@@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
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.
@@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.
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.
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 👍👍
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 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.
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.
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...
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!!!!!
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.
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.
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).
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
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.
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.
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.
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..."
@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!
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.
Division operator is the most important because it has implied limits and the fundamental theorem of calculus requires approaching the divisor’s limit. Considering quantum physics the lepton defining factor is spin which is dependent on measuring real values where again there’s a quotient involved so somehow electricity implies reality via neurons.
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.
This was inspiring but I feel I’m no closer to actually understanding anything. When reversing the Ricci flow, why do the spheres form necks and connect with each other smoothly removing a hole (ie two spheres turn into one)? Wouldn’t spheres just keep increasing (under time reversal) until they touch? And then what are the rules when two disconnected manifolds touch, does one need to use the time reversal of the surgery step before the spheres meet at the point of intersection?
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?
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
great explanation, thank you. But I an confused about something, if the singularities are manually removed, then we are leaving the part of the set out of the proof, how is that addressed?
When we "hit the rewind button" and go back to look for necks- those necks are what we had right before manually removing them and replacing them with caps. So, without the surgery the neck would stretch out until we had a singularity but we performed surgery on the neck to get two balls- by rewinding back to when the necks existed we've rewound to before we removed parts of the set. Does that make sense?
@@austinalderete2730 I see. I looked up surgery theory and it makes perfect sense. Do you know if the choice of the manifold to replaces the necks is unique? like instead of unconnected cups, he could have used a single cylinder like manifold - what's the problem with that? Also, seems like the surgery manifold must be of positive curvature, or is that not necessary condition?
@@austinalderete2730 that clarified a lot for me, thanks! It's often a question of semantics.. When he said "surgery" in the video, I thought to myself "what are the rules of that game, what he is even talking about, surgery?". Turns out it's a whole theory!
i've got an astrophysics degree and worked as a seismic processor for a decade...i find this analogous to deconvolution... they set off a charge of dynamite down a hole and a whole array of detectors record the sound waves and by modeling the original explosion, they can deconvolute the signal to produce a picture of the earth below... in this case, the sphere is the final product and you're deconvoluting it from the initial condition to define the characteristics of the vector space?
Can anybody help explain to me why it's not just completely arbitrary and kind of just cheating to chop it in two and add a couple of half spheres? And how do the two separated manifolds just magically join back together? Wouldn't the definition of the 'surgery' tactic already presuppose success because half spheres were chosen and not, say, the top of a pyramid? This is hard for my brain
Wouldn't removing the problem-areas manually change the underlying structure of the manifold, just like a ricci flow would with singularities? Why is it allowed to just substitute problem-areas with spheres' sections?
As explained in the video, you're right, it does change the structure of the manifold. In fact it disconnects the manifold into a number of parts, where on each part Ricci Flow can be done properly. The last step of the proof, then, is to show that when you put these parts back together in the same way you took them apart, you still end up with a sphere.
@@avz1865 ah, I think I grasp that concept now, thank you. I still feel the need to look deeper into the mathematics of those last two steps, but I'm afraid that, even with my engineering degree, the proof would be way out of my reach ':)
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!
@John Cavanaugh Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. There is a 4th law of thermodynamics based upon the fact that your mind is predicting reality into existence, entangled entropy or mutual information is correlated information or syntropy. Predictions are used to track targets -- teleology. Teleological physics (syntropy) is dual to non-teleological physics (entropy). "Through imagination and reason we turn experience into foresight (prediction)" -- Spinoza describing syntropy. The word "spammy" is incorrect according to Spinoza, he would have disagreed with you.
