Topics in Model Theory

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ABSTRACTLaskowski is continuing his research on three projects within the realm of model theory. First, in collaboration with Hart and Hrushovski, the investigator is seeking to determine the `fine structure' of models of classifiable theories. This amounts to determining the amount of flexibility one has in choosing a decomposition tree for a model of such a theory. Recently, Laskowski has discovered that any shallow, strictly stable theory that does not have the dimensional order property interprets an infinite group. In collaboration with Shelah, Laskowski is seeking to understand the extent to which this group determines the dependence relation of dividing. Finally, in collaboration with Dolich, Laskowski is seeking to identify a class of theories that contains the simple theories, yet also a satisfactory notion of independence. Model theory is concerned with the interplay between theories (i.e., sets of sentences in a very formal language) and the classes of algebraic structures (models) that satisfy these sentences. Most of Laskowski's research to date concerns broad classes of theories, as opposed to concentrating on a specific theory or structure. Because of the breadth of the context, applications have been found in other areas of mathematics and in computer science, specifically in the analysis of structured data bases. These classes of theories are described in terms of the embeddability or non-embeddability of certain configurations of elements into models of the theory. Laskowski is continuing to analyze three classes of theories. In all three cases, the combinatorial assumptions on the class of theories imply that the models of such theories contain some algebraic content. The investigator is studying the extent to which this algebraic component determines the behavior of a model of the theory.