The proposal addresses problems in model theory of o-minimal structures. The principal investigator will continue further developing model theory and analytic geometry associated with o-minimal structures. In developing model theory of o-minimal structures the principal investigator is planning to use differential-geometric techniques. In turn the principal investigator hopes to apply model-theoretic techniques to questions in complex and real analytic geometries. Model theory of compact complex manifolds has had remarkable success in the recent years and the proposer plans, among the other things, to extend these advances, using methods developed by him and his collaborators.
The proposal is in in a branch of mathematical logic called model theory. Model theory studies mathematical structures by considering the first-order sentences true in those structures, and the family of alternate structures that also satisfy all of those first-order sentences. (Sentences in logic are built out of a small repertoire of elements and constructions. "First-order" refers to the number of quantifiers in a sentence, a measure of complexity.) In many cases these alternative structures illuminate some properties of the original mathematical objects. A good example is the non-standard analysis. A part of this proposal is an extension on ideas of non-standard analysis to complex-analytic functions.
