Mathematical Sciences: Model Theory for Analytic Structures

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The investigator will continue his research on the model theory of structures related to real and complex numbers. In particular, he will focus on definability questions in expansions with extra analytic structure. Most of his work over the past five years has been concentrated in this area. He will attempt to prove further results of the following character. (1) Marker, Macintyre, and van den Dries have shown that the structure of the real numbers with exponentiation and restricted analytic functions is O-minimal and admits elimination of quantifiers. O-minimality implies that there is a rich structure theory for the definable sets. (2) Marker and Pillay have shown that any non-trivial reduct of an algebraically closed field that contains addition is strong enough to recover the full field structure. (3) Marker has shown that if any non-trivial real structure is added to the complex numbers, then the real numbers are definable. These latter two results can be thought of as attempts to examine the boundary of algebraic geometry. The first says that weakening algebraic geometry leads to little more than linear geometry. The second says that extending algebraic geometry leads to the full complications of real algebraic geometry. While "definability" is a logical notion, often it turns out that the definable sets are ones which arise naturally and are of great importance. Model theoretic methods provide a new tool for their study. For example, the sets definable in the structure of the real numbers with exponentiation and restricted analytic functions arise naturally in studying analytic geometry, dynamical systems, statistics, and control theory.