Combinatorial algebraic topology I
This course will discuss topological modelling of combinatorial objects such as (di-)graphs and hypergraphs. The course will guide the students to read the relevant references on the topic for learning how to apply the theories in algebraic topology to the combinatorial objects. The course will also display some applications of algebraic topology in data analytics through topological modelling on complex network. The students in the course are encouraged to explore topological questions arising complex networks or the applications of algebraic topology in data.
Lecturer
Date
10th October, 2023 ~ 9th January, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday | 17:05 - 20:55 | A3-4-101 | ZOOM A | 388 528 9728 | BIMSA |
Prerequisite
Algebraic Topology
Syllabus
1.Simplicial complex
1.1 Geometric and abstract simplicial complex
1.25.∆-set and ∆-complex
1.3Simplicial set and nerve complex
2.Simplicial complex construction on graphs
2.1 neighborhood complex
2.2 Hom complex
2.3 Independance complex
3.Simplicial homology
3.1 definition and basic properties
3.2 Mayer-Vietoris sequence
3.3 Künneth theorem
4.Embedded homology of hypergraphs and super-hypergraphs
4.1 Super-hypergraph
4.2 Embedded homology
4.3 path complexes and GLMY homology of digraphs
4.4 Super-hypergraph structures on (di-)graphs and quivers
4.5 scoring scheme and super-persistent homology
5.Homotopy theory of digraphs
5.1 path digraph and loop digraphs
5.2 fundamental group and homotopy groups
1.1 Geometric and abstract simplicial complex
1.25.∆-set and ∆-complex
1.3Simplicial set and nerve complex
2.Simplicial complex construction on graphs
2.1 neighborhood complex
2.2 Hom complex
2.3 Independance complex
3.Simplicial homology
3.1 definition and basic properties
3.2 Mayer-Vietoris sequence
3.3 Künneth theorem
4.Embedded homology of hypergraphs and super-hypergraphs
4.1 Super-hypergraph
4.2 Embedded homology
4.3 path complexes and GLMY homology of digraphs
4.4 Super-hypergraph structures on (di-)graphs and quivers
4.5 scoring scheme and super-persistent homology
5.Homotopy theory of digraphs
5.1 path digraph and loop digraphs
5.2 fundamental group and homotopy groups
Reference
1. Dmitry Kozlov, Combinatorial Algebraic Topology, Springer Berlin, Heidelberg, eBook ISBN 978-3-540-71962-5 DOI https://doi.org/10.1007/978-3-540-71962-5
2. Alexander Grigor'yan, Yong Lin, Yuri Muranov, Shing-Tung Yau, Homologies of path complexes and digraphs, arXiv:1207.2834, 2012.
3. Jelena Grbić, Jie Wu, Kelin Xia, and Guo-Wei Wei. Aspects of topological approaches for data science. Foundations of Data Science. 2022, 4(2), 165-216.
2. Alexander Grigor'yan, Yong Lin, Yuri Muranov, Shing-Tung Yau, Homologies of path complexes and digraphs, arXiv:1207.2834, 2012.
3. Jelena Grbić, Jie Wu, Kelin Xia, and Guo-Wei Wei. Aspects of topological approaches for data science. Foundations of Data Science. 2022, 4(2), 165-216.
Audience
Graduate
Video Public
Yes
Notes Public
Yes
Language
Chinese
Lecturer Intro
Jie Wu received a Ph.D. degree in Mathematics from the University of Rochester and worked as a postdoc at Mathematical Sciences Research Institute (MSRI), University of California, Berkeley. He was a former tenured professor at the Department of Mathematics, National University of Singapore. In December 2021, he joined the Yanqi Lake Beijing Institute of Mathematical Sciences and Applications (BIMSA). His research interests are algebraic topology and applied topology. The main achievements in algebraic topology are to establish the fundamental relations between homotopy groups and the theory of braids, and the fundamental relations between loop spaces and modular representation theory of symmetric groups. In terms of applied topology, he has obtained various important results on topological approaches to data science. He has published more than 90 academic papers in top mathematics journals such as “the Journal of American Mathematical Society”, “Advances in Mathematics”, etc. In 2007, he won the "Singapore National Science Award”. In 2014, his project was funded by the “Overseas Joint Fund of National Natural Science Foundation” (Jieqing B).