Poincare Conjecture and Ricci Flow | A Million Dollar Problem in Topology
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- Опубліковано 20 тра 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.
The question is: WHO CARES?
Well, Poincares.
Lame..
This is amazing lmao
pada...bushhhhh.....
:D
Funny
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.
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
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.
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 UA-cam; 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.
ua-cam.com/video/idr3C3lMoAQ/v-deo.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.
3b1b:
Finally, a worthy opponent, our battle will be legendary.
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.
He solved "Thurston Geometrization Theorem", Poincare Conjecture is just one case of it.
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!
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
A new gem has emerged on UA-cam!
Thanks for the video :)
That's a great explanation of a surreal complex topic. I'm amazed.
I guess the comparisons with 3b1b are warranted.
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 :)
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!
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.
I have never seen such an understood video about something so complicated. Congratulations.
Best visualisation on the topic i've ever seen! Thank you!
thanks for the explanation. I have started this topic countless times but every time I'm drowning in details. good stuff sir.
This video deserves million views. Quality content.
Wow... this channel gives exceptionally well made explanations.. please keep going!
The visual effects are awesome 🤩It really offers me a invitation into learning Ricci flow 🥳
What an awesome channel; I hope you'll get the publicity that you deserve.
The UA-cam algorithm just recommended me your channel and man it is simply amazing, can't wait for more videos to come out :)
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 :)
Pretty cool, I actually vaguely comprehended that - thanks for the great explanation and visualization.
What an explanation! Amazing! Thank you so much for this brilliant content.
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.
What an incredible video, your channel deserves to be huge
Thank you! That's very kind :)
Very interesting, informative and worthwhile video.
Sick animations man! Good job
another awesome video buddy!
Absolutely amazing and concise explanation!
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 :)
This video is a triumph of modern mathematics
Thanks man. Outstandingly clear.
excellent explanation, thank you so much!
Excellent video .. thanks UA-cam thanks Aleph 0 . Please keep making more
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
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.
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 😊
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.
Awesome videos Man!!
This was helpful. Thank you much
Iam happy that I found this channel
Thank you for this wonderful video! :-)
This is so cool! Thank you for the video.
Amazing explanation!
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.
Great video. Thanks for the presentation.
Thanks! Appreciate it :)
Amazing Vid .. Thank you Creating it..
Excellent!!!
This video is super well done! Hope you'll get more subs.
Thanks so much! Glad you enjoyed it :)
Well done! Nice channel!
This concept reminds me of how glass condenses when its heated up; regardless of its initial shape it always wants to end up as a sphere with enough time and heat.
A very nice Video, thank you for explaining the great ideas from Gregori Perelman 😀
The most daunting problems in mathematics oftentimes have the most elegant solutions.
Simplicity at its core 💯💯
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.
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 UA-cam 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
That proof is quite the elegant one, if I would say so myself! I had already seen an outline of the proof while I was reading a book about the conjecture, but it still amazes me quite a bit. Another thing related to it that find interesting is Perelman himself, he's quite the interesting character, pretty much refusing both a Fields Medal and the 1 million dollar prize, and that if I remember correctly, he treated on his first paper about it, he presented it as just an afterthought, just a corollary of his proof of the geometrization conjecture (which, admittedly, was a very big result, but c'mon, we are talking here about the Poincaré's conjecture!). Speaking of famous conjectures, I wonder if you could do a video on Fermat's Last Theorem, but not just talking about it, but do something similar to this one, giving a bit of insight and a general overview of how the proof goes, talking about the actual theorem that was proved by Andrew Wiles, the modularity of elliptic curves, or you think that's just way too out of your league? Anyway, great video, looking forward to the next one.
That's a BRILLIANT idea! Thanks for suggesting it. I can't exactly claim to be the world's expert on modular forms :P but I guess that's a chance to learn something new and present it! I'd love to take that on.
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.
Great video! Love it.
Thank u for this one...💯❤
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.
The inversion/eversion of the circle is best model for our manifold.
n=3 is one of the greatest parts of mathematics
Agreed.
