The proposal addresses problems in the model theory of o-minimal structures. The principal investigator plans to develop Borel-Moore homology for sets definable in o-minimal expansions of fields. One of the goals is to study characteristic cycles of definable sets. The principal investigator will also consider model theoretical aspects of non-archimedean amoebas of algebraic varieties.
The proposal is 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 analytic-geometric categories.