Mathematical Sciences: Model Theory of Groups

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The concept of Morley rank was defined by Morley in 1965. Since then many mathematicians have applied this notion to algebraic structures and obtained an analogue of "finite dimensional algebras." Given a structure M of finite Morley rank, one associates to each definable subset of n-tuples from M, two integers, called the rank and degree of the subset. These definable subsets and their associated integers satisfy some number of axioms which may be thought of as an "abstract dimension theory." Making an analogy with algebraic geometry: definable subsets correspond to constructible subsets; the rank corresponds to the dimension; the degree corresponds to the number of irreducible components (if the given set is closed). If M is an algebraically closed field, the above correspondences become equalities. From these correspondences one may guess that groups of finite Morley rank must look like geometrical groups, but Macintyre's result, that infinite fields of Morley rank are algebraically closed, reduces the possibilities. Cherlin's conjecture: An infinite simple group G of finite Morley rank is an algebraic group over some algebraically closed field, and the set G, its multiplication, and its inverse function are defined by polynomials. On the other hand, Pillay and Steinhorn defined the concept of o-minimal structures, and Pillay began studying groups G defined in an o-minimal structure M. This new class of groups includes many real Lie groups (e.g. algebraic ones). They behave like groups of finite Morley rank: one has a notion of dimension and connectedness. The principal investigator of this project, Huseyin Ali Nesin, intends to work on these two classes of groups for at least the next two years. Many problems remain to be solved before their relation to the classical groups is fully understood.