Mathematical Sciences: Model Theory and Its Geometric Applications

在研究带幂的实数时，van den Dries、Macintyre和Marker给出了一个非标准模型的代数构造。这个模型已被证明对理解函数的渐近行为是有用的。马克的工作集中于在这个模型中正式求解微分方程和在无穷远处寻找实际解的问题之间的关系。马克还将研究微分场的模型理论问题。这是一个迷人的领域，需要模型理论、微分代数和代数几何的复杂混合。（通过Buium和赫鲁晓夫斯基的工作，微分代数有了重要的数论应用。）马克研究几何解的代数微分方程，集中在使用模型理论的维度来发展交集理论。模型理论是数学逻辑的一个分支，人们通过观察用形式语言表达的数学结构的性质来研究数学结构。例如，在实数中，我们可以说每个数都是立方体，但我们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它们所有的形式性质。研究非标准模型通常会让我们对实际数字本身有新的认识。最近，模型理论家一直在研究指数函数。这些研究对涉及指数方程解集的几何有重要的新见解。这项工作已经在分析、控制理论和神经网络中得到了应用。＊＊＊

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. ***