Model Theory for Metric Structures

公制结构的模型理论

• 批准号: 0555904
• 负责人: C. Ward Henson
• 金额: $ 21万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-05-15 至 2010-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555904&HistoricalAwards=false
• 关键词: Model; Theory; Metric; Structures

The research to be done in this project involves a continuous extension of first order logic that is suitable for structures based on metric spaces. The main emphasis of the project is on the usefulness of this theory in relation to a wide variety of structures from analysis and geometry.Previous work indicates that the potential role of this logic in analysis and geometry is parallel to the role played by ordinary first order logic in the more algebraic aspects of mathematics. Settings in which the features of this logic will be further developed include: highly homogeneous metric structures and their automorphism groups, probability measure algebras and their expansions, operator algebras and other structures based on Hilbert spaces, and the geometry of Banach spaces.This project involves new and effective ways to use model theory (a branch of mathematical logic) within analysis and geometry. During the past 40 years or more, techniques from logic have not only clarified the foundations of mathematics, but they have contributed new tools for solving mathematical problems. These tools have been especially effective in areas of mathematics close to algebra and number theory. In this project, C. Ward Henson aims to make those tools from logic equally effective in the areas of mathematics that are close to topology, geometry, and analysis.

在这个项目中要做的研究涉及一阶逻辑的连续扩展，适用于基于度量空间的结构。 该项目的主要重点是该理论与分析和几何中各种结构相关的有用性。以前的工作表明，该逻辑在分析和几何中的潜在作用与普通一阶逻辑在分析和几何中所发挥的作用平行。数学的更多代数方面。 该逻辑的特征将进一步发展的设置包括：高度齐次度量结构及其自同构群，概率测度代数及其扩展，算子代数和基于希尔伯特空间的其他结构，以及Banach空间的几何。该项目涉及在分析和几何中使用模型论（数学逻辑的分支）的新的有效方法。 在过去的40多年里，逻辑技术不仅澄清了数学的基础，而且为解决数学问题提供了新的工具。 这些工具在接近代数和数论的数学领域特别有效。 在本项目中，C. Ward Henson的目标是使这些逻辑工具在接近拓扑、几何和分析的数学领域同样有效。