Model Theory, Algebra and Geometry

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AbstractAward: DMS-0303618Principal Investigator: Thomas W. ScanlonThe research supported by this award is to be performed byMatthias Aschenbrenner, under the sponsorship of ThomasW. Scanlon. These projects in model theory and its applicationsto algebra and geometry are concerned with asymptoticdifferential algebra, o-minimal geometry, and bounds andalgorithms in algebra. The project on asymptotic differentialalgebra will pursue model-theoretic and algebraic properties ofHardy fields and transseries, and the relationship between them.A Hardy field is an ordered differential field of germs ofreal-valued, once-differentiable functions defined onneighborhoods of positive infinity in the real line; they areimportant in the asymptotic theory of differential equations andappear naturally in connection with o-minimal expansions of thereal field. An example of a field of transseries is the field oflogarithmic-exponential series over the real numbers, which hasbeen explored by analysts as well as model-theorists. Among thealgorithmic issues in algebra that will be investigated arealgorithms with performance bounds for polynomial rings over theintegers and for rings of power series; problems for which goodalgorithms are sought include tests for ideal membership.The idea of the branch of logic called model theory is, roughly, thatif we know all of the simply-stated truths about an object then eitherwe should know how to recognize that object uniquely, or anything elsesharing the same collection of first-order properties should berevealing like the original and might sometimes be easier to study.To be more precise, model theory studies mathematical structures byconsidering the first-order sentences true in those structures, andthe family of alternate structures that also satisfy all of thosefirst-order sentences. (Sentences in logic are built out of a smallrepertoire of elements and constructions. "First-order" refers to thenumber of quantifiers in a sentence, a measure of complexity.)A model for the algorithms and bounds sought in some of these projectsis long division: if you are given two whole numbers to divide by handthen you can estimate the number of steps required by long divisionby comparing the number of digits in the decimal expansionsof the dividend and divisor.