Model Theory for Metric Structures

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

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.