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.
The project involved research on the interface between group theory and mathematical logic by Gregory Cherlin and Simon Thomas at Rutgers University, and their collaborators and students, and included support for an international research visitor program led by the distinguished logician Saharon Shelah, as well as postdoctoral support for Adrien Deloro, and support for several graduate students. Intellectual merit 1. Using methods applied previously to finite groups, a large class of infinite groups of finite dimension (even type) were classified, and this was shown to have broad implications for this class of groups as a whole. 2. Deep connections among geometrical group theory, representation theory, descriptive set theory, and major conjectures in computability theory were uncovered, as well as connections between finite group theory and the theory of large cardinals. 3. In combinatorics, a new class of highly symmetric graphs was discovered, using the methods of model theory. With these examples in hand we may possibly have identified all of the maximally symmetric graphs, but that question remains open. Broader Impacts: The study of symmetry via group theory controls a wide range of mathematical and physical phenomena. Results under the present grant demonstrate that methods of mathematical logic can be effectively combined with a variety of techniques to shed light on group theoretic problems and other mathematical problems arising in the systematic classification of fundamental mathematical structures. This involved finding new ways to apply methods originating in dynamics, descriptive set theory, or the classical theory of finite groups. Other results under the grant deal with the use of mathematical logic to analyze certain decision problems in combinatorics.
