Model Theory for Metric Structures

本课题的研究涉及一阶逻辑的连续扩展，该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在各种分析和几何结构中的有用性。先前的工作表明，这种逻辑在分析和几何中的潜在作用与普通一阶逻辑在数学中更多的代数方面所起的作用是平行的。该逻辑的特征将进一步发展的环境包括：高度齐次度量结构及其自同构群，概率测度代数及其展开，基于Hilbert空间的算子代数和其他结构，以及Banach空间的几何。这个项目涉及在分析和几何中使用模型理论（数学逻辑的一个分支）的新的和有效的方法。在过去的40多年里，逻辑技术不仅阐明了数学的基础，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。在这个项目中，C. Ward Henson的目标是使这些来自逻辑的工具在接近拓扑、几何和分析的数学领域同样有效。

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.