Mathematical Sciences: Model Theory and Analysis

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DMS 9626628 C. Ward Henson C. Ward Henson seeks to use methods from logic to solve problems within other areas of mathematics and computer science, and to use logic as a basis for formulating new concepts and posing new problems within them. This project has the following main aspects: (a) the further development of Henson's logic for models based on Banach spaces (or, more generally, on metric spaces), and the continued application of this theory to problems of functional analysis; (b) the study of exponentiation and other elementary functions from the model-theoretic point of view; objectives of this study include determining the computational complexity of the first order theory of the real exponential field and axiomatizing the identities satisfied by such other elementary functions as the Gamma function; (c) continued investigation of the computational complexity of logical decision problems, especially those arising from the most central areas of mathematics. This project emphasizes the relations between symbolic logic and other areas of mathematics and computer science. During the past 40 years techniques from logic have not only clarified the foundations of mathematics, but have contributed new tools for solving mathematical problems. These tools have been particularly effective in those areas of mathematics close to algebra. This project concentrates on making those same logic tools equally effective in the areas of mathematics closer to topology and geometry.