Model-theoretic tree properties and their applications

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Model theory is a branch of mathematical logic dedicated to studying definable sets in mathematical structures. Model theorists aim to isolate certain combinatorial features enjoyed by definable sets in tame structures and to develop a general theory of structures satisfying these properties. This unifies and explains the tameness of a structure or family of structures, but also provides an engine for proving deep structure theorems by exploiting the combinatorial similarity of a structure under analysis to well-understood mathematical objects. The model-theoretic notion of simplicity was a distillation of the tameness of sets definable in random graphs, pseudo-finite fields, and algebraically closed difference fields and has served as a key ingredient in applications of model theory to combinatorics and algebraic dynamics. Simplicity was the first of the model-theoretic tree properties to be isolated and studied, but since then there have been a number of model-theoretic properties that have received intensive study, generalizing and deepening the achievements of simplicity theory. The PI will continue to develop the theory of model-theoretic tree properties and pursue applications in algebra and combinatorics. The project also provides research training opportunities for graduate students. The simple theories were defined by Shelah as the class of theories without the tree property, a combinatorial property of a formula that he isolated in the study of independence in stable theories. The first applications were set-theoretic, but the core tools of simplicity theory also led to a consolidation of the techniques at play in the analysis of concrete structures coming from algebra and combinatorics, most notably in the case of pseudo-finite fields and smoothly approximable structures built out of classical geometries over finite fields. The development of the theory, then, sparked a two-way exchange between the theory and the examples, leading to refined understanding of core notions in model theory. The template established by simple theories, of a theory for class of theories defined by a combinatorial condition on trees of definable sets built on a theory of independence, was successfully replicated for a variety of dividing lines. Moreover, this extension has revealed striking connections and applications to combinatorics, allowed for the analysis of core new algebraic examples, and led to the resolution of difficult open problems in model theory. This project takes a systematic approach to model-theoretic tree properties, consolidating the core techniques to address both internal questions within model theory and applications to a diverse array of structures coming from algebra, geometry, and combinatorics. 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.