Baldwin proposes work on three topics in model theory: abstract elementary classes, generic expansions of fields, and the preservation of stability under expansion by naming predicates. The last two are in some sense dual. The twenty year old search for a bad-field tries to construct a sufficiently generic subgroup of the complex numbers that omega stability is preserved. Both refined model theoretic and algebraic tools are essential. In the other direction Baldwin is exploring with Baizhanov, the most general conditions under which can hope to preserve stability when a subset is named. Baldwin has been writing a monograph summarizing the current state of `Morley's theorem' for Abstract Elementary Classes. This monograph, tries to clarify and organize the general work in this topic, underline the connections with both Zilber's work on complex geometry and the `Hrushovski construction, and lay the foundation for a more general stability theory in this context.
The general perspective of logic in understanding the foundations of various mathematical topics has provided in Baldwin's work links among fields as diverse as database theory (computer science), random graphs, abstract algebra and complex analysis. In many cases the role of logic is place limits on what can be proved and direct research into more productive endeavors. Baldwin has served as a link between the logic and mathematics education communities for the last fifteen years. He is currently director of the Office of Mathematics Education at UIC. He aims to provide future teachers with a profound understanding of fundamental secondary school mathematics. A teacher with profound understanding of school mathematics is not only aware of the conceptual structure of secondary mathematics, but also understands the common strategies that students use to solve problems along with common student misconceptions. A background in logic is invaluable in helping future teachers develop the ability to communicate mathematics clearly.
