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.
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{'Code': '126800', 'Text': 'FOUNDATIONS'}
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