Mathematical Sciences: Model Theory of Groups

莫利秩的概念是由莫利在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，并得到了类似的"有限 维代数"给定有限莫利结构M 秩，一个关联到来自M的n元组的每个可定义子集， 两个整数，称为子集的秩和度。 这些 可定义子集及其相关整数满足某些 可以被认为是"抽象"的公理的数量 维度理论"用代数几何做个类比： 可定义子集对应于可构造子集;秩 对应维度;度对应 不可约成分的数量（如果给定的集合是封闭的）。 若M是代数闭域，则上述对应 变成平等。 从这些对应关系中，人们可以猜测， 有限莫利秩的群必须看起来像几何群， 但麦金太尔的结果，即莫利秩的无限域是 代数封闭，减少了可能性。 Cherlin猜想：有限的无限单群G 莫利秩是代数闭群上的代数群 域，以及集合G，它的乘法和它的逆 函数由多项式定义。 另一方面，皮莱和斯坦霍恩定义了这个概念， o-最小结构，皮莱开始研究G群 定义在O-极小结构M中。 这类新的群体 包括许多真实的李群（例如代数李群）。 它们的行为 像有限莫利秩的群：人们有维数的概念 和连通性。 这个项目的首席研究员， Huseyin Ali Nesin打算研究这两类群体， 至少在未来两年。 许多问题有待解决 在它们与经典群的关系完全 明白

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.