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.***

9971417标记器的对数指数序列的领域是由Van Den Dries，MacIntyre和Marker介绍的，以研究具有点的实数领域。 该领域具有自然推导，并为渐近分析提供了可爱的代数框架。 对于对实数的差异化，在对数指数系列中具有形式的解决方案之间似乎存在着很强的联系，而无穷大则具有实际非振荡的解决方案。 标记将研究这种联系。 标记还将在差异领域模型理论中处理昆虫问题。 这个令人着迷的领域，需要从稳定理论，差异代数和代数几何形状中复杂的思想混合。 他希望在鉴别式的Galois理论和应用Hrushovski的几何思想中研究问题的问题，以了解通用方程解决方案的几处行为。 近年来，来自数学逻辑的工具为经典数学的问题提供了新的视觉。 模型理论方法Shave在证明对指数代数方程和通用级别微分方程的几何溶液集上的新结果方面非常成功。 这项工作已经在渐近分析，控制理论，微核分析，数字理论和神经网络中找到了应用。 标记工作的一个方面试图使用逻辑和代数工具来理解解决方案解决方案的静脉响应行为。 另一个方面是试图理解解决方案的几何形状。***