Research in Model Theory

模型理论研究

• 批准号: 0500841
• 负责人: John Baldwin
• 金额: --
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500841&HistoricalAwards=false
• 关键词: Research; Model; Theory

Baldwin proposes work on three topics in model theory: abstractelementary classes, generic expansions of fields, and thepreservation of stability under expansion by naming predicates.The last two are in some sense dual. The twenty year old searchfor a bad-field tries to construct a sufficiently generic subgroupof the complex numbers that omega stability is preserved. Bothrefined model theoretic and algebraic tools are essential. In theother direction Baldwin is exploring with Baizhanov, the mostgeneral conditions under which can hope to preserve stability whena subset is named. Baldwin has been writing a monographsummarizing the current state of `Morley's theorem' for AbstractElementary Classes. This monograph, tries to clarify and organizethe general work in this topic, underline the connections withboth Zilber's work on complex geometry and the `Hrushovskiconstruction, and lay the foundation for a more general stabilitytheory in this context.The general perspective of logic in understanding the foundationsof various mathematical topics has provided in Baldwin's worklinks among fields as diverse as database theory (computerscience), random graphs, abstract algebra and complex analysis. Inmany cases the role of logic is place limits on what can be provedand direct research into more productive endeavors. Baldwin hasserved as a link between the logic and mathematics educationcommunities for the last fifteen years. He is currently directorof the Office of Mathematics Education at UIC. He aims to providefuture teachers with a profound understanding of fundamentalsecondary school mathematics. A teacher with profoundunderstanding of school mathematics is not only aware of theconceptual structure of secondary mathematics, but alsounderstands the common strategies that students use to solveproblems along with common student misconceptions. A backgroundin logic is invaluable in helping future teachers develop theability to communicate mathematics clearly.

Baldwin在模型理论中提出了三个主题：抽象类，域的一般扩展，以及通过命名谓词保持扩展下的稳定性。后两者在某种意义上是双重的。对坏场的研究已经进行了20年，试图构建一个足够一般的复数子群，以保持稳定性。精细模型理论和代数工具都是必不可少的。在另一个方向上，Baldwin正在与Baizhanov探索，当一个子集被命名时，可以希望保持稳定性的最一般条件。鲍德温一直在写一本专著，总结了《抽象基础课程》中“莫利定理”的现状。这个专著，试图澄清和组织在这个主题的一般工作，强调与齐尔伯的工作在复杂几何和赫鲁晓夫斯基结构的联系，并奠定了在这种情况下更一般的稳定性理论的基础。在理解各种数学主题的基础逻辑的一般观点提供了鲍德温的工作领域之间的联系，如数据库理论（计算机科学），随机图，抽象代数和复杂分析。在许多情况下，逻辑的作用是限制可以证明的东西，并将研究指导为更有成效的努力。在过去的15年里，鲍德温一直是逻辑学和数学教育界的桥梁。他目前是UIC数学教育办公室主任。他的目标是为未来的教师提供对基础中学数学的深刻理解。一个对学校数学有深刻理解的老师不仅知道中学数学的概念结构，而且还了解学生用来解决问题的常见策略以及学生常见的误解。逻辑背景对于帮助未来的教师发展清晰地沟通数学的能力是无价的。