Real analytic geometry and model theory

实解析几何与模型理论

• 批准号: 9988855
• 负责人: Christopher Miller
• 金额: $ 7.82万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9988855&HistoricalAwards=false
• 关键词: Real; analytic; geometry; model; theory

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.

米勒建议继续他的研究模型理论的扩展领域的真实的号码，集中在进一步发展的模型理论和解析几何与o-最小的，和某些其他良好的行为，扩展领域的真实的号码。他打算这样做的应用技术从描述集理论和几何测量理论除了模型理论和分析几何技术通常与o-极小。反过来，米勒希望将模型理论技术应用于描述集理论和几何测度理论中的问题。米勒还开始与应用数学家和控制工程师合作，研究模型理论在混合控制系统中的应用，并打算继续下去。所谓经典数学的许多结果是非常普遍的，可以说，它们适用于各种各样的输入，所以我们必须处理相应的各种各样的输出。 然而，我们可以希望，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、更有建设性的证明，以及对输入的良好性质的更深入的理解。然而，在我们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）应该被认为是行为良好的，哪些应该被认为是麻烦的。这可能是一个困难的问题。真实的域上的o-极小结构理论作为数理逻辑的一个分支学科，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，并进行了合作。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领域得到了广泛的应用。