NSF-BSF: Simple Theories and Keisler's Order

Over the last half-century, investigations in model-theoretic classification theory, within mathematical logic, have led to the discovery of certain fundamental properties of classes of mathematical structures and to a theory explaining them, with applications across mathematics and theoretical computer science. The topic of the present NSF-BSF project (joint with BSF PI Shelah) can roughly be described as investigating properties of classes of mathematical structures with a certain amount of randomness. This area was first intensely studied in the 1990s and it became clear that central questions were quite difficult. Recently, the investigators have made progress on the problem of Keisler's order for these classes and have developed a new tool, called the canonical Boolean algebra, which changes their picture of this area. This award will build on these discoveries to advance the field. This project studies the so-called simple unstable theories in model theory, a class including random graphs and hypergraphs, pseudofinite fields, extraspecial p-groups, and algebraically closed fields with an automorphism. The name 'simple' originally reflected an idea that these theories extended the foundational class of stable theories in a simple way, by a certain addition of randomness, but in the decades since this randomness has proved quite challenging to understand. Recent work of the investigators has established a productive approach to studying the complexity of simple theories using regular ultrapowers and Keisler's order, which classifies theories by means of infinite averages (essentially, the amount of saturation of their regular ultrapowers). Under prior NSF support, some key results were obtained. Among them, Keisler's order has the maximum number of classes within the simple theories, which informally involved proving that these infinite averages could be very carefully calibrated to give different outcomes based on small changes in the theory's randomness. Very recently, it was shown there is a canonical object, called the canonical Boolean algebra, which can in particular be used to study any class of simple theories. The present project builds on these advances, aiming to develop the canonical Boolean algebras and to further our understanding of Keisler's order on simple theories, and our understanding of simple unstable theories in general. The award will also allow for the training of students at multiple levels. 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.