FRG: Collaborative Research: Model Theory of Differential and Difference Equations with Applications

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

This project is concerned with differential equations, which relate a quantity to its rate of change with respect to continuous time; difference equations, which relate a quantity to its rate of change with respect to discrete time; and combinations of these. Systems of such equations describe or model behavior and phenomena throughout the sciences; epidemiology, population dynamics, mechanics etc. The project will use methods from model theory, a branch of mathematical logic, to develop or improve procedures for showing consistency (existence of solutions) of systems of such equations, for identifying auxiliary parameters in systems of such equations, and for eliminating unknowns from such systems. The project will also extend the existing classification of sets of solutions from the case of over-determined (or finite-dimensional) systems to under-determined (or infinite-dimensional systems). In this focused research group project, the team of researchers studies differential, difference, and differential-difference equations, from the point of view of model theory in the sense of mathematical logic. Among the aims is to provide decision procedures or improve existing decision procedures regarding consistency and elimination. Applications to a variety of modelling problems for physical systems are expected. The proposed research is threefold. The first is to develop efficient algorithms in cases where the existing methods from theories of differential or difference fields provide algorithms in principle. The second is to extend the existing model-theoretic classification of finite-dimensional solution spaces of systems to infinite-dimensional solution spaces, as well as developing methods to apply the classification in practice. The third is to introduce and study new first order theories of rings with operators, whose decision procedures will apply to new examples arising in applications. 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.