Mathematical Sciences: Topics in Model Theory

小行星9504788 该项目的一个重要部分涉及以下方面的相互作用： 逻辑和微分代数 一方面，一个新的伽罗瓦理论 微分场的开发，使用模型理论的方法（和 Picard-Vessiot理论）和各种逆伽罗瓦 问题就来了。 另一方面，有人问， 支持模型论维度理论的领域必须是 差分闭合（即包含所有代数方程的解 域上的微分方程）。 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 完整理论模型的数量;（2）研究群体 在O-最小理论中定义，这相当于试图开发一个 任意真实的闭域上李群的合理理论。 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 项目涉及各种情况下产生的对称性：微分 方程，代数方程，以及更抽象的上下文。 他将利用 从逻辑（特别是模型理论）的方法来分析，识别， 在这些对称群之间建立联系。的表现力 这样应用的逻辑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. ***