Mathematical Sciences: Geometric Stability Theory

希拉的分类理论赋予每一个 “可分类”结构同质组合系统 几何体，由常规类型表示。 根本的一 几何稳定性理论的思想是 组合几何、线性几何和几何类型的几何。 一 满足结构理论存在于线性域，对于 几何体本身和它们协调的结构。 的 第三种类型的常规类型非常不了解;直到 最近他们被认为是完全由代数 几何对象（在有限秩），但一系列的建设 显示出这个阶级要大得多。 进行的研究 这里将尝试创建一个强烈的非线性映射 极小集 其中一个目标是Cherlin的猜想，即aleph-one- 范畴群是代数上的单代数群 封闭的领域 另一方面，有迹象表明， 几何剖分法在现有边界之外有效， 稳定性理论 调查人员希望将这些 边界，创建分叉理论，常规类型， 线性理论的平行，对于一类结构， 包括所有的经典几何（射影的，酉的， 正交，辛）之间的（线性）定期类型。 这 可以应用于伪有限结构， 与有限群理论的相互作用 这个研究领域的一个值得注意的事情是 它表明大量关于代数的信息 结构可以从密集的研究， 用数学语言来处理它们。 希拉， Hrushovski和其他人推动了这种模型理论方法 比大多数人以前认为的要远得多 可能

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.