持续同调与GLMY同调
丘成桐先生及其合作者于2012年引入的有向图的新型拓扑理论（GLMY理论），该理论以更加抽象的图结构数据为研究目标，发展出了GLMY同调、超图同调，该同调的相关理论研究已经非常丰富，且其应用已经在诸多领域取得很好的成效。有向图的同伦论已经在丘先生及其合作者在的理论中被引入，目前还在迅速发展之中。该课程主要以论文研读的方式梳理GLMY同调与同伦，以帮助学生们熟悉和掌握此理论。
日期
2023年02月27日 至 06月02日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周一 | 13:30 - 16:55 | A3-2a-201 | ZOOM 02 | 518 868 7656 | BIMSA |
| 周五 | 10:40 - 15:05 | A3-2a-201 | ZOOM 01 | 928 682 9093 | BIMSA |
修课要求
代数拓扑基础理论
课程大纲
第一到六周：持续同调、持续图与持续模及其在生物学的应用。
第七-十一周：侧重介绍GLMY同调理论及由GLMY同调发展起来的超图同调和Quiver的道路同调等。第十二周进展报告。
第七-十一周：侧重介绍GLMY同调理论及由GLMY同调发展起来的超图同调和Quiver的道路同调等。第十二周进展报告。
参考资料
Carlsson, Gunnar. "Topology and data." Bulletin of the American Mathematical Society 46.2 (2009): 255-308.
Oudot, Steve Y. Persistence theory: from quiver representations to data analysis. Vol. 209. American Mathematical Soc., 2017.
Edelsbrunner, Herbert, and Dmitriy Morozov. "Persistent homology." Handbook of Discrete and Computational Geometry. Chapman and Hall/CRC, 2017. 637-661.
Qaiser, Talha, et al. "Persistent homology for fast tumor segmentation in whole slide histology images." Procedia Computer Science 90 (2016): 119-124.
Petri, Giovanni, et al. "Homological scaffolds of brain functional networks." Journal of The Royal Society Interface 11.101 (2014): 20140873.
Zhu, Xiaojin. "Persistent homology: An introduction and a new text representation for natural language processing." IJCAI. 2013.
A. Grigor’yan, Y. Lin, Y. Muranov and S.-T. Yau, Homologies of path complexes and digraphs, preprint, 2012. arXiv:1207.2834v4.
Grigor'yan, A., Muranov, Yu., Yau, S.-T., "Homologies of digraphs and Künneth formulas", Comm. Anal. Geom., 25 (2017) 969-1018.
Grigor'yan, A., Lin Yong, Muranov, Yu., Yau, S.-T., "Path complexes and their homologies", J. Math. Sciences, 248 (2020) 564-599.
A. Grigor’yan, Y. Muranov and S.-T. Yau, Graphs associated with simplicial complexes, Homology Homotopy Appl. 16 (2014), 295–311.
A. Grigor’yan, R. Jimenez, Y. Muranov and S.-T. Yau, On the path homology theory of digraphs and Eilenberg–Steenrod axioms, Homology Homotopy Appl. 20 (2018), no. 2, 179–205.
A. Grigor’yan, R. Jimenez, Y. Muranov, and S.-T. Yau, Homology of path complexes and hypergraphs, Topology Appl.267(2019), no. 2, 106877.
A. Grigor’yan, Y. Lin, Y. Muranov and S.-T. Yau, Cohomology of digraphs and (undirected) graphs, Asian J. Math. 19 (2015),887–932.
Grigor'yan, Alexander; Muranov, Yuri; Vershinin, Vladimir; Yau, Shing-Tung； Path homology theory of multigraphs and quivers. Forum Math. 30 (2018), no. 5, 1319–1337.
Sergei Ivanov and Fedor Pavutnitskiy， SIMPLICIAL APPROACH TO PATH HOMOLOGY OF QUIVERS, SUBSETS OF GROUPS AND SUBMODULES OF ALGEBRAS，preprint.
Chen, Beifang, Yau, Shing-Tung, and Yeh, Yeong-Nan, Graph homotopy and Graham homotopy, Discrete Math., 241 (2001) 153-170.
