CAREER: Model Theory and Homogeneous Structures

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Model theory is a branch of mathematical logic that studies combinatorial properties of mathematical structures (that is, sets equipped with certain operations). Since most mathematical objects can be represented as a structure, its subject matter is very broad and it interacts with several areas of mathematics. However, whereas other fields of mathematics are interested in specific structures, or classes of structures (for instance arithmetic studies the structure of the integers with addition and multiplication), model theory takes a step back and studies structures in general, looking for dividing lines and common properties. It singles out large classes of structures defined by combinatorial tameness conditions (which imply that the structure is in some sense not too complicated) and then attempts to understand the structures in those classes. One such class is that of NIP structures, which can be thought of as structures that have a geometric flavor. This class has received a lot of attention in the past decade. Another class is that of homogeneous structures, which are structures that have a lot of symmetries. Those are objects that belong to combinatorics and have been largely studied outside, or at the border of model theory. The main goal of this project is the investigation of new model-theoretic tools to understand homogeneous structures and in particular those that are NIP. There are few general theorems on homogeneous structures and at the same time no evidence that there cannot be any: we do not know how complicated homogeneous structures can be and this project hopes to shed light on this.The starting point of this project is the classification of the primitive rank 1 NIP homogeneous structures obtained by the PI. One goal is naturally to generalize this result to all finite rank NIP homogeneous structures. A more long-term goal is the classification of all NIP homogeneous structures. This would require understanding omega-categorical linear orders, which is an interesting project in its own right. A more general goal of the project is the development of model-theoretic tools to study homogeneous structures. This will build on the work mentioned above as well as on a previous work with Itay Kaplan on finitely generated dense subgroups of automorphism groups. Projects include finding model-theoretic consequences of the Ramsey property and clarifying its link with distality, finding necessary and sufficient conditions for admitting homogeneous expansions with nice properties (including having a stationary or canonical independence relation, being linearly ordered, Ramsey etc.). Finally, a more speculative project is the understanding of random-like behaviors in homogeneous structures and finding ways to measure the complexity of homogeneous structures, in particular those that lie outside the usual model-theoretic tame classes.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.