Mathematical Sciences: Topics in Model Theory

9504788 Pillay这个项目的很大一部分是关于逻辑和微分代数之间的相互作用。一方面，利用模型理论方法（并推广了Picard-Vessiot理论）建立了新的微分场伽罗瓦理论，并提出了各种反伽罗瓦问题。另一方面，一个支持模型论维度理论的微分场是否必须是微分闭的（即包含该场上所有代数微分方程的解）。该项目的其他组成部分涉及(1)模型理论中更传统的问题，例如确定完全理论的模型数量，以及(2)研究在o -极小理论中可定义的群，这相当于试图在任意实闭域上建立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、代数方程和更抽象的环境。他将使用逻辑方法（特别是模型理论）来分析、识别并在这些对称组之间建立联系。这样运用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. ***