Table of Contents

• 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

About the Author

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).