Mathematical Sciences: Model Theory for Analytic Structures

数学科学：解析结构的模型论

• 批准号: 9306150
• 负责人: David Marker
• 金额: $ 7.8万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-06-01 至 1996-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9306150&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Model; Theory; Analytic

The investigator will continue his research on the model theory of structures related to real and complex numbers. In particular, he will focus on definability questions in expansions with extra analytic structure. Most of his work over the past five years has been concentrated in this area. He will attempt to prove further results of the following character. (1) Marker, Macintyre, and van den Dries have shown that the structure of the real numbers with exponentiation and restricted analytic functions is O-minimal and admits elimination of quantifiers. O-minimality implies that there is a rich structure theory for the definable sets. (2) Marker and Pillay have shown that any non-trivial reduct of an algebraically closed field that contains addition is strong enough to recover the full field structure. (3) Marker has shown that if any non-trivial real structure is added to the complex numbers, then the real numbers are definable. These latter two results can be thought of as attempts to examine the boundary of algebraic geometry. The first says that weakening algebraic geometry leads to little more than linear geometry. The second says that extending algebraic geometry leads to the full complications of real algebraic geometry. While "definability" is a logical notion, often it turns out that the definable sets are ones which arise naturally and are of great importance. Model theoretic methods provide a new tool for their study. For example, the sets definable in the structure of the real numbers with exponentiation and restricted analytic functions arise naturally in studying analytic geometry, dynamical systems, statistics, and control theory.

研究者将继续研究与实数和复数相关的结构模型理论。 特别是，他将重点关注具有额外分析结构的扩展中的可定义性问题。 过去五年他的大部分工作都集中在这个领域。 他将尝试进一步证明以下性质的结果。 (1) Marker、Macintyre 和 van den Dries 已经证明，具有求幂和受限解析函数的实数结构是 O 最小的，并且允许消除量词。 O-极小性意味着可定义集合存在丰富的结构理论。 (2) Marker 和 Pillay 已经证明，包含加法的代数闭域的任何非平凡约简都足以恢复完整的域结构。 (3) 标记表明，如果任何非平凡的实数结构添加到复数中，则实数是可定义的。 后两个结果可以被认为是检验代数几何边界的尝试。 第一个观点认为，弱化代数几何只会导致线性几何。 第二个说法是，扩展代数几何会导致实代数几何的完全复杂化。 虽然“可定义性”是一个逻辑概念，但事实证明，通常可定义的集合是自然产生的并且非常重要。 模型理论方法为他们的研究提供了新的工具。 例如，在研究解析几何、动力系统、统计学和控制理论时，自然会出现可在具有求幂和受限解析函数的实数结构中定义的集合。