Persistent homology and GLMY-homology
In applied mathematics, topological based data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. In this course we will concentrate mainly on two such techniques: persistent homology and GLMY-homology.
Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization of data into a topological formalism. In our lectures we will give some exposition of mathematical theory and show some applications in biology.
Prof Yau and his collaborators introduced the new topology theory (GLMY theory) of digraph in 2012,this theory takes more abstract graph structured data as the research target,developed GLMY homology, hypergraph homology，etc. The relevant theory of this homology has been very rich, and its application has achieved good results in many fields.This course mainly sorts out GLMY homology to help students become familiar with and master the theory.
Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization of data into a topological formalism. In our lectures we will give some exposition of mathematical theory and show some applications in biology.
Prof Yau and his collaborators introduced the new topology theory (GLMY theory) of digraph in 2012,this theory takes more abstract graph structured data as the research target,developed GLMY homology, hypergraph homology，etc. The relevant theory of this homology has been very rich, and its application has achieved good results in many fields.This course mainly sorts out GLMY homology to help students become familiar with and master the theory.
Lecturers
Date
27th February ~ 2nd June, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday | 13:30 - 16:55 | A3-2a-201 | ZOOM 02 | 518 868 7656 | BIMSA |
| Friday | 10:40 - 15:05 | A3-2a-201 | ZOOM 01 | 928 682 9093 | BIMSA |
Prerequisite
The elementary theory about algebraic topology
Syllabus
The 1-6th weeks: Persistent modules, persistent diagrams, persistent homology, with application to biology. The 7-11th weeks:mainly talk about the GLMY homology theory and the relative GLMY homology theory,the last week will introduce some new results.
Reference
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.
Audience
Graduate
Video Public
Yes
Notes Public
Yes
Language
Chinese
Lecturer Intro
Prof. Sergei Ivanov is a mathematician from St. Petersburg, Russia. His research interests include homological algebra, algebraic topology, group theory, simplicial homotopy theory, simplicial groups.
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.
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.