代数拓扑 | 与未来人工智能息息相关的年轻数学分支
世界数学领域最高奖“菲尔兹奖”得主、国际著名微分几何大师哈佛大学的丘成桐教授与他的博士生高徒、目前任教于纽约州立大学石溪分校的顾险峰教授，将微分几何、代数拓扑、黎曼面理论、偏微分方程与计算机科学相结合，创立了跨领域的新学科——“计算共形几何”。计算共形几何特别适用于提取几何特征、寻找曲面映射、进行几何分类，其方法已经被广泛地应用于各种计算机与人工智能领域，包括人脸识别、电影特效、动漫游戏、数字制造、物联网协议、医疗导航、美容手术、癌症诊断、阿尔茨海默病预防等。
世界图书出版公司最新出版的2卷本《代数拓扑简明教程》是美国芝加哥大学著名教授乔·彼得·梅（Jon Peter May）所著的代数拓扑经典教材，在全世界广为流行。
（▲乔·彼得·梅（Jon Peter May））
乔·彼得·梅教授是代数拓扑与范畴论领域国际闻名的数学家与教育家，他是抽象同伦论的先驱之一，提出了operads以及梅谱序列。
以下是乔·彼得·梅教授讲授代数拓扑导论课的视频：
代数拓扑导论课（上）
代数拓扑导论课（中）
代数拓扑导论课（下）
《代数拓扑简明教程（第1卷）》目录
Introduction
Chapter 1. The fundamental group and some of its applications
Chapter 2. Categorical language and the van Kampen theorem
Chapter 3. Covering spaces
Chapter 4. Graphs
Chapter 5. Compactly generated spaces
Chapter 6. Cofibrations
Chapter 7. Fibrations
Chapter 8. Based cofiber and fiber sequences
Chapter 9. Higher homotopy groups
Chapter 10. CW complexes
Chapter 11. The homotopy excision and suspension theorems
Chapter 12. A little homological algebra
Chapter 13. Axiomatic and cellular homology theory
Chapter 14. Derivations of properties from the axioms
Chapter 15. The Hurewicz and uniqueness theorems
Chapter 16. Singular homology theory
Chapter 17. Some more homological algebra
Chapter 18. Axiomatic and cellular cohomology theory
Chapter 19. Derivations of properties from the axioms
Chapter 20. The Poincaré duality theorem
Chapter 21. The index of manifolds; manifolds with boundary
Chapter 22. Homology, cohomology, and K(π, n)s
Chapter 23. Characteristic classes of vector bundles
Chapter 24. An introduction to K-theory
Chapter 25. An introduction to cobordism
Introduction
PART 1. Preliminaries: Basic Homotopy Theory and Nilpotent Spaces
Chapter 1. Cofibrations and fibrations
Chapter 2. Homotopy colimits and homotopy limits;lim1
Chapter 3. Nilpotent spaces and Postnikov towers
Chapter 4. Detecting nilpotent groups and spaces
PART 2. Localizations of Spaces at Sets of Primes
Chapter 5. Localizations of nilpotent groups and spaces
Chapter 6. Characterizations and properties of localizations
Chapter 7. Fracture theorems for localization: groups
Chapter 8. Fracture theorems for localization: spaces
Chapter 9. Rational H-spaces and fracture theorems
PART 3. Completions of Spaces at Sets of Primes
Chapter 10. Completions of nilpotent groups and spaces
Chapter 11. Characterizations and properties of completions
Chapter 12. Fracture theorems for completion: groups
Chapter 13. Fracture theorems for completion: spaces
PART 4. An Introduction to Model Category Theory
Chapter 14. An introduction to model category theory
Chapter 15. Cofibrantly generated and proper model categories
Chapter 16. Categorical perspectives on model categories
Chapter 17. Model structures on the category of spaces
Chapter 18. Model structures on categories of chain complexes
Chapter 19. Resolution and localization model structures
PART 5. Bialgebras and Hopf Algebras
Chapter 20. Bialgebras and Hopf algebras
Chapter 21. Connected and component Hopf algebras
Chapter 22. Lie algebras and Hopf algebras in characteristic zero
Chapter 23. Restricted Lie algebras and Hopf algebras in characteristic p
Chapter 24. A primer on spectral sequences
把世界介绍给中国
把中国介绍给世界