Model Theory and Differential Equations

9971417标记 幂指数级数域是由货车den Dries，Macintyre和Marker为了研究具有幂指数的真实的数域而引入的。 这个领域有一个自然的推导，并提供了一个有用的代数框架的渐近分析。 对于真实的数上的微分方程，在无穷指数级数的形式解和无穷远处的真实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。 马克还将致力于更一般的问题，在模型理论的微分场。 这伊萨个迷人的领域，需要从稳定性理论，微分代数和代数几何的思想复杂的混合。 特别是，他希望工作的问题，微分伽罗瓦theory和应用几何思想的Hrushovski了解thegeometric行为的解决方案的一般方程。 近年来，数理逻辑工具为研究经典数学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。 这一工作已在渐近分析、控制理论、微观局部分析、数论和神经网络等领域得到应用。 一个方面的标记的工作试图使用逻辑和代数工具来理解的渐近行为的解决方案微分方程。 另一方面是试图理解非常一般的微分方程的解集的几何形状。

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.***