Aschenbrenner proposes to contribute to the fields of model theory and algebra. He proposes to continue his collaborative efforts to create a synthesis between the fields of real algebraic geometry and differential algebra, in the form of asymptotic differential algebra. This will lead to a deeper understanding of the model-theoretic and algebraic properties of Hardy fields and fields of transseries, and the relationship between them, and will open up new perspectives in the asymptotic theory of algebraic differential equations. He also proposes to continue his investigations into non-standard methods and degree bounds in commutative algebra. For example, Aschenbrenner proposes to increase our knowledge about Grobner bases for ideals in the ring of polynomials with integral coefficients.
Mathematical logic, although it originated in the late 19th century with philosophical investigations into the foundations of mathematics, has in recent decades found many applications in other parts of mathematics, in computer science, and even in engineering (quantifier elimination as it relates to robotics). This project deals mainly with pushing the applicability of the methods of mathematical logic into as of yet unexplored territory, namely, asymptotic analysis. Our hope is that these investigations will lead to a deeper understanding of the behavior of solutions to differential equations. Another part of the proposed project involves the construction and analysis of algorithms for algebraic problems. Finding degree bounds as suggested in this project is of central importance for applications of computer algebra in science and engineering.
