Model Theory and Differential Equations

模型理论和微分方程

基本信息

批准号：
0200393
负责人：
David Marker
金额：
$ 51.65万
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：
2002
资助国家：
美国
起止时间：
2002-06-01 至 2008-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200393&HistoricalAwards=false
关键词：
Model Theory Differential Equations

项目摘要

These research projects address problems in the model theory ofdifferential fields and the model theory of the field of realnumbers with exponentiation. The model theory of differentialfields is a fascinating area requiring a sophisticated mixture ofideas from stability theory, differential algebra and algebraicgeometry. The work of Buium and Hrushovski has shown that theseideas have important consequences in Diophantine geometry. Inparticular, the principal investigator will study the modeltheoretic behavior of solution sets of families of algebraicdifferential equations. This line of research on the modeltheory of the field of real numbers with exponentiation isexpected to concentrate on the relationship between globalsolutions to differential equations at infinity and formalsolutions in the field of logarithmic-exponential series.Model theory is a branch of logic that explores mathematicalstructures such as the real numbers together with theirarithmetic operations and ordering relation, analyzing the degreeto which the basic rules or axioms for a collection of objectsand their operations determine the shape of that collection. Themethods and results of model theory are cast in terms ofdefinable sets and functions, where a construction is definableexactly when it can be expressed by first-order logical formulasin the language of the structure. Model theoretic methods for thefield of real numbers with exponentiation have been remarkablysuccessful in proving new results on the geometry of exponentialvarieties and sets defined from them. This work has already foundapplications in asymptotic analysis, control theory, microlocalanalysis, and neural networks. The model theory of differentialfields has had significant applications in number theory.

这些研究项目解决了不同领域的模型理论中的问题，以及具有指数化的实数领域的模型理论。差异场的模型理论是一个引人入胜的区域，需要从稳定理论，差异代数和代数测定学中获得复杂的混合物。 Buium and Hrushovski的工作表明，这些虫在二磷剂几何形状中产生了重要的后果。主要研究者将研究代数分化方程组的溶液集的模型理论行为。 This line of research on the modeltheory of the field of real numbers with exponentiation isexpected to concentrate on the relationship between globalsolutions to differential equations at infinity and formalsolutions in the field of logarithmic-exponential series.Model theory is a branch of logic that explores mathematicalstructures such as the real numbers together with theirarithmetic operations and ordering relation, analyzing the degreeto which the basic rules or axioms for a对象的收集及其操作决定了该集合的形状。模型理论的方法和结果是用可定义的集合和功能施加的，当构造可以通过一阶逻辑公式表示结构的语言时，构造是可定义的。具有凸起数的实数的模型理论方法在证明有关指数变化的几何形状和从中定义的集合的新结果方面取得了极大的成果。这项工作已经在渐近分析，控制理论，微层分析和神经网络中找到了应用。差异场的模型理论在数量理论中具有重要的应用。