What is...algebraic topology?

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Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What is...algebraic topology? Or: Translating from topology to algebra.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
These videos are concerned with algebraic topology, and not general topology. (These two are not to be confused.) I assume that you know bits and pieces about general topology, but not too much, I hope.
Slides.
www.dtubbenhauer.com/youtube.html
Website with exercises.
www.dtubbenhauer.com/lecture-a...
Material used.
en.wikipedia.org/wiki/Topology
The four main fields (I also stole some pictures from there).
en.wikipedia.org/wiki/General...
en.wikipedia.org/wiki/Algebra...
en.wikipedia.org/wiki/Differe...
en.wikipedia.org/wiki/Geometr...
Awesome fundamental group animations (and other great stuff).
math.unice.fr/~indira/Mygifs....
Fundamental group.
www.math3ma.com/blog/the-fund...
www.math3ma.com/blog/the-fund...
mathworld.wolfram.com/Fundame...
en.wikipedia.org/wiki/Fundame...
Fundamental theorem of algebra using pi_1 (I also stole some pictures from there).
www.homepages.ucl.ac.uk/~ucahj...
math.stackexchange.com/questi...
Topological data analysis.
brickisland.net/DDGSpring2019...
towardsdatascience.com/topolo...
community.wolfram.com/groups/...
arxiv.org/abs/1710.04019
en.wikipedia.org/wiki/Topolog...
Venn diagrams in latex.
latexdraw.com/how-to-draw-ven...
Hatcher’s book (I sometimes steal some pictures from there).
pi.math.cornell.edu/~hatcher/...
Always useful.
en.wikipedia.org/wiki/Counter...
#algebraictopology
#topology
#mathematics

Опубликовано:

30 июл 2021

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Комментарии : 38

@francescabeard8804 Год назад

You've inspired my daughter to look into studying topology at university - we thoroughly enjoyed your video!

@VisualMath Год назад

I am flattered! Glad that you liked the videos, and if they even turn out to be useful, then that is awesome. We certainly need more people learning and promoting one of the greatest and mostly unnoticed ideas of the previous century. So for everyone who is up for the task, including you + family: Good luck studying topology, I hope you will enjoy it!

@apokryphos_ Год назад

Your videos are extraordinarily helpful building intuition and really helped me along my AT studies. By pure accident I today stumbled over the fact that you very well know my lecturer and bachelor thesis mentor, Prof. Dr. Thomas Schick :D Greetings from Göttingen and keep up the amazing work!!

@VisualMath Год назад

Awesome, thank you so much for the feedback! And, of course: Gruss zurück nach Göttingen ;-)

@HDQuote Год назад

Thanks for the video! Wow, I would have never thought that there would be such a close connection between topology and data analysis. This really motivates me to look closer into it

@VisualMath Год назад

You're very welcome! Same here by the way: "Wow" was my first thought when I heard about topology+data analysis going hand-in-hand. Absolutely fascinating and powerful ;-) en.wikipedia.org/wiki/Topological_data_analysis

@NoNTr1v1aL 2 года назад

Absolutely amazing playlist and channel!

@VisualMath 2 года назад

Thanks for the kind words and the feedback! AT is so nice, and I hope you enjoy it. If the videos are helpful, then that's awesome.

@valentinussofa4135 Год назад

This lecture enrich my insight about algebraic topology with visualization. Thank you very much🙏

@VisualMath Год назад

Glad it was helpful! And thanks for the kind feedback, that is very much appreciated! I hope you will enjoy AT in general.

@JackLee-dp3hm 3 месяца назад

Please continue to update this series. I am studying AT and your videos truly give me a good understanding of AT and some interesting ideas.

@VisualMath 3 месяца назад

I am glad that you like AT and that the videos are helpful for you 😀 I hope you enjoy them and AT as well! What kind of topic for AT do you have in mind, i.e. what is it that you are studying at the moment?☺

@JackLee-dp3hm 3 месяца назад

@@VisualMath Maybe the word study is a little misleading. Learn is better. I am a beginner on AT. The book I am using is Algebraic Topology by Hatcher. I will appreciate it if you can give some suggestions on my learning.

