
Preface  Preview Material  Table of Contents  Index  Supplementary Material 
Graduate Studies in Mathematics 2014; 338 pp; hardcover Volume: 153 ISBN10: 147041564X ISBN13: 9781470415648 List Price: US$69 Member Price: US$55.20 Order Code: GSM/153 See also: Compactness and Contradiction  Terence Tao Mathematics of Probability  Daniel W Stroock Topics in Random Matrix Theory  Terence Tao  Winner of the 2015 Prose Award for Best Mathematics Book! In the fifth of his famous list of 23 problems, Hilbert asked if every topological group which was locally Euclidean was in fact a Lie group. Through the work of Gleason, MontgomeryZippin, Yamabe, and others, this question was solved affirmatively; more generally, a satisfactory description of the (mesoscopic) structure of locally compact groups was established. Subsequently, this structure theory was used to prove Gromov's theorem on groups of polynomial growth, and more recently in the work of Hrushovski, Breuillard, Green, and the author on the structure of approximate groups. In this graduate text, all of this material is presented in a unified manner, starting with the analytic structural theory of real Lie groups and Lie algebras (emphasising the role of oneparameter groups and the BakerCampbellHausdorff formula), then presenting a proof of the GleasonYamabe structure theorem for locally compact groups (emphasising the role of Gleason metrics), from which the solution to Hilbert's fifth problem follows as a corollary. After reviewing some modeltheoretic preliminaries (most notably the theory of ultraproducts), the combinatorial applications of the GleasonYamabe theorem to approximate groups and groups of polynomial growth are then given. A large number of relevant exercises and other supplementary material are also provided. Readership Graduate students and research mathematicians interested in lie groups, topological groups, geometric group theory, and approximate groups. 


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