O-Minimal Expansions of the Real Field, and Model Theory of Hardy Fields

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This project concerns model-theory---a branch of mathematical logic---and its interaction with other areas of mathematics, especially algebra and analysis. Van den Dries will continue the investigation and construction of o-minimal expansions of the real field. He will give special attention to possible applications to real analytic geometry and to analytic differential equations. In connection with the latter he intends to develop the model theory of Hardy fields and of fields of logarithmic-exponential series. The motivation for this proposal is to investigate in great detail certain fundamental mathematical structures where the solutions of differential equations automatically have a definite asymptotic behavior. (Differential equations are a ubiquitous mathematical tool in the sciences.) The novelty is that this study is undertaken from the point of view of model theory, a branch of mathematical logic which has made enormous strides in recent years. The techniques of model theory have never been applied to such asymptotic questions before, and it will be extremely interesting to see what this new perspective will reveal in this area of algebraic analysis.