Zilber's Trichotomy and its Applications

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

This project aims to study interactions between model theory, geometry, and combinatorics. Here, model theory refers to a branch of mathematical logic: roughly speaking, model theorists study mathematical structures by studying the expressive power of abstract languages that describe them. Over the last few decades, the abstract perspective of model theory has led to the discovery of certain surprising patterns across different areas of mathematics. A key such pattern is Zilber's trichotomy: this is a general observed phenomenon in which sufficiently well-behaved mathematical structures (in the sense of model theory) tend to have one of just three basic forms. This is an imprecise phenomenon -- but recent work has increased our understanding of when and how it can be formally studied. This project aims to expand on such recent work, using the latest techniques to find more precise and true instances of the trichotomy. The PI then plans to use new cases of the trichotomy to expand applications of model theory in other areas of mathematics, with a particular focus on certain aspects of geometry and combinatorics. The project will provide research opportunities for graduate students. More specifically, the project has four main goals. First, building on recent work, the PI will seek to prove new instances of the Zilber trichotomy for relics of structures of geometric interest. The main questions of this form concern relics of o-minimal structures and relics of algebraically closed valued fields; in each of these cases, the status of the trichotomy has seen substantial recent progress but remains open. Second, the PI will use new instances of the trichotomy to study reconstruction problems in geometry -- focusing particularly on reconstructing algebraic groups over algebraically closed fields from partial data. This will generalize the work of Zilber on recovering a curve from its Jacobian variety: the goal is to prove stronger and more general statements of the same type, using the latest trichotomy results as a tool. Third, the PI will investigate broader interpretations of the trichotomy, weakening the usual assumption of strong minimality to similar notions of tameness, such as geometric structures—the hope being that a wider array of trichotomy-style theorems will allow for a broader array of geometric applications. Finally, the PI will investigate pseudo-finite variants of the trichotomy in the hope of finding new applications to incidence problems in 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.