Geometric Model Theory

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AbstractAward: DMS-0301771Principal Investigator: Thomas W. ScanlonThese projects study the connections between model theory andgeometry. In this context, model theory includes the search foraxiomatizations and quantifier elimination results for concretestructures and the development of general stability and dimensiontheories. The geometric problems to be studied arise inalgebraic, complex analytic, rigid analytic, and differentialalgebraic geometry. Specific projects include the extension ofthe model theory of compact complex manifolds to nonstandardanalytic spaces and rigid analytic spaces. Some of this workinvestigates the extent to which Kaehler manifolds are free frommodel theoretic pathologies. In diophantine geometry newuniformity and finiteness principles are being sought asreflections of similar results for enriched fields. Inparticular, a program based on the model theory of separablyclosed fields aims to resolve a function field variant of Denis'analogue of the Mordell-Lang conjecture for Drinfeld modules.The fine structure of definable sets over D-fields will beinvestigated, aiming to extend relative completeness andquantifier elimination theorems to valued D-fields with richeranalytic structure and to produce effective versions ofquantifier elimination theorems in this setting. Differentialjet spaces and related constructions will be used to uncover thecombinatorial geometry of regular types in partial differentialfields. Abstract Euler characteristics and Grothendieck rings offirst-order structure will be investigated.The idea of the branch of logic called model theory is, roughly,that if we know all of the simply-stated truths about an objectthen either we should know how to recognize that object uniquely,or anything else sharing the same collection of first-orderproperties should be revealing like the original and mightsometimes be easier to study. To be more precise, model theorystudies mathematical structures by considering the first-ordersentences true in those structures, and the family of alternatestructures that also satisfy all of those first-order sentences.(Sentences in logic are built out of a small repertoire ofelements and constructions. "First-order" refers to the numberof quantifiers in a sentence, a measure of complexity.) A modelfor the algorithms and bounds sought in some of these projects islong division: if you are given two whole numbers to divide byhand then you can estimate the number of steps required by longdivision by comparing the number of digits in the decimalexpansions of the dividend and divisor.