Mathematical Sciences: Research in Model Theory

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9308768 Baldwin Baldwin intends to investigate stable structures derived as the homogeneous universal model for classes of finite structures with dimension functions. These structures have significantly broadened the class of stable structures in recent years. In particular, he asks whether structures of this sort can possibly satisfy the Vaught or Lachlan conjectures, i.e. whether the theory of such a structure can have finitely many or aleph-one countable models. He will also study the relation between `generic' models for classes of finite structures and recent advances in the study of `random' models by Spencer. During the last twenty years stability theory has developed a technical apparatus for detailed investigation of a large class of mathematical structures. Zil'ber suggested that in fact all the structures involved were tied very closely to classical structures. Hrushovski provided a counterexample to this conjecture. Baldwin refined his construction to find a new non-Desarguesian projective plane. Baldwin also discovered a close connection between these structures and the theory of random graphs. Baldwin intends to study this new class of structures: to see whether they provide counterexamples to some other long standing conjectures and to explore further the connection with random graphs. ***