Poincare Conjecture and Ricci Flow | A Million Dollar Problem in Topology
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- Опубликовано: 28 июн 2024
- How do we use Riemannian Geometry and Surgery Theory to crack a million-dollar problem in topology? Ricci flow, that's how. In this video, we tackle the only Millennium Prize Problem that's been solved so far, and find the deep mathematics uncovered in the process.
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Official Problem Statement:
www.claymath.org/millennium-p...
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Music Info: Documentary - AShamaluevMusic.
Music Link: www.ashamaluevmusic.com
Intro: (0:00)
Poincare Conjecture: (0:45)
Riemannian Geometry: (2:31)
Ricci Flow: (4:17)
Surgery Theory: (7:10)
Proof of Poincare Conjecture: (7:26)
0:20 “By the end of this video you’ll understand exactly what it is” - how dare you overestimate me, sir
Like, I don't even know what the hell this is supposed to be
@@zaxarrrr3659 There's a reason the guy got $1,000,000 for it! Lol
@@67hoursAndCounting actually he did not took the money.
Look up positive curvature and negative curvature then this video will make perfect sense. Ricci flow is the process of the manifold evolving according to the rule R=-dg/dt as explained.
This guy's just really confusing, don't worry if you don't get it because it's really not a clear explanation.
Disclaimer: there are easier ways to make a million dollars
But there's no easier way to make an original contribution. I mean no matter how much we celebrate cutthroat salesmen, they are still feeding off the innovations of these people who made original discoveries.
He didn't take the million dollar...
My favorite way to make a million dollars is to start with ten million dollars. Then spend nine.
@@arthdubey Yeah, but can somebody else claim it?
@@DJVARAO No, because the prize was for someone who solves any one of the millenium problems. Since it's well established that Perelman solved it, they can't.
The question is: WHO CARES?
Well, Poincares.
Lame..
This is amazing lmao
pada...bushhhhh.....
:D
Funny
Ricci Flow always sounded like a rapper name to me...
If so, he should be Italian, no?
Here's a rap about Ricci Flow (credit to ChatGPT):
(Verse 1)
Yo, it’s a geometric flow, call it Ricci,
Transformin’ metrics, smooth and tricky.
In the realm of manifolds, it’s a revolution,
Aimin’ for the shape that’s the best solution.
(Hook)
Ricci Flow, where the curves align,
Evolvin’ shapes, through space and time.
With Hamilton's touch, it starts to glow,
This ain't just math, it's a dynamic show.
(Verse 2)
From the streets to the sheets of a complex map,
It smooths the curves, no gaps, no trap.
Curvature decreasin’, it’s a smooth operator,
Reshaping the universe, like a skilled innovator.
(Bridge)
Three dimensions, spheres get rounder,
Topology’s king, no bounds to flounder.
Grigori Perelman, he dropped the mic,
Solved Poincaré, and he did it right.
(Outro)
So, when you hear Ricci, think of the flow,
Mathematical rhythms that uniquely grow.
It's not just equations or abstract art,
It's the poetry of science, where change is the heart.
As an engineer, I can tell you with cerainty that it's true for small angles.
cerainty
Please spill some explanation for us regular folks.
@@aniksamiurrahman6365 Engineers often cut corners in solving problems for the sake of simplicity. This particular joke might be a reference to the small angle approximation, often used in engineering courses for solving harmonic motion problems and such. en.wikipedia.org/wiki/Small-angle_approximation
If you truncate the Taylor expansion at the linear term, almost anything is possible.
Sin(x) = x
Man your effort is appreciated. I hope your channel grows.
Thanks!!
@@Aleph0 just FYI, in Putin's Russia mathematicians are tortured with a screwdriver. Google Azat Miftakhov for details. So the general advise is : whenever you see name Putin - spit, curse and hit dislike.
@@BgAndrew100 shut up lmao it's a joke account name
Andrei Kalinin I looked up Azat. The Russian police might have viewed him as a rich boy and so they arrested him and expected his family to bribe police for his release. Police in Russia are horridly corrupt.
