Mathematical Sciences: Logic and Analysis

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The project will concern mathematical logic and its connections with other areas of mathematics, especially algebra and analysis. C. Ward Henson will continue developing his model theory for Banach spaces and will investigate topics in nonstandard analysis, including descriptive set theory on hyperfinite sets. In addition, he will study the computational complexity of logical decision problems, including the theory of the real exponential field. Carl Jockusch will work in classical recursion theory and its connections with other areas. In particular, he will study degrees of unsolvability, both in their own right and in connection with model-theoretic structures (especially Boolean algebras and certain groups). L. van den Dries will investigate definability in algebraic and analytic structures. In particular, he will investigate O-minimality in expansions of the ordered field of real numbers and questions connected with A. Wilkie's recent proof that the elementary theory of the expansions of this ordered field by exponentiation is model complete. He will also study the model theory of local and Henselian fields. Mathematical logic is a natural tool for studying any discipline expressed in a formal language, which should bring most of mathematics itself under its focus. Algebra and number theory and certain parts of analysis are particularly natural subjects for this type of treatment. Large parts of the instant project bear this out. A typical result of the investigators' recent work is the proof of a completeness conjecture raised by Tarski: Is there a finite list of true identities combining polynomial formation and exponentiation of constants from which all other such true identities can be derived? (Counterexamples are known to exist if exponentiation of variables is allowed.) Another area in which a natural role for logic can be found is the complexity of algorithms. The investigators have been ingenious in finding such niches as well as in exploring them, to the obvious benefit of both logic and the other discipline involved.