Model Theory and Differential Equations

模型理论和微分方程

• 批准号: 9971417
• 负责人: David Marker
• 金额: $ 10.77万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971417&HistoricalAwards=false
• 关键词: Model; Theory; Differential; Equations

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.***

9971417Marker 对数指数级数域由 van den Dries、Macintyre 和 Marker 引入，用于研究具有指数的实数域。 该域具有自然的推导性，并为渐近分析提供了有用的代数框架。 对于实数上的微分方程，对数指数级数的形式解与无穷大的实数非振荡解之间似乎存在密切的联系。 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普遍的问题。 这是一个令人着迷的领域，需要稳定性理论、微分代数和代数几何的思想的复杂结合。 特别是，他希望致力于解决微分伽罗瓦理论中的问题，并应用赫鲁索夫斯基的几何思想来理解泛方程解的几何行为。 近年来，数理逻辑工具为经典数学问题提供了新的见解。 模型理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巨大成功。 这项工作已经在渐近分析、控制理论、微观局部分析、数论和神经网络中得到应用。 马克的工作一方面尝试使用逻辑和代数工具来理解微分方程解的渐近行为。 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。***