Research in Model Theory

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Baldwin proposes work on three topics in model theory: abstractelementary classes, generic expansions of fields, and thepreservation of stability under expansion by naming predicates.The last two are in some sense dual. The twenty year old searchfor a bad-field tries to construct a sufficiently generic subgroupof the complex numbers that omega stability is preserved. Bothrefined model theoretic and algebraic tools are essential. In theother direction Baldwin is exploring with Baizhanov, the mostgeneral conditions under which can hope to preserve stability whena subset is named. Baldwin has been writing a monographsummarizing the current state of `Morley's theorem' for AbstractElementary Classes. This monograph, tries to clarify and organizethe general work in this topic, underline the connections withboth Zilber's work on complex geometry and the `Hrushovskiconstruction, and lay the foundation for a more general stabilitytheory in this context.The general perspective of logic in understanding the foundationsof various mathematical topics has provided in Baldwin's worklinks among fields as diverse as database theory (computerscience), random graphs, abstract algebra and complex analysis. Inmany cases the role of logic is place limits on what can be provedand direct research into more productive endeavors. Baldwin hasserved as a link between the logic and mathematics educationcommunities for the last fifteen years. He is currently directorof the Office of Mathematics Education at UIC. He aims to providefuture teachers with a profound understanding of fundamentalsecondary school mathematics. A teacher with profoundunderstanding of school mathematics is not only aware of theconceptual structure of secondary mathematics, but alsounderstands the common strategies that students use to solveproblems along with common student misconceptions. A backgroundin logic is invaluable in helping future teachers develop theability to communicate mathematics clearly.