Mathematical Sciences: Semigroups and Endormorphisms

数学科学：半群和自同态

• 批准号: 8902132
• 负责人: Geoffrey Price
• 金额: $ 2.29万
• 依托单位: United States Naval Academy
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-01 至 1991-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8902132&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Semigroups; Endormorphisms

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.

普莱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，可观测量产生算子代数。 时间演化由指定每个时刻代数变换的连贯规则给出。数学运算允许人们从正在建模的系统随时间演变的方式中读出其属性。在目前的工作中，将研究一种相当普遍的时间演化，称为自同态半群，目的是获得分类方案。该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。 将研究比最近发现的这些半群的 Arveson 和 Powers 的量化 Fredholm 指数更精细的共循环共轭不变量。 Price教授还将考虑希尔伯特空间上满足一定交换关系的等距流对生成的算子代数的表示论。本科生部分涉及移位寄存器序列的研究，这是一项密码分析技术，似乎与超有限因子的移位自同态有关。