Descriptive Set Theory, Geometric Group Theory, and Combinatorial Model Theory

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The research proposed involves the development and application of techniques of mathematical logic to a broad range of mathematical problems. First, the classical subject of descriptive set theory has been found, in combination with techniques of ergodic theory and superrigidity theory, to have extensive applications to the analysis of problems of classification in algebra and elsewhere. In particular the subject of Borel equivalence relations will be applied to problems in geometric group theory, revolving around the central notion of quasi-isometry. The now classical theory of Borel reductions will be sharpened by considering continuous reductions, which correspond in a number of cases to "natural" or "invariant" constructions. Second, the techniques of model theory will be applied to settle certain problems in combinatorics, notably the problem of the existence of universal graphs subject to one forbidden subgraph, and the classification of the metrically homogeneous graphs, first proposed in 1998 by Cameron and subsequently seen to have connections with topological dyamics in work of Kechris, Pestov, and Todorcevic. In both cases, new examples of homogeneous structures will be identified (in an appropriate refinement of the category of graphs), and then the relation to topological dynamics will involve the further study of structural Ramsey theory in the associated classes of finite structures. The proposal includes a visitors' program based on extended visits by Saharon Shelah, one of the world's preeminent mathematical logicians, and a variety of his collaborators, who will collaborate on aspects of the proposed research, both on the side of descriptive set theory and in the applications to combinatorics.The thrust of the research is interdisciplinary. The notion of classification is fundamental to many branches of mathematical research, and techniques of mathematical logic, in combination with more specialized tools from other branches of mathematics, some the result of very recent breakthroughs, make it possible to analyze the scope and limitations of this method, revealing a wealth of structure. The techniques of mathematical logic also provide a very broad perspective on problems of combinatorics, raising issues of computability and providing new and systematic techniques for the resolution of concrete problems regarding the complexity of classes of finite structures, and the problem of the existence of a single infinite limit of such a class with a tractable structure, which provides a way for distinguishing "chaotic" from "highly structured" classes, and the identification of new cases which can be analyzed structurally.