Investigations in Model Theory

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Baldwin proposes investigations in stability theory, the model theory of algebra and finite model theory. Baldwin will continue to investigate the the construction of homogeneous universal models with respect to a notion of strong submodel. One application of this method links stability theory with probability on finite models by providing a technique for not only proving 0-1 laws but obtaining model theoretic properties of the almost sure theory. These model theoretic properties of the almost sure theory can be applied to problems in finite model theory. Baldwin has been a leader in the application of stability theory to the development of logic with finitely many variables. This work has, for the first time, provided serious links between the deep work of Cherlin, Harrington, Lachlan, and Zilber on homogeneous structures and the mainstream of finite model theory. Baldwin and Baizhanov are studying expansions of stable structures by arbitrary predicates; this is motivated by the earlier work with Benedikt on embedded finite model theory. Baldwin's earlier work with Holland has included applications in the area of algebra, specifically group theory, and in expansions of the complex numbers by an arbitrary subset. The present project will try to find some specific new structures which expand the complex numbers by subgroups of the multiplicative group. Most of Baldwin's work has been in model theory, a branch of mathematical logic. The general aim of this work is to understand `ordinary mathematics' at a higher level of abstraction. This abstraction allows the discovery of common features in widely different areas of mathematics, ranging from probability theory to combinatorics to algebraic geometry. This kind of work has been fruitful both in a better understanding of algebraic structures and in providing a background for investigations in database theory. In particular, Baldwin's work with his co-author Michael Benedikt of Lucent Technologies on `embedded finite model theory' has found limits on the expressibility of database queries. Baldwin will continue his educational work - primarily focusing on developing innovative and effective ways to prepare mathematics teachers. The educational work is connected with two other NSF sponsored programs.