As a tourist, you would need to carry lot and lot of money because when you get stopped on the traffic and be falsely accused of speed driving offences, you must pay the fines to the self-serving Russian police. Sometimes, you can get stopped up to 10 times a day by Russia police, if they see how rich you are. It happened to my sister's neighbour. I am from Ireland and you would not believe how often that Russian police stop Irish tourists and other foreigners on the roads. My sister's neighbour swears that the Russian police are getting rich off foreigners.
Andrei Kalinin PS, you should hide your name, when you are politically disagreeable.
I do wonder what Perelman is up to these days. Again, supremely good content! I've seen a lot of the Poincaré videos on RUclips; your effort exceeds them all! Great presentation 👍👍
Haha yes - we all wonder that! Thank you for the kind words :)
Last I heard of him, which was like more than 5 years ago, that he took some kind of job in Finland and moved there with his mother.
@@jabhutt1013 he still lives in Russia.
ruclips.net/video/idr3C3lMoAQ/видео.html
Sometimes he goes to Sweden.
He hated the publicity so who knows if he ever works on anything again. Would love to see him work with Tao.
It was suspected he was working on Navier-Stokes now.
He solved "Thurston Geometrization Theorem", Poincare Conjecture is just one case of it.
THANK YOU. Geez it’s impossible to get someone to give a straightforward answer about what this even is lol
Holy shit. I didn’t think I’d ever see a video that explains a millennium problem this well, let alone problem + solution 🤯 My new favorite math channel for sure 👏
This is definitely my favorite channel on youtube. Thank you for your hard work.
You are literally the only youtuber to whom i let the ads play full length. Amazing content, keep it up!
Best math video I have seen in a very long time! If you keep delivering this quality videos you will have a big success. Totally subscribed!
3b1b:
Finally, a worthy opponent, our battle will be legendary.
I can't describe how glad I am that I found the channel. Thx for the content bro!
Beautiful, absolutely beautiful. Thank you so much. I wish you nothing but the highest orders of success because you’re helping more humans than you could ever imagine with this
I have never seen such an understood video about something so complicated. Congratulations.
This was the best video explaining the poincaré conjecture that I've found, awesome!! Of course I'll have to watch it some more 3 times to get a better grasp of the math, but I got the chills in the end nevertheless. Pretty elegant proof, that surgery thing is a great insight, never heard of it before.
Thank you! I totally got the chills too (that is, when I finally understood the proof :P). Glad you enjoyed it :)
A new gem has emerged on RUclips!
Thanks for the video :)
This video deserves million views. Quality content.
Absolutely killer video man that was awesome. Really felt like understood it after watching it and was thinking the whole time about how it might relate to physics.
thanks for the explanation. I have started this topic countless times but every time I'm drowning in details. good stuff sir.
Best visualisation on the topic i've ever seen! Thank you!
Amazing. Lots of effort were put into this, truly a great video; thank you!
Finally a nice video about this topic! Thank you so much!
What an awesome channel; I hope you'll get the publicity that you deserve.
That's a great explanation of a surreal complex topic. I'm amazed.
I guess the comparisons with 3b1b are warranted.
Wow... this channel gives exceptionally well made explanations.. please keep going!
What an incredible video, your channel deserves to be huge
Thank you! That's very kind :)
Pretty cool, I actually vaguely comprehended that - thanks for the great explanation and visualization.
Thanks man. Outstandingly clear.
another awesome video buddy!
The visual effects are awesome 🤩It really offers me a invitation into learning Ricci flow 🥳
Awesome videos Man!!
Brilliantly simple explanation. The video does it all, at least for us with less expertise in the field. I could not imagine those shapes in this context without the video. Indeed an images is worth 1000 words...
Thanks @Bogdan! Glad you liked the video :)
Very interesting, informative and worthwhile video.
Sick animations man! Good job
Absolutely amazing and concise explanation!
What an explanation! Amazing! Thank you so much for this brilliant content.
