Mathematical Sciences: Real Analytic Geometry and Model Theory

米勒将继续他对o-极小结构的研究，主要集中在进一步发展与实数序域(及其相应的几何范畴)的o-极小展开相关的模型理论和解析几何。在过去的几年里，这是一个快速发展的领域，模型理论家和分析几何学者做出了许多贡献，并进行了合作。米勒希望通过研究o-极小性在控制理论和变分中的可能联系和应用来建立进一步的跨学科联系。所谓的经典数学分析和几何的许多结果都是非常普遍的；它们适用于各种各样的“输入”，所以人们必须预料到必须处理相应的各种各样的“输出”。然而，人们可能希望，如果输入在某些方面表现得特别好，那么输出也会表现得同样好。事实证明，这在许多重要情况下都是正确的，但要想看到这一点，通常需要对经典结果进行新的、更具建设性的证明，以及对投入的“好”性质有更深的理解。然而，在开始这样的项目之前，有必要有一些方法来确定哪些数学对象(输入)应该被认为行为良好，哪些应该被认为是麻烦的。(这可能是一件困难的事情。)作为数理逻辑的一个分支，实域上的o-极小结构理论在很大程度上就是为了解决这个问题而发展起来的。近年来，该学科发展迅速。在神经网络学习理论和理论经济学以及纯数学的几个领域中都发现了应用和联系。

Miller will continue his research on o-minimal structures, concentrating mainly on further developing the model theory and analytic geometry associated with o-minimal expansions of the ordered field of real numbers (and their corresponding geometric categories). This has been a rapidly-developing area for the last several years, with many contributions from---and cooperation between---model theorists and analytic geometers. Miller hopes to build further interdisciplinary links by investigating possible connections and applications of o-minimality to control theory and variational calculus. Many results of so-called classical mathematical analysis and geometry are very general; they apply to a wide variety of 'input', so one must expect to have to deal with a correspondingly wide variety of 'output'. However, one could hope that if the input is, in some respect, particularly well behaved, then the output would be similarly well behaved. This turns out to be true in many important cases, but to see this usually requires new, more constructive, proofs of classical results, as well as a deeper understanding of the 'good' properties of the input. Before such projects can be begun, though, it is necessary to have some way of deciding which mathematical objects (inputs) should be considered as well behaved, and those which should be considered as troublesome. (This can be a difficult matter.) The theory of o-minimal structures on the real field, a sub-discipline of mathematical logic, has been developed in large part to deal with this issue. Recent years have seen rapid growth of the subject. Applications and connections have been found in neural-net learning theory and theoretical economics, as well as several areas of pure mathematics.