Classification of Unstable Theories

University Of Chicago, Chicago IL

Investigators

Abstract

The proposed research focuses on unstable theories, primarily theories with the independence property. One direction of this work involves studying unstable theories from an asymptotic (ultrapower) point of view. Specifically, Malliaris plans to work on the structure of Keisler's order, a long standing program of comparing the complexity of theories, whose structure on the unstable theories has remained open since 1967. Keisler's order directly compares theories independent of language, and it was recognized early on that this order would likely give significant model-theoretic information. A complementary direction of this work involves developing further connections between model theory and graph theory. For instance, Malliaris plans to further develop the theory of characteristic sequences of hypergraphs (associated to formulas), a framework which brings ideas and techniques from graph theory and finite combinatorics such as Szemeredi regularity to bear on the analysis of types in unstable theories, and on various tradeoffs between independence and order in unstable theories. Informally, both the connections to graph theory and to ultrapowers give a perspective in which local noise is smoothed out and the nature of the significant jumps in the complexity of pseudofinite structure can be more clearly seen. This is an ideal context for studying the independence property, in which it is crucial to see finer distinctions between 'noise' or inconsistency arising from certain random structure, and more global compatibility. Broadly speaking, the proposed research program builds toward a more unified structure theory for the unstable theories, and aims to discover and further develop productive notions of complexity inherent to unstable theories. The proposed work is in model theory, specifically the classification of unstable theories. A model is a structure given by the data of an underlying set along with an interpretation for all function, relation, and constant symbols in a fixed background language L; a theory is a complete consistent set of L-sentences of first-order logic; and an elementary class is precisely the class of models of some theory, e.g. algebraically closed fields of characteristic 0. A major interest of model theory is studying theories via elementary classes of models, that is, studying a theory at hand by classifying the possible variations within its associated elementary class. One of the most fertile contexts for model theoretic analysis has been stability theory, the study of so-called stable or 'tame' theories, developed by Shelah in the 1970s. While the much larger class of non-stable or 'unstable' theories are quite complex from the point of view of stability, they are of significant interest. Many mathematical structures are unstable, so provide both theoretical motivation and examples; and there are many conjectures and problems about unstable theories which naturally attract research. The proposed work aims to build tools and a perspective from which to identify and analyze jumps in model-theoretic complexity among the unstable theories from a more uniform point of view, as well as to develop certain productive analogies and interactions with finite and asymptotic combinatorics. At the same time, these tools will be tested, in part, via the large scale structural conjecture of Keisler's order.