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-极小性和 o-极小性扩展的问题。有关 o 极小性的一些问题与原型 o 极小结构和其域以实数的阶类型作为其阶类型的结构的扩展有关。其他人则将焦点小组定义为 o-最小结构，或者开发更复杂的拓扑方法和工具的 o-最小类似物。与 o 极小性扩展有关的问题特别涉及弱 o 极小性和局部 o 极小性的概念。现在已知一系列弱最小结构的自然例子，并且在该主题中已经有大量结果。希望局部 o 极小性可以为开发亚解析集的一些模型理论提供合适的框架。上述研究属于模型理论的范畴，模型理论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，这些结构可以用谓词逻辑等形式数学语言来描述。模型理论的独特观点可以提供对此类结构的见解和理解，否则这些结构可能不容易实现。该项目重点关注包含并在重要方面表现的结构，例如实数的有序域，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。这十年见证了重大进步，模型理论在其中发挥了至关重要的作用。 所获得的结果加深​​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。