Mathematical Sciences: Semigroups and Endormorphisms

普赖斯教授的项目涉及由量子力学推动的数学研究。量子力学的观点是用希尔伯特空间上的线性算符而不是标量来表示能观量。在某些模型中，可见项产生了一个算符代数。时间演化是由一个连贯的规则给出的，该规则规定了代数在每个时刻的变换。数学处理使人们能够从系统随时间演变的方式中读出被建模的系统的属性。在目前的工作中，将研究一种相当一般的时间演化，称为自同态半群，目的是获得一种分类方案。该项目由本科生研究活动资助；研究中有一个组成部分将由本科生进行。比最近发现的Arveson和幂的量子化Fredholm值更精细的循环共轭不变量将被研究。普赖斯教授还将考虑由满足某种对易关系的希尔伯特空间上的等距流对生成的算子代数的表示理论。本科生的部分涉及移位寄存器序列的工作，这是一种密码分析技术，似乎与超有限因子的移位自同态有关。

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.