Foundations of Asymptotic Differential Algebra

Aschenbrenner建议在模型理论和代数领域做出贡献。他建议继续他的合作努力，以渐近微分代数的形式，在实代数几何和微分代数领域之间建立一个综合。这将导致对哈代场和反列场的模型论和代数性质以及它们之间的关系的更深入的理解，并将在代数微分方程的渐近理论中开辟新的视角。他还建议继续研究交换代数中的非标准方法和阶界。例如，Aschenbrenner建议增加我们对具有积分系数的多项式环中理想的Grobner基的知识。数学逻辑虽然起源于19世纪晚期对数学基础的哲学研究，但在最近几十年里，它在数学的其他部分，计算机科学，甚至工程（与机器人相关的量词消除）中发现了许多应用。该项目主要涉及将数理逻辑方法的适用性推向尚未探索的领域，即渐近分析。我们的希望是，这些研究将导致对微分方程解的行为有更深的理解。该计划的另一部分涉及代数问题算法的构建和分析。在这个项目中提出的寻找度界对于计算机代数在科学和工程中的应用是至关重要的。

Aschenbrenner proposes to contribute to the fields of model theory and algebra. He proposes to continue his collaborative efforts to create a synthesis between the fields of real algebraic geometry and differential algebra, in the form of asymptotic differential algebra. This will lead to a deeper understanding of the model-theoretic and algebraic properties of Hardy fields and fields of transseries, and the relationship between them, and will open up new perspectives in the asymptotic theory of algebraic differential equations. He also proposes to continue his investigations into non-standard methods and degree bounds in commutative algebra. For example, Aschenbrenner proposes to increase our knowledge about Grobner bases for ideals in the ring of polynomials with integral coefficients.Mathematical logic, although it originated in the late 19th century with philosophical investigations into the foundations of mathematics, has in recent decades found many applications in other parts of mathematics, in computer science, and even in engineering (quantifier elimination as it relates to robotics). This project deals mainly with pushing the applicability of the methods of mathematical logic into as of yet unexplored territory, namely, asymptotic analysis. Our hope is that these investigations will lead to a deeper understanding of the behavior of solutions to differential equations. Another part of the proposed project involves the construction and analysis of algorithms for algebraic problems. Finding degree bounds as suggested in this project is of central importance for applications of computer algebra in science and engineering.