The Million Dollar Problem that Went Unsolved for a Century - The Poincaré Conjecture
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- Опубліковано 20 тра 2024
- Topology was barely born in the late 19th century, but that didn't stop Henri Poincaré from making what is essentially the first conjecture ever in the subject. And it wasn't any ordinary conjecture - it took a hundred years of mathematical development to solve it using ideas so novel that they were worth at least a million dollars.
Here I talk about the Poincaré Conjecture by introducing fundamental topological concepts like homeomorphisms and homology, and eventually talk about its solution through Ricci Flow.
Note: At 12:25, it's 3-Manifold, not 3D Manifold. I don't know what possessed me to say 3D.
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Music Credits:
Inspired by Kevin MacLeod
Link: incompetech.filmmusic.io/song...
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Wind Of The Rainforest Preview by Kevin MacLeod
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Wholesome by Kevin MacLeod
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The Forest and the Trees by Kevin MacLeod
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Music: www.purple-planet.com
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Image Credits:
Tesseract, 24-cell and Tetrahedral Compound: Robert Webb's Stella software is the creator of this image. A link to the website: www.software3d.com/Stella.php.CC BY-SA 3.0 (creativecommons.org/licenses/..., via Wikimedia Commons
Klien Bottle Homology: Steelpillow, CC BY-SA 4.0 (creativecommons.org/licenses/..., via Wikimedia Commons
John Stallings Photograph: George M. Bergman, CC BY-SA 4.0 (creativecommons.org/licenses/..., via Wikimedia Commons
Saddle Shape: Nicoguaro, CC BY 3.0 (creativecommons.org/licenses/..., via Wikimedia Commons
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Sources and Citations:
1) people.math.osu.edu/fiedorowi...
2) www.math.unl.edu/~mbrittenham...
3) Evelyn Lamb, 'What We Talk about When We Talk about Holes,' Scientific American (blogs.scientificamerican.com/...)
4) infoshako.sk.tsukuba.ac.jp/~ha...
5) enry (math.stackexchange.com/users/..., Why is $S_1$ said to "enclose" a 1-dimensional hole if the region it encloses is a (2-dimensional) disk?, URL (version: 2017-06-06): math.stackexchange.com/q/2311769
6) Wikipedia contributors, "Homology (mathematics)," Wikipedia, The Free Encyclopedia, en.wikipedia.org/w/index.php?...
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Chapters:
00:00 - The Euler Characteristic
03:00 - Intro
03:10 - What Even is a Homeomorphism?
05:07 - The Euler Characteristic, Reprise
07:50 - Introducing Homology
09:20 - The need for a Fundamental Group
11:55 - The Conjecture
13:50 - MORE Dimensions
15:05 - Deforming Surfaces with Ricci Flow
17:02 - Ricci Flow 2.0
18:24 - The Million Dollars
I have a PhD in Theoretical physics and I enjoy your videos. Not sure what you are doing or planning on doing with your career but you have a great way of explaining things. I will be following for sure.
Robert , Have you ever heard of Phase Field theory?
I’m getting my PhD in theoretical physics and I agree with this. There’s a desperate need for those who can explain concepts well
Phd = Pure Heart Divided ymonkeys ;)
"I'll leave that one for you to figure out for yourself"
You have wildly over-estimated me, sir!
Video version of "left as an exercise for the reader".
It's all around the tourus, as is "the outside", then you also can't do anything to change it to the other shapes, because the line will go around the insides of the tourus and come out to the same place it was before.
@@Lufernaal thank you!
@@Lufernaal couldn't the same be said if you put it "inside" the hole?
@@roman111117 they're both equivalent, I think
This channel deserves so many more views and subscribers! Great content!
Glad you think so!
I am currently a high school senior taking taking AP Calc BC. I plan on being a math major and it’s videos like yours that motivate me every day
There is something deeply beautiful in topology, where the shapes defining feature is not as much the area shaping the hole, or any such property that can be expressed as positively being and thus causing the hole to appear. That all can be abstracted away, morphed and scaled, but the hole itself and crucially only the hole survives as defining feature that gives raise to unique characteristics. A negative being, lack of being, is what defines the object or objects that fall under the category.
The fact that we humans feel often so keen and connected to something we project as being and notice the lack off, or absence, and that it makes deep and complex sense mathematically, is to me beautiful. After all that is required to even notice a hole: you project something that is not there, and give particular meaning to the lack of it. Else it would be a tunnel; maybe set of low ceiling, walls and floor. There is no definite need to notify the absence or mark it with meaning: all things lack almost everything. Yet we gravitate towards negations and not only give them meaning but define and assign meaning to whole things based on what negations they include.
What a beautiful comment
You make me sick
Where do you hide your red sashes Ringo
Detestable thuggie strawdog pontoon
I just realised why odd dimensional shapes have Euler characteristic 2 and even dimensional shapes have Euler characteristic 0. Because each shape also has 1 of itself and 1 null set (negative 1 dimensional thing). This makes all Euler characteristics 0 (without holes).
That's the first time I've been told, why he rejected the price. Up until now I liked to believe that he valued math so much higher than anything else that he could simply not see the necessity of having a million dollars...
Just sen this channel. That was excellent. That was by miles the best explanation of a very difficult topic that I have ever seen. Excellent delivery too. I don’t understand the maths involved in topology but now I at least understand what the Poincaré conjecture was all about. Brilliant thanks.
Can I request a video on P vs Np particularly if P = NP with a fairly practical algorithm to solve NP problem in practical times?
To shut yourself away and focus on something until you solve it, that's dedication.
Excellent vid!
Beautifully well done explanations and top-notch quality!
This was really well done! Kudos to you man!
This is great! Thank you for giving me at least an intuitive notion of what Ricci flow is!
