NSF-BSF: Independent Theories in Model Theory

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Model theory, a branch of mathematical logic, asks what we can say about arbitrary mathematical theories. Surprisingly, for such a general context, we can say meaningful things, discovering new general laws on the one hand and giving new methods for concrete mathematical theories on the other. Model-theoretic classification theory gives an approach to discovering the general laws: informally, it aims to isolate so-called dividing lines where a significant structural change in the class of theories may be detected, and in parallel to develop the internal theory to understand what is producing the phenomena observed. One such dividing line has to do with whether theories have nontrivial randomness. Current methods have been successful in many ways on the non-random side. This project concerns independent theories, those on the random side. The randomness complicates to varying degrees the ability of current methods to gauge the full structure. This research works towards a more fundamental understanding of such theories, using a range of classical and newly developed tools able to detect average behavior and changes in the complexity of model-theoretic randomness. Along with the research developments, the project will support the training of students at various levels.More precisely, the project aims to study independent theories, those with the independence property. The following interrelated test problems guide the work in this project. The first concerns characterizing saturated models of simple theories in a way analogous to the known characterization for stable theories, as a test problem for understanding of the building blocks of simple unstable theories. An approach to this question, of independent interest, involves developing a recently introduced generalization of forking. A second problem involves characterizing and developing a structure theory for a certain interesting class of simple theories 'near' the random graph. A third is to investigate the structure of quotients of Keisler's order. A fourth is to find true dividing lines within the class of independent theories and in parallel to develop the internal theory justifying these as dividing lines. These questions themselves connect to the classical problem of understanding the so called Keisler order, whose structure on dependent theories is known but whose structure on the independent theories remains in some senses mysterious.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.