代数拓扑在数据分析中的应用
本课程主要介绍单纯同调理论的基础知识、GLMY同调理论，超图的嵌入同调理论及奇异同调理论，然后结合实际案例，介绍单纯同调和GLMY同调在数据分析中的应用。
Lecturer
Date
10th September ~ 10th December, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 10:40 - 12:15 | A3-4-301 | ZOOM 08 | 787 662 9899 | BIMSA |
Prerequisite
近世代数
Syllabus
1. Homology groups of a simplicial complex.
2. Topological invariance of the homology groups.
3. Relative homology and the Eilenberg-Steenrod Axioms.
4. The GLMY homology of digraphs.
5. The embedded homology of hypergraphs.
6. Persistent Homology.
7. Singular Homology Theory
8. Applications of Homology Theories.
2. Topological invariance of the homology groups.
3. Relative homology and the Eilenberg-Steenrod Axioms.
4. The GLMY homology of digraphs.
5. The embedded homology of hypergraphs.
6. Persistent Homology.
7. Singular Homology Theory
8. Applications of Homology Theories.
Reference
1. James R. Munkres，Elements of Algebraic Topology.
2. A. Grigor’yan, Y. Lin, Y. Muranov and S.-T. Yau, Homologies of path complexes and digraphs, preprint, 2012. arXiv:1207.2834v4(2013).
3. Jingyan Li，Y. Muranov，Jie Wu and S.-T. Yau, On singular homology theories of digraphs and quivers, Journal of Combinatorics, Priprint.
4. Bressan, Stephane; Li, Jingyan; Ren, Shiquan; Wu, Jie The embedded homology of hypergraphs and applications. Asian J. Math. 23 (2019), no. 3, 479–500.
5. Jingyan Li，Y. Muranov，Jie Wu and S.-T. Yau, On homology of circuits, BJPAM, Priprint.
2. A. Grigor’yan, Y. Lin, Y. Muranov and S.-T. Yau, Homologies of path complexes and digraphs, preprint, 2012. arXiv:1207.2834v4(2013).
3. Jingyan Li，Y. Muranov，Jie Wu and S.-T. Yau, On singular homology theories of digraphs and quivers, Journal of Combinatorics, Priprint.
4. Bressan, Stephane; Li, Jingyan; Ren, Shiquan; Wu, Jie The embedded homology of hypergraphs and applications. Asian J. Math. 23 (2019), no. 3, 479–500.
5. Jingyan Li，Y. Muranov，Jie Wu and S.-T. Yau, On homology of circuits, BJPAM, Priprint.
Audience
Undergraduate
, Graduate
Video Public
No
Notes Public
No
Language
Chinese
Lecturer Intro
Assistant Reserch fellow Jingyan Li received a PhD degree from the Department of Mathematics of Hebei Normal University in 2007. Before joining BIMSA in September 2021, she has taught in the Department of Mathematics and Physics of Shijiazhuang Railway University and the School of Mathematical Sciences of Hebei Normal University as an associate professor. Her research interests include topology data analysis and simplicial homology and homotopy.