Simplicial homotopy theory I
Simplicial approach is an important tool for establishing potential connections between algebraic topology and other areas of mathematics. In this course, we will give an introduction to simplicial homotopy theory. After introducing some fundamental notions in the theory of simplicial sets, we will provide the proofs of the fundamental theorems in algebraic topology in simplicial setting, such as Poincare Theorem, Hurewicz theorem, Whitehead theorem, Homotopy Addition Theorem. The course will also discuss simplicia fibration theory, Moore-Postnikov system and simplicial group theory.
This course is designed for helping the students and junior researchers who are interested in GLMY theory and algebraic topology of digraphs, where the students and junior researchers can learn the ideas of simplicial homotopy theory with aiming to develop the homotopy theory of digraphs. The course is also suitable for the graduate students and junior researchers who are working on simplical approaches to data and deep learning with being interested in upgrading their topological skills.
The lecture notes will be available before the program starts. The prerequisite is Allen Hatcher’s book: Algebraic Topology, Cambridge University Press, 2002. However most materials in the lecture notes are self-contained and so the smart students can still follow the lecture notes without much background on algebraic topology.
This course is designed for helping the students and junior researchers who are interested in GLMY theory and algebraic topology of digraphs, where the students and junior researchers can learn the ideas of simplicial homotopy theory with aiming to develop the homotopy theory of digraphs. The course is also suitable for the graduate students and junior researchers who are working on simplical approaches to data and deep learning with being interested in upgrading their topological skills.
The lecture notes will be available before the program starts. The prerequisite is Allen Hatcher’s book: Algebraic Topology, Cambridge University Press, 2002. However most materials in the lecture notes are self-contained and so the smart students can still follow the lecture notes without much background on algebraic topology.
讲师
日期
2024年03月21日 至 06月20日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周四 | 17:10 - 21:00 | A3-4-301 | ZOOM 01 | 928 682 9093 | BIMSA |
修课要求
Algebraic Topology
课程大纲
simplicial set and simplicial objects, nerve complexes of categories, geometric realization, homotopy and fibrant simplicial sets, homotopy groups, Homotopy Addition Theorem, minimal simplicial sets, Whitehead theorem, simplicial fibration, Moore paths and Moore loops, Moore-Postnikov system, simplicial H-sets and simplicial groups, Moore chain complex and Moore homotopy groups, Hurewicz Theorem, Poincare Theorem, Kan construction, Milnor construction.
参考资料
1.Wu, Jie, Simplicial objects and homotopy groups.(English summary)Braids, 31–181. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 19 World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010 ISBN:978-981-4291-40-8
2.Curtis, Edward B. Simplicial homotopy theory.Advances in Math.6(1971), 107–209.
2.Curtis, Edward B. Simplicial homotopy theory.Advances in Math.6(1971), 107–209.
视频公开
公开
笔记公开
公开
语言
中文
讲师介绍
BIMSA研究员，美国罗切斯特大学数学系博士，加州大学伯克利分校数学研究所博士后，前新加坡国立大学数学系终身教授，2021年12月入职北京雁栖湖应用数学研究院（BIMSA）。研究方向为代数拓扑与应用拓扑，在代数拓扑理论研究上的主要成就是建立了同伦群与辫子群理论的基础性关系，以及回路空间同伦论与置换群模表示论的基础性关系；在应用拓扑方面，将代数拓扑理论应用在大数据等领域并取得了一系列成果。在Journal of American Mathematical Society, Advances in Mathematics等数学顶尖期刊发表学术论文90余篇。2007年获得新加坡国家科学奖。2014年获得国家自然科学基金海外联合基金（杰青B）的资助。