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.

该项目涉及的问题有关的O-极小和O-极小的扩展。关于o-极小的一些问题涉及到原型o-极小结构和结构的扩展，其域具有作为其序类型的真实的数。其他人有他们的焦点群体定义在o-最小的结构或发展的o-最小的类似物更复杂的拓扑方法和工具。问题与O-极小的扩展处理特别是与弱O-极小和局部O-极小的概念。一系列自然的例子弱o-极小结构现在是已知的，已经有一个显着的身体的结果在这个问题上。人们希望，局部o-极小可能提供一个合适的框架，发展一些模型理论的次解析集。上述研究福尔斯属于模型论的范畴，模型论是数理逻辑的主要分支之一。模型理论家研究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）来描述。模型论的独特观点可以提供对这种结构的洞察和理解，否则可能不容易实现。这个项目的重点是结构，包括和行为的重要方面，如有序领域的真实的数字，即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。这十年见证了重大的进展，模型理论发挥了至关重要的作用。 所获得的结果加深了我们的理解熟悉的数学系统的兴趣在这样的不同领域的数学科学的分析和几何的真实的功能，神经网络和关系数据库理论。