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

小行星9501176 该项目涉及群论（有限的和代数的）与模型论和集合论以及一些 涉及不寻常的组合几何的组合问题。 切尔林 将研究一类在模型论中出现的群，其简单的 截面被证明是代数群。 在这个类中， 一类“驯服的Ω稳定群”提供了一个测试案例， Borovik哲学认为， 在有限单群的分类中使用的策略将 足以证明这个猜想。 这实际上等于做了 比简单代数群的分类更多一些 代数闭域，使用有限群理论的方法， 除了维数的基本性质之外，很少有代数几何。 的 博罗维克的哲学在《奇 特征，并要求在特征二类似的审查。 （请参阅 偶数特征和奇数特征的定义是群论的，但相当 不同于有限群理论中使用的那些）。 托马斯会继续 对置换群的共尾性的研究。 这次调查 回到塞尔的一个问题，其中的确切答案是沉重的 依赖于集合论假设。 这一领域的成果往往依赖于 大量地使用精细集合论强制构造，还涉及 与共轭快速生成有关的特征标理论计算 班 这些群体之间也存在着一些关系， 理论上的共尾性和更标准的集合论基数不变量 连接到连续体。 最近，人们发现， 也与pcf理论有着坚实的联系，pcf理论是由 撒哈拉希拉。 这一理论被看作是一种新的方法基数 这是一种受独立现象影响较小的算术， 经典的应用程序蟑螂。 模型论试图对数学理论进行分类。 “最好的”理论是那些模型（实现）可以被给出的理论。 结构分类 二十年前，Zilber证明，所有这些 理论可以从熟悉的代数结构中建立起来：群。 切尔林 然后，Zilber为这些群体制定了一个分类方案， 以这种方式出现。一个有希望的途径是挖掘有限单群的文献。 由于约100名数学家的共同努力，在约10，000页高度浓缩的期刊文章中，有限单群被分类。 一个平行的战略发展的一个大类的无限群中出现的模型论。 这应该需要大大少于100名数学家或10，000页的期刊，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，这些有限群在无限情况下不涉及任何重要的程度;（2）在有限情形下，原始证明在二十多年的时间里发生了多少有些偶然的演变，这是可以根据从该证明中吸取的教训来避免的。 尽管如此，这是一个大规模的项目，大规模的规划将是 这对它的成功至关重要。 目前，该计划虽然 不完整，目前正在实施。 Cherlin将解决紧急问题 需要一个“顶级”分析，作为指导数学家在美国部分地区以及南美和欧洲从事这一战略。 自从保罗·科恩在1963年的突破中解决了集合论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集合论中令人惊讶的深刻问题。 最近这已表明，由托马斯的情况下，在研究的结构，这些无限群是最好的近似有限群。这一领域的工作有两个完全不同的方面， 好. 这个主题作为一个网站，其中开发新的集合论方法，提供了一个新的计算方法之间的联系，代数和集合论。 ***

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. ***