Logic, Group theory, Combinatorics and Ergodic theory

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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.