RUI: Classification of quantum dynamical systems

算子代数理论起源于20世纪30年代冯·诺伊曼和其他人为发展量子力学系统的严格数学模型所做的努力。该理论的一个基本原则是，这种数学模型的时间演化是由一组单参数自同构规定的。如果考虑到系统的进化可以向前移动，但不一定能在时间上向后移动，就会得到一个更笼统的图景。这个概念导致了对自同构的单参数半群的研究，而不是（时间可逆的）自同构的单参数群。Price希望与该领域的两位领导者Alexis Alevras和robert T. Powers继续他的工作，对作用于hilbert空间上所有有界算子的算子代数的自同态半群族进行分类。Price还计划继续他在称为超有限II_1因子的算子代数上的自同态研究。超有限代数R在算子代数的研究中具有重要的基础意义，它本身与自旋系统的某些数学模型有关。在此之前，他已经利用数论思想对这种称为二元移位的代数上的自同态族进行了完全的分类。他目前在这一领域的工作主要集中在获得R上所有像在R中极大的自同态的完整分类。这项工作使用了组合学和线性代数以及数论的思想。本研究的线性代数与组合方面，可望带来一些资优本科生可触及的研究问题。price建议继续与对与这项工作相关的荣誉项目感兴趣的数学专业学生合作。

The theory of operator algebras originated in the 1930s with theefforts of J. von Neumann and others to develop rigorous mathematicalmodels of quantum mechanical systems. A fundamental precept of thistheory is that the time evolution of such a mathematical model isprescribed by a one-parameter group of automorphisms. A more generalpicture is obtained if one considers evolutions of systems that can moveforward but not necessarily backward in time. This notion leads to thestudy of one-parameter semigroups of endomorphisms, as opposed to (timereversible) one-parameter groups of automorphisms. Price hopes tocontinue his work with two leaders in the field, Alexis Alevras andRobert T. Powers, on classifying families of semigroups of endomorphismsthat act on the operator algebras of all bounded operator on a Hilbertspace. Price also plans to continue his study of endomorphisms on theoperator algebra called the hyperfinite II_1 factor. The hyperfiniteII_1 algebra R is of fundamental importance in the study of operatoralgebras and is itself related to certain mathematical models of spinsystems. Previously he has used number-theoretic ideas to arrive at acomplete classification of the family of endomorphisms on this algebraknown as binary shifts. His current work in this area is focused onobtaining a complete classification of all endomorphisms on R whoseimage is maximal in R. This work uses ideas from combinatorics andlinear algebra as well as number theory. The linear algebraic andcombinatoric aspects of this study are expected to lead to some researchproblems that are accessible to gifted undergraduate students. Priceproposes to continue working with mathematics majors who are interestedin honors projects related to this work.