Mathematical Sciences: Geometric Stability Theory

在过去的20年里，一个非常强大的结构理论 出现了完全的分类理论。 结果表明， 都是建立在有限域上的射影空间上的 它 人们认为，存在一个类似但更深刻的理论， aleph-one范畴理论;这里对应的几何 包括代数闭域 最近，一个非- 经典几何控制一个Alph-one范畴结构 已经建成。 拟议的研究将分两个阶段进行。 方向 第一种方法是直接研究 这样的建设，并试图创造一个新的图片的 话题吧 这里最重要的测试问题是， Cherlin猜想有限莫利的单群 秩是代数闭域上的代数群， 仍然可行。 第二个方向涉及给定的几何形状 在一个环境结构内;在最简单的情况下， 是从一个强极小集通过吹每个点 到一个有限的集合;问题是如何结构可能是 在不影响基础的情况下进行扩展。 这将 应用程序都对分类的总 分类理论和一般理论。 立即 这样一个项目在抽象模型理论中的结果可能会 主要是基础性的兴趣，但精致的 为了证明它们而开发的组合技术可以 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.