Topics in o-minimal structures

o-最小结构中的主题

• 批准号: 0701364
• 负责人: Sergei Starchenko
• 金额: $ 27.13万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701364&HistoricalAwards=false
• 关键词: Topics; minimal; structures

The proposal addresses problems in the model theory of o-minimal structures.The principal investigator plans to develop Borel-Moore homology for sets definable in o-minimal expansions of fields. One of the goals is to study characteristic cycles of definable sets. The principal investigator will also consider model theoretical aspects of non-archimedean amoebas of algebraic varieties.The proposal is in a branch of mathematical logic called model theory.Model theory studies mathematical structures by considering the first-order sentences true in those structures, and the family of alternate structures that also satisfy all of those first-order sentences. (Sentences in logic are built out of a small repertoire of elements and constructions."First-order" refers to the number of quantifiers in a sentence, a measure of complexity.) In many cases these alternative structures illuminate some properties of the original mathematical objects.A good example is the non-standard analysis. A part of this proposal is an extension on ideas of non-standard analysis to analytic-geometric categories.

该建议解决了o-最小结构模型理论中的问题。主要研究者计划为可在o-最小域扩展中定义的集合开发Borel-Moore同调。目标之一是研究可定义集合的特征循环。主要研究者还将考虑代数簇的非阿基米德变形虫的模型理论方面。该提议属于数理逻辑的一个分支，称为模型理论。模型理论通过考虑在这些结构中为真的一阶句子以及满足所有这些一阶句子的替代结构家族来研究数学结构。（逻辑中的句子是由一小部分元素和结构组成的。“一阶”指的是句子中量词的数量，是衡量复杂性的一个标准。在许多情况下，这些替代结构阐明了原始数学对象的某些性质，一个很好的例子是非标准分析。这个建议的一部分是对非标准分析的概念扩展到分析几何范畴。