Algebraic Topology WS 2019/20
Lectures
Time and place:
Mon 14:15--16:00, Room 00.09.022
Tue 8:30--10:00, Room 00.07.014
Some links to videos illustrating the material covered in class:
- Hopf fibration: https://nilesjohnson.net/hopf.html
- Covering spaces and many more: http://neil-strickland.staff.shef.ac.uk/courses/MAS435/demos/index.php
- Cutting the Klein bottle in half: https://www.youtube.com/watch?v=I3ZlhxaT_Ko&list=PLt5AfwLFPxWJeBhzCJ_JXdaIXi_YJl7Bh&index=4
- Dance my PhD, surfaces on polygons: https://imaginary.org/film/cutting-sequences-on-the-double-pentagon
- Möbius bagel : https://www.youtube.com/watch?v=NRvK_07KRV8
- Torus games: http://www.geometrygames.org/TorusGames/
- Deck transformations of the universal cover of the wedge of two circles: https://globberingmattress.wordpress.com/2017/12/26/deck-transformations-revisted/
- Quaternions and 3D rotations: https://eater.net/quaternions
- "A hole in a hole in a hole": https://www.youtube.com/watch?v=k8Rxep2Mkp8&t=95s
Examination
NOTE: Due to the current situation with COVID-19 it is not yet clear whether the exam can take place. We are waiting fo official news from TUM. I will notify all registered students as soon as we receive any information.
First exam: 14.00-16.00 the 24.02 in MI HS 2 - 00.04.011
Second exam: first week of summer semester
Additional Q&A session: 14.00-16.00 the 19.02 in 00.07.014.
Exercise classes
Exercise class: given by Eilind Karlsson
Fri 8:30--10:00, Room 00.07.014
Exercise sheets
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5 NEW version: problem 4 updated
- Exercise 6 NEW version of Problem 6 and fixed typo in exercise 4 (from smash product to wedge product)
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10 Note: Changed direction of arrows in Ex 1 and corrected Ex 4.
- Exercise 11
- Exercise 12
- Exercise 13
- Sketch of solutions - Sheet 13 Updated version (February 18): changes in exposition -- please justify your arguments using things we have seen in the lectures or exercises. Please send questions and/or corrections to eilind.karlsson@ma.tum.de !
Further reading
Some texts for further exploration
These articles are just for exploration and are not meant to be read (and understood) in full detail. Topics we have seen or will see in class such as the fundamental group, covering spaces, triangulations, (semi-)simplicial complexes, and homology appear naturally and have neat applications.
On triangulations of surfaces - proof of (non-)existence
- Doyle, Moran, Inventiones, A Short Proof that Compact 2-Manifolds Can Be Triangulated
- Hartnett, Quanta magazine, A proof that some spaces can't be cut
On links between topology and phases of matter - uses fundamental group and categories
On links between topology and crystals - uses covering spaces and homology
- Baez, Topological crystals
- Chen et al, Generating the Hopf Fibration Experimentally in Nematic Liquid Crystals - look at the pictures of how the Hopf fibration appears when looking at liquid crystals
Topological Data Analysis - uses homology
- Deshmuk in medium.com, Topological data analysis - a very short introduction
- Hess, TED talk on Digital Neuroscientist of the Future
- Carlsson, Bulletin of the AMS, Topology and Data
- Ghrist, Bulletin of the AMS, Barcodes: the persistent homology of data
Higher categories - uses the fundamental groupoid and simplicial sets
- Lurie, AMS Notices, 2008, What is... an infinity-category?
Distributed Computing - uses simplicial complexes and homology
- Herlihy, Rajsbaum, Computer Science Today, Algebraic Topology and Distributed Computing A Primer