Simplicial homotopy theory I
Simplicial approach is an important tool for establishing potential connections between algebraic topology and other areas of mathematics. In this course, we will give an introduction to simplicial homotopy theory. After introducing some fundamental notions in the theory of simplicial sets, we will provide the proofs of the fundamental theorems in algebraic topology in simplicial setting, such as Poincare Theorem, Hurewicz theorem, Whitehead theorem, Homotopy Addition Theorem. The course will also discuss simplicia fibration theory, Moore-Postnikov system and simplicial group theory.
This course is designed for helping the students and junior researchers who are interested in GLMY theory and algebraic topology of digraphs, where the students and junior researchers can learn the ideas of simplicial homotopy theory with aiming to develop the homotopy theory of digraphs. The course is also suitable for the graduate students and junior researchers who are working on simplical approaches to data and deep learning with being interested in upgrading their topological skills.
The lecture notes will be available before the program starts. The prerequisite is Allen Hatcher’s book: Algebraic Topology, Cambridge University Press, 2002. However most materials in the lecture notes are self-contained and so the smart students can still follow the lecture notes without much background on algebraic topology.
This course is designed for helping the students and junior researchers who are interested in GLMY theory and algebraic topology of digraphs, where the students and junior researchers can learn the ideas of simplicial homotopy theory with aiming to develop the homotopy theory of digraphs. The course is also suitable for the graduate students and junior researchers who are working on simplical approaches to data and deep learning with being interested in upgrading their topological skills.
The lecture notes will be available before the program starts. The prerequisite is Allen Hatcher’s book: Algebraic Topology, Cambridge University Press, 2002. However most materials in the lecture notes are self-contained and so the smart students can still follow the lecture notes without much background on algebraic topology.
Lecturer
Date
21st March ~ 20th June, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Thursday | 17:10 - 21:00 | A3-4-301 | ZOOM 01 | 928 682 9093 | BIMSA |
Prerequisite
Algebraic Topology
Syllabus
simplicial set and simplicial objects, nerve complexes of categories, geometric realization, homotopy and fibrant simplicial sets, homotopy groups, Homotopy Addition Theorem, minimal simplicial sets, Whitehead theorem, simplicial fibration, Moore paths and Moore loops, Moore-Postnikov system, simplicial H-sets and simplicial groups, Moore chain complex and Moore homotopy groups, Hurewicz Theorem, Poincare Theorem, Kan construction, Milnor construction.
Reference
1.Wu, Jie, Simplicial objects and homotopy groups.(English summary)Braids, 31–181. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 19 World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010 ISBN:978-981-4291-40-8
2.Curtis, Edward B. Simplicial homotopy theory.Advances in Math.6(1971), 107–209.
2.Curtis, Edward B. Simplicial homotopy theory.Advances in Math.6(1971), 107–209.
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).