@@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
@@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.
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.
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.
@@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.
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.
I do wonder what Perelman is up to these days. Again, supremely good content! I've seen a lot of the Poincaré videos on TH-cam; your effort exceeds them all! Great presentation 👍👍
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 👏
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.
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.
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...
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
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.
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
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!!!!!
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 TH-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
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.
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.
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.
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.
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.
Thank you for the detailed explanation of this amazing conjecture! Watching this made me want to dive more into what these mathematical geniuses were thinking about
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.
The part I did not understand is the transition from "surgically removing the problem part". This happens a lot in my self-study of math problems and some theorems. My question is, how is that allowed? Not sure if that is because they usually don't go into the details of the rigorous proof (treating that part of the proof as "trivial"). What then happens to the surgically-removed "problem part"? How was it exactly rejoined to the manifold? I get that one can "cut it out" since it is a closed manifold (therefore becoming a sub-manifold and is therefore continuous as well and by itself can be reduced to a sphere and then a point). But how is that different from trying to first cut a torus or pasting a sphere in the hole of a doughnut to "remove" the problem part? Why is the surgery done in this case allowed but not for the old torus? If that is allowed by virtue of that "problem part" being a sub-manifold, then why even go through the trouble of cutting it out? A better proof should be to find a metric that you can apply to different sections and still converge the manifold to a point. Or, is that exactly how Perelman did it? That is, is the surgery theory just a way to apply multiple metric tensor values following different time scales (to adjust the shrinking/expansion of each section such that they reach a smooth curve at a specified time t) to different areas of the same manifold?
There are no holes in the “problem area” we see. It’s just a cobordism or something. So you can cut it out without changing the homotopy class or something, some invariant of the manifold which will witness that it didn’t really matter the way we cut it (as long as we fixed it afterwards). But it’s a good question
It seems that the Surgery Theory was used to show that the "problem part" was never a problem to begin with. It seems that, based the video's explanation, that everyone was approaching it in a wrong way, i.e. applying the metric across the manifold at the same time. Surely, if you try to squish something with a pointlike region/a neck, then that will shrink faster than the rest of manifold. So yeah, while Perelman did show the proof I feel unsatisfied because I think a more beautiful proof would be finding a metric that adjust itself to the shape of a given given (acting as an if-else algorithm) such that the metric self-adjust the speed at which it shrinks faster or slower to accommodate the shrinking without needing to cut out something. The surgery part still seems clunky to although I get the part where it shows that it is indeed possible to shrink any continuous manifold to a point as long as they are homeomorphic to a sphere (and why a torus with a hole cannot be shrinked to a sphere then a point because it is not homeomorphic to a sphere).
Some guy on reddit gas-lit me so hard on this problem. The post was, "What conjecture seems intuitively obvious, but turns out to be incredibly complicated." - My answer was the Poincare conjecture. Dude went on a tirade about how there's nothing intuitive about this problem. That there's absolutely no intuition behind this problem, and that I have absolutely no idea what I'm talking about. Small wins come from random videos, where other people are like, "Yeah, this problem seems obviously true..."
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.
Nobody is talking about the supreme discover and the benefits for the humanity. But thank you very much for the effort and the outstanding explanation. So, as a result we must say this is a sphere and this repeats every time a new string of matter borns. Am I correct or just daydreaming?
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 😊
5:22 Weird the video does not mention the original inventor's name: Richard Hamilton. He was _very well aware_ already around 1985 that his method could be used to prove the conjecture. Without Hamilton there would be no Perelman.
@Aleph 0 I like you channel and your videos have aided me, but I do have questions about this one. As a non-mathematician (a philosopher interested in topology and manifolds) I feel like you papered over how surgery isn't just cheating, since it appears like one is just collaging shapes together to get what one wants rather than deriving the desired results. Why should one be allowed to attached parts of a sphere to make something spherical, and how does it say anything about the topology of the object before the surgery? It's kind of like pulling a rabbit out of a hat, and then saying, "since I pulled this rabbit out of this hat, this rabbit must have been in this hat at all times before I pulled it out" when everyone watched you put the rabbit in the hat just before saying that. I realize in making it not appear like a slight of hand, it may become it too technical for the lay person, but then that is kind of the challenge you've set yourself here. If you or anyone could help a non-mathematician better understand surgery it would be appreciated!
