RUI: Research in O-minimality and Related Topics

RUI：O-极小性及相关主题的研究

• 批准号: 0070743
• 负责人: Charles Steinhorn
• 金额: $ 8.7万
• 依托单位: Vassar College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-15 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070743&HistoricalAwards=false
• 关键词: RUI; Research; minimality; Related; Topics

The project deals with questions concerning o-minimality, extensions ofo-minimality, and classes of finite structures. Some of the problems having to do with o-minimality relate to expansions of archetypal o-minimal structures and structures whose domain has as its order type that of the real numbers. Other have as their focus abelian groups definable in o-minimal structures or the development of o-minimal analogues of differential and algebraic topological methods and tools. Problems concerning extensions of o-minimality have to do in particular with weak o-minimality, local o-minimality, and, in analogy with Morley rank, the development of a model theory for ordered structures of finite rank. The third main topic of the project involves classes of finite structures with dimension and measure. This work has as its aim the development of a model theory for classes of finite structures that is in analogy with mainstream model theory for infinite structures. The results obtained to date and the examples that have been found suggest that there is much to be done.The research outlined above concerns model theory, one of the principal subfields of mathematical logic. Model theorists study properties of familiar mathematical structures that can be expressed in a formal mathematical language such as predicate logic. This distinctive point of view can provide insights and understanding into such structures that otherwise might prove elusive. One aspect of this project focuses on structures that include and behave in important ways like the ordered field of real numbers, that is, the real numbers together with the polynomial and algebraic functions that are studied in first-year calculus and describe many phenomena. Model theory has played a key role in many of the significant advances that have been made in the last ten years. These have deepened our understanding of familiar mathematical systems in such diverse areas of the mathematical sciences as the analysis and geometry of real functions, neural nets, and relational database theory. Applications also have been made in economics. A second principal aspect of the project deals with classes of finite structures. Finite structures in general are central to computer science: any database can be construed as a finite structure in the sense in which they are studied in here, and a particular class of finite structures called finite fields are especially important in cryptology.

该项目涉及的问题有关O-极小，扩展ofo-极小，和类的有限结构。与o-极小有关的一些问题涉及原型o-极小结构的扩展，以及其域具有真实的数的序类型的结构。其他人的重点是可在o-极小结构中定义的阿贝尔群或微分和代数拓扑方法和工具的o-极小类似物的开发。有关问题的扩展o-极小必须做特别是与弱o-极小，地方o-极小，并在类比莫利秩，发展一个模型理论的有序结构的有限秩。第三个主要课题的项目涉及类有限结构的尺寸和措施。这项工作的目的是发展一个模型理论类的有限结构，这是在类比与主流模型理论的无限结构。迄今为止所取得的结果和已经发现的例子表明，还有许多工作要做。上述研究概述涉及模型论，数理逻辑的主要子领域之一。模型理论学家研究的是熟悉的数学结构的性质，这些结构可以用形式化的数学语言（如谓词逻辑）来表达。这种独特的观点可以提供对这种结构的见解和理解，否则可能会被证明是难以捉摸的。这个项目的一个方面集中在结构，包括和行为的重要方式一样，有序领域的真实的数字，也就是说，真实的数字连同多项式和代数函数，在第一年的微积分研究，并描述了许多现象。模型理论在过去十年中取得的许多重大进展中发挥了关键作用。这些都加深了我们对熟悉的数学系统在不同领域的数学科学，如分析和几何的真实的功能，神经网络和关系数据库理论的理解。在经济学中也有应用。该项目的第二个主要方面涉及有限结构的类别。有限结构通常是计算机科学的核心：任何数据库都可以被解释为有限结构，在这里研究它们的意义上，一类特殊的有限结构称为有限域，在密码学中特别重要。