Model Theory of Valued Fields

我的研究项目是关于模型理论在估值理论中的应用。赋值理论是代数的一个领域，它起源于一个领域中绝对值的概念到赋值的概念的推广。一个绝对值由一个满足某些性质的从域到真实的数的加法群的映射组成。如果我们考虑满足相同性质的映射，但我们允许用任意的阿贝尔群代替真实的数，我们就得到了更一般的赋值概念。值域理论是在过去的世纪中发展起来的，它在数学的其他分支，如数论和代数几何中也有一些应用。模型论是数理逻辑的一个领域。它可以被认为是对数学结构的研究，而这些数学结构是关于满足它们的公式的，或者是对公式和公式集的研究，而这些公式和公式集是关于满足它们的结构的。所考虑的公式通常用一阶逻辑或与一阶逻辑相关的逻辑来表示。数学结构涉及到数学的许多分支，这就是为什么模型论是数理逻辑中与其他数学联系最多的部分。特别是，由于代数结构是满足一阶理论的结构的特殊情况，模型论自然有几个应用代数。在过去的衰变模型理论在代数理论的研究中发挥了重要作用。例如，代数闭域的理论已经被更深入地理解，这要归功于它已经从模型论的角度进行了研究。由于定义赋值的性质也可以在一阶逻辑中表达，因此它们的研究也适用于模型论的设置。在过去的几年里，现代模型理论在研究有价值的领域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。这将增加我们对有价值的领域的理解，并且在我的工作中将被证明的结果可能有助于获得更大数量的应用到其他数学领域。这个方向的第一步包括解决经典的模型理论问题，如可定义性或可判定性的价值领域。此外，我的研究可能涉及进一步的模型理论工具的发展，这将有助于在调查有价值的领域的追求，但这也可能在其他情况下是有用的。这个项目属于EPSRC数理逻辑研究领域福尔斯

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