I just found your Channel and i am amazed of the quality of your Content.it's Really extremly interesting And well explained. Keep it up! :)
Thanks! Glad that you found us :)
Amazing video. Glad I found this gem of a channel!
Awesome, thanks!
@@Aleph0 its rare finding high quality channels with little views. Keep up the good work! You’ll get a large audience in no time.
Great work and clearly explained 👍👍
This is just awesome, I wish you the best for your channel as this video is as beautiful as the idea behind the proof it presents. =D
Excellent video .. thanks RUclips thanks Aleph 0 . Please keep making more
This was helpful. Thank you much
excellent explanation, thank you so much!
The RUclips algorithm just recommended me your channel and man it is simply amazing, can't wait for more videos to come out :)
This was brilliant and deserves a lot more views the one thing I didn't understand was actually the very ending I'm not a mathematician I can barely add and subtract but this was a beautiful and intuitive proof
Don't worry adding and subtracting are useless in this age unless you are a cashier.
This is so cool! Thank you for the video.
This video is super well done! Hope you'll get more subs.
Thanks so much! Glad you enjoyed it :)
This video is a triumph of modern mathematics
Thank you for this wonderful video! :-)
Great video. Thanks for the presentation.
Thanks! Appreciate it :)
Great video! Love it.
Did he solved N=3 or N=4? I’m confused.
Amazing explanation!
Well done! Nice channel!
Excellent!!!
The most daunting problems in mathematics oftentimes have the most elegant solutions.
Iam happy that I found this channel
Incredible video! Very illuminating even to a layperson like myself! Can't thank you enough for the effort!
Is it possible to explain (at the level of this video, of course) whether this argument generalizes to other dimensions, and if not, why?
Once again, thank you very much for creating such wonderful educational content! I wish you all the best 😊
Great video!
Thanks man, that was a great explanation, though I would’ve loved more details on the surgery part.
The idea of "surgery theory" comes from Richard Hamilton as he proved that you could use it to fix the curvature of those objects which result in unwanted singularities under Ricci flow. The conference he presents his proof at is on RUclips and provides an in-depth explanation of how it works. Perelman actually stated, when asked why he didn't accept the $1mil awarded to his proof, that his[Perelman's] proof was no more impressive than Hamilton's proof
Thank u for this one...💯❤
Man that was awesome
This video is awesome
tip for the animation: you can add a sort of "smoothing" when combining two objects. Of course with low level access to the renderer it's easy, but even in programs like blender they have metaballs and stuff which will make the spheres combining looks smooth.
Loved it truly👍
Thank you!
Amazing Vid .. Thank you Creating it..
A very nice Video, thank you for explaining the great ideas from Gregori Perelman 😀
SPELLBINDINGLY BEAUTIFUL. Thank you Aleph. If it is possible for Schrodinger's wave function of quantum sates to clump up like Ricci flows, then it might be possible to define how classical objects (planets, suns, black holes etc.) can evolve from quantum states and Hawking's theory of the unitary evolution of the entire universe, maybe correct.
Sir your presentation is amazing , you are an inspiration for me and I hope I will be able to learn a lot from your channel .
Thank you!! That's very kind. (btw: I love your channel picture; very classy.)
@@Aleph0 thank you sir
incredible video, you guys are amazing
Thanks for stopping by!
Simplicity at its core 💯💯
I understood that! Even though not good at understanding complex maths/problems. Thank you :)
Thanks! Glad you liked it.
Amazing video and content! What program do you use to make the animations?
Brilliant!
Fantastic explanation, especially considering I'm not a mathematician yet I understood precisely what you were trying to get across. Thank you that's the first video I've watched on the subject that made it clear.
Can you point me to an exact equation for the pointcare that may help explain movements and financial markets that go beyond the rational explanation? Market topology seems to be one of the fields that could do this. There is a group of traders that supposedly used Perelman's algorithm in their AI to achieve tremendous results. Thank you.
The inversion/eversion of the circle is best model for our manifold.
