Intro to the Fundamental Group // Algebraic Topology with
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- čas přidán 27. 07. 2024
- In this video I teach the amazing @TomRocksMaths a little bit of algebraic topology, specifically the fundamental group. Tom also taught me some really cool fluid dynamics and you can find our collab over at his channel here:
►►► • Helmholtz Principle an...
0:00 What is Algebraic Topology?
4:01 The alphabet to a topologist
8:20 The algebra of loops about a ring
14:50 Defining Homotopy Equivalence
18:54 The Fundamental Group
23:58 Fundamental Group of R^2
25:50 Fundamental Group of a Sphere
28:32 Fundamental Group of a Circle
31:45 Fundamental Group of a Torus
34:18 Proof of Brouwer's Fixed Point Theorem
We begin by talking about what connotations the words "algebraic" and "topology" have; "algebra" has a certain concreteness to it as we can add or multiply things, have explicit formulas, etc while "topology" is all about considering spaces that are thought of the same even if we continuously deformed like they were playdoh. Under this perspective, the alphabet only has three letters to a topologist, a single point, a single circle, and a double circle (the letter B).
We then define more precisely the notion of a homotopy equivalence between two maps into a space X. There is an operation we call multiplication on such paths which captures the idea of doing one path followed by the next. It turns out that the homotopy equivalence classes of loops in a space X starting and finishing from a basepoint x_0 with this notion of multiplication form the fundamental group which we often write is Pi_1(X, x_0).
Tom then computes the fundamental group of many spaces, the plane, the 2-sphere, the 1-sphere or circle, and finally - triumphantly - the torus! Finally we finish with a nice proof of Brouwer's Fixed Point theorem that uses the power of the fundamental group to arrive at a contradiction.
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Check out our second video on @TomRocksMaths's channel where he teaches me some Fluid Dynamics, it is some pretty awesome stuff. Great to do this pair of videos with you Tom! ►►►czcams.com/video/bpeCfwY4qa0/video.html&ab_channel=TomRocksMaths
wait......he said it was part 1....
This was such fun! Can you teach me more topology please? :)
@@TomRocksMaths of course( he can)
@Tomrocksmaths Omg yes please
If you swap the words in algebraic topology, you get topological algebra, which is also a topic worth checking out! One of my favourite kinds of theorems in math are duality/representation theorems, and Pontryagin duality is one particularly cool example.
Indeed Pontryagin duality gives, (S^1)^PD = Hom_{cts}(S^1,S^1) = Z, while in algebraic topology, pi_1(S^1) = [S^1,S^1] = Z where [X,Y] denotes homotopy classes of maps from X to Y. It's somehow the "same" formula!
Is this a thing that mathematicians do a lot? Having the field "Xic Y" be distinct from the field "Yic X"? "Algebraic Geometry" is very different from "Geometric Algebra". Or is this just a thing with "Algebra"?
I loved this lecture!
I'm quite impressed that I was able to follow along with it knowing only a little bit of group theory.
Sir you did a excellent job at presenting this topic.
Sorry any mistakes, my english is a work in progress.
Thank you both of you so much. I was confused in Topology a lot. Now, my messy loops are untied. Bravo, brilliant !
This is like a highly anticipated crossover 😎😎
Math Avengers:D
@@DrTrefor mathevengers check that channel :)
Love that you and Tom hooked up on this. Great topic too. Both of you are great teachers.
This is fun: its somewhere in between a question requiring user input (maybe youtube will develop that feature in the future) and a fixed presentation :-).
Not going to lie, it took me 2-3 watchings and some notebook action to actually understand it, but it was certainly worth it!
After a ton of request finally prof will cover some snippets from his area of research. Feeling blessed.
Can’t wait, Dr. Trefor!
Thanks a lot for the vivid explanation 🙏
very useful, insightful, & kind: thank you!! ❤️❤️
The audio is much better in this video, great work!
Worth binge watching!
That was really great! thanks so much to you both :-D
Glad you enjoyed!
cool! me cant wait too!
I’d be upset if tom doesn’t discuss the topology of rocks
... future video idea :)
Damn I am really here procrastinating studying by studying huh
Awesome! Loved it!
Glad you enjoyed it!
Awesome!
You 2 are my favorite
Topology is the best!
Hey dr!! Very Interesting!! Im thinking of getting a minor in mathematics and I still have Abstract Algebra, so I would like to ask you when you will do a course on that? Thank You so much, you never fail to impress us! And Im pleased to be an alumni of yours, you taught me Discrete Maths, Calculus and Differential Equations and Markov Chains and Graph Theory and now Game Theory
That's cool! I do want to do a series on Abstract Algebra one of these days:D
Nice video guys 😍
Great Video Dr!! I would like to take abstract algebra maybe this spring or fall 2022, and I will self tutor myself first, so I would like to ask you if you will post anything related to that at the end of 2021 or in 2022. Thank You so much for your effort and god bless you.
Cool! I do mean to do some abstract algebra at some point!
This is going to be epic 😎🤙
If you include non capital letters there is also a two-point with i and j.
