9401723 Steinhorn This project focuses upon questions having to do with o-minimality and its extensions. Some of these questions relate to expansions of archetypal o-minimal structures and structures whose domain has as its order type that of the real numbers. Other problems concerning o-minimality are intended to develop the structure of groups definable in o-minimal structures as far as possible in analogy with locally Euclidean groups. Among the variants and extensions of o-minimality to be considered are weak o-minimality and local o-minimality. It is hoped that the latter will provide a suitable framework for developing some model theory for subanalytic sets. Another goal of the research is to begin to develop a general model theory for ordered structures, patterned after stability theory. The research described above falls under the heading of model theory within the branch of mathematics called logic. Model theory attempts to deal with the properties of familiar mathematical structures which can be described by a formal mathematical language such as predicate logic. The focus of this project is a class that includes and behaves very much like the polynomial and algebraic functions based on the real numbers that are studied in freshman calculus and are used to describe many phenomena. There has been much exciting progress during the past five years, in which model theory has played a central role in deepening our understanding of familiar mathematical systems of interest in diverse areas of mathematics outside of logic. ***
