Model theory and o-minimal structures

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The proposal addresses problems in model theory of o-minimal structures. Theprincipal investigator will continue further developing model theory andanalytic geometry associated with o-minimal structures. In developing modeltheory of o-minimal structures the principal investigator is planning to usedifferential-geometric techniques. In turn the principal investigator hopesto apply model-theoretic techniques to questions in complex and realanalytic geometries. Model theory of compact complex manifolds has hadremarkable success in the recent years and the proposer plans, among theother things, to extend these advances, using methods developed by him andhis collaborators. The proposal is in in a branch of mathematical logic called model theory.Model theory studies mathematical structures by considering the first-ordersentences true in those structures, and the family of alternate structuresthat also satisfy all of those first-order sentences. (Sentences in logicare built out of a small repertoire of elements and constructions."First-order" refers to the number of quantifiers in a sentence, a measureof complexity.) In many cases these alternative structures illuminate someproperties of the original mathematical objects. A good example is the non-standard analysis. A part of this proposal is anextension on ideas of non-standard analysis to complex-analytic functions.