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Large finite structures often display very regular behavior when small, sporadic phenomena are ignored. Examples include Szemeredi’s regularity lemma in graph theory, the classification of highly transitive finite group actions, and the Lang-Weil estimates for algebraic varieties over large finite fields. Model theory proposes an infinite limit object where the asymptotic rules governing the situation can be clearly seen, in terms of certain measure and dimension theories. Notions and tools developed for quite different questions – initially, to study categoricity in uncountable powers – appear to be relevant for analyzing these structures. Most recently, the basic ideas of this approach have proved useful in the study of approximate subgroups, pointing to a new connection to Lie groups.

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