Research in Model Theory: Generic Structures

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AbstractAward: DMS-0203747Principal Investigator: Kitty L. HollandHolland's proposed work centers on applying the Hrushovski-styleamalgamation technique to produce novel examples of omega-stablestructures. Together with Baldwin, Holland is engaged in anattempt to use this technique to produce a bad field. Holland'swork toward this end has produced algebro-geometric results ofindependent interest on the relationship between transcendencedegree and multiplicative group rank in characteristic zerofields. Holland proposes to exploit this work to complete thebad field project with Baldwin and to extend thealgebro-geometric results. She further proposes to settlestability questions for certain structures produced by thetechnique; to analyze the geometries of structures produced bythe technique and to extend her joint work with Baldwin toward anabstract model theoretic analysis of the technique and itsproducts.Since its introduction in the early 90s, Hrushovski'samalgamation technique has been applied with great success tosettle a variety of old existence conjectures central toalgebraic model theory. Holland's ambition, embodied in thisproposal, is threefold: To continue breaking new ground inapplication of the technique, to explore the limitations of thetechnique by characterizing common properties of its products,and to make the technique more widely and efficiently applicableby extracting common difficulties arising in its implementationand dealing with them uniformly on an abstract level.Holland's proposed work centers on applying the Hrushovski-styleamalgamation technique to produce novel examples of omega-stablestructures. Together with Baldwin, Holland is engaged in anattempt to use this technique to produce a bad field. Holland'swork toward this end has produced algebro-geometric results ofindependent interest on the relationship between transcendencedegree and multiplicative group rank in characteristic zerofields. Holland proposes to exploit this work to complete thebad field project with Baldwin and to extend thealgebro-geometric results. She further proposes to settlestability questions for certain structures produced by thetechnique; to analyze the geometries of structures produced bythe technique and to extend her joint work with Baldwin toward anabstract model theoretic analysis of the technique and itsproducts.Since its introduction in the early 90s, Hrushovski'samalgamation technique has been applied with great success tosettle a variety of old existence conjectures central toalgebraic model theory. Holland's ambition, embodied in thisproposal, is threefold: To continue breaking new ground inapplication of the technique, to explore the limitations of thetechnique by characterizing common properties of its products,and to make the technique more widely and efficiently applicableby extracting common difficulties arising in its implementationand dealing with them uniformly on an abstract level.