Model Theory and Analysis

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This project concerns the further development of Henson's logicfor structures based on metric spaces and, especially, thecontinued application of this theory to problems in functionalanalysis and in the geometry of metric spaces. The syntax ofthis logic is restricted to the class of positive boundedformulas and the semantics of the logic uses an approximatesatisfaction relation. Its potential role in analysis andgeometry is parallel to the role of ordinary first order logic inthe more algebraic aspects of mathematics.During the past 40 years or more, techniques from logic have notonly clarified the foundations of mathematics, but they havecontributed new tools for solving mathematical problems. Thesetools have been especially effective in those areas ofmathematics that are close to algebra and number theory. In thisproject, C. Ward Henson aims to make those tools from logicequally effective in the areas of mathematics that are close totopology, geometry, and analysis.