Logic, Group theory, Combinatorics and Ergodic theory

有限Morley秩群在模型论中是作为一类代数群的实质性推广出现的。人们猜测这个范畴中的单群都是代数的，Cherlin正在与一组国际合作者一起研究这个猜想。这涉及到有限单群分类中使用的技巧以及来自黑箱群理论的一些想法。在图论中，通过使用模型论和组合技术的混合，Cherlin和Shelah正在开发技术，以确定对于给定的有限组禁用图，是否存在满足约束的普适图。这里的最终问题是，整个问题是否在算法上是可决定的。托马斯将描述集合论的方法与超刚性相关的技术结合起来，致力于可数Boreleequence关系理论的研究。描述集合论的方法对经典分类问题有相当大的启发，而来自群论的强有力的方法则是对一般理论的启发和发展。谢林和托马斯还在罗格斯大学主持了一个动态访客计划，与谢拉的年度访问相协调。无限群论为研究和利用数学模型或物理系统的对称性提供了一个工具。Cherlin和他的合作者致力于对与行为良好的代数系统相关联的群进行分类，而Simon Thomas则从无限群的动作和一个动作对另一个动作的分析的角度来研究无限群。来自动力系统理论的思想在这里扮演着特别重要的角色。图一般是网络的数学抽象，所考虑的问题涉及对以禁止的固定模式集为特征的图类的分析，最好是通过通用(可计算的)算法。

Groups of finite Morley rank arise in model theory as a substantialgeneralization of the class of algebraic groups. It has been conjecturedthat the simple groups in this category are all algebraic, and Cherlin isworking on this conjecture with a team of international collaborators.This involves techniques used in the classification of the finite simplegroups as well as some ideas from black box group theory. In graph theory,using a mix of model theoretic and combinatorial techniques, Cherlin andShelah are developing techniques to determine, for a given finite set offorbidden graphs, whether there is a universal graph meeting theconstraints. The ultimate question here is whether the entire problem isalgorithmically decidable. Thomas works on the theory of countable Borelequivalence relations, combining the methods of descriptive set theory withtechniques related to superrigidity. The methods of descriptive set theorycast considerable light on classical classification problems, andconversely powerful methods coming from group theory illuminate and advancethe general theory. Cherlin and Thomas also host a dynamic visitor programat Rutgers, coordinated with annual visits by Shelah.Infinite group theory provides a tool for studying and exploiting thesymmetries of a mathematical model or a physical system. Cherlin and hiscollaborators are aiming at the classification of the groups associatedwith well-behaved algebraic systems, while Simon Thomas approaches thestudy of infinite groups from the point of view of their actions and theanalysis of one action in terms of another. A particularly strong role isplayed here by ideas coming from the theory of dynamical systems. Graphsare the mathematical abstraction of networks in general, and the problemsunder consideration relate to the analysis, preferably by a general(computable) algorithm, of classes of graphs characterized by forbidding afixed set of patterns.