Hilbert's Fifth Problem and Related Topics
American Mathematical Society (Verlag)
978-1-4704-1564-8 (ISBN)
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.
Terence Tao was the winner of the 2014 Breakthrough Prize in Mathematics. He is the James and Carol Collins Chair of mathematics at UCLA and the youngest person ever to be promoted to full professor at the age of 24. In 2006 Tao became the youngest ever mathematician to win the Fields Medal. His other honours include the George Polya Prize from the Society for Industrial and Applied Mathematics (2010), the Alan T Waterman Award from the National Science Foundation (2008), the SASTRA Ramanujan Prize (2006), the Clay Research Award from the Clay Mathematical Institute (2003), the Bocher Memorial Prize from the American Mathematical Society (2002) and the Salem Prize (2000).
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
| Reihe/Serie | Graduate Studies in Mathematics |
|---|---|
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 766 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 1-4704-1564-X / 147041564X |
| ISBN-13 | 978-1-4704-1564-8 / 9781470415648 |
| Zustand | Neuware |
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