Hilbert第五问题及相关论题（影印版）

　　Hilbert著名的23个问题的第5个问题为：是否每个局部Euclid拓扑群实际上都是Lie群。通过Gleason、Montgomery-Zippin、 Yamabe等人的工作，这个问题得到了肯定的回答；更一般地，他们建立了局部紧群令人满意的（介观）结构理论。随后，这种结构理论被用来证明Gromov关于多项式增长群的定理，也用在最近Hrushovski、Breuillard、Green和作者关于近似群结构的工作中。 本书所有材料以统一的方式呈现，从实Lie群和Lie代数的分析结构理论（强调单参数群的作用和Baker-Campbell-Hausdorff公式）开始，然后给出局部紧群的Gleason-Yamabe结构定理的证明（强调Gleason度量的作用），由此得到Hilbert第五问题的解答。在回顾了一些模型论基础知识（特别是超积理论）之后，作者给出了Gleason-Yamabe定理在多项式增长群和近似群中的组合应用。本书还提供了大量相关练习和其他补充材料供读者参考。

暂缺《Hilbert第五问题及相关论题（影印版）》作者简介

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