Structure Theorems in Model Theory

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Laskowski is continuing his research in model theory, which is a branch of mathematical logic. The bulk of his research is organized around the theme of obtaining better structure theorems in three distinct settings. He will continue his investigations of the fine structure of models of classifiable theories with an eye on extensions to uncountable languages, look for a generalization of Shelah's notion of excellence that is applicable to more classes of structures, and seek more instances where stability theoretic assumptions automatically bound the quantifier complexity of the formulas. Additionally, Laskowski will attempt to answer specific questions about dependent formulas, elementary extensions of expansions of Archimedean ordered abelian groups, the Schroeder-Bernstein property of a theory, and a long standing decidability question from symbolic dynamics.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. There is a well established taxonomy of theories, which is based on the embeddability or non-embeddability of certain configurations of elements into models of the theory. Laskowski has identified certain classes of theories that have direct connections to limitations on data compression and computational learning theory, and will continue his examination of these theories.