MODEL THEORY AND ALGEBRA

模型论是数学逻辑的一个分支，它本身是一个相当新的学科，起源于19世纪后期对数学基础的哲学研究。然而，近几十年来，它在数学的其他部分，计算机科学，甚至在工程（与机器人相关的量词消除）中得到了许多应用。另一方面，代数是数学中最古老、最成熟的学科之一。Aschenbrenner建议在这两个学科的交叉点上继续他的研究，在过去半个世纪的大部分时间里，这两个学科一直在进行富有成效的交流。该项目围绕几个主题进行组织，包括延续Aschenbrenner？S合作努力发展渐近分析的模型理论处理，并用组合几何的方法研究了一些模型理论问题。这些主题不仅与数学中的其他领域有关，而且与微分方程在工程中的应用以及计算学习理论有关。该提案的另一个方面涉及寻找代数问题的算法并研究其复杂性，这一主题对于计算机代数在科学和工程中的应用至关重要。

Model theory is a branch of mathematical logic, which itself is a fairly new subject that originated in the late 19th century with philosophical investigations into the foundations of mathematics. However, in recent decades it has found many applications in other parts of mathematics, in computer science, and even in engineering (quantifier elimination as it relates to robotics). Algebra, on the other hand, is one of the oldest and most mature disciplines of mathematics. Aschenbrenner proposes to continue his research in the intersection of these two disciplines, where a fruitful exchange has been going for the better part of the last half-century.The project is organized around several themes, including a continuation of Aschenbrenner?s collaborative efforts to develop a model-theoretic treatment of asymptotic analysis, and investigations into some model-theoretic questions with the means of combinatorial geometry. These topics have relationships not only to other fields within mathematics but also to applications of differential equations in engineering, and to computational learning theory. Another aspect of the proposal concerns finding algorithms for algebraic problems and studying their complexity, a topic which is of central importance for applications of computer algebra in science and engineering.