Andrews proposes a program of research spanning classical computability theory, classical model theory, and especially the interplay between computability theory and model theory. More specifically, the proposed research will develop the emerging field of Computable Stability Theory, where the methods of stability theory plays a key role is the study of effective properties of first order theories. Stability theory is the part of model theory which focuses on the underlying geometrical or structural nature of the mathematical objects. Generally speaking, the more underlying structure the mathematical object respects, the easier it is to compute information about the object. This forms a back-and-forth relationship between stability and computability, and this interplay is the focus of Computable Stability Theory. Questions arising from Computable Stability Theory lead to new questions in classical model theory as well as computability, which sheds light on both subjects. Among these questions is when quantifier elimination to some level can be derived from geometrical properties.
Computable Stability Theory endeavors to explore the basic statement that "structurally simple objects ought to be easier to compute." Model theory, a branch of mathematical logic, offers tools to analyze structural simplicity by studying objects in the context of their first order language. The understanding of the language associated to an object or class of objects can often be translated to a deeper understanding of the objects themselves. Computability theory studies questions of computation and relative computation with regard to mathematical objects. By combining tools from both fields, Andrews will examine the question of when structural simplicity does and when it does not translate into computational simplicity.
