《Hilbert第五問題及相關論題（影印版）》是高等教育出版社出版圖書。
基本介紹
- 中文名：Hilbert第五問題及相關論題（影印版）
- 作者：孫淼
- 出版時間：2021年2月1日
- 出版社：高等教育出版社
- 頁數：338 頁
- ISBN：9787040556292
- 開本：16 開
- 裝幀：精裝
內容簡介,目錄,
內容簡介
《Hilbert第五問題及相關論題（影印版）》所有材料以統一的方式呈現，從實Lie群和Lie代數的分析結構理論（強調單參數群的作用和Baker-Campbell-Hausdorff公式）開始，然後給出局部緊群的Gleason-Yamabe結構定理的證明（強調Gleason度量的作用），由此得到Hilbert第五問題的解答。在回顧了一些模型論基礎知識（特別是超積理論）之後，作者給出了Gleason-Yamabe定理在多項式增長群和近似群中的組合套用。
《Hilbert第五問題及相關論題（影印版）》還提供了大量相關練習和其他補充材料供讀者參考。
目錄
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 Birkhof-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
Chapter 8. Models of ultra approximate groups
8.1 Ultralimits of metric spaces (Optional)
8.2 Sanders-Croot-Sisask theory
8.3 Locally compact models of ultra approximate groups
8.4 Lie models of ultra approximate groups
Chapter 9. The microscopic structure of approximate groups
9.1 Gleason's lemma
9.2 A cheap version of the structure theorem
9.3 Local groups
Chapter 10. Applications of the structural theory of approximate groups
10.1 Sets of bounded doubling
10.2 Polynomial growth
10.3 Fundamental groups of compact manifolds (optional)
10.4 A Margulis-type lemma
Part 2. Related Articles
Chapter 11. The Jordan-Schur theorem
11.1 Proofs
Chapter 12. Nilpotent groups and nilprogressions
12.1 Some elementary group theory
12.2 Nilprogressions
Chapter 13. Ado's theorem
13.1 The nilpotent case
13.2 The solvable case
13.3 The general case
Chapter 14. Associativity of the Baker-Campbell-Hausdorff-Dynkin law
Chapter 15. Local groups
15.1 Lie's third theorem
15.2 Globalising a local group
15.3 A nonglobalisable group
Chapter 16. Central extensions of Lie groups, and cocycle averaging
16.1 A little group cohomology
16.2 Proof of theorem
Chapter 17. The Hilbert-Smith conjecture
17.1 Periodic actions of prime order
17.2 Reduction to the p-adic case
Chapter 18. The Peter-Weyl theorem and nonabelian Fourier analysis
18.1 Proof of the Peter-Weyl theorem
18.2 Nonabelian Fourier analysis
Chapter 19. Polynomial bounds via nonstandard analysis
Chapter 20. Loeb measure and the triangle removal lemma
20.1 Loeb measure
20.2 The triangle removal lemma
Chapter 21. Two notes on Lie groups
