Model theory and o-minimal structures

该方案解决了0最小结构模型理论中的问题。首席研究员将继续进一步发展与0最小结构相关的模型理论和解析几何。在发展0最小结构的模型理论时，首席研究员计划使用微分几何技术。反过来，首席研究员希望将模型理论技术应用于复杂和实际解析几何中的问题。紧凑复杂流形的模型理论在最近几年取得了显著的成功，而该提议者计划，除其他外，使用他和他的合作者开发的方法来扩展这些进展。这个建议属于数学逻辑的一个分支，叫做模型论。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族来研究数学结构。逻辑学中的句子是由少量的元素和结构构成的。“一阶”指的是句子中量词的数量，衡量句子的复杂程度。)在许多情况下，这些替代结构阐明了原始数学对象的某些特性。非标准分析就是一个很好的例子。这一建议的一部分是将非标准分析的思想推广到复解析函数。

The proposal addresses problems in model theory of o-minimal structures. Theprincipal investigator will continue further developing model theory andanalytic geometry associated with o-minimal structures. In developing modeltheory of o-minimal structures the principal investigator is planning to usedifferential-geometric techniques. In turn the principal investigator hopesto apply model-theoretic techniques to questions in complex and realanalytic geometries. Model theory of compact complex manifolds has hadremarkable success in the recent years and the proposer plans, among theother things, to extend these advances, using methods developed by him andhis collaborators. The proposal is in in a branch of mathematical logic called model theory.Model theory studies mathematical structures by considering the first-ordersentences true in those structures, and the family of alternate structuresthat also satisfy all of those first-order sentences. (Sentences in logicare built out of a small repertoire of elements and constructions."First-order" refers to the number of quantifiers in a sentence, a measureof complexity.) In many cases these alternative structures illuminate someproperties of the original mathematical objects. A good example is the non-standard analysis. A part of this proposal is anextension on ideas of non-standard analysis to complex-analytic functions.