Logic, Group theory, Combinatorics and Ergodic theory

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

Groups of finite Morley rank arise in model theory as a substantial generalization of the class of algebraic groups. It has been conjectured that the simple groups in this category are all algebraic, and Cherlin is working on this conjecture with a team of international collaborators. This involves techniques used in the classification of the finite simple groups as well as some ideas from black box group theory. In graph theory, using a mix of model theoretic and combinatorial techniques, Cherlin and Shelah are developing techniques to determine, for a given finite set of forbidden graphs, whether there is a universal graph meeting the constraints. The ultimate question here is whether the entire problem is algorithmically decidable. Thomas works on the theory of countable Borel equivalence relations, combining the methods of descriptive set theory with techniques related to superrigidity. The methods of descriptive set theory cast considerable light on classical classification problems, and conversely powerful methods coming from group theory illuminate and advance the general theory. Cherlin and Thomas also host a dynamic visitor program at Rutgers, coordinated with annual visits by Shelah. Infinite group theory provides a tool for studying and exploiting the symmetries of a mathematical model or a physical system. Cherlin and his collaborators are aiming at the classification of the groups associated with well-behaved algebraic systems, while Simon Thomas approaches the study of infinite groups from the point of view of their actions and the analysis of one action in terms of another. A particularly strong role is played here by ideas coming from the theory of dynamical systems. Graphs are the mathematical abstraction of networks in general, and the problems under consideration relate to the analysis, preferably by a general (computable) algorithm, of classes of graphs characterized by forbidding a fixed set of patterns.