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.

该项目处理有关 o-minimality、ofo-minimality 的扩展和有限结构类的问题。与 o 极小性有关的一些问题涉及原型 o 极小结构和其域具有实数顺序类型的结构的扩展。其他人则以可在 o 最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 最小类似物的开发为重点。有关 o 极小性扩展的问题尤其与弱 o 极小性、局部 o 极小性以及与莫利等级类似的有限等级有序结构模型理论的发展有关。该项目的第三个主要主题涉及具有维度和度量的有限结构类。这项工作的目标是开发一种与无限结构的主流模型理论类似的有限结构类模型理论。迄今为止获得的结果和已发现的例子表明还有很多工作要做。 上述研究涉及模型理论，它是数理逻辑的主要子领域之一。模型理论家研究熟悉的数学结构的属性，这些结构可以用形式数学语言（例如谓词逻辑）来表达。这种独特的观点可以提供对此类结构的见解和理解，否则这些结构可能难以捉摸。该项目的一个方面重点关注包含实数有序域并以重要方式表现的结构，即实数以及第一年微积分中研究的多项式和代数函数，并描述许多现象。模型理论在过去十年取得的许多重大进展中发挥了关键作用。这些加深了我们对数学科学不同领域中熟悉的数学系统的理解，例如实函数的分析和几何、神经网络和关系数据库理论。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。一般来说，有限结构是计算机科学的核心：任何数据库都可以被解释为这里研究的有限结构，并且一类称为有限域的特殊有限结构在密码学中尤其重要。