到了重要的作用，但非对易环境下的熵相对较新，还没有得到很好的理解。这项研究旨在研究数学的这一方面，目的是为未来的考试实现具体的计算。***

9802696 Dadarlat This award supports a twelve month NSF Dissertation Enhancement Award for Dr. Marius Dadarlat at the University of Tokyo, Tokyo, Japan. Dr. Dadarlat's student, Mr. Nathaniel P. Brown, will work with Dr. Yasuyuki Kawahigashi, Professor in the Department of Mathematics, on a research project entitled, "Noncummutative Dynamical Systems." A dynamical system is a mathematical model through which one attempts to understand the behavior of physical phenomena. The study of classical, or commutative, dynamical systems is closely related to many aspects of modern physics including both chaos and quantum theories. The proposed research will attempt to apply classical dynamical systems theory to the noncommutative setting and will investigate a series of geometric and algebraic questions concerning crossed product C*-algebras. The first component of the research will examine when certain classes of crossed product C*-algebras are embedded into AF algebras, allowing deduction concerning the specific geometric and algebraic properties of the crossed product C*-algebra. The second component of the research will build on some of the known classification results for crossed product C*-algebras; specifically, it will attempt to determine when certain crossed products are essentially the same. Finally, the research will attempt to develop an innovative notion of topological entropy for these noncommutative systems. Entropy has played an important role in the study of commutative systems, but entropy in the noncommutative setting is relatively new and not yet well understood. The research is designed to study this facet of mathematics with the goal of achieving concrete calculations for future examination. ***