Mathematical Sciences: Groups of Finite Morley Rank

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9501415 Nesin Nesin's work concerns the classification of simple groups of finite Morley rank. Morley rank, a 30-year-old concept due to Michael Morley, measures in some sense the complexity of the structure. In a structure of finite Morley rank, certain subsets (the so-called definable subsets) are endowed with a natural number, and these natural numbers behave like the "dimension" of the set to which they are attached. For example, every variety over an algebraically closed field is a structure of finite Morley rank. In particular, every algebraic group over an algebraically closed field is a group of finite Morley rank. The Cherlin-Zil'ber conjecture states a partial converse: A simple group of finite Morley rank is an algebraic group over an algebraically closed field. If true, the conjecture will show that the concept of algebraic group is independent of the base field when the latter is known to be algebraically closed: just a concept of dimension satisfying certain simple axioms is enough to insure that a simple group is an algebraic group, i.e., a geometric object. Apart from its use in model theory, this almost philosophical (even metaphysical!) consequence of the conjecture suffices to make the subject worthwhile. If the conjecture is false, it will give rise to new groups that are not algebraic but that look very much like both infinite algebraic and finite groups. Nesin's research is a fundamental part of the program of showing the Cherlin-Zil'ber conjecture. The primary purpose of mathematics is the understanding of geometry and natural numbers. Algebra was invented to understand better the nature of geometry and numbers. The concept of dimension is, of course, very important in geometry; in general, any geometric object has dimension. One can ask the reverse problem: suppose each object in a class has a dimension and that these dimensions behave well, i.e. they satisfy some natural axioms. Is it then true that th ese objects are geometric? The work of Nesin concerns this problem in a particular setting. His objects are groups. His research is part of a large project whose goal is to show that groups on which a certain concept of dimension is defined are groups of transformations of a geometric object. ***