Winner of the 2015 Prose Award for most sensible arithmetic booklet! within the 5th of his well-known record of 23 difficulties, Hilbert requested if each topological crew which used to be in the community Euclidean was once actually a Lie crew. throughout the paintings of Gleason, Montgomery-Zippin, Yamabe, and others, this query used to be solved affirmatively; extra often, a passable description of the (mesoscopic) constitution of in the community compact teams used to be tested. for that reason, this constitution thought was once used to turn out Gromov's theorem on teams of polynomial development, 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 concept of genuine Lie teams and Lie algebras (emphasising the function of one-parameter teams and the Baker-Campbell-Hausdorff formula), then proposing an explanation of the Gleason-Yamabe constitution theorem for in the neighborhood 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 progress are then given. a lot of proper workouts and different supplementary fabric also are supplied.
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