You Could Have Invented Homology, Part 2: Some Simple Spaces | Boarbarktree
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- čas přidán 8. 09. 2024
- If it looks like this video increases dramatically in production quality over its runtime that's because this thing took hundreds of hours so I genuinely just got better at animating this kind of thing
P.S., I make a slight mistake in the voice-over. First person to find it gets a special prize
Patronise me: patreon.com/boarbarktree
If you want to see me both get extremely political with completely unprompted and also post sentences that I like, follow me on twitter at @boarbarktree. Oh also you can get updates on my work (mathematical/youtubological) there too i guess. Basically I won't shut up
P.P.S., isn't it amazing how my voice still breaks at 24 years old
"R^n has no surprises", well, except for R^4.
Lmao tru, tbh ℝ^n is endlessly surprising, I just liked how that sentence sounded 😅
@@Boarbarktree Well, and what you said *is* true for the context of the video.
No need to bring up the non-unique differential structure(s) of R^4 :p
Congrats on the acceptance to your PhD btw! Hoping to receive the same news in the next couple months.
Hope you're still able to keep these amazing videos goin!
You'll have to drag me kicking and screaming out of the homotopy category before I'll acknowledge your "exotic ℝ^4"s 🤣 cheers and good luck!!
Love finding channels that are going places before it got big.
Me too
"The Euclidian spaces are the perfect topological spaces. You may not like it, but this is what peak performance looks like" 😂😂
Lmfao, just came down to comment this
I love the fluid algebraic manipulations, as visually pleasing manipulations.
"Mathematician's hate making choices", proceeds to make the choices anyway. I see you hiding your choice of basis behind the structure of R^n but that is already a choice, and a bigger one at that ;p
If you choose something with the word "standard" on it it doesn't count as a choice ;)
You might say that mathematicians prefer to steal choices!
this video is creepy, spooky, as should topology videos be, topology gives me the chills, its the weirdest thing we have invented I think
Loved this introduction to simplices! It’s a very natural presentation of a very important topic. Thanks for all your hard work; I saw from your Twitter that you struggled with this video, but it really paid off :)
Congratulations on the PhD! I’m also starting mine at my dream school in April. :)
Congrats on the PhD! I hope you can finish this series before April!
the easiest patreon support of my life. You have a phenomenal thing going here. I wish you the best in your big move, and in your studies! Purely inspiring work. 3b1b wasn't messing around by praising you on Twitter, I'm so glad I clicked through.
Thank you 😁😁😁😁
All of the descriptions of spacetime in mathematics rely on orthogonal parameters. Algebraic topology extends the mapping of a flat earth on to a sphere in three dimensions which has perturbations on it's surface making a fourth dimension.
Man... I just finished my course on algebraic topology this January. These video's would have greatly helped me with the intuition surrounding the subject. To bad I only found them now, this is golden stuff. Great Job!!
Man these videos are so nicely paced and the explanations and the way the ideas flow into each other is great, and the sometime spooky music works really well places me in abstract space world, an absolute pleasure to watch
Congratulations on getting into PhD. programs! I just finished one right before the pandemic start. Really love the way you shuffe those algebraic terms around, guess I could told others now what it looks like when we're doing math stuff. Looking forward for your next one!
Beautiful stuff! I feel like I deeply understand triangles now
Even tho I'm a math major I still enjoy a lot your videos because it is a nice way to process concepts keep up the good work! Congrats for your PhD
I'm SO glad to have found you, we are not even 2k! Can't wait to force everyone I know watch the series :)
Well.. that was an experience... Got to write a lot of notes and learned a lot of stuff! Thank you!!
How would co(S) be defined if S is infinite? Is it similar to how you would define a linear combination of an infinite set? That is, x\in co(S) if for some finite A\subseteq S, x\in co(A)? Or, more explicitly x in co(S) means that x in {t_1x_1+t_2x_2\ldots t_nx_n: t_1+t_2+t_3+\ldots t_n=1, {x_i}\subset S} (co(S)={t_1x_1+t_2x_2\ldots t_nx_n: t_1+t_2+t_3+\ldots t_n=1, {x_i}\subset S})
This is great! May I suggest, when you are animating equations, to use different colours for different variables? This would make things clearer! Keep up the good work!
