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.

该项目涉及有关O最小性，最小程度的扩展和有限结构类别的问题。与O最小性有关的一些问题与原型O-最低结构和结构的扩展有关，其域作为其实数的顺序类型。其他人则是在O最低结构中定义的Abelian群体，或者开发了差异和代数拓扑方法和工具的O-Winimal类似物。有关O-最低限度扩展的问题，尤其是在O最小性，局部O最小程度的较弱的问题上，并且与Morley等级类似，是有限等级有序结构的模型理论的发展。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别。这项工作的目的是开发用于有限结构类别的模型理论，该理论与无限结构的主流模型理论类似。迄今为止获得的结果以及发现的示例表明，有很多事情要做。模型理论家研究熟悉的数学结构的属性，可以用正式的数学语言（例如谓词逻辑）表示。这种独特的观点可以提供对否则可能难以捉摸的结构的见解和理解。该项目的一个方面集中于包括和行为的结构，例如实际数字的有序领域，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。模型理论在过去十年中取得的许多重大进步​​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，例如对真实功能，神经网和关系数据库理论的分析和几何形状。也已在经济学中提出了申请。该项目的第二个主要方面涉及有限结构的类别。通常，有限的结构是计算机科学的核心：在此处研究的任何数据库都可以解释为有限结构，并且在密码学中尤其重要。