Mathematical Sciences: Real Analytic Geometry and Model Theory

米勒将继续他的研究o-最小的结构，主要集中在进一步发展模型理论和解析几何与o-最小的扩展有序领域的真实的号码（及其相应的几何类别）。 这是一个快速发展的领域，在过去的几年里，有许多贡献-和合作-模型理论家和解析几何学家。米勒希望建立进一步的跨学科的联系，调查可能的连接和应用o-极小控制理论和变分。 所谓经典数学分析和几何学的许多结果是非常一般的;它们适用于各种各样的“输入”，因此人们必须预期必须处理相应的各种各样的“输出”。 然而，人们可以希望，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。 这在许多重要的情况下都是正确的，但是要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，以及对输入的“好”属性有更深入的理解。 然而，在开始这样的项目之前，有必要有某种方法来决定哪些数学对象（输入）应该被认为是行为良好的，哪些应该被认为是麻烦的。(This这可能是一个困难的问题）。真实的域上的o-极小结构理论是数理逻辑的一个分支学科，它的发展在很大程度上就是为了解决这个问题。近年来，这一主题迅速发展。 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.