Del 0 - Graduate Studies in Mathematics

Hilbert's Fifth Problem and Related Topics

Inbunden, Engelska, 2014

1 469 kr

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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.

Produktinformation

Utgivningsdatum2014-07-30
Mått178 x 254 x undefined mm
Vikt766 g
FormatInbunden
SpråkEngelska
SerieGraduate Studies in Mathematics
Antal sidor338
FörlagAmerican Mathematical Society
ISBN9781470415648

Tillhör följande kategorier

Algebra inom Naturvetenskap och teknik
Matematik inom Naturvetenskap och teknik

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