Topics in Model Theory

模型理论主题

• 批准号: 0653484
• 负责人: David Marker
• 金额: $ 51.43万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653484&HistoricalAwards=false
• 关键词: Topics; Model; Theory

Marker plans to continue his research in model theory, focusing on the connections with other areas of mathematics. One direction of his current research focuses on trying to understand definable sets in the complex numbers with exponentiation. He also remains very interested in the model theory of differential fields--a fascinating area requiring a sophisticated mixture of ideas from stability theory, differential algebra and algebraic geometry. In a different direction Marker will continue his work on connections between model theory and descriptive set theory. In particular, he will try to understand more about theories where the isomorphism relation on countable models is Borel.In model theory one studies mathematical structures by looking at solution sets to systems of equations and more complicated sets you can build from these basic sets. In some situations, like the real or complex numbers with algebraic operations, the sets constructed are geometrically simple-for example there are only finitely many connected pieces. In the real numbers if you add the exponential function, the definable sets are still geometrically simple, but in the complex numbers you can construct infinite discrete sets like the integers. In this case it is unknown if the sets constructed can be arbitrarily complicated. Possibly, all the sets arise from simple pieces. Marker will investigate this problem.The sets studied arise naturally in many applications in dynamical systems and control theory and it would be useful to understand their constraints.

马克计划继续他在模型理论方面的研究，专注于与其他数学领域的联系。他目前研究的一个方向是试图理解复数中的可定义集。他还对微分场的模型理论非常感兴趣--这是一个迷人的领域，需要将稳定性理论、微分代数和代数几何的思想复杂地混合在一起。在不同的方向上，马克将继续他关于模型理论和描述集合论之间的联系的工作。特别是，他将试图更多地了解可数模型上的同构关系是Borel.在模型理论中，人们通过观察方程系统的解集和可以从这些基本集合建立的更复杂的集合来研究数学结构。在某些情况下，例如具有代数运算的实数或复数，构造的集合在几何上是简单的--例如，只有有限多个连通部分。在实数中，如果你加上指数函数，可定义的集合在几何上仍然是简单的，但在复数中，你可以像整数一样构造无限离散的集合。在这种情况下，构造的集合是否可以任意复杂是未知的。可能，所有的布景都是由简单的片段组成的。Marker将研究这个问题，所研究的集合自然地出现在动力系统和控制理论的许多应用中，理解它们的约束将是有用的。