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

9501176 Cherlin该项目涉及群论（有限和代数）与模型论和集合论的相互作用，以及涉及不寻常组合几何的一些组合问题。Cherlin将研究模型理论中出现的一类群，其简单部分被推测为代数群。在这个类中，“驯服-稳定群”类提供了一个测试案例，在这个案例中，人们推测——Borovik哲学——在有限单群分类中使用的策略的修改将足以证明这个猜想。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，并且除了基本的维数性质之外，很少涉及代数几何。博罗维克哲学在其奇特的特征上已经相当清楚地充实了，在特征二上也需要加以同样的考察。（奇偶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共结性。这个调查回到了Serre的一个问题，这个问题的精确答案在很大程度上依赖于集合论假设。这一领域的结果往往严重依赖于微妙的集合理论强迫结构，也涉及与共轭类快速生成有关的特征理论计算。在这些群论的共性和附加在连续体上的更标准的集合论的基数不变量之间，也有很大程度上模糊的关系。最近已经看到，它也与pcf理论有坚实的联系，pcf理论是由撒哈拉on Shelah发展的集合理论的一个领域。这个理论被看作是基数算术的一种新方法，它比经典的应用方法受独立性现象的影响更小。模型理论试图对数学理论进行分类。“最好的”理论是那些模型（实现）可以被给予结构性分类的理论。20年前，Zilber展示了所有这些理论都可以从熟悉的代数结构——群——中建立起来。然后，Cherlin和Zilber推测了一个以这种方式出现的群体的分类方案。一个有希望的途径是挖掘关于有限简单群的文献。由于大约100位数学家的共同努力，在期刊文章中填满了大约10,000页高度浓缩的期刊，有限单群被分类了。针对模型理论中出现的一类无限群，提出了一种并行策略。这需要的数学家不超过100人，也不超过1万页期刊，原因有二：(1)有限情况下的大部分工作涉及26个奇异有限群的性质，而这些性质在无限情况下没有多大意义；(2)原来的证明在有限的情况下，经过二十多年的偶然演变，根据从证明中吸取的教训，是可以避免的。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。目前这个计划虽然还不完整，但已经开始实施了。Cherlin将阐述对“顶层”分析的迫切需求，为美国部分地区、南美和欧洲从事这一战略的数学家提供指导。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奥问题。最近托马斯在研究有限群最能逼近的无限群的结构时，证明了这一点。这个领域的工作有两个完全不同的方面，它们很好地结合在一起。本课题是发展新的集合论方法的场所，在代数计算方法和集合论之间提供了新的联系。＊＊＊

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. ***