Mathematical Sciences: Model Theory of Groups

数学科学：群模型论

• 批准号: 8801021
• 负责人: Huseyin Nesin
• 金额: $ 1.26万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-15 至 1989-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8801021&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Model; Theory; Groups

The concept of Morley rank was defined by Morley in 1965. Since then many mathematicians have applied this notion to algebraic structures and obtained an analogue of "finite dimensional algebras." Given a structure M of finite Morley rank, one associates to each definable subset of n-tuples from M, two integers, called the rank and degree of the subset. These definable subsets and their associated integers satisfy some number of axioms which may be thought of as an "abstract dimension theory." Making an analogy with algebraic geometry: definable subsets correspond to constructible subsets; the rank corresponds to the dimension; the degree corresponds to the number of irreducible components (if the given set is closed). If M is an algebraically closed field, the above correspondences become equalities. From these correspondences one may guess that groups of finite Morley rank must look like geometrical groups, but Macintyre's result, that infinite fields of Morley rank are algebraically closed, reduces the possibilities. Cherlin's conjecture: An infinite simple group G of finite Morley rank is an algebraic group over some algebraically closed field, and the set G, its multiplication, and its inverse function are defined by polynomials. On the other hand, Pillay and Steinhorn defined the concept of o-minimal structures, and Pillay began studying groups G defined in an o-minimal structure M. This new class of groups includes many real Lie groups (e.g. algebraic ones). They behave like groups of finite Morley rank: one has a notion of dimension and connectedness. The principal investigator of this project, Huseyin Ali Nesin, intends to work on these two classes of groups for at least the next two years. Many problems remain to be solved before their relation to the classical groups is fully understood.

Morley等级的概念是由Morley在1965年定义的。从那时起，许多数学家将这一概念应用于代数结构，得到了类似于“有限维代数”的概念。给出一个有限Morley阶的结构M，从M到每个可定义的n元组子集都有两个整数，称为子集的秩和度。这些可定义的子集及其相关的整数满足一定数量的公理，这些公理可以被认为是一种“抽象维度理论”。用代数几何作类比：可定义子集对应于可构造子集；排名对应于维度；次数对应于不可约分量的数目(如果给定的集合是封闭的)。如果M是代数闭域，则上述对应变为等式。从这些对应关系可以推测，有限Morley秩群看起来一定像几何群，但Macintyre的结果，即Morley秩无穷域在代数上是封闭的，减少了这种可能性。Cherlin猜想：有限Morley秩无限单群G是代数闭域上的代数群，集合G及其乘法和逆函数由多项式定义。另一方面，Pillay和Steinhorn定义了o-极小结构的概念，Pillay开始研究定义在o-极小结构M中的群G。这类新的群包括许多实李群(例如代数李群)。它们的行为就像有限Morley秩群：一个人有一个维度和连通性的概念。该项目的首席调查员侯赛因·阿里·内森打算至少在未来两年内研究这两类群体。在完全了解它们与经典群的关系之前，还有许多问题需要解决。