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.
This guy's just really confusing, don't worry if you don't get it because it's really not a clear explanation.
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.
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.
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.
I do wonder what Perelman is up to these days. Again, supremely good content! I've seen a lot of the Poincaré videos on TH-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.
@@Jab_hutt he still lives in Russia.
th-cam.com/video/idr3C3lMoAQ/w-d-xo.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.
@user-yb5cn3np5q I thought they said he'd come to be disgusted by mathematics.
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 👏
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.
You are literally the only youtuber to whom i let the ads play full length. Amazing content, keep it up!
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 :)
I have never seen such an understood video about something so complicated. Congratulations.
3b1b:
Finally, a worthy opponent, our battle will be legendary.
A new gem has emerged on TH-cam!
Thanks for the video :)
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 :)
That's a great explanation of a surreal complex topic. I'm amazed.
I guess the comparisons with 3b1b are warranted.
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
This is definitely my favorite channel on youtube. Thank you for your hard work.
THANK YOU. Geez it’s impossible to get someone to give a straightforward answer about what this even is lol
thanks for the explanation. I have started this topic countless times but every time I'm drowning in details. good stuff sir.
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.
Best visualisation on the topic i've ever seen! Thank you!
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
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!
He solved "Thurston Geometrization Theorem", Poincare Conjecture is just one case of it.
What an awesome channel; I hope you'll get the publicity that you deserve.
The TH-cam algorithm just recommended me your channel and man it is simply amazing, can't wait for more videos to come out :)
The visual effects are awesome 🤩It really offers me a invitation into learning Ricci flow 🥳
Very interesting, informative and worthwhile video.
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.
Finally a nice video about this topic! Thank you so much!
What an incredible video, your channel deserves to be huge
Thank you! That's very kind :)
Wow... this channel gives exceptionally well made explanations.. please keep going!
Excellent video .. thanks TH-cam thanks Aleph 0 . Please keep making more
This video is a triumph of modern mathematics
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. Lots of effort were put into this, truly a great video; thank you!
A very nice Video, thank you for explaining the great ideas from Gregori Perelman 😀
another awesome video buddy!
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 :)
The most daunting problems in mathematics oftentimes have the most elegant solutions.
Great video. Thanks for the presentation.
Thanks! Appreciate it :)
What an explanation! Amazing! Thank you so much for this brilliant content.
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 TH-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
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.
Pretty cool, I actually vaguely comprehended that - thanks for the great explanation and visualization.
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
The sound is messed up near the end
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.
Awesome videos Man!!
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.
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.
This was a fun watch. I think it was so easy to understand because in some sense it would be really strange if the poincare conjecture was false.
Great work and clearly explained 👍👍
well explained for something difficult for many to grasp
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.
The inversion/eversion of the circle is best model for our manifold.
Thank you for the detailed explanation of this amazing conjecture! Watching this made me want to dive more into what these mathematical geniuses were thinking about
Thank you! I'm glad you enjoyed it :)
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.
Iam happy that I found this channel
This video is super well done! Hope you'll get more subs.
Thanks so much! Glad you enjoyed it :)
Absolutely amazing and concise explanation!
Wow glad I discovered your channel.
You’re going to be the next 3blue1brown.
Aw thanks!! Glad to have you join us :)
Sick animations man! Good job
Thanks man. Outstandingly clear.
Simplicity at its core 💯💯
That's a very simple and cute video for beginners. I hope you will get more attentions.
I hope to become such a fantastic mathematician in the future!
I understood that! Even though not good at understanding complex maths/problems. Thank you :)
Thanks! Glad you liked it.
The part I did not understand is the transition from "surgically removing the problem part". This happens a lot in my self-study of math problems and some theorems. My question is, how is that allowed? Not sure if that is because they usually don't go into the details of the rigorous proof (treating that part of the proof as "trivial"). What then happens to the surgically-removed "problem part"? How was it exactly rejoined to the manifold? I get that one can "cut it out" since it is a closed manifold (therefore becoming a sub-manifold and is therefore continuous as well and by itself can be reduced to a sphere and then a point). But how is that different from trying to first cut a torus or pasting a sphere in the hole of a doughnut to "remove" the problem part? Why is the surgery done in this case allowed but not for the old torus? If that is allowed by virtue of that "problem part" being a sub-manifold, then why even go through the trouble of cutting it out? A better proof should be to find a metric that you can apply to different sections and still converge the manifold to a point. Or, is that exactly how Perelman did it? That is, is the surgery theory just a way to apply multiple metric tensor values following different time scales (to adjust the shrinking/expansion of each section such that they reach a smooth curve at a specified time t) to different areas of the same manifold?
