Mathematical Sciences: Group Definable in o-minimal Structures

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Abstract DMS 9626377 Sergei Starchenko, Vanderbilt University The project concerns the classification of groups definable in o-minimal structures. While "an o-minimal structure" is a notion of model theory, it turns out that many important expansions of real numbers are o-minimal, and this model theoretical notion provides a natural framework for the study of "analytic-geometric" categories. Tools of model theory have been successfully applied to these categories yielding interesting theorems about the real numbers. The starting point for this project is the observation that groups definable in o-minimal structures can be viewed as Lie groups over real closed fields. Certain aspects of these groups are easily overlooked when treated only over the field of real numbers. These aspects come into full view, however, when we regard the groups as definable in model-theoretically well-behaved structures. The main goal of Starchenko's project is to obtain a reasonable classification of o-minimal groups, in order to develop a model-theoretical analogue of Lie theory. He conjectures that locally o-minimal groups are algebraic and Lie morphisms are, essentially, Nash morphisms. If this is the case, then many properties of algebraic groups can be transferred to o-minimal groups, thus providing new tools to study such groups. Since the field of algebraic real numbers is o-minimal, these tools can then be applied to study groups definable in expansions of number fields. On the other hand, any negative example will provide new information about interactions between geometric and algebraic properties of Euclidean groups. The second part of the project concerns some open questions in stability theory. It has been shown that many ideas and tools of model theory have natural geometric analogues in o-minimal structures. The investigator hopes that the presence of a nice geometry can help to solve o-minimal versions of the two most famous questions in stability theory. This project employs l ogical tools of model theory, and specifically a notion called o-minimality, to deepen the mathematical understanding of several important number systems, including the real numbers. An understanding of the real numbers and of their geometric properties is basic to mathematics, and also to many applications of mathematics, e.g. to manufacturing. Model theory provides a valuable perspective in this understanding.