A. Grigor’yan, Y. Lin, Y. Muranov and S.-T. Yau, Homotopy theory for digraphs, Pure Appl. Math. Q. 10 (2014), 619–674.
Oudot, Steve Y. Persistence theory: from quiver representations to data analysis. Vol. 209. American Mathematical Soc., 2017.
Edelsbrunner, Herbert, and Dmitriy Morozov. "Persistent homology." Handbook of Discrete and Computational Geometry. Chapman and Hall/CRC, 2017. 637-661.
Qaiser, Talha, et al. "Persistent homology for fast tumor segmentation in whole slide histology images." Procedia Computer Science 90 (2016): 119-124.
Petri, Giovanni, et al. "Homological scaffolds of brain functional networks." Journal of The Royal Society Interface 11.101 (2014): 20140873.
Zhu, Xiaojin. "Persistent homology: An introduction and a new text representation for natural language processing." IJCAI. 2013.
A. Grigor’yan, Y. Lin, Y. Muranov and S.-T. Yau, Homologies of path complexes and digraphs, preprint, 2012. arXiv:1207.2834v4.
Grigor'yan, A., Muranov, Yu., Yau, S.-T., "Homologies of digraphs and Künneth formulas", Comm. Anal. Geom., 25 (2017) 969-1018.
Grigor'yan, A., Lin Yong, Muranov, Yu., Yau, S.-T., "Path complexes and their homologies", J. Math. Sciences, 248 (2020) 564-599.
A. Grigor’yan, Y. Muranov and S.-T. Yau, Graphs associated with simplicial complexes, Homology Homotopy Appl. 16 (2014), 295–311.
A. Grigor’yan, R. Jimenez, Y. Muranov and S.-T. Yau, On the path homology theory of digraphs and Eilenberg–Steenrod axioms, Homology Homotopy Appl. 20 (2018), no. 2, 179–205.
A. Grigor’yan, R. Jimenez, Y. Muranov, and S.-T. Yau, Homology of path complexes and hypergraphs, Topology Appl.267(2019), no. 2, 106877.
A. Grigor’yan, Y. Lin, Y. Muranov and S.-T. Yau, Cohomology of digraphs and (undirected) graphs, Asian J. Math. 19 (2015),887–932.
Grigor'yan, Alexander; Muranov, Yuri; Vershinin, Vladimir; Yau, Shing-Tung； Path homology theory of multigraphs and quivers. Forum Math. 30 (2018), no. 5, 1319–1337.
Sergei Ivanov and Fedor Pavutnitskiy， SIMPLICIAL APPROACH TO PATH HOMOLOGY OF QUIVERS, SUBSETS OF GROUPS AND SUBMODULES OF ALGEBRAS，preprint.
Chen, Beifang, Yau, Shing-Tung, and Yeh, Yeong-Nan, Graph homotopy and Graham homotopy, Discrete Math., 241 (2001) 153-170.
A. Grigor’yan, Y. Lin, Y. Muranov and S.-T. Yau, Homotopy theory for digraphs, Pure Appl. Math. Q. 10 (2014), 619–674.
听众
Graduate
视频公开
公开
笔记公开
公开
语言
中文
讲师介绍
Sergei Ivanov教授是来自俄罗斯圣彼得堡的数学家。 他的研究兴趣包括同调代数、代数拓扑、群论、单纯同伦论和单纯群。
李京艳，BIMSA助理研究员，2007年获河北师范大学数学系博士学位，先后执教于石家庄铁道大学数理系和河北师范大学数学科学学院，职称副教授，2021年9月入职北京雁栖湖应用数学研究院（BIMSA）。主要研究兴趣在拓扑数据分析和单纯同调与同伦方面。
李京艳，BIMSA助理研究员，2007年获河北师范大学数学系博士学位，先后执教于石家庄铁道大学数理系和河北师范大学数学科学学院，职称副教授，2021年9月入职北京雁栖湖应用数学研究院（BIMSA）。主要研究兴趣在拓扑数据分析和单纯同调与同伦方面。