Tameness in expansions of the real field

真实领域扩张中的驯服

• 批准号: 1001176
• 负责人: Christopher Miller
• 金额: $ 17.2万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001176&HistoricalAwards=false
• 关键词: Tameness; expansions; real; field

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.

米勒将继续他的研究一阶结构领域的真实的数字，集中在进一步发展模型理论和解析几何与o-最小和某些其他类别的良好表现的结构领域的真实的数字。他打算这样做的应用技术从描述集理论和几何措施理论除了模型理论和分析几何技术通常与o-极小。反过来，米勒希望应用模型理论的技术来获得控制理论（具体地说，通过可定义向量场的轨迹对真实的域上的结构展开进行分类）、描述集合论和几何测度论中的结果。经典数学的许多结果都非常普遍：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。但是，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，但通常需要对经典结果进行新的、更有建设性的证明，以及更深入地理解哪些输入应该被视为行为良好。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。这是一个快速发展的领域，在过去的二十五年里，有许多贡献，并合作，研究人员从几个分支的数学和逻辑。应用领域广泛，如理论经济学、神经网络学习理论、混合控制系统以及纯数学。然而，o-极小性有一个缺点：它只允许对局部连通行为进行建模，因此在理解噪声或振荡设置方面的用途相当有限。米勒建议开发的O-极小的扩展，可以处理至少有一些这些非O-极小的现象。