Collaborative Research: Generic Properties of Groups, Geometric Invariants and Algorithms

协作研究：群的泛性、几何不变量和算法

• 批准号: 0405105
• 负责人: Vladimir Shpilrain
• 金额: --
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-15 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405105&HistoricalAwards=false
• 关键词: Collaborative; Research; Generic; Properties; Groups

The idea of genericity in geometric group theory was introduced byGromov and Ol'shanskii and is now the subject of very active research.Genericity exhibits itself on many different levels in algebraic,geometric and algorithmic properties of ``random'' algebraic objectsand in the generic-case behavior of their natural geometric invariantsand decision problems. As already demonstrated by the proposers'work, a generic approach often leads to the discovery of objects withgenuinely new and interesting properties. For example, genericityprovides a totally new source of group-theoretic rigidity,quite different from the standard source provided by lattices insemi-simple Lie groups.Much of the prior research on probabilistic aspects of infinite groups has concentrated on mostly analytic questions, such as amenability, asymptotic properties of random walks, Poisson boundary, and so on. This project will focus on understanding the algebraic and algorithmic properties of random group-theoretic objects as well as probabilistic properties of various traditionally studied geometric invariants of groups. Specific topics, where the proposers have already made substantial inroads, will include: the isomorphism rigidity of generic groups, analysis of ``random'' van Kampen diagrams and generic Dehn functions, generic subgroup distortion and generic stretching factors of automorphisms, and the action of the outer automorphism group of a free group on the ``frequency space'' of a free group.With the rapid development of modern computers, understanding the practical behavior of the performance of various algorithms is becoming increasingly important. Yet most of theoretical research thus far deals with the worst-case analysis of algorithms, which often has little to do with their practical performance. On the other hand, in many real-life applications, in particular, to public key cryptography, there is a great deal of experimental data on the practical performance of algorithms that has not been adequately explained theoretically. The current project should provide a number of new benchmark ideas and theoretical tools for studying and explaining probabilistic properties and practical behavior of algorithms both in group theory and in the broad field of computational complexity.

在几何群体理论中的一般概念引入了Bygromov和Ol'shanskii，现在已成为非常积极的研究的主题。代理性在代数，几何，几何和算法属性中以“随机”'Algebraic对象的代数，几何和算法性能在其自然图中的通用case行为中的``Algebraic''a algebraic objects and shergebraic objects和自然的自然基础群体中的sandemants andsandsandsandsant invarialsandsandsandsandsandsandsandsandsandsandsand sandsandsandsandsandsandsandsand sand。 正如提案者工作已经证明的那样，一种通用的方法通常会导致发现具有新的新事物属性的对象。例如，GenericityPreved为群体理论刚性的全新来源提供了与Lattices Insemi-Simple Lie组提供的标准来源完全不同的。该项目将着重于理解随机组理论对象的代数和算法特性以及各种传统研究组的几何不变的概率特性。提议者已经进行了大量侵害的具体主题将包括：一般群体的同构刚性，分析``随机''''van kampen图和通用的dehn功能，通用亚组扭曲和自动形态的快速范围的一般性延伸因素，以及一组自动形态组的快速发展群体的动作，并在``''''''''''''''''''''''''''''''''''''''''''''计算机，了解各种算法的性能的实际行为变得越来越重要。然而，到目前为止，大多数理论研究都涉及对算法的最坏情况分析，这通常与它们的实际性能无关。另一方面，在许多现实生活中，尤其是在公共密钥密码学上，有很多关于算法实践性能的实验数据，这些数据尚未在理论上得到充分解释。当前的项目应提供许多新的基准观念和理论工具，以研究和解释算法的概率属性和算法在群体理论和计算复杂性的广泛领域中的实际行为。