Preface
Notation
Acknowledgments

Part 1． Hilbert's Fifth Problem
Chapter 1． Introduction
§1．1． Hilbert's fifth problem
§1．2． Approximate groups
§1．3． Gromov's theorem
Chapter 2． Lie groups， Lie algebras， and the Baker-Campbell-Hausdorff formula
§2．1． Local groups
§2．2． Some differential geometry
§2．3． The Lie algebra of a Lie group
§2．4． The exponential map
§2．5． The Baker-Campbell-Hausdorff formula
Chapter 3． Building Lie structure from representations and metrics
§3．1． The theorems of Cartan and von Neumann
§3．2． Locally compact vector spaces
§3．3． From Gleason metrics to Lie groups
Chapter 4． Haar measure， the Peter-Weyl theorem． and compact or abelian groups
§4．1． Haar measure
§4．2． The Peter-Weyl theorem
§4．3． The structure of locally compact abelian groups
Chapter 5． Building metrics on groups， and the Gleason-Yamabe theorem
§5．1． Warmup： the Birkhoff-Kakutani theorem
§5．2． Obtaining the commutator estimate via convolution
§5．3． Building metrics on NSS groups
§5．4． NSS from subgroup trapping
§5．5． The subgroup trapping property
§5．6． The local group case
Chapter 6．The structure of locally compact groups
§6．1． Van Dantzig's theorem
§6．2． The invariance of domain theorem
§6．3． Hilbert's fifth problem
§6．4． Transitive actions
Chapter 7． Ultraproducts as a bridge between hard analysis and soft analysis
§7．1． Ultrafilters
§7．2． Ultrapowers and ultralimits
§7．3． Nonstandard finite sets and nonstandard finite sums
§7．4． Asymptotic notation
§7．5． Ultra approximate groups
……

Part 2． Related Articles

Bibliography
Index