Model Theory of Valued Differential Fields

Ohio State University, The, Columbus OH

Investigators

Abstract

Differential equations, which express a relationship between a function and its rates of change, are used to understand many phenomena in science and engineering. Often, the eventual behavior of these functions is of interest: Do they stabilize or tend to infinity? How fast do they grow? Valued differential fields are mathematical structures that abstractly formalize these questions and possible answers. This project will investigate valued differential fields from the perspective of model theory, a branch of mathematical logic which involves studying properties of mathematical structures that can be expressed in formal languages, in turn revealing new insights into these structures and their complexity. In recent years the ideas and tools of model theory have had fruitful applications in other areas of mathematics, a trend this project will continue in the setting of valued differential fields. The project includes the training and mentoring of undergraduate students. Valued differential fields are algebraic structures that formalize the asymptotic comparison of solutions to differential equations. This project aims to deepen our understanding of such structures and their model theory in two primary ways. First, it will investigate unordered valued differential fields, most ambitiously complex transseries, which have been much less studied than ordered valued differential fields. These latter structures describe only non-oscillating behavior, while oscillating functions play an important role in mathematics and in describing the real world, from water waves to wave functions in quantum physics. This project will search for universal domains for the kinds of asymptotic differential algebra in which oscillation is permitted and study their complexity. Second, the project will investigate certain ordered valued differential fields that can arise from transexponential extensions of transseries. The work will then address natural next steps, most significantly the study of certain quotient structures, called imaginaries, associated to such ordered valued differential fields. 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.