Record: Yes

Level: Graduate

Language: Chinese

Prerequisite

Algebraic Topology

Abstract

Topological data analysis is a new-born research area that explores topological approaches in data science, where persistent homology has been proved as an effective mathematical tool in data analytics with various successful applications. This course will discuss the mathematical foundations of (higher) topological structures on graphs, aiming to explore new topological approaches beyond the classical persistent homology.

Reference

1.   J. Wu, Simplicial objects and homotopy groups//Braids: introductory lectures on braids, configurations and their applications, World Scientific Publishing Company, 2010, 31-181.

2.   G. Carlsson, Topology and data. AMS Bulletin , 46(2) (2009), 255-308.

3.   Jelena Grbic, Jie Wu, Kelin Xia, Guo-Wei , Aspects of topological approaches for data science, Foundations of Data Science, Vol. 4, No. 2, June 2022, pp. 165–216.

4.   Grigor'yan, A., Lin, Y., Muranov, Y., Yau, S.-T., Cohomology of digraphs and (undirected) graphs, Asian J. Math., 19(5) (2015), 887–931.

5.   Stephen Bressan, Jingyan Li, Shiquan Ren, Jie Wu, The embedded homology of hypergraphs and applications. Asian J. Math. 23 (2019), no. 3, 479-500.

6.   Mengmeng Zhang, Jingyan Li and Jie Wu, the twisted homology of simplicial set, preprint.

Syllabus

1. simplicial complex and Δ-complex,

2. (persistent) simplicial homology,

3. persistent modules,

4. weighted simplicial complex and (persistent) weighted simplicial homology,

5. path complex and (persistent) path homology,

6. hypergraph and (persistent) hypergraph homology,

7. super-hypergraph and super-persistent homology theory.

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).

Lecturer Email: wujie@bimsa.cn

TA: Dr. Yingjie Zhang, zyj.crypto@outlook.com