Mathematical Sciences: Geometric Stability Theory

在过去的二十年里，一个非常强大的结构理论出现了。证明了它们都是围绕有限域上的射影空间建立的。人们期望有一个类似但更深刻的理论存在于α - 1范畴理论；这里相应的几何将包括代数闭场。最近构造了一个控制α - 1范畴结构的非经典几何。拟议的研究将从两个方向进行。第一种方法将直接调查这种结构的局限性，并试图创造一幅关于主体的新图景。这里最重要的检验问题是Cherlin关于每一个有限Morley秩的单群是代数闭域上的代数群的猜想是否仍然成立。第二个方向涉及环境结构内部的给定几何形状；在最简单的情况下，通过将每个点扩展为有限集，从强极小集得到结构；问题是如何在不影响基础的情况下扩大结构。这既适用于完全直言理论的分类也适用于一般理论。抽象模型理论中这样一个项目的直接结果可能主要是基础兴趣，但是为了证明它们而开发的精细组合技术可能具有更广泛的吸引力和实用性。

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.