Mathematical Sciences: Topics in Model Theory

9504788皮莱该项目的很大一部分涉及逻辑和微分代数之间的相互作用。一方面，利用模型论方法(推广了Picard-Vessiot理论)，发展了一种新的微分场Galois理论，并提出了各种Galois逆问题。另一方面，一个支持模型论维度理论的微分域是否必须是差分闭的(即包含域上所有代数微分方程解)。该项目的其他部分涉及(1)模型理论中更传统的问题，如确定完备理论的模型数量，以及(2)研究O-极小理论中可定义的群，这相当于试图在任意实闭域上发展一个合理的李群理论。对对称集合(也称为群)的理解是数学及其应用的重要组成部分。皮莱的项目涉及各种情况下产生的对称性：微分方程式、代数方程式和更抽象的背景。他将使用逻辑学中的方法(特别是模型理论)来分析、识别这些对称组，并在它们之间建立联系。这样应用的逻辑的表达能力将获得具体的数学信息。***

9504788 Pillay A substantial part of the project concerns the interaction between logic and differential algebra. On the one hand a new Galois theory for differential fields is developed, using model-theoretic methods (and generalising the Picard-Vessiot theory), and various inverse Galois problems are posed. On the other hand, it is asked whether a differential field which supports a model-theoretic dimension theory must be differentially closed (namely contain solutions to all algebraic differential equations over the field). Other components of the project involve (1) more traditional questions in model theory such as determining the number of models of complete theories, and (2) studying groups definable in O-minimal theories, which amounts to trying to develop a reasonable theory of Lie groups over arbitrary real closed fields. The understanding of collections of symmetries (also called groups) is an important part of mathematics and of its applications. Pillay's project concerns symmetries arising from various situations: differential equations, algebraic equations, and more abstract contexts. He will use methods from logic (specifically model theory) to analyse, identify, and make connections between such groups of symmetries. The expressive power of logic thus applied will gain concrete mathematical information. ***