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子而不是标量来表示可观测量。在某些模型中，可观测量产生了算子代数。时间演化是由一个连贯的规则给出的，该规则规定了在每个时刻的代数变换。数学操作允许人们从系统随时间演变的方式中读出被建模系统的属性。在目前的工作中，研究一种相当一般的时间演化，称为自同态的半群，目的是获得一种分类方案。本项目由“本科院校研究”活动资助；该研究的一部分将由本科生进行。对于这些半群，将研究比最近发现的量化的Arveson和Powers的Fredholm指数更精细的环共轭不变量。Price教授还将考虑Hilbert空间上满足某种交换关系的等距流对所产生的算子代数的表示理论。本科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.