Real analytic geometry and model theory

米勒建议继续他对实数领域展开的模型理论的研究，集中精力进一步发展与最小和某些其他表现良好的实数领域展开相关的模型理论和解析几何。他打算通过应用描述集合论和几何测度论的技术，以及通常与极小性相关的模型论和分析几何技术来做到这一点。反过来，米勒希望将模型理论技术应用于描述集理论和几何测量理论中的问题。米勒还开始与应用数学家和控制工程师合作，将模型理论应用于混合控制系统，并打算继续下去。许多所谓经典数学的结果是非常普遍的；可以说，它们适用于各种各样的输入，因此我们必须相应地处理各种各样的输出。然而，我们希望如果输入在某些方面表现得特别好，那么输出也会同样表现得很好。在许多重要的情况下，这被证明是正确的，但要看到这一点，通常需要对经典结果进行新的、更有建设性的证明，以及对输入的良好性质有更深入的理解。然而，在我们甚至可以开始这样的项目之前，我们需要一些方法来决定哪些数学对象（输入）应该被认为是表现良好的，哪些应该被认为是麻烦的。这可能是一件困难的事情。实数领域的0极小结构理论，即数理逻辑的一门分支学科，在很大程度上就是为了解决这个问题而发展起来的。在过去的十年里，这是一个快速发展的领域，数学和逻辑的几个分支做出了许多贡献，并相互合作。这些发展的应用和潜在应用已经在理论经济学、神经网络学习理论和混合控制系统等不同领域被发现。

Miller proposes to continue his research on the model theory ofexpansions of the field of real numbers, concentrating on furtherdeveloping the model theory and analytic geometry associated witho-minimal, and certain other well-behaved, expansions of the field ofreal numbers. He intends to do this by applying techniques fromdescriptive set theory and geometric measure theory in addition to themodel-theoretic and analytic-geometric techniques usually associatedwith o-minimality. In turn, Miller hopes to apply model-theoretictechniques to questions in descriptive set theory and geometric measuretheory. Miller has also begun to collaborate with applied mathematiciansand control engineers on applications of model theory to hybrid controlsystems, and intends to continue.Many results of so-called classical mathematics are very general; theyapply to a wide variety of input, so to speak, so we must expect to haveto deal with a correspondingly wide variety of output. However, wecould hope that if the input is, in some respect, particularly wellbehaved, then the output would be similarly well behaved. This turns outto be true in many important cases, but to see this usually requiresnew, more constructive proofs of classical results, as well as a deeperunderstanding of the good properties of the input. Before we can evenbegin such projects, though, we need some way of deciding whichmathematical objects (inputs) should be considered as well behaved, andwhich should be considered as troublesome. This can be a difficultmatter. The theory of o-minimal structures on the real field, asub-discipline of mathematical logic, has been developed in large partto deal with this issue. This has been a rapidly-developing area for thelast decade, with many contributions from---and cooperationbetween---several branches of mathematics and logic. Applications, andpotential applications, of these developments have been found in areasas diverse as theoretical economics, neural-net learning theory, andhybrid control systems.