RUI: Geometry and Complexity in the Model Theory of Groups

University Enterprises, Incorporated, Sacramento CA

Investigators

Abstract

This project is organized around the long-standing "Algebraicity Conjecture" for groups of finite Morley rank, which arise naturally in model theory and are crucial to understanding a fundamental class of structures. The project investigates key remaining obstructions to the Algebraicity Conjecture by developing a newly found connection with approximately classical geometries. The project will also apply the existing partial solution to the Algebraicity Conjecture to establish natural limits to how much symmetry a group of finite Morley rank may encode, with the added goal of classifying those at the extreme. Additionally, the project explores the notion of "relational complexity" for finite symmetry groups, a key component of a classification theory for certain highly symmetric structures. This research addresses core problems about the complexity of certain natural families of groups and develops computational tools for further exploration. Finally, the project provides new opportunities and support for undergraduate and Masters students at California State University, Sacramento, to engage with and build skill in the model theory of groups. The first thread of this project addresses remaining obstructions to the Algebraicity Conjecture with the goal of exploiting a connection between certain groups of small 2-rank and generically defined projective geometries. Specific aims include the elimination of a long-standing pathological configuration, a clarification of a second, and a significant expansion of the existing techniques for analyzing groups of small, but nonzero, 2-rank. The second thread of research studies permutation groups of finite Morley rank. The focus is on Borovik and Cherlin's guiding problem of classifying those groups with a sufficiently high degree of generic transitivity as being of a single form that arises naturally in projective geometry. The final thread investigates the relational complexity of finite permutation groups. The study of relational complexity is currently anchored by Cherlin's Binary Conjecture, which proposes a classification of the primitive groups of complexity 2. This project aims to broaden the scope of research on relational complexity by analyzing various natural families of permutation groups, focusing on the complexity of the symmetric and alternating groups acting on partitions. Moreover, this thread further develops algorithms and refines exiting code for computing relational complexity, with the additional goals of creating a public repository for the code and manual to support its use. 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.