Logic, Group Theory, Combinatorics and Ergodic Theory

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The notion of classification is fundamental to many branches of mathematical research: one looks for the deep underlying structure common to many widely varying topics, with particular attention to underlying symmetries. Techniques of modern mathematical logic, in combination with more specialized tools from other branches of mathematics, make it possible to analyze the scope and limitations of such classifications and reveal a wealth of additional structure. The techniques of mathematical logic also provide a broad perspective on combinatorial problems. Here one must first separate the "chaotic" problems which cannot be neatly categorized and identify the subclass of more tractable problems, looking ultimately for an algorithm to carry out this part of the analysis at a completely general level. This research project investigates such classification problems. In particular, issues of computability are an essential component of the program. The research is interdisciplinary, establishing new relationships between mathematical logic and classical mathematics.This award supports a group research project. Using methods of topological dynamics, combinatorial group theory, model theory, and algebra, the two investigators will investigate a variety of classification problems in analysis, algebra, and combinatorics. Simon Thomas will combine these methods with modern descriptive set theory to analyze classification problems in algebra, particularly those associated with representation theory and the subject of "invariant random subgroups" in the sense of Vershik, while Gregory Cherlin will pursue joint work with visitor Saharon Shelah on the existence of universal graphs with forbidden subgraphs and the associated structural Ramsey theory, as well as the classification of all metrically homogeneous graphs and analogous problems for finite permutation groups. A further component of the project is an active visitor's program under the direction of Shelah when he is in residence at Rutgers.