Model theory is a branch of mathematical logic, which itself is a fairly new subject that originated in the late 19th century with philosophical investigations into the foundations of mathematics. However, in recent decades it has found many applications in other parts of mathematics, in computer science, and even in engineering (quantifier elimination as it relates to robotics). Algebra, on the other hand, is one of the oldest and most mature disciplines of mathematics. Aschenbrenner proposes to continue his research in the intersection of these two disciplines, where a fruitful exchange has been going for the better part of the last half-century.
The project is organized around several themes, including a continuation of Aschenbrenner?s collaborative efforts to develop a model-theoretic treatment of asymptotic analysis, and investigations into some model-theoretic questions with the means of combinatorial geometry. These topics have relationships not only to other fields within mathematics but also to applications of differential equations in engineering, and to computational learning theory. Another aspect of the proposal concerns finding algorithms for algebraic problems and studying their complexity, a topic which is of central importance for applications of computer algebra in science and engineering.
