Winner of the 2015 Prose Award for top arithmetic e-book! within the 5th of his recognized record of 23 difficulties, Hilbert requested if each topological crew which was once in the community Euclidean was once in reality a Lie crew. during the paintings of Gleason, Montgomery-Zippin, Yamabe, and others, this query was once solved affirmatively; extra in general, a passable description of the (mesoscopic) constitution of in the community compact teams used to be validated. hence, this constitution thought was once used to turn out Gromov's theorem on teams of polynomial progress, and extra lately within the paintings of Hrushovski, Breuillard, eco-friendly, and the writer at the constitution of approximate teams. during this graduate textual content, all of this fabric is gifted in a unified demeanour, beginning with the analytic structural idea of genuine Lie teams and Lie algebras (emphasising the function of one-parameter teams and the Baker-Campbell-Hausdorff formula), then proposing an evidence of the Gleason-Yamabe constitution theorem for in the community compact teams (emphasising the function of Gleason metrics), from which the answer to Hilbert's 5th challenge follows as a corollary. After reviewing a few model-theoretic preliminaries (most particularly the speculation of ultraproducts), the combinatorial functions of the Gleason-Yamabe theorem to approximate teams and teams of polynomial development are then given. a good number of suitable workouts and different supplementary fabric also are supplied.

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