Tameness in expansions of the real field

米勒将继续他对实数领域一阶结构的研究，专注于进一步发展与 o 极小相关的模型理论和解析几何以及实数领域某些其他类型的良好结构。他打算通过应用描述性集合论和几何测度论以及通常与 o 极小性相关的模型理论和解析几何技术来实现这一目标。反过来，米勒希望应用模型论技术来获得控制论（具体来说，通过可定义向量场的轨迹对实数域上结构的展开进行分类）、描述性集合论和几何测度论方面的结果。经典数学的许多结果都非常通用：可以说，它们适用于广泛的输入，因此往往会产生广泛的输出。但人们可能希望，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。在许多重要的情况下事实证明这是正确的，但通常需要对经典结果进行新的、更具建设性的证明，以及对哪些输入应被视为行为良好的更深入的理解。实数域上的最小结构理论（数理逻辑的一个子学科）在很大程度上是为了解决这个问题而发展起来的。在过去的二十五年里，这是一个快速发展的领域，来自数学和逻辑多个分支的研究人员做出了许多贡献并相互合作。其应用已广泛应用于理论经济学、神经网络学习理论、混合控制系统以及纯数学等领域。然而，o-极小性有一个缺点：它只允许对局部有限连接的行为进行建模，因此在理解噪声或振荡设置方面的用途相当有限。米勒建议开发 o-极小性的扩展，至少可以处理其中一些非 o-极小现象。

Miller will continue his research on first-order structures on the field of real numbers, concentrating on further developing the model theory and analytic geometry associated with o-minimal and certain other classes of well-behaved structures on the field of real numbers. He intends to do this by applying techniques from descriptive set theory and geometric measure theory in addition to the model-theoretic and analytic-geometric techniques usually associated with o-minimality. In turn, Miller hopes to apply model-theoretic techniques to obtain results in control theory (specifically, classifying expansions of structures on the real field by trajectories of definable vector fields), descriptive set theory, and geometric measure theory.Many results of classical mathematics are very general: They apply to a wide range of input, so to speak, and thus tend to produce a wide range of output. But one could hope that if the input is particularly well behaved in some respect, then the output would be similarly well behaved. This turns out to be true in many important cases, but usually requires new, more constructive, proofs of classical results, as well as a deeper understanding of which inputs should be regarded as well behaved. 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. This has been a rapidly-developing area for the last twenty-five years, with many contributions from, and cooperation between, researchers from several branches of mathematics and logic. Applications have been found in areas as diverse as theoretical economics, neural-net learning theory, and hybrid control systems, as well as in pure mathematics. However, o-minimality has a drawback: It allows only for the modelling of locally finitely connected behavior, and thus has rather limited use in understanding noisy or oscillatory settings. Miller proposes to develop extensions of o-minimality that can deal with at least some of these non-o-minimal phenomena.