Model Theory for Metric Structures

本课题将对一阶逻辑进行连续扩展，使其适用于基于度量空间的结构。该项目的主要重点是这一理论在分析和几何的各种结构中的实用性。以前的工作表明，这种逻辑在分析和几何中的潜在作用与普通一阶逻辑在更代数的数学方面所起的作用平行。这一逻辑的特征将进一步发展的背景包括：高度齐次度量结构及其自同构群，概率度量代数及其展开，算子代数和基于Hilbert空间的其他结构，以及Banach空间的几何。这个项目涉及到在分析和几何中使用模型理论(数理逻辑的一个分支)的新的和有效的方法。在过去的40多年里，来自逻辑的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尤其有效。在这个项目中，C.Ward Henson的目标是使那些来自逻辑的工具在接近拓扑学、几何学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.