Tame Groups, Universal Graphs, Automorphism Towers, and Cofinalities of Infinite Groups

Cherlin建议与一个由8名研究人员组成的团队合作，对有限莫利秩的简单驯服群进行分类，使用有限群经验所建议的方法：预计所有这些群都是代数的。 此外，切尔林，与施，旨在引进模型理论的方法到研究的普遍图，并显示作用的模型理论代数封闭运营商这类问题。托马斯将研究集合论和群论的相互作用与自同构塔和有限群的无限类似物的共尾性，利用强迫，pcf理论和大基数，以及特征理论。一个突出的问题是无限对称群的共尾性与Blass不变量，群密度的关系。 矩阵群在现代数学中的作用是公认的，在纯数学和应用数学中都占有中心地位。 已经证实，这些群在许多显然不相关的一般类型结构的详细分析中也起着主导作用。 本项目的目标之一是在所谓的"驯服案件"中确认这一点，因为这种案件更容易立即获得。一个单独的目标是证明逻辑（模型论）的标准思想可以为组合学（图论）中现有的问题带来新的启发。 希望图论学家在这种情况下也能采用这些方法。托马斯的工作揭示了以前未知的关系之间的代数和集理论的问题，并应导致新的"强迫"的工具，这是最强大的基础工具之一，现代集理论。该提案的主旨是展示如何在一个领域自然产生的技术可以卓有成效地转移到其他学科领域。

Cherlin proposes to collaborate with a team of eight researchers on the classification of connected simple tame groups of finite Morley rank, using methods suggested by experience with finite groups: it is anticipated that all such groups are algebraic. In addition Cherlin, working with Shi, seeks to introduce model theoretic methods into the study of universal graphs, and to show the role of the model theoretic algebraic closure operator for this class of problems. Thomas will study interactions of set theory and group theory in connection with automorphism towers and the cofinalities of infinite analogs of finite groups, making use of forcing, pcf theory, and large cardinals, as well as character theory. An outstanding problem is the relationship of the cofinality of the infinite symmetric group to Blass' invariant, the groupwise density. The role of matrix groups in modern mathematics is well established and occupies a central place in both pure and applied mathematics. It has been conjectured that these groups also play a predominant role in the detailed analysis of many apparently unrelated structures of general type. One of the goals of the present project is to confirm this in the so-called ``tame case'', which is more immediately accessible. A separate goal is to demonstrate that standard ideas of logic (model theory) can cast new light on existing problems in combinatorics (graph theory). It is hoped that graph theorists will themselves adopt these methods in such cases. The work of Thomas uncovers previously unsuspected relations between algebraic and set theoretical issues, and should lead to new ``forcing'' tools, which are among the most powerful foundational tools of modern set theory. The general thrust of the proposal is to show how techniques arising naturally in one area can be transported fruitfully to other subject areas.