In the last two decades a very powerful structure theory has emerged for totally categorical theories. It was shown that they are all built around projective spaces over finite fields. It was expected that a similar but much deeper theory exists for aleph-one categorical theories; here the corresponding geometries would include algebraically closed fields. Recently a non- classical geometry controlling an aleph-one categorical structure has been constructed. The proposed research would proceed in two directions. The first would directly investigate the limits of such constructions and attempt to create a new picture of the subject. The most important test-question here is whether Cherlin's conjecture that every simple group of finite Morley rank is an algebraic group over an algebraically closed field is still viable. The second direction involves a given geometry inside an ambient structure; in the simplest case the structure is obtained from a strongly minimal set by blowing up each point to a finite set; the question is how the structure may be expanded without disturbing the base. This would have applications both to the classification of the totally categorical theories and to the general theory. The immediate results of such a project in abstract model theory are likely to be mainly of foundational interest, but the refined combinatorical techniques developed in order to prove them may have much wider appeal and utility.
