RUI: Research in O-minimality and Related Topics

RUI：O-极小性及相关主题的研究

• 批准号: 9704869
• 负责人: Charles Steinhorn
• 金额: $ 8.22万
• 依托单位: Vassar College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704869&HistoricalAwards=false
• 关键词: RUI; Research; minimality; Related; Topics

The project involves questions concerning o-minimality and extensions of o-minimality. Some of the problems concerning o-minimality relate to expansions of archetypal o-minimal structures and structures whose domain has as its order type that of the real numbers. Others have as their focus groups definable in o-minimal structures or the development of o-minimal analogues of more sophisticated topological methods and tools. Problems having to do with extensions of o-minimality deal in particular with the notions of weak o-minimality and local o-minimality. A range of natural examples of weakly o-minimal structures is now known and there already is a significant body of results in the subject. It is hoped that local o-minimality may provide a suitable framework for developing some model theory for subanalytic sets. The research described above falls under the heading of model theory, one of the principal subfields of mathematical logic. Model theorists study properties of familiar mathematical structures which can be described by a formal mathematical language such as predicate logic. The distinctive point of view of model theory can provide insights and understanding into such structures that otherwise might not be easily achieved. This project focuses on structures that include and behave in important respects 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. This decade has witnessed significant advances in which model theory has played a crucial role. The results obtained have deepened our understanding of familiar mathematical systems of interest in such diverse areas of the mathematical sciences as the analysis and geometry of real functions, neural nets, and relational database theory.

该项目涉及有关O-极小和O-极小的扩展的问题。与o-极小有关的一些问题涉及原型o-极小结构的展开，以及其定义域具有实数的阶类型的结构。另一些人则以o-极小结构或开发更复杂的拓扑方法和工具的o-极小类似物来定义他们的重点群体。与O-极小的扩张有关的问题特别涉及到弱O-极小和局部O-极小的概念。弱o-极小结构的一系列自然例子现在已为人所知，并且在这个主题中已经有了大量的结果。希望局部o-极小性能够为次解析集的某些模型理论的发展提供一个合适的框架。上述研究属于模型理论，它是数理逻辑的主要分支领域之一。模型理论家研究可用谓词逻辑等形式数学语言描述的常见数学结构的性质。模型理论的独特观点可以提供对此类结构的洞察和理解，否则这些结构可能不容易实现。这个项目专注于包含并表现在重要方面的结构，如实数的有序域，即实数以及在第一年微积分中研究并描述许多现象的多项式和代数函数。这十年见证了巨大的进步，模型理论在其中发挥了关键作用。所获得的结果加深了我们对在数学科学的不同领域中感兴趣的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。