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

这个项目涉及模型理论——数学逻辑的一个分支——以及它与其他数学领域的相互作用，尤其是代数和分析。Van den Dries将继续对实际领域的o-极小扩展进行调查和建设。他将特别关注实际解析几何和解析微分方程的可能应用。与后者相联系，他打算发展哈代场和对数指数级数场的模型理论。这一建议的动机是非常详细地研究某些基本的数学结构，其中微分方程的解自动具有确定的渐近行为。（微分方程是科学中无处不在的数学工具。）新颖之处在于，这项研究是从模型论的角度进行的，模型论是数理逻辑的一个分支，近年来取得了巨大的进步。模型理论的技术以前从未被应用于这样的渐近问题，这将是非常有趣的，看看这个新的视角将揭示在这个代数分析领域。

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.