Model Theory of Valued Fields

值域模型理论

• 批准号: 1790750
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2016
• 资助国家: 英国
• 起止时间: 2016 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1790750
• 关键词: Model; Theory; Valued; Fields

My research project is concerned with application of model theory to valuation theory. Valuation theory is an area of algebra which stems from the generalization of the concept of absolute value in a field to the concept of a valuation. An absolute value consists of a map, satisfying certain properties, from a field to the additive group of real numbers. If we consider maps satisfying the same properties, but we allow to replace the real numbers with an arbitrary abelian group, we come to the more general notion of valuation. The theory of valued fields has been developed during the past century, and it has several applications to other branches of mathematics, such as number theory and algebraic geometry. Model theory is an area of mathematical logic. It can be considered as the study of mathematical structures with respect to the formulas which are satisfied in them, or as the study of formulas and sets of formulas with regard to the structures in which those are satisfied. The formulas in consideration are usually expressed in first-order logic, or in logics related to the latter. Mathematical structures are involved in plenty of branches of mathematics, and this is the reason why model theory is the part of mathematical logic which has most connections with the rest of mathematics. In particular, as algebraic structures are particular cases of structures satisfying first-order theories, model theory has naturally several applications to algebra. In the past decays model theory has played an important role in the study of algebraic theories. For example, the theory of algebraically closed fields has been understood in much more depth thanks to the fact that it has been studied from a model theoretic point of view. Since the properties which define valuations are also expressible in first-order logic, their study is also suitable in a model theoretic setting. Over the last years, there have been some advances in modern model theory for applications to the study of valued fields. In light of these developments, the aim of my research is to enhance the bridge between model theory and valuation theory. This would increase our understanding of valued field, and the results which will be proved in my work could help obtaining a larger amount of applications to other areas of mathematics. A first step into this direction consists of addressing classical model theoretic questions, such as definability or decidability of valued fields. Further, my research may involve the development of further model theoretic tools, which would help in the pursue of investigating valued fields, but which might also turn out to be useful in other contexts.This project falls within the EPSRC Mathematical Logic research area

我的研究项目是关于模型理论在估值理论中的应用。估值理论是代数的一个领域，它源于一个领域的绝对价值概念的推广到估值的概念。一个绝对值由一个从域到实数加性群的映射构成，该映射满足一定的性质。如果我们考虑映射满足相同的性质，但我们允许用任意阿贝尔群替换实数，我们就得到了更一般的估值概念。值域理论是在过去的一个世纪中发展起来的，它在数学的其他分支中有几个应用，如数论和代数几何。模型论是数理逻辑的一个领域。它可以被认为是研究数学结构中满足的公式，或者是研究公式和公式集中满足这些公式的结构。所考虑的公式通常用一阶逻辑或与后者相关的逻辑表示。数学结构涉及数学的许多分支，这就是为什么模型论是数学逻辑中与数学其他部分联系最多的部分。特别是，由于代数结构是满足一阶理论的结构的特殊情况，模型理论在代数中自然有几种应用。在过去的几年中，模型理论在代数理论的研究中起着重要的作用。例如，由于从模型理论的角度对代数闭场理论进行了研究，因此人们对它的理解更加深入。由于定义赋值的性质在一阶逻辑中也是可表达的，因此它们的研究也适用于模型理论的设置。近年来，现代模型理论在应用于数值领域研究方面取得了一些进展。鉴于这些发展，我的研究目的是加强模型理论和估值理论之间的桥梁。这将增加我们对有值场的理解，在我的工作中所证明的结果将有助于在其他数学领域获得更大的应用。进入这个方向的第一步包括解决经典的模型理论问题，例如值域的可定义性或可判定性。此外，我的研究可能涉及进一步的模型理论工具的发展，这将有助于调查有价值的领域，但也可能在其他情况下是有用的。该项目属于EPSRC数理逻辑研究领域