Mathematical Sciences: Groups of Finite Morley Rank

9501415 Nesin Nesin的工作涉及有限Morley秩单群的分类。莫利排名是迈克尔·莫利提出的一个有30年历史的概念，在某种意义上衡量了结构的复杂性。在有限Morley阶的结构中，某些子集(所谓的可定义子集)被赋予一个自然数，这些自然数的行为类似于它们所依附的集合的“维”。例如，代数闭域上的每一个簇都是有限Morley秩的结构。特别地，代数闭域上的每个代数群都是有限Morley秩群。Cherlin-zil‘ber猜想说明了一个部分逆：有限Morley秩单群是代数闭域上的代数群。如果为真，则该猜想将证明，当基场是代数闭域时，代数群的概念与基场无关：只要满足某些简单公理的维度概念就足以确保一个单群是代数群，即几何对象。除了在模型理论中的使用之外，这几乎是哲学的(甚至是形而上学的！)猜想的结果足以使这个主题值得一试。如果猜想是错误的，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。内森的研究是展示切尔林-齐伯猜想的计划的基本部分。数学的主要目的是理解几何和自然数。发明代数是为了更好地理解几何和数字的本质。尺寸的概念在几何学中当然是非常重要的；一般来说，任何几何物体都有尺寸。人们可以问相反的问题：假设一个类中的每个对象都有一个维度，并且这些维度表现良好，即它们满足一些自然公理。那么，这些物体真的是几何的吗？奈森的工作在一个特定的背景下关注这个问题。他的对象是团体。他的研究是一个大型项目的一部分，该项目的目标是证明定义了某个维度概念的群是几何对象的变换群。***

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. ***