By Terence Tao
Winner of the 2015 Prose Award for top arithmetic publication! within the 5th of his well-known record of 23 difficulties, Hilbert requested if each topological crew which was once in the neighborhood Euclidean was once in reality a Lie staff. during the paintings of Gleason, Montgomery-Zippin, Yamabe, and others, this question was once solved affirmatively; extra normally, a passable description of the (mesoscopic) constitution of in the neighborhood compact teams used to be tested. accordingly, this constitution idea was once used to end up 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 actual Lie teams and Lie algebras (emphasising the position of one-parameter teams and the Baker-Campbell-Hausdorff formula), then proposing an explanation 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 purposes of the Gleason-Yamabe theorem to approximate teams and teams of polynomial development are then given. plenty of correct routines and different supplementary fabric also are supplied.