Mathematical Sciences: Group Theoretic Problems in Model Theory and Set Theory

9501176谢林这个项目涉及群论(有限和代数)与模型论和集合论的相互作用，以及一些涉及不寻常的组合几何的组合问题。Cherlin将研究模型理论中出现的一类群，其简单截面被猜想为代数群。在这一类中，“驯服omega-稳定群”类提供了一个测试案例，在这个测试案例中，它被猜想--Borovik哲学--在有限单群的分类中使用的策略的修改将足以证明该猜想。在实践中，这相当于使用有限群论的方法对代数闭域上的简单代数群进行分类，而除了维度的基本性质之外，几乎没有做什么代数几何。波罗维克哲学已经在奇怪的特征中得到了相当明确的充实，并需要在特征二中进行类似的审查。(偶数和奇数特征的定义是群论的，但与有限群论中使用的定义有很大不同。)托马斯将继续研究置换群的余定性。这一研究回到了Serre的一个问题上，这个问题的准确答案在很大程度上依赖于集合论的假设。这一领域的结果往往严重依赖于精细的集合论强制构造，也涉及与共轭类快速生成相关的特征理论计算。在这些群论余定性和更多的标准集合论基数不变量之间也有关系，这些关系在很大程度上是模糊的。最近，人们看到与PCF理论也有密切的联系，PCF理论是由撒哈拉·谢拉发展起来的集合论领域。这一理论被视为一种新的基数算术方法，与经典的应用程序蟑螂相比，它受独立现象的影响较小。模型理论试图对数学理论进行分类。“最好的”理论是那些其模型(实现)可以被给予结构分类的理论。20年前，齐尔伯证明了所有这样的理论都可以建立在熟悉的代数结构上：群。然后，Cherlin和Zilber猜测了以这种方式出现的群体的分类方案。一条很有希望的途径是挖掘关于有限单群的文献。大约100名数学家在大约10,000个高度浓缩的期刊页面上发表的期刊论文的共同劳动的结果，有限的简单群被分类。针对模型理论中出现的一大类无限群，提出了一种并行策略。这应该需要不到100名数学家或10,000页期刊页，原因有两个：(1)有限情况下的大部分工作与26个奇怪的有限群的性质有关，这些群在无限情况下没有涉及到任何重要程度；(2)在有限情况下，原始证明在20多年的时间里多少有些随意的演变，可以根据从该证明中学到的教训来避免。尽管如此，这是一个大规模的项目，大规模的规划将是其成功的关键。目前，该计划虽然仍未完成，但已经在实施中。切尔林将阐述“顶层”分析的迫切需要，以此作为美国部分地区以及南美和欧洲从事这一战略的数学家的指南。自从保罗·科恩在1963年的突破中解决了该学科中一些最古老的问题以来，集合论得到了迅速而稳定的发展。数学的许多领域都涉及到集合论中令人惊讶的深刻问题。最近，Thomas在研究那些最接近于有限群的无限群的结构时就证明了这一点。这一领域的工作有两个完全不同的方面，这两个方面配合得很好。这门学科是开发新的集合论方法的场所，提供了代数中的计算方法和集合论之间的新联系。***

9501176 Cherlin The project concerns the interaction of group theory (finite and algebraic) with model theory and set theory, as well as with some combinatorial issues that involve unusual combinatorial geometries. Cherlin will investigate a class of groups arising in model theory, whose simple sections are conjectured to be algebraic groups. Within this class, the class of "tame omega-stable groups" provides a test case in which it is conjectured -- the Borovik philosophy -- that a modification of the strategy used in the classification of the finite simple groups will suffice to prove the conjecture. This amounts in practice to doing somewhat more than the classification of simple algebraic groups over algebraically closed fields, using methods of finite group theory, and very little algebraic geometry beyond rudimentary properties of dimension. The Borovik philosophy has been fleshed out fairly clearly in odd characteristic, and requires similar scrutiny in characteristic two. (The definitions of even and odd characteristic are group theoretic but quite different from the ones used in finite group theory.) Thomas will continue the study of the cofinalities of permutation groups. This investigation goes back to a question of Serre, the precise answer of which is heavily dependent on set theoretic hypotheses. Results in this area tend to rely heavily on delicate set theoretic forcing constructions and also involve character theoretic computations relating to rapid generation by conjugacy classes. There are also relationships, largely obscure, between these group theoretic cofinalities and more standard set theoretic cardinal invariants attached to the continuum. Most recently it has been seen that there are also solid connections to pcf theory, an area of set theory developed by Saharon Shelah. This theory is seen as a new approach to cardinal arithmetic that is less affected by independence phenomena than the classical app roach. Model theory attempts to classify mathematical theories. The "best" theories are those whose models (realizations) can be given structural classifications. Twenty years ago Zilber showed that all such theories can be built from familiar algebraic structures: groups. Cherlin and Zilber then conjectured a classification scheme for the groups that arise in this way. One promising route is to mine the literature on finite simple groups. As a result of the combined labors of about one hundred mathematicians in journal articles filling about 10,000 highly condensed journal pages, the finite simple groups were classified. A parallel strategy is developed for a large class of the infinite groups arising in model theory. This should require considerably less than 100 mathematicians or 10,000 journal pages, for two reasons: (1) much of the work in the finite case relates to the properties of 26 bizarre finite groups which are not involved to any significant extent in the infinite case; (2) the somewhat haphazard evolution of the original proof in the finite case, over more than two decades, can be avoided on the basis of lessons learned from that proof. It is nonetheless a large-scale project, and large-scale planning will be critical to its success. At the present time the plan, though still incomplete, is already being implemented. Cherlin will address the urgent need for a "top-level" analysis to serve as a guide for the mathematicians working on this strategy in parts of the U.S. and also in South America and Europe. Set theory has developed rapidly and steadily since Paul Cohen settled some of the oldest questions in the subject in his 1963 breakthrough. Many areas of mathematics involve surprisingly deep questions in set theory. Recently this has been shown by Thomas to be the case in the study of the structure of those infinite groups which are best approximated by finite groups. Work in this area has two quite different aspects which mesh toge ther well. This subject serves as a site in which to develop new set theoretic methods, providing a new link between computational methods in algebra and set theory. ***