Mathematical Sciences: Geometric Stability Theory

数学科学：几何稳定性理论

• 批准号: 9106711
• 负责人: Ehud Hrushovski
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-08-01 至 1995-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9106711&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometric; Stability; Theory

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.

希拉的分类理论赋予每一个“可分类”结构一个齐次组合几何系统，由正则类型表示。几何稳定性理论的一个基本思想是组合几何、线性几何和几何型几何的三分法。对于几何本身及其所协调的结构，在线性领域中存在着令人满意的结构理论。对第三种类型的常规类型了解甚少；直到最近，它们被认为完全由代数几何对象（有限秩）组成，但一系列的构造表明，这类对象要大得多。这里进行的研究将尝试创建一个非线性强极小集的映射。一个目标是Cherlin的猜想，即α -单范畴群是代数闭域上的简单代数群。在另一个方向上，有迹象表明几何三分法在稳定性理论的现有边界之外是有效的。研究者希望扩展这些边界，创建一个分岔理论，正则型，线性理论的平行，为一类结构，包括所有经典几何（射影，酉，正交，辛）在其（线性）正则型。这将应用于伪有限结构和可能与有限群论的相互作用。这个研究领域的一个值得注意的事情是，它表明，通过深入研究处理代数结构的数学语言，可以获得大量关于代数结构的信息。近年来，Shelah、Hrushovski等人将这种模型理论方法推进得比大多数人之前认为的要远得多。