Mathematical Sciences: Geometric Stability Theory

在过去的二十年里，出现了一种非常强大的结构理论，它支持完全的直言理论。证明了它们都是围绕有限域上的射影空间建立的。对于Aleph-One范畴理论，人们期望存在一个类似但更深层次的理论；在这里，相应的几何将包括代数闭域。最近，一种控制Aleph-one范畴结构的非经典几何被构造出来。拟议的研究将从两个方向进行。第一种将直接调查这种结构的局限性，并试图创造出一幅主题的新图景。这里最重要的测试问题是，Cherlin关于有限Morley秩单群是代数闭域上的代数群的猜想是否仍然成立。第二个方向涉及周围结构中的给定几何；在最简单的情况下，结构是通过将每个点爆炸到有限集来从强极小集获得的；问题是如何在不干扰底座的情况下扩展结构。这将既适用于完全范畴理论的分类，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性的兴趣，但为证明它们而开发的精致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.