Methods of Algebraic Topology in Graph Theory. Homology theories.
Currently the problem of transferring results of algebraic topology to various categories of digraphs and graphs is widely investigated. The main technical tools are given by the homotopy theory and various homology theories. Now we present a collection of homology theories on various categories in graph theory. We describe main constructions and basic properties of these theories and describe relations between them. The examples which illustrate the computation of homology will be given.
Lecturer
Date
9th October ~ 18th December, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday | 09:30 - 12:00 | A3-4-312 | ZOOM 2 | 638 227 8222 | BIMSA |
Prerequisite
basic knowledge of linear algebra and algebra, but all relevant notions will be defined in the class.
Syllabus
1. Chain complexes.
2. Simplicial complexes and simplicial homology.
3. Cliques homology of (di)graphs.
4. Singular simplicial and singular cubical homology for various categories of graphs.
5. Homotopy graph theory.
6. Cell cubical homology for various categories of graphs.
7. Path homology theory of di(graphs)
8. Path homology theory for various categories of graphs.
2. Simplicial complexes and simplicial homology.
3. Cliques homology of (di)graphs.
4. Singular simplicial and singular cubical homology for various categories of graphs.
5. Homotopy graph theory.
6. Cell cubical homology for various categories of graphs.
7. Path homology theory of di(graphs)
8. Path homology theory for various categories of graphs.
Reference
1. Hatcher A.. Algebraic Topology, Cambridge University Press, 2002..
2. Chartrand G., Lesniak L., and Zhang P. Graphs and digraphs, CRC Press: Boca Raton, 2011.
3. Chen B., Yau Sh.-T. , Yeh Ye.-N.. Graph homotopy and Graham homotopy. Discrete Mathematics, 241 (2001), 153-170.
4. Grigor’yan A., Muranov Yu., and Jimenez R. Homology of Digraphs. Mathematical Notes, 2021, V. 109, P. 712–726.
5. Grigor’yan A., Lin Y., Muranov Yu., Yau Sh.-T. Homologies of path complexes and digraphs. arXiv:1207.2834.
6. Grigor’yan A., Rolando J., Muranov Yu., Yau Sh.-T. On the path homology theory of digraphs and Eilenberg-Steenrod axioms. Homology, Homotopy and Appl., 2018. V. 20. P. 179–205.
7. Grigor’yan A., Muranov Yu., Yau Sh.-T. Graphs associated with simplicial complexes. Homology, Homotopy, and Appl., 2014. V. 16. 295–311.
2. Chartrand G., Lesniak L., and Zhang P. Graphs and digraphs, CRC Press: Boca Raton, 2011.
3. Chen B., Yau Sh.-T. , Yeh Ye.-N.. Graph homotopy and Graham homotopy. Discrete Mathematics, 241 (2001), 153-170.
4. Grigor’yan A., Muranov Yu., and Jimenez R. Homology of Digraphs. Mathematical Notes, 2021, V. 109, P. 712–726.
5. Grigor’yan A., Lin Y., Muranov Yu., Yau Sh.-T. Homologies of path complexes and digraphs. arXiv:1207.2834.
6. Grigor’yan A., Rolando J., Muranov Yu., Yau Sh.-T. On the path homology theory of digraphs and Eilenberg-Steenrod axioms. Homology, Homotopy and Appl., 2018. V. 20. P. 179–205.
7. Grigor’yan A., Muranov Yu., Yau Sh.-T. Graphs associated with simplicial complexes. Homology, Homotopy, and Appl., 2014. V. 16. 295–311.
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English