There are no holes in the “problem area” we see. It’s just a cobordism or something. So you can cut it out without changing the homotopy class or something, some invariant of the manifold which will witness that it didn’t really matter the way we cut it (as long as we fixed it afterwards). But it’s a good question
It seems that the Surgery Theory was used to show that the "problem part" was never a problem to begin with. It seems that, based the video's explanation, that everyone was approaching it in a wrong way, i.e. applying the metric across the manifold at the same time. Surely, if you try to squish something with a pointlike region/a neck, then that will shrink faster than the rest of manifold. So yeah, while Perelman did show the proof I feel unsatisfied because I think a more beautiful proof would be finding a metric that adjust itself to the shape of a given given (acting as an if-else algorithm) such that the metric self-adjust the speed at which it shrinks faster or slower to accommodate the shrinking without needing to cut out something. The surgery part still seems clunky to although I get the part where it shows that it is indeed possible to shrink any continuous manifold to a point as long as they are homeomorphic to a sphere (and why a torus with a hole cannot be shrinked to a sphere then a point because it is not homeomorphic to a sphere).
I love your explanation and understanding madness genius of Grigori Perelman
Some guy on reddit gas-lit me so hard on this problem. The post was, "What conjecture seems intuitively obvious, but turns out to be incredibly complicated." - My answer was the Poincare conjecture. Dude went on a tirade about how there's nothing intuitive about this problem. That there's absolutely no intuition behind this problem, and that I have absolutely no idea what I'm talking about. Small wins come from random videos, where other people are like, "Yeah, this problem seems obviously true..."
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.
Simplifying such intricate math takes some grt effort ..... kinda what this video does here.
Aleph 0 is a great channel name btw.
Thank you for this wonderful video! :-)
Wow great channel .. Nice explanation
how do you make these geometry animations? 3b1b does it but less with the dough style shape you used.
hey! can you make videos on all the millenium problems?
That's the plan!
Why have you removed your learning undergraduate maths video, pleas reupload
excellent explanation, thank you so much!
Amazing video ..... 👍👍👍👍
Thanks!
Did he solved N=3 or N=4? I’m confused.
N=3
Man that was awesome
This was helpful. Thank you much
Nobody is talking about the supreme discover and the benefits for the humanity.
But thank you very much for the effort and the outstanding explanation.
So, as a result we must say this is a sphere and this repeats every time a new string of matter borns. Am I correct or just daydreaming?
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 😊
Amazing explanation!
incredible video, you guys are amazing
Thanks for stopping by!
5:22 Weird the video does not mention the original inventor's name: Richard Hamilton. He was _very well aware_ already around 1985 that his method could be used to prove the conjecture. Without Hamilton there would be no Perelman.
I understood nothing.
You were born and will die with nothing. What else is there to say?
n=3 is one of the greatest parts of mathematics
Agreed.
It reminds me a little the equations from general relativity for the Minkowski spacetime mass and energy interaction
It's only a matter of time before the 3b1b collab.
Very nice presentation dude.
Thank u for this one...💯❤
@Aleph 0 I like you channel and your videos have aided me, but I do have questions about this one. As a non-mathematician (a philosopher interested in topology and manifolds) I feel like you papered over how surgery isn't just cheating, since it appears like one is just collaging shapes together to get what one wants rather than deriving the desired results. Why should one be allowed to attached parts of a sphere to make something spherical, and how does it say anything about the topology of the object before the surgery? It's kind of like pulling a rabbit out of a hat, and then saying, "since I pulled this rabbit out of this hat, this rabbit must have been in this hat at all times before I pulled it out" when everyone watched you put the rabbit in the hat just before saying that.
I realize in making it not appear like a slight of hand, it may become it too technical for the lay person, but then that is kind of the challenge you've set yourself here. If you or anyone could help a non-mathematician better understand surgery it would be appreciated!
Well done! Nice channel!
Wow this channel is good!