Geometric Model Theory

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Abstract Award: DMS-0301771 Principal Investigator: Thomas W. Scanlon These projects study the connections between model theory and geometry. In this context, model theory includes the search for axiomatizations and quantifier elimination results for concrete structures and the development of general stability and dimension theories. The geometric problems to be studied arise in algebraic, complex analytic, rigid analytic, and differential algebraic geometry. Specific projects include the extension of the model theory of compact complex manifolds to nonstandard analytic spaces and rigid analytic spaces. Some of this work investigates the extent to which Kaehler manifolds are free from model theoretic pathologies. In diophantine geometry new uniformity and finiteness principles are being sought as reflections of similar results for enriched fields. In particular, a program based on the model theory of separably closed 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 be investigated, aiming to extend relative completeness and quantifier elimination theorems to valued D-fields with richer analytic structure and to produce effective versions of quantifier elimination theorems in this setting. Differential jet spaces and related constructions will be used to uncover the combinatorial geometry of regular types in partial differential fields. Abstract Euler characteristics and Grothendieck rings of first-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 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.