Topics in Model Theory

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Laskowski is continuing his research into model theory. In the first set of problems he investigates several distinct combinatorial situations that arise in different contexts, including recursive model theory and classification theory. In each of these situations a definable group is known to be present as well as an action of the group on the combinatorial configuration. Laskowski will continue his investigations into what specific properties such a group must possess. As the group is acting on the configuration, such results would immediately yield restrictions on the complexity of the configuration. In many different settings, Laskowski has been able to apply some of the tools from Descriptive Set Theory to answer questions in model theory, primarily for theories in a countable language. Recent developments give him hope that the Main Gap for omega1-saturated models is within reach. A third problem is to continue his investigations into the algebraic underpinnings of various fuzzy logics. Specifically, he is aiming for a sharp bound on the computational complexity of determining which sentences are logically valid in various fuzzy logic systems. By building on previous results, this question is cast in terms of various embedding questions between certain ordered abelian semigroups.Model theory is concerned with the interplay between theories (i.e., sets of sentences in a very formal language) and classes of algebraic structures (models). As one strengthens the theory, the class of models of the theory decreases. In particular, some theories are strong enough to obviate specific configurations from appearing in any model of the theory. Much of Laskowski's prior research to date, as well as much of the proposed research, concerns understanding the mechanisms by which this elimination can occur. Fuzzy logics have been useful in computer science and operations research for some time, but only recently have significant attempts been made to provide a firm mathematical basis for them. Laskowski and his graduate students have recently made progress in this regard, and the investigator will continue working in this area.