Winner of the 2015 Prose Award for most sensible arithmetic e-book! within the 5th of his recognized checklist of 23 difficulties, Hilbert requested if each topological crew which was once in the neighborhood Euclidean used to be in reality a Lie team. during the paintings of Gleason, Montgomery-Zippin, Yamabe, and others, this question was once solved affirmatively; extra commonly, a passable description of the (mesoscopic) constitution of in the neighborhood compact teams used to be verified. as a result, this constitution thought was once used to end up 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 conception 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 idea of ultraproducts), the combinatorial functions of the Gleason-Yamabe theorem to approximate teams and teams of polynomial progress are then given. various appropriate workouts and different supplementary fabric also are supplied.

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