Model Theory and Differential Equations

对数-指数级数域是由van den Dries、Macintyre和Marker引入的，用于研究实数域的指数运算。这个场有一个自然的导数，并为渐近分析提供了一个有用的代数框架。对于实数上的微分方程，在对数-指数级数中有形式解和在无穷远处有真正的非振荡解之间似乎有很强的联系。马克将继续研究这种联系。马克还将研究微分场模型理论中更一般的问题。这是一个迷人的领域，需要将稳定性理论、微分代数和代数几何的思想复杂地结合在一起。特别是，他希望研究微分伽罗瓦理论中的问题，并应用赫鲁晓夫斯基的几何思想来理解一般方程的解的几何行为。近年来，数理逻辑的工具为经典数学中的问题提供了新的视角。模型论方法在证明指数代数方程和一般代数微分方程解集几何的新结果方面取得了显著的成功。这项工作已经在渐近分析、控制理论、微局部分析、数论和神经网络中得到了应用。马克的工作的一个方面是试图使用逻辑和代数工具来理解微分方程解的渐近行为。另一个方面是试图了解非常一般的微分方程解的几何。*

9971417Marker The field of logarithmic-exponential series was introduced by van den Dries, Macintyre, and Marker to study the field of real numbers withexponentiation. This field has a natural derivation and provides auseful algebraic framework for asymptotic analysis. For differentialequations over the real numbers, there seems to be a strong connectionbetween having formal solutions in the logarithmic-exponential seriesand having real non-oscillating solutions at infinity. Marker willcontinue to study this connection. Marker will also work on moregeneral problems in the model theory of differential fields. This isa fascinating area requiring a sophisticated mixture of ideas fromstability theory, differential algebra, and algebraic geometry. Inparticular, he hopes to work on problems in differential Galois theoryand on applying geometric ideas of Hrushovski to understand thegeometric behavior of solutions of generic equations. In recent years, tools from mathematical logic have provided newinsights into problems in classical mathematics. Model theoretic methodshave been remarkably successful in proving new results on the geometryof solution sets to exponential algebraic equations and generalalgebraic differential equations. This work has already foundapplications in asymptotic analysis, control theory, micro-localanalysis, number theory and neural networks. One aspect of Marker'swork attempts to use logical and algebraic tools to understand theasymptotic behavior of solutions to differential equations. Anotheraspect is attempting to understand the geometry of the solution setsto very general differential equations.***