So that's why the fucking Simplex Method for solving Linear Problems is called that! I could definitely see a relation between the Convex Hull and the Feasible Region of a Linear Problem while you where explaining.
Ah so we're headed for singular homology. Neat!
Also congrats on getting into your PhD!
i love boarbarktree
I really should be getting to work
but I *really want to watch this*
Nice way for simplicial homology
Congrats on your acceptance into your dream PhD! I am also applying this fall, so wish I could get as lucky as you watching your videos! They are natural to me and offered tremendous help on my current research project:)
Intro ends at 1:29.
Keep the videos coming while you still can!
You got the makings of a great Math Expositor
HE'S GOT MEMES, I LOVE IT
You could have invented Homology ≅ Euclid have invented Homology.
And you are homeomorphic to legend (;
Thank you.
Amazing! Great Animation, great explanation. It is good to use other animations techniques besides manim
I typically get more political during graduate school... you must be in a top program!
such a well made video & cool series, excited to see where you’re taking it! (polyhedron?)
Thank you! and nice spotting! Unfortunately that is not the mistake I was thinking of - turns out I had also forgotten the difference between a polyhedron and a polygon LOL
Amazing videos
Fantastic work. This channel is going to grow!
Awesome work! Good luck with the phd!!
Love this content, i hope you keep it up!
Such a good video!! Congrats!!
Congrats on the PhD acceptance. I hope you still have time to finish this series.
Your videos are fantastic!!!
These videos are awesome
Congrats on PhD! I’m doing mine now too. Also, you said t2 v3 instead of t3 v3 around the 11:00 mark :)
2:00 When you use an openly theorized problem offhand for a simple example
This is so good!
I see that the simplex of dimension n is defined in R^(n+1). Is there any reason (other than definitional symmetry) not to take the other special point, the origin, so we can define each standard simplex in its tightest embedding Euclidean space?
Great question! Yes there is! Firstly, the fact that the points of the standard n-simplex is given by the points with non-negative coordinated x_i satisfying the equation Σx_i=1 is convenient pretty often, but a more important reason is that we're going to want to define linear maps between our simplices. Having their vertices form a basis for the space they're embedded in means we can do this by just describing where the vertices are mapped to, which extends to a linear map on all of ℝ^(n+1). For instance, we can define a map that includes the standard 1-simplex as a face in the standard 2-simplex by defining
F(e_1)=e_1, F(e_2)=e_3. This gives a linear map ℝ² → ℝ³ via F(xe_1+ye_2)=xe_1+ye_3 which in particular defines the map on the whole 1-simplex.
If the dimension of the space the simplices are embedded in is less than the number of vertices they have, or if one of the vertices is 0, then we can't define linear maps so easily, since their vertices don't form a basis.
Hope this helps 😁
@@Boarbarktree Ahh okay, fascinating! Feels like there's shadows of homogeneous coordinates here... using a higher dimensional space lets your linear maps do "more" than they otherwise could (translation in homogeneous coordinates, for instance).
That's a good way to think about it!
Amazing video! Congrats on the PhD as well. If I might ask, how do you make these videos e.g what software etc do you use?
amazinnn
Really enjoyed this video, thanks! :)
I have a silly question:
I think the Hadamard Conjecture says that if and only if n=4, we can inscribe a regular n-simplex in a regular n-hypercube. Does this have any nice interpretation or consequences for algebraic topology?
Can't think of any off the top of my head, but it's very much in line with the pattern of dimension 4 being different to all the other dimensions !
The voiceover mistake is t_2v_3 @11:05
Also! Great video series, good luck in your PhD
Congrats you got it! Special prize coming up! No further action is necessary. Your patience is appreciated
I think I found where you made a slight error in the voice over. You said t2 v3 instead of t3 v3 when talking about the set of points inside the filled in triangle.
You beat me to it lol.
SQUAREGOLIEK
fuck yeah
Congratz on getting accepted to a PhD programm, you already have some Idea what you will be studying for your thesis?
∞-category theory 😎
Interesting, the standard simplex is just called the probability simplex in statistics.
Commenting to boost the algorithm :)
By wich app you have done this?, thank you
Where's the music from?
Like before watching!
Homology and cohomology is not used any more. There are lots of definitions and they do not know the homology group even of a sphere in height dimensions. Waist of work and time.
Bruuuuuuuh PLEASE do videos on your research at some point! 🤓