The project involves questions concerning o-minimality and extensions of o-minimality. Some of the problems concerning o-minimality relate to expansions of archetypal o-minimal structures and structures whose domain has as its order type that of the real numbers. Others have as their focus groups definable in o-minimal structures or the development of o-minimal analogues of more sophisticated topological methods and tools. Problems having to do with extensions of o-minimality deal in particular with the notions of weak o-minimality and local o-minimality. A range of natural examples of weakly o-minimal structures is now known and there already is a significant body of results in the subject. It is hoped that local o-minimality may provide a suitable framework for developing some model theory for subanalytic sets. The research described above falls under the heading of model theory, one of the principal subfields of mathematical logic. Model theorists study properties of familiar mathematical structures which can be described by a formal mathematical language such as predicate logic. The distinctive point of view of model theory can provide insights and understanding into such structures that otherwise might not be easily achieved. This project focuses on structures that include and behave in important respects like the ordered field of real numbers, that is, the real numbers together with the polynomial and algebraic functions that are studied in first-year calculus and describe many phenomena. This decade has witnessed significant advances in which model theory has played a crucial role. The results obtained have deepened our understanding of familiar mathematical systems of interest in such diverse areas of the mathematical sciences as the analysis and geometry of real functions, neural nets, and relational database theory.
