Interactions of Logic with Group Theory and Combinatorics

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Cherlin and Thomas will pursue interactions of the techniques of logic with problems in algebra and combinatorics. Cherlin will work with groups of finite Morley rank using methods modeled heavily on finite group theory, aiming particularly at an approach to the odd characteristic case compatible with the existence of bad fields, and on problems in graph theory susceptible to model theoretic analysis (universal graphs and problems of wqo). Thomas will work on Borel equivalence relations, particularly with those associated with natural classification problems in algebra, which may well provide the examples needed to settle some problems presently open in full generality, as well as providing information on the relative difficulty (according to a very robust system of measurement) of the algebraic problems, some very classical and open, for what can now seen to be essential reasons. Thomas will also pursue his work on the automorphism tower problem, using set theoretic techniques.Mathematical logic provides tools of great generality which can be applied to various areas of mathematics. In combinatorial contexts the model theoretic point of view provides methods that can be used to handle specific problems very uniformly, rather than on the case by case basis sometimes encountered in the literature. Descriptive set theory provides methods for analyzing the relative difficulty of both solved and unsolved problems in algebra, and in particular provides concrete information as to how detailed an answer one may usefully seek in a classification problem, making it possible to distinguish dead ends from fruitful lines of inquiry on an a priori basis.