Mathematical Sciences: Semigroups and Endormorphisms

普莱斯教授的项目涉及数学研究 是由量子力学激发的。量子力学 一种观点是用线性表示可观测量 希尔伯特空间中的算子，而不是标量。在一些 模型中，观测值产生了一个代数的运营商。 时间演化是由一个连贯的规则给出的， 代数的变换。数学 操纵允许一个读出系统的属性 是根据它随时间演变的方式来建模的。本 一种相当普遍的时间演化，称为 半群的自同态，是要研究的目标是 获得分类方案。该项目已获得资助 在本科院校的研究活动;有 是研究的一个组成部分， 本科生。 循环共轭不变量比最近的 发现了量子化Fredholm指数的Arveson和权力， 将研究这些半群。普赖斯教授还将 考虑生成算子代数的表示理论 通过满足a的Hilbert空间上的等距流对 一定的对易关系。本科部分 涉及移位寄存器序列的工作， 密码分析技术，似乎与转移有关， 超有限因子的自同态

Professor Price's project involves mathematical research that is motivated by quantum mechanics. The quantum mechanical point of view is to represent observable quantities by linear operators on Hilbert space rather than by scalars. In some models, the observables give rise to an algebra of operators. Time evolution is given by a coherent rule specifying a transformation of the algebra at each moment. Mathematical manipulations allow one to read off properties of the system being modeled from the way it evolves over time. In the present undertaking, a rather general sort of time evolution, called a semigroup of endomorphisms, is to be studied with the goal of obtaining a scheme of classification. This project is funded under the Research in Undergraduate Institutions activity; there is a component of the research that will be carried out by undergraduate students. Cocycle conjugacy invariants finer than the recently discovered quantized Fredholm index of Arveson and Powers for these semigroups will be investigated. Professor Price will also consider the representation theory of operator algebras generated by pairs of isometric flows on Hilbert space that satisfy a certain commutation relation. The undergraduate component involves work on shift register sequences, a piece of cryptanalytic technology that seems to be related to shift endomorphisms of the hyperfinite factor.