Beautiful video! Unbelievable work I literally grasped everything in a single pass. Intuitively it's a very simple solution in retrospect - but that's the thing with these asymmetric types of problems.
I assume there is a close connection with NP problems here - it's hard to find the solution (exponential complexity) easy to verify it's correct (polynomial complexity).
Who knows maybe one of these tools will be used to crack the P=NP problem.
Best explanation!
This was a fantastic video, thanks! One thing I was missing, however, is a reasoning for why n=3 was so much more difficult.
I'm no where near understanding the maths behind this whole question, but from what I've heard the difference is that the techniques used in higher dimensions require "moving stuff around" in such a way that they could not be applied either in 4 or 3 dimensions. Hence entirely different proofs for both of those cases.
Amazing explanation can you do similar tutorials on other clay institute problems
Awesome video
Great video and great channel! You illustrate the idea of Perelman‘s proof very nicely.
What you don’t mention, however, is where the real „hard work“ in his proof had to be done: namely to control the geometry of the evolving necks in such a way that one knows that after surgery the next singularity will occur only after a controlled amount of time. This is necessary in order to guaranty that only finitely many surgeries happen before extinction.
By the way, the regions near the surgery look much more like very long tubes and not like cones, but I admit that this is really hard to illustrate.
When I see Ricci Flow in the title, I subscribe :D
Excellent
That's a very simple and cute video for beginners. I hope you will get more attentions.
Wow great channel .. Nice explanation
i love watching these kinds of videos and confusing myself
Oh man!!! Beautiful indeed, isnt´it. Let me tell you that I didn´t have any idea about Perelman contribution. Great!! Now I think I understand why he didn´t accepted the million dollar prize. Ricci flow was really relevant for him. As for me I think it is a quiestion of humbleness. But what a humble guy!!!!!
amazing content, wow
im a clueless of math stuff, but i like them.. its somehow inspiring...it pushes me to think on boundaries of human mind and its working principles.. math is a human creation, boundaries of math are the form of pure human mind; everything we create, problems or solutions, everything we find in searching for answers is just an reflection of our mind field.. and we all can go there and search, its just that someone who doesnt know math LANGUAGE practically cannot do it in the same way someone who knows can, but intuitively its very possible.. boundaries of our language are boundaries of our world
nah, the notations of maths was invent but maths itself is a product of nature, language and nature doesn't describe maths, maths describes THEM, maths is intrinsically the language of nature itself.
I hope to become such a fantastic mathematician in the future!
Wow this channel is good!
Amazing video ..... 👍👍👍👍
Thanks!
hey! can you make videos on all the millenium problems?
That's the plan!
Take a pan of water and try heating the pan, drop some oil randomly, and observe the motion of oil blobs as the temperature increases. Discrete oil bubbles coalesce to form larger circular blobs of oil, eventually to largest possible. If the container size is large, the oil drop not only maintains circular shape, but keeps on increasing in size.
This seems like a nice physical process for Ricci flow. The g and R properties can be shown to be preserve the flow equations, until turbulence destroys everything to make it point like oil drops.
Such a simple explanation for so enormous an undertaking. Thank you very much for posting this video.
Where can I go to see every proof leading up to the Poincaré Conjecture itself?
Sorry for the late reply! Here are the links to a detailed explanation of Perelman's Proof by Terrence Tao (though, I must say, I understood approximately 0% of this paper, so you're very brave for looking through it!)
arxiv.org/abs/math/0610903
Enjoy!
@@Aleph0 You are a Godsend! A few years back I tried to go through Terence's course on Ricci Flow, and could understand very little. Your video is so beautiful that I want to see the beauty of the entire exposition from the minds of those who were graced with its discovery.
I can't wait for your new videos!
The sound is messed up near the end
How'd you create those computer visuals? I like your paper/pen/hands-on approach a lot...there is a lot of charm and delight in simplicity. I sometimes get jealous watching say 3b1b videos because I could never make those animations, and now I'm jealous of your animations too!