Mathematical Sciences: Model Theory and Its Geometric Applications

数学科学：模型论及其几何应用

• 批准号: 9626856
• 负责人: David Marker
• 金额: $ 11.5万
• 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: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626856&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Model; Theory; Its

9626856 Marker In studying the real numbers with exponentiation, van den Dries, Macintyre and Marker gave an algebraic construction of a nonstandard model. This model has proved useful in understanding the asymptotic behavior of functions. Marker's work concentrates on the relationship between formally solving differential equations in this model and the problem of finding actual solutions at infinity. Marker will also work on problems in the model theory of differential fields. This is a fascinating area, requiring a sophisticated mixture of ideas from model theory, differential algebra and algebraic geometry. (Through the work of Buium and Hrushovski, differential algebra has had significant number theoretic applications.) Marker studies the geometry of solutions to algebraic differential equations, concentrating on the use of model theoretic dimensions to develop intersection theory. Model theory is a branch of mathematical logic where one studies mathematical structures by looking at their properties expressed in formal languages. For example, in the real numbers one can say that every number is a cube, but one cannot say that every bounded set has a least upper bound. Nonstandard models of the real numbers share all their formal properties. Studying nonstandard models often leads to fresh insights into the real numbers themselves. Recently, model theorists have been studying the exponential function. These investigations have led to important new insights into the geometry of solution sets of equations involving exponentials. This work has already found applications in analysis, control theory and neural networks. ***

9626856标记在研究实数时，MacIntyre和Marker为非标准模型提供了代数结构。 事实证明，该模型可用于理解功能的渐近行为。 Marker的工作集中在该模型中正式求解微分方程与在Infinity中找到实际解决方案的问题之间的关系。 标记还将在差异字段模型理论中解决问题。 这是一个引人入胜的领域，需要模型理论，差异代数和代数几何形状的精致思想混合。 （通过Buium和Hrushovski的工作，差异代数具有大量理论应用。）标记研究解决方案解决方案的几何形状，集中于使用模型理论维度来发展交叉理论。 模型理论是数学逻辑的一个分支，其中一个人通过以形式语言表达的属性来研究数学结构。 例如，在实数中，每个数字都是一个立方体，但是不能说每个有限的集合都具有最小的上限。 实数的非标准模型共享其所有正式属性。 研究非标准模型通常会导致对实数本身的新见解。 最近，模型理论家一直在研究指数函数。 这些研究导致了对涉及指数的方程组的几何组的重要新见解。 这项工作已经在分析，控制理论和神经网络中找到了应用。 ***