One of the best, clearest videos on any topic, let alone maths or topology. So nice I watched it twice!
By far the BEST exposition I've yet seen on The P.C., Topology & Ricci Flow!
I've 2 Math Degrees & was able to follow quite easily & enjoyably...LOL
U deserve more views & subscribers my young, versatle friend!
Keep up the excellent work indeed...
3:00 what's the tune name, btw liked the video 👍, make one on yang miller's and reimann too
The best video I’ve seen on this abstruse proof, well done. You are talented at this!
The quality of your videos are amazing man!
I appreciate that!
In 2010 Gregory Perelman proved that we live in 4-D Universe ( X^2+Y^2+Z^2+t^2=1). Now he is working on the problem of " Holes ". Possibly, will prove how this Universe was born.
Excellent job buddy. I love the way you explained. The no. of views on your videos shows that very few viewers are interested in such specialized topics. I wish more people get motivated to take up advanced mathematics.
I disagree with your conclusion. The number of viewers is primarily determined by the algorithmic shuffling of these videos, not the number who would enjoy if such videos were presented in their intray, as happened to me. Just saying.
I am surprised this doesn't have more views. Great video!
Great to see more videos, keep it up!
inb4 10k subs for the people who are coming from 1 mil! This channel is going places fs
Brilliant content with smooth explanation ! Please keep it up.
Thanks a lot!
I never made it past high school Trig, and I am able to follow along with you. You are an excellent communicator.
Great piece of work, nice job
Let me echo what others have said. This video is so very excellent for explaining and making accessible a complex and difficult and abstract topic. The creator and host of this video is a gifted teacher and orator. So glad I found this video. Adding it to my playlist for future reference!
The third path on the torus is where you draw it on top and encircling the hole, right?
Wonderful visuals. What a superb video.
Your animations are so beautiful ! What softwares do you use to make them ?
Normally Adobe After Effects, but there's a couple of 3D effects in this video which I used C4D to make.
well i am a biology student and came here just out of curiosity and i was awestruck by the way you explained it .. SUBSCRIBED
The other way is all around the tourus in a horizontal line. The first shape is a square, right? The second is a "ring" that circles the tourus like a ring circles a finger in a sort of vertical way and it can move around horizontally. The third is a straight line going all around the tourus, in a sort of horizontal way, and connecting it around it. You can then go around vertically, reaching the inside of the tourus. You can never make that line become the other two shapes.
Brilliant mann just keep going UA-cam takes some time to reach you to the people who actually loves it... Don't stop keep going ❤️
Awesome video, but little bit more info on Perelman solution would have been great 🙏
One of the finest and apt explanation on the problem statement BTW.
Noted. Thanks!
You are a very gifted teacher indeed
Smale came *before* Stallings, not after - but their work was independent of each other.
Rad! Best explanation on UA-cam bar-none.
I really like your videos, keep it up
Is a manifold the set of points
(x,y) €R^n*R such that y=f(x) with f being differentiable and with a non zero jacobian?
Great job btw, keep it up!
Great explanation
Oh my god you explained it so well, thank you!
great work....
Great video
You are so genius💖
Bhai bhai!
People like you make indians proud!
my guess for the 3rd unique type of loop on the surfuse of a turso is a circle around the girth of the shape.
Your videos are amazing 👍👍
Glad you like them!
Wonderful!
Glad you think so!
Hey, i love the content of your videos! You deserve much more attention. Only thing that makes it sometimes hard to unterstand and fallow is your accent. I know that is hard to change. I am from Germany so my accent isnt perfect neither. You can use an app called Elsa Speak to exercise your accent, it uses AI and is very good. I know this may sound stupid to you because you are probably a native speaker but it could help grow your channel a lot.
Thanks for the tips! This is not my normal accent, it's one that I'm putting on by popular demand. Hopefully as I get more used to speaking in a more Americanized accent (I'm also getting a new mic, so the entirety of the audio will eventually improve) the issue gets sorted. Meanwhile I've put subtitles on the video.
What’s the third way to draw the line around the thing???? I’m going insane I can’t figure it out
Very nice job my friend...
Glad you liked it!
Thank you
Good stuff...
Really good
Math truly has no limits
I still did not get why the man rejected the prize? What was his reason?
Waooo keep it going 🙌
Great content .. What are u doing brother ???
Your explanation is very good but background music is very distracting
I am a student in this subject and my research interest is in low dimension topology. You made it good but one thing I want to say that if you were bit slow it would be better..
Best EXPLAINed
man
Second video, aaand subbed.
Please bhaiya, make a playlist on topology and string theory... Please please please accept my request 🙏🙏🙏🙏
Where do you go to college
Did you make a massive error? [look at the frames between 1:39 and 1:50]
cool channel
I see you were interested in the poincaré conjecture
And here I thought topology was the same as topography
👍
cool
Still waiting on a video to explain this properly...let's see how yours fares....it's really not that complex, just needs to be explained visually
CUT CUT CUT CUT CUT
My one year old can make a spherical shape out of a doughnut shape. No problem.
"Vanilla circle"
Fetish language. Interesting.
13:50 - So they did what most mathematicians do when they do when they can't solve something: Make a prize for someone else to do it.
Great content, but mediocre minds will never catch it .That's why few subscribers and less views.
бля я извиняюсь, но меня прорвало, как же сложно объяснить кому то , что 3d сфера это двумерный объект... это походу просто за гранью интеллекта
A PERFECT WOMAN.
Really nice 👌 😍💋 💝💖❤️
Speak s l o w e r , t h a n k s
You are doing anything original or just cut copy and paste
Hamilton lost!, Perelann won! Why cannot Hamilton relise his loss?
Ohhhhh, so Trump was just doing topology! 8:23
dont call me homo😡