Principal Investigator: Thomas W. Scanlon
The research supported by this award is to be performed by Matthias Aschenbrenner, under the sponsorship of Thomas W. Scanlon. These projects in model theory and its applications to algebra and geometry are concerned with asymptotic differential algebra, o-minimal geometry, and bounds and algorithms in algebra. The project on asymptotic differential algebra will pursue model-theoretic and algebraic properties of Hardy fields and transseries, and the relationship between them. A Hardy field is an ordered differential field of germs of real-valued, once-differentiable functions defined on neighborhoods of positive infinity in the real line; they are important in the asymptotic theory of differential equations and appear naturally in connection with o-minimal expansions of the real field. An example of a field of transseries is the field of logarithmic-exponential series over the real numbers, which has been explored by analysts as well as model-theorists. Among the algorithmic issues in algebra that will be investigated are algorithms with performance bounds for polynomial rings over the integers and for rings of power series; problems for which good algorithms are sought include tests for ideal membership.
The idea of the branch of logic called model theory is, roughly, that if we know all of the simply-stated truths about an object then either we should know how to recognize that object uniquely, or anything else sharing the same collection of first-order properties should be revealing like the original and might sometimes be easier to study. To be more precise, model theory studies mathematical structures by considering the first-order sentences true in those structures, and the family of alternate structures that also satisfy all of those first-order sentences. (Sentences in logic are built out of a small repertoire of elements and constructions. "First-order" refers to the number of quantifiers in a sentence, a measure of complexity.) A model for the algorithms and bounds sought in some of these projects is long division: if you are given two whole numbers to divide by hand then you can estimate the number of steps required by long division by comparing the number of digits in the decimal expansions of the dividend and divisor.
