Simplicial Homology and GLMY-Homology
本课程从单纯复形的相关概念开始，介绍单纯复形的单纯同调，持续单纯同调和Combinatorial Laplacian；进而介绍了有向图的GLMY同调，超图的嵌入同调以及相应的持续同调和算法、Hodge Laplacian，最后介绍twisted 同调。
Lecturer
Date
19th September ~ 14th December, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 13:30 - 15:05 | A3-4-101 | ZOOM 04 | 482 240 1589 | BIMSA |
Prerequisite
Linear Algebra
Syllabus
1. Simplicial complex.
2. Homology groups of a simplicial complex and topological invariance of the homology groups, persistent homology and Combinatorial Laplacian.
3. The GLMY homology of digraphs, persistent homology and Hodge Laplacian.
4. The embedded homology of hypergraphs.
5. Twisted Homology.
2. Homology groups of a simplicial complex and topological invariance of the homology groups, persistent homology and Combinatorial Laplacian.
3. The GLMY homology of digraphs, persistent homology and Hodge Laplacian.
4. The embedded homology of hypergraphs.
5. Twisted Homology.
Reference
1. Simplicial Objects and Homotopy Groups, Lecture notes of Jie Wu.
2. Elements of Algebraic Topology，James R. Munkres.
3. A. Grigor’yan, Y. Lin, Y. Muranov and S.-T. Yau, Homologies of path complexes and digraphs, preprint, 2012. arXiv:1207.2834v4(2013).
4. A. Grigor’yan, Y. Lin, Y. Muranov and S.-T. Yau, Cohomology of digraphs and (undirected) graphs, Asian J. Math. 19 (2015),887–932.
5. A. Grigor’yan, Y. Muranov and S.-T. Yau, Graphs associated with simplicial complexes, Homology, Homotopy Appl., 16:1 (2014), pp. 295–311.
6. Bressan, Stephane; Li, Jingyan; Ren, Shiquan; Wu, Jie The embedded homology of hypergraphs and applications. Asian J. Math. 23 (2019), no. 3, 479–500.
2. Elements of Algebraic Topology，James R. Munkres.
3. A. Grigor’yan, Y. Lin, Y. Muranov and S.-T. Yau, Homologies of path complexes and digraphs, preprint, 2012. arXiv:1207.2834v4(2013).
4. A. Grigor’yan, Y. Lin, Y. Muranov and S.-T. Yau, Cohomology of digraphs and (undirected) graphs, Asian J. Math. 19 (2015),887–932.
5. A. Grigor’yan, Y. Muranov and S.-T. Yau, Graphs associated with simplicial complexes, Homology, Homotopy Appl., 16:1 (2014), pp. 295–311.
6. Bressan, Stephane; Li, Jingyan; Ren, Shiquan; Wu, Jie The embedded homology of hypergraphs and applications. Asian J. Math. 23 (2019), no. 3, 479–500.
Audience
Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
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.