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Hilbert’s Fifth Problem and Related Topics

Terence Tao University of California, Los Angeles, CA
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Hardcover ISBN: 978-1-4704-1564-8
Product Code: GSM/153
List Price: $74.00 MAA Member Price:$66.60
AMS Member Price: $59.20 Electronic ISBN: 978-1-4704-1856-4 Product Code: GSM/153.E List Price:$69.00
MAA Member Price: $62.10 AMS Member Price:$55.20
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AMS Member Price: $88.80 Click above image for expanded view Hilbert’s Fifth Problem and Related Topics Terence Tao University of California, Los Angeles, CA Available Formats:  Hardcover ISBN: 978-1-4704-1564-8 Product Code: GSM/153  List Price:$74.00 MAA Member Price: $66.60 AMS Member Price:$59.20
 Electronic ISBN: 978-1-4704-1856-4 Product Code: GSM/153.E
 List Price: $69.00 MAA Member Price:$62.10 AMS Member Price: $55.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$111.00 MAA Member Price: $99.90 AMS Member Price:$88.80
• Book Details

Volume: 1532014; 338 pp
MSC: Primary 22; 11; 20;
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.

Graduate students and research mathematicians interested in lie groups, topological groups, geometric group theory, and approximate groups.

• Part 1. Hilbert’s Fifth Problem
• Chapter 1. Introduction
• Chapter 2. Lie groups, Lie algebras, and the Baker-Campbell-Hausdorff formula
• Chapter 3. Building Lie structure from representations and metrics
• Chapter 4. Haar measure, the Peter-Weyl theorem, and compact or abelian groups
• Chapter 5. Building metrics on groups, and the Gleason-Yamabe theorem
• Chapter 6. The structure of locally compact groups
• Chapter 7. Ultraproducts as a bridge between hard analysis and soft analysis
• Chapter 8. Models of ultra approximate groups
• Chapter 9. The microscopic structure of approximate groups
• Chapter 10. Applications of the structural theory of approximate groups
• Part 2. Related articles
• Chapter 11. The Jordan-Schur theorem
• Chapter 12. Nilpotent groups and nilprogressions
• Chapter 14. Associativity of the Baker-Campbell-Hausdorff-Dynkin law
• Chapter 15. Local groups
• Chapter 16. Central extensions of Lie groups, and cocycle averaging
• Chapter 17. The Hilbert-Smith conjecture
• Chapter 18. The Peter-Weyl theorem and nonabelian Fourier analysis
• Chapter 19. Polynomial bounds via nonstandard analysis
• Chapter 20. Loeb measure and the triangle removal lemma
• Chapter 21. Two notes on Lie groups

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Volume: 1532014; 338 pp
MSC: Primary 22; 11; 20;
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.

Graduate students and research mathematicians interested in lie groups, topological groups, geometric group theory, and approximate groups.

• Part 1. Hilbert’s Fifth Problem
• Chapter 1. Introduction
• Chapter 2. Lie groups, Lie algebras, and the Baker-Campbell-Hausdorff formula
• Chapter 3. Building Lie structure from representations and metrics
• Chapter 4. Haar measure, the Peter-Weyl theorem, and compact or abelian groups
• Chapter 5. Building metrics on groups, and the Gleason-Yamabe theorem
• Chapter 6. The structure of locally compact groups
• Chapter 7. Ultraproducts as a bridge between hard analysis and soft analysis
• Chapter 8. Models of ultra approximate groups
• Chapter 9. The microscopic structure of approximate groups
• Chapter 10. Applications of the structural theory of approximate groups
• Part 2. Related articles
• Chapter 11. The Jordan-Schur theorem
• Chapter 12. Nilpotent groups and nilprogressions