Unstable Model Theory

University Of Chicago, Chicago IL

Investigators

Abstract

The proposed research will apply methods from model theory to study the generic interactions between families of finite combinatorial objects in models of unstable theories. These questions naturally admit, and benefit from, ideas and methods from graph theory and finite combinatorics. More precisely, Malliaris will work to further develop the theory of characteristic sequences, which are countable sequences of hypergraphs defined on the parameter space of a first-order formula. Malliaris has shown that a deep collection of techniques from graph theory and combinatorics, including Szemeredi regularity, can be brought to bear on model-theoretic structure via the characteristic sequence. Malliaris proposes to further explore these promising connections, which are relevant to many structural questions in unstable theories, including the fine structure of order and independence and the classification of theories with the independence property, as well as to understanding the model-theoretic significance of graph-theoretic phenomena such as edge density and hypergraph regularity in this context. Moreover, these investigations shed light on the ways in which families of types are realized and omitted in regular ultrapowers, and are therefore relevant to the longstanding open problem of the structure of Keisler's order on unstable theories (a preorder on countable theories which, roughly speaking, measures the difficulty of producing saturated regular ultrapowers). Model theory is a branch of mathematics which studies the fundamental structure of certain classes of mathematical objects, the models of a given theory. The structural variation which occurs within a given class (or between comparable models of different classes) sheds light on the inherent simplicity, or complexity, of the theory itself. The work in this proposal arises from new indications that certain theories may have deep and previously undetected structural similarities. These similarities have to do with possible limit behavior, that is, with the complexity of infinite configurations which arise from families of finite combinatorial objects within models of the theory. One broad aim of this work is the development of finer tools to detect this behavior in certain invariants of the theory (e.g. the persistence and distribution of graphs in the characteristic sequence of certain formulas of T). These ideas would give new leverage in the ongoing program of classifying unstable theories. Model-theoretic classification theory has, over the last forty years, developed a richly informative toolbox for analyzing the complexity of first-order theories and isolated many useful indicators of complexity, but much remains to be done. In particular, a second underlying goal of this work is to give a language in which to precisely ask, and to explore, the largely open questions of distribution and density of these indicators: how and where they cluster and how they interact with other objects in the model, e.g. base sets for types.