Mathematical Sciences: Geometric Stability Theory

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Shelah's classification theory assigns to each "classifiable" structure a system of homogeneous combinatorial geometries, represented by regular types. One of the fundamental ideas of geometric stability theory is a trichotomy between geometries of combinatorial, linear, and geometric type. A satisfying structure theory exists in the linear domain, for the geometries themselves and the structures they coordinatize. The regular types of the third type are very poorly understood; until recently they were believed to consist entirely of algebraic geometry objects (in finite rank), but a series of constructions has shown the class to be much larger. The research undertaken here will attempt to create a map of the nonlinear strongly minimal sets. One goal is Cherlin's conjecture, that aleph-one- categorical groups are simple algebraic groups over algebraically closed fields. In another direction, there are indications that the geometric trichotomy is valid beyond the existing borders of stability theory. The investigator hopes to extend these borders, creating a theory of forking, regular types, and the parallel of the linear theory, for a class of structures that includes all the classical geometries (projective, unitary, orthogonal, symplectic) among its (linear) regular types. This would have applications to pseudo-finite structures and possible interactions with finite group theory. One of the remarkable things about this research area is that it shows that a great deal of information about algebraic structures can be obtained from intensive study of the mathematical language in which one deals with them. Shelah, Hrushovski, and others have pushed this model-theoretic approach much further in recent years than most people previously thought possible.