Mathematical Sciences: Topics in Model Theory

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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. ***