Hilbert's fifth problem and related topics


Hilbert's fifth problem and related topics

Terence Tao

(Graduate studies in mathematics, v. 153)

American Mathematical Society, c2014

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Includes bibliographical references (p. 329-333) and index



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, Montgomery-Zippin, 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 one-parameter groups and the Baker-Campbell-Hausdorff formula), then presenting a proof of the Gleason-Yamabe 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 model-theoretic preliminaries (most notably the theory of ultraproducts), the combinatorial applications of the Gleason-Yamabe 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.


Hilbert's fifth problem: Introduction Lie groups, Lie algebras, and the Baker-Campbell-Hausdorff formula Building Lie structure from representations and metrics Haar measure, the Peter-Weyl theorem, and compact or abelian groups Building metrics on groups, and the Gleason-Yamabe theorem The structure of locally compact groups Ultraproducts as a bridge between hard analysis and soft analysis Models of ultra approximate groups The microscopic structure of approximate groups Applications of the structural theory of approximate groups Related articles: The Jordan-Schur theorem Nilpotent groups and nilprogressions Ado's theorem Associativity of the Baker-Campbell-Hausdorff-Dynkin law Local groups Central extensions of Lie groups, and cocycle averaging The Hilbert-Smith conjecture The Peter-Weyl theorem and nonabelian Fourier analysis Polynomial bounds via nonstandard analysis Loeb measure and the triangle removal lemma Two notes on Lie groups Bibliography Index

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