Mathematical Sciences: Model Theory and Its Geometric Applications

9626856 Marker在研究带指数的实数时，van den Dries、Macintyre和Marker给出了一个非标准模型的代数构造。该模型已被证明对理解函数的渐近行为很有用。马克的工作集中在这个模型中形式求解微分方程和在无穷远处寻找实际解之间的关系。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复杂混合思想。(通过Buium和Hrushovski的工作，微分代数已经有了大量的理论应用。)马克研究代数微分方程解的几何，专注于使用模型理论维度来发展交集理论。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性质来研究数学结构。例如，在实数中，人们可以说每个数字都是一个立方体，但不能说每个有界集都有一个最小上界。实数的非标准模型共享它们的所有形式性质。研究非标准模型往往会带来对真实数字本身的新见解。最近，模型理论家一直在研究指数函数。这些研究为涉及指数的方程解集的几何问题带来了重要的新见解。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. ***