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.
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
Amazing video and content! What program do you use to make the animations?
Great video!
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!!!!!
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.
Man that was awesome
Hey really enjoyed the vid!! btw im wondering if you can give me the name of the name of the soundtrack used in the video(i went to the website you posted but could not find it ) thanks for advance!
I finally found it after weeks lmao
if anyone wants it here is the link to the video : ua-cam.com/video/d77QqiM5Mzc/v-deo.html
I hope to become such a fantastic mathematician in the future!
Aleph 0 is a great channel name btw.
This, super short explanation of the ideas behind the proof of the Poincare conjectute, is brilliant and also deserves a prize. I had written a short explanation of the proof of the Poincare conjecture for physicists. It got 1325 reads at arXiv.org. Knowing what this is all about in details, I freely admit that for the unassumed readers it is best among the best. This is an example of how books on math should be written for school children. Only late Martin Gardner would be able to come close to this gem.
Thesis is dual to anti-thesis creates converging thesis or synthesis -- the time independent Hegelian dialectic.
The Ricci tensor is dual to the Weyl tensor synthesizes Riemann geometry -- Sir Roger Penrose.
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature or gravitation is therefore dual.
Convergence is dual to divergence.
Apples fall to the ground because they are conserving duality.
Potential energy is dual to kinetic energy.
Action is dual to reaction -- Sir Isaac Newton.
Gravitation is equivalent or dual to acceleration - Einstein's happiest thought, the principle of equivalence (duality).
Energy is dual to mass -- Einstein.
Dark energy is dual to dark matter.
The big bang is a Janus hole (point) or composed of two faces = Duality.
"Always two there are" -- Yoda.
Duality creates reality!
@@hyperduality2838 You had your chance to prove Poincare hypothesis with your duality ideas.However, you did not! I was talking about presentation.The guy did his job fantastically.The rest is up to readers. You see it using duality as an universal cure...Fine.Then write a paper for Annals of Mathematics and publish it. And then, send me a pdf file of your paper. After that we shall talk again
@@SuperArkleo I do not have to prove anything.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
Your mind/soul is a dual concept.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Mind is dual to matter -- Descartes.
Mind duality is the next level beyond Descartes.
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
@@hyperduality2838 You surely do not need to prove anything to anybody. But I am surely entitled also to read works of scientists whom I respect as well as to write my own scientific papers for common good. Incidentally, what your duality is saying about masses? Is Higgs mechanism of mass generation has its dual?
@@SuperArkleo The Higgs duality field.
Symmetry is dual to anti-symmetry.
Bosons (symmetric wave functions) are dual to Fermions (anti-symmetric wave functions).
The Higgs field is actually dual, nothing is dual to something.
Wow glad I discovered your channel.
You’re going to be the next 3blue1brown.
Aw thanks!! Glad to have you join us :)
I understood that! Even though not good at understanding complex maths/problems. Thank you :)
Thanks! Glad you liked it.
Brilliant!
Awesome video
PlayDoh proved this long before Grisha.
pretty good description.
Excellent
Wow great channel .. Nice explanation
incredible video, you guys are amazing
Thanks for stopping by!
amazing content, wow
It is important to mention that Perelman completed Hamilton's program. The idea of Ricci Flow with Surgery was already floating around, Perelman just finished the job.
"just finished the job"
@@MIbra96 Perelman himself even felt this way, a contributing element in the drama of his rejection of the fields medal.
@@jessemoeller8557 Perelman gave several reasons for his refusal, but you took only one and brought it to the point of absurdity. What an ignorance!
For those, who want to understand the reasons for Perelman's refusal from numerous awards, I suggest reading the article Manifold Destiny (The New Yorker)
@@alex_grothendieck9701 I said explicitly that this was merely a contributing element. There's nothing absurd, or even remotely false, about what I said. You are the one bringing things to the point of absurdity.
He rounded it out.
i love watching these kinds of videos and confusing myself
Loved it truly👍
Thank you!