@VisualMath 3 месяца назад

@@JackLee-dp3hm It depends what you are looking for. I love Hatcher, but some people think its not “rigorous enough”, or “too hard” for beginners. There are plenty of opinions about Hatcher out there; I think that shows how popular the book is 🥰 Anyway, I can also recommend this free text: friedl.app.uni-regensburg.de/papers/1at-uptodate.pdf The only “problem” is its length 🤨 but its fairly complete.

@JackLee-dp3hm 3 месяца назад

@@VisualMath Thank you. I will give a try to read. By the way, For the book by Hatcher, when I read the Chapter 2, I just feel that he is telling some stories. Sometimes, he says that we complete the proof of sth suddenly. I even don't realize where he starts the proof. This is the biggest trouble for me. But in general it is a good book.

@yamansanghavi Год назад

Your videos are extremely helpful. I can't thank you enough.

@VisualMath Год назад

Thank you for the feedback: Happy to hear that! I always hope that the videos are helpful; that is my main motivation. I am super glad that the explanations worked for you. And enjoy your topology journey, I hope it will be a blast!

@mohitdas4065 Год назад

ur videos are amazing and enlightening . Please do something on differential topology also in intuitive and visual way

@VisualMath Год назад

Thanks for the kind feedback! Right now I am doing geometric topology, which is pretty close to differential topology already - let us see what the future holds ;-)

@speedbird7587 Месяц назад

Very nice introduction, Good tutorial, I really enjoyed it, Thanks!

@VisualMath Месяц назад

Thanks, I hope you will enjoy AT ☺

@shadowlord4625 2 года назад

Amazing playlist that I discovered today.I have studied only upto homology.Can you suggest good book for cohomology groups?

@VisualMath 2 года назад

Thanks: I hope some of the videos will be helpful. Book: That depends, of course, on your taste. I am a fan of Hatcher's "Algebraic topology" and I think Hatcher does a good job regarding cohomology. Not quite how I would write it, but I am not in position to complain before writing a book on AT myself ;-) With contrast, tom Dieck "Algebraic topology" for example takes the axiomatic approach. That, imho, does accompany Hatcher book well.

@DanillaDaniela Год назад

thank you thank you thank you !!!

@VisualMath Год назад

You are so welcome! I hope you will enjoy AT.

@paramsiva1515 2 года назад

shape matters

@VisualMath 2 года назад

It does ;-)

@motherisape Год назад

@@VisualMath ok

@englandcricketshorts9094 Год назад

Hey Sir. Asking something we asked today as Mathematics Student. Why Topology is Called Abstract Branch of Mathematics? Kindly explain

@VisualMath Год назад

Mathematics is historically divided in pure math (the study of mathematical concepts independently of any application; topology, algebra, analysis, …) and applied math (aiming at applications; numerical analysis, optimization, …) This distinction gets more and more questionable by the second, so I would personally ignore it. Topology is for example very applicable, while numerical analysis is often only considered “for the fun of it”. Some people might even divide pure math further into “abstract” and “the rest (sometimes this just means analysis or number theory)”, but that feels very artificial to me. Topology for example is very explicit and applicable and yet counts as “abstract”. Again, I would not worry about this - its artificial anyway. I hope that helps! P.S.: I go by they/them, so please no “sir”.

@47lokeshkumar74 2 года назад

kindly give combinatorics lecture

@VisualMath 2 года назад

Thank you for your comment! Combinatorics is certainly on my list. But it is a broad topic, with very ill-defined and very fuzzy borders. (Even more than other fields of math and the sciences.) I have to think about how to cover and organize the material. This will take a while, sorry for being slow!

@47lokeshkumar74 2 года назад

Kindly give some lust of book for reference in your video. Thanks

@VisualMath 2 года назад

@@47lokeshkumar74 Please have a look into the description. I hope the collection of links given there is helpful.

@tayzrayz3797 8 месяцев назад

Very excellent explanation, never fails also r u russian

@VisualMath 8 месяцев назад

No I am not from Russia. You are in fact the first one that thinks I have a Russian accent 😆 (No worries, its all good ☺) And thanks for the feedback - that is very much appreciated!