RUI: Research in O-minimality and Related Topics

Vassar College, Poughkeepsie NY

Investigators

Abstract

The project deals with questions concerning o-minimality, extensions of o-minimality, and classes of finite structures. Some of the problems having to do with o-minimality relate to expansions of archetypal o-minimal structures and structures whose domain has as its order type that of the real numbers. Other have as their focus abelian groups definable in o-minimal structures or the development of o-minimal analogues of differential and algebraic topological methods and tools. Problems concerning extensions of o-minimality have to do in particular with weak o-minimality, local o-minimality, and, in analogy with Morley rank, the development of a model theory for ordered structures of finite rank. The third main topic of the project involves classes of finite structures with dimension and measure. This work has as its aim the development of a model theory for classes of finite structures that is in analogy with mainstream model theory for infinite structures. The results obtained to date and the examples that have been found suggest that there is much to be done. The research outlined above concerns model theory, one of the principal subfields of mathematical logic. Model theorists study properties of familiar mathematical structures that can be expressed in a formal mathematical language such as predicate logic. This distinctive point of view can provide insights and understanding into such structures that otherwise might prove elusive. One aspect of this project focuses on structures that include and behave in important ways 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. Model theory has played a key role in many of the significant advances that have been made in the last ten years. These have deepened our understanding of familiar mathematical systems in such diverse areas of the mathematical sciences as the analysis and geometry of real functions, neural nets, and relational database theory. Applications also have been made in economics. A second principal aspect of the project deals with classes of finite structures. Finite structures in general are central to computer science: any database can be construed as a finite structure in the sense in which they are studied in here, and a particular class of finite structures called finite fields are especially important in cryptology.