Mathematical Sciences: Geometric Stability Theory

谢拉的分类理论为每个“可分类”的结构分配了一个由规则类型表示的同质组合几何系统。几何稳定性理论的基本思想之一是组合几何、线性几何和几何类型几何之间的三分法。一个令人满意的结构理论存在于线性域中，因为几何本身和它们所协调的结构。第三种类型的规则类型理解得很少；直到最近，人们还认为它们完全由代数几何对象组成(按有限阶)，但一系列结构表明，这类对象要大得多。这里所进行的研究将试图建立非线性强极小集的映射。一个目的是Cherlin的猜想，即Aleph-One-范畴群是代数闭域上的简单代数群。另一方面，有迹象表明，几何三分法在现有的稳定性理论边界之外是有效的。研究人员希望扩展这些边界，创建一种分叉理论、正则类型和线性理论的并行，用于一类结构，该结构包括其(线性)正则类型中的所有经典几何(射影几何、酉几何、正交几何、辛几何)。这将适用于伪有限结构以及可能与有限群论的相互作用。这一研究领域值得注意的一点是，它表明，通过对处理代数结构的数学语言的深入研究，可以获得大量关于代数结构的信息。近年来，谢拉、赫鲁绍夫斯基和其他人将这种模型理论方法推向了比大多数人之前认为可能的更远的距离。

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.