Great
So if I get this right in the Proof of Brouwer's Fixed Point Theorem you're saying that if every x is not equal to f(x), then there is a mapping from the disk to the s1 right? but I still don't understand where the contradiction is. Is it because the mapping changes the fundamental group?
I’ve shown a way that every path on the circle can in fact be sent to the trivial loops through the composition of the two maps. But that is impossible as the fundamental group is Z
@@DrTrefor In 1 dimension, the proof of Brouwer's Fixed Point Theorem is easily proven by the Intermediate Value Theorem. Is there an analog of the Intermediate Value Theorem in a higher dimension? and thanks for the reply Dr. Bazett.
@@yonathan4194
I think what you may be looking for as a generalization of the intermediate value theorem is the fact that a continuous image of a connected set is connected. This is a theorem of topology.
In the proof at the end, I think it's not clear that f continuous implies that r f_t also a homotopy
I love seeing functions as they exist. forms in the field of mathematics.
Q? Surface(cos(u/2) cos(v/2), cos(u/2)sin(v/2),sin(u)/2) 0>u>4π 0>v>2π
Klein or not? it requires 4pi to complete the surface (electron half spin) but the node is problematic. opinions? proofs?
♥️♥️♥️♥️
Ok, I feel like it's a dumb question. I guess we didn't prove the fixed point theorem on arbitary metric, or did we? If that's the case then why are we using only the fundamental group of S1, is it because ultimately metric gives a real output and the generalisation balls and spheres correspond to a real number.
Ya we have some work to do to understand something in arbitrary metric spaces, but the basic arguments ultimately will work out the same
Why is pi1(S^2) = 0? You can loop in the opposite direction or stay put, wouldn't it be equal to Z also?
Imagine you had a rope around the equator. You can sort of pull that rope up up up to towards the north pole making a smaller and smaller circle all the time. THat is the sense in which it collapses to that constant path at the north pole every time.
@@DrTrefor if i loop the loop at the equator will it still collapse to the north pole? or do i get a unique construct?
@@sdsa007 it can still be "slid", or continuously deformed into a trivial loop on the surface of the sphere
But what we want to determine the fundamental group of line
It’s just 0!
@@DrTrefor I understand that it's not necessary to the figure to be closed if we want to find it's fundamental group?
20:09 multiplacruon can always be viewed as repeated adding like 1 * 200 = 200 = sigma k[0;200] k
31:00 so if I loop around twice in S^2 (north pole to south to north), it is the same loop as looping around once. But if I do that in S^1, these are different loops? Why in the living hell is that!
Late to the party, but while capital letters only have three forms, "letters" (English ones at least =) have four: there's one circle, two circles, one point, but also two points (e.g. "i" or "j")
This adds a level of topological complexity, in that these letters firm disconnected spaces.
Love the shirt, professor. Where can I buy it?
The theories of correct Mathematical communication are represented in the presence of the sender, who is the Mathematics teacher, who believes that Numbers have an end, and the receiver, who is a Mathematics student who receives a Mathematics education free from the illusion of infinity.
Numbers have NO END.......
!!!!!
INFINITY IS THE NUMBER OF NATURAL NUMBERS!!! THERE ARE INFINITELY MANY INFINITIES OF INFINITIES AND SO ON........;!!!!!!!! STOP IT U WHO KNOW NOTHING ABOUT MATHEMATICS!!!
YOU CANT APPROXIMATE USING TAYLOR SERIES!!!! U KNOW NOTHING ABOUT MATHEMATICS AND WILL NEVER SUCCEED YOU BEAT THE JEWISH PEOPLE .......
قفلتوني من أم الفيديو
True and scientific Mathematics is Mathematics devoid of the illusion of infinity...Mathematics is an exact sciences, not an abstract one.
Go away. Comment on another channel. Don't spam. I just needed to reply cause how idiotic you were .
I've been told by scientists that mathematica isn't science because it is not falsifiable. I prefer my mathematics to not be falsifiable...
Everyone here forgot how involuntarily funny you can be...
It is not possible to build a correct Mathematical educational framework without abandoning the illusion of infinity and its symbol.
Get out of here . If u believe it then don't comment. Don't tell others. Tell yourself .
You're infinitely ridiculous.
Why do you say that? Do you take issue with Cantor's thm?
The unwillingness to believe that Numbers have an end and and the denial ofillusion of infinity, cannot be a natural characteristic emanating from an ordinary human being... Rather, it is a condition that expresses the existence of either a deliberate desire to spread Mathematical ignorance or a psychiatric condition.
Numbers don't have an end. Infinity is the number of integers there are.
You don't even understand mathematics .
Please, sir, what's the largest positive integer?
These comments of yours are the dumbest thing ever.
It's no denial of infinity. We can both argue it's existence but one is more probable and useful than the other. The concept of counting via integers alone is an abstract concept created by humans. For what is the number 1? We find it useful to describe some groups of objects. Math is always an approximation of our reality, not reality itself. As a mathematician you're living in an "illusion," and infinity is a useful/proper tool that helps describe spaces/thms in a useful and predictable way.