Logic, Group theory, Combinatorics and Ergodic theory

有限Morley秩群作为代数群的实质推广出现在模型理论中。据推测，这一类中的单群都是代数的，切尔林正在与一个国际合作者团队研究这一猜想。这涉及到有限单群的分类技术以及黑箱群论中的一些思想。在图论中，Cherlin和shelah使用模型理论和组合技术的混合，开发了一种技术来确定给定的有限禁图集是否存在满足约束的通用图。这里的终极问题是，整个问题是否可以由算法决定。托马斯致力于可数波勒等价关系的理论，将描述集合论的方法与超刚性相关的技术相结合。描述集理论的方法对经典的分类问题给出了相当多的启示，反过来，来自群论的强大方法照亮和推进了一般理论。切尔林和托马斯还在罗格斯大学主持了一个动态访客项目，与希拉每年的访问进行协调。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和他的合作者的目标是对与表现良好的代数系统相关的群进行分类，而Simon Thomas从它们的行为的角度来研究无限群，并根据另一个行为来分析一个行为。来自动力系统理论的思想在这里发挥了特别重要的作用。图通常是网络的数学抽象，所考虑的问题与分析有关，最好是通过一般（可计算）算法，分析以禁止固定模式集为特征的图类。

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.