Introduction to combinatorial homotopy theory

This course aims to help the students whose future research plan would be in studying homotopy theory or applying the techniques of homotopy theory in their areas. The course will intensively discuss some important notions in homotopy theory using combinatorial approaches, aiming to established the fundamental connections between the modular representation theory and the homotopy theory of the loop-suspension functor. The course will discuss functorial decompositions of some basic constructions in homotopy theory such as self-smashes and the loop-suspensions using representation theory and combinatorial group approaches, with concentrating on the homotopy theory of concrete examples of finite complexes, so that the students may apply the techniques learned from the course to the homotopy theory of more abstract categories in their future development.

21st September, 2026 ~ 15th January, 2027

Weekday	Time	Venue	Online	ID	Password
Friday	14:20 - 16:55	Shuimo	ZOOM 01	928 682 9093	BIMSA

Algebraic Topology

H-spaces and co-H-spaces, Geometric representation of symmetric groups and homotopy decompositions of self-smashes, Hopf algebra; James construction, Whitehead products, Samelson products, Cohen group, natural self-transformations of the loop suspension functor, functor A^{min} , functor B^{max}，functor Q^{max}，homotopy decompositions of loops on co-H-spaces; EHP sequences and homotopy groups.

1. Selick, Paul; Wu, Jie, On natural coalgebra decompositions of tensor algebras and loop suspensions, Mem. Amer. Math. Soc. 148, no. 701, viii+109 pp.(2000).
2. Jie Wu, Homotopy theory of the suspensions of the projective plane, Mem. Amer. Math. Soc. , 162(769), x+130pp. (2003)
3. Jie Wu, On maps from loop suspensions to loop spaces and the shuffle relations on the Cohen groups, Mem. Amer. Math. Soc. 180, no. 851, vi+64 pp.(2006)

Advanced Undergraduate , Graduate

Yes

Chinese

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. His 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 applied topology, he has obtained various important results on topological approaches to data science. Currently, he leads his research team for bi-directional applications of topology aiming to topological foundation of high-order interactions in complex systems, developing algebraic topology of directed graphs and hypergraphs, and applying topological approaches beyond persistent homology to practical problems. He has published more than 100 academic papers in top journals such as “the Journal of American Mathematical Society”, “Proceedings of the National Academy of Sciences of the United States of America”,“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).