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 Marker 在研究实数取幂时，van den Dries、Macintyre 和 Marker 给出了非标准模型的代数构造。 事实证明，该模型对于理解函数的渐近行为非常有用。 马克的工作集中于该模型中形式求解微分方程与寻找无穷远实际解的问题之间的关系。 马克还将研究微分场模型理论中的问题。 这是一个令人着迷的领域，需要模型理论、微分代数和代数几何的思想的复杂结合。 （通过 Buium 和 Hrushovski 的工作，微分代数在数论中得到了重要的应用。）Marker 研究代数微分方程解的几何形状，专注于使用模型理论维数来发展交集理论。 模型论是数理逻辑的一个分支，人们通过观察用形式语言表达的数学结构的属性来研究数学结构。 例如，在实数中，我们可以说每个数字都是一个立方体，但不能说每个有界集合都有最小上界。 实数的非标准模型共享其所有形式属性。 研究非标准模型通常会带来对实数本身的新见解。 最近，模型理论学家一直在研究指数函数。 这些研究对涉及指数的方程解集的几何产生了重要的新见解。 这项工作已经在分析、控制理论和神经网络中找到了应用。 ***