Mathematical Sciences: Groups of Finite Morley Rank

数学科学：有限莫利秩群

• 批准号: 9501415
• 负责人: Huseyin Nesin
• 金额: $ 5.8万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9501415&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Groups; Finite; Morley

9501415 Nesin Nesin's work concerns the classification of simple groups of finite Morley rank. Morley rank, a 30-year-old concept due to Michael Morley, measures in some sense the complexity of the structure. In a structure of finite Morley rank, certain subsets (the so-called definable subsets) are endowed with a natural number, and these natural numbers behave like the "dimension" of the set to which they are attached. For example, every variety over an algebraically closed field is a structure of finite Morley rank. In particular, every algebraic group over an algebraically closed field is a group of finite Morley rank. The Cherlin-Zil'ber conjecture states a partial converse: A simple group of finite Morley rank is an algebraic group over an algebraically closed field. If true, the conjecture will show that the concept of algebraic group is independent of the base field when the latter is known to be algebraically closed: just a concept of dimension satisfying certain simple axioms is enough to insure that a simple group is an algebraic group, i.e., a geometric object. Apart from its use in model theory, this almost philosophical (even metaphysical!) consequence of the conjecture suffices to make the subject worthwhile. If the conjecture is false, it will give rise to new groups that are not algebraic but that look very much like both infinite algebraic and finite groups. Nesin's research is a fundamental part of the program of showing the Cherlin-Zil'ber conjecture. The primary purpose of mathematics is the understanding of geometry and natural numbers. Algebra was invented to understand better the nature of geometry and numbers. The concept of dimension is, of course, very important in geometry; in general, any geometric object has dimension. One can ask the reverse problem: suppose each object in a class has a dimension and that these dimensions behave well, i.e. they satisfy some natural axioms. Is it then true that th ese objects are geometric? The work of Nesin concerns this problem in a particular setting. His objects are groups. His research is part of a large project whose goal is to show that groups on which a certain concept of dimension is defined are groups of transformations of a geometric object. ***

小行星9501415 Nesin的工作涉及分类简单的群体有限莫利秩。 莫利排名，一个30岁的概念，由于迈克尔莫利， 在某种意义上衡量了结构的复杂性。 的结构中 有限莫利秩，某些子集（所谓的可定义子集）是 被赋予一个自然数，这些自然数的行为就像它们所连接的集合的“维数”。 例如，代数闭域上的每个簇都是有限莫利秩的结构。特别地，代数闭域上的每个代数群都是有限莫利秩群。 Cherlin-Zil'ber猜想指出， 部分匡威：有限莫利秩的单群是代数群， 代数闭域上的群。 如果是真的，这个猜想将证明 代数群的概念与基域无关， 已知后者是代数封闭的：只是维数的概念 满足某些简单公理就足以确保一个简单群是 代数群，即，几何物体 除了在模型中使用 理论，这几乎哲学（甚至形而上学！）后果 推测足以使这个问题有价值。 如果这个推测是 假，它将产生新的群，这些群不是代数的，但看起来非常像无限代数群和有限群。 内辛氏 研究是展示计划的一个基本部分， Cherlin-Zil'ber猜想 数学的主要目的是理解几何和自然数。 代数的发明是为了更好地理解 几何和数字的本质 维度的概念，当然， 在几何中很重要;一般来说，任何几何对象都有尺寸。 一 我可以提出相反的问题：假设类中的每个对象都有一个 尺寸，并且这些尺寸表现良好，即它们满足一些 自然公理 那么这些物体是几何的吗？ 工作 Nesin在一个特定的环境中关注这个问题。 他的目标是 组 他的研究是一个大型项目的一部分，该项目的目标是表明， 定义了某个维数概念的群是 几何对象的变换。 ***