@Electro_blob That is because it is correct. Mind (intangible) is dual to matter (tangible) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality. Active matter (life) is dual to passive matter (atoms, forces) -- matter duality. Mind duality is dual to matter duality. There is a second level of duality beyond Descartes according to Kant, Aquinas and physics. Energy is duality, duality is energy, an apple converts potential energy into kinetic energy when falling in a gravitational field or duality is being conserved. Electro is dual to magnetic, light or photons are dual -- Maxwell's equations. Positive charge is dual to negative charge -- Electric fields. North poles are dual to south poles -- Magnetic fields. Electro-magnetic energy is therefore dual. Everything in physics is made out of energy or duality! Duality creates reality, the big bang is a Janus hole/point, all holes in topology are dual -- the 1st homology group of the real projective plane is a finite group with two elements [0,1]. The bad news is that "philosophy is dead" -- Stephen Hawking. If you think that your mind/soul is dual (Kant, Aquinas) then Stephen Hawking is wrong!
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.
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
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.
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.
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.
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.
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.
@@Jab_hutt 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.
@user-yb5cn3np5q I thought they said he'd come to be disgusted by mathematics.
You are literally the only youtuber to whom i let the ads play full length. Amazing content, keep it up!
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 👏
3b1b:
Finally, a worthy opponent, our battle will be legendary.
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.
A new gem has emerged on UA-cam!
Thanks for the video :)
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.
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.
He solved "Thurston Geometrization Theorem", Poincare Conjecture is just one case of it.
Best visualisation on the topic i've ever seen! Thank you!
That's a great explanation of a surreal complex topic. I'm amazed.
I guess the comparisons with 3b1b are warranted.
thanks for the explanation. I have started this topic countless times but every time I'm drowning in details. good stuff sir.
The sound is messed up near the end
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 :)
I can't describe how glad I am that I found the channel. Thx for the content bro!
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
What an awesome channel; I hope you'll get the publicity that you deserve.
THANK YOU. Geez it’s impossible to get someone to give a straightforward answer about what this even is lol
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!
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!
Wow... this channel gives exceptionally well made explanations.. please keep going!
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!!!!!
What an incredible video, your channel deserves to be huge
Thank you! That's very kind :)
The UA-cam algorithm just recommended me your channel and man it is simply amazing, can't wait for more videos to come out :)
another awesome video buddy!
The visual effects are awesome 🤩It really offers me a invitation into learning Ricci flow 🥳
Excellent video .. thanks UA-cam thanks Aleph 0 . Please keep making more
Awesome videos Man!!
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.
The most daunting problems in mathematics oftentimes have the most elegant solutions.
Very interesting, informative and worthwhile video.
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 :)
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.
Amazing. Lots of effort were put into this, truly a great video; thank you!
What an explanation! Amazing! Thank you so much for this brilliant content.
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).
The inversion/eversion of the circle is best model for our manifold.
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
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
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.
Pretty cool, I actually vaguely comprehended that - thanks for the great explanation and visualization.
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. Thanks for the presentation.
Thanks! Appreciate it :)
Why have you removed your learning undergraduate maths video, pleas reupload
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
This video is super well done! Hope you'll get more subs.
Thanks so much! Glad you enjoyed it :)
A very nice Video, thank you for explaining the great ideas from Gregori Perelman 😀
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.
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.
Thanks man. Outstandingly clear.
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..."
Great work and clearly explained 👍👍
@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!
how do you make these geometry animations? 3b1b does it but less with the dough style shape you used.
Sick animations man! Good job
It reminds me a little the equations from general relativity for the Minkowski spacetime mass and energy interaction
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.
Wow glad I discovered your channel.
You’re going to be the next 3blue1brown.
Aw thanks!! Glad to have you join us :)
well explained for something difficult for many to grasp
Absolutely amazing and concise explanation!
Iam happy that I found this channel
Division operator is the most important because it has implied limits and the fundamental theorem of calculus requires approaching the divisor’s limit. Considering quantum physics the lepton defining factor is spin which is dependent on measuring real values where again there’s a quotient involved so somehow electricity implies reality via neurons.
Thank you for this wonderful video! :-)
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.
This was inspiring but I feel I’m no closer to actually understanding anything. When reversing the Ricci flow, why do the spheres form necks and connect with each other smoothly removing a hole (ie two spheres turn into one)? Wouldn’t spheres just keep increasing (under time reversal) until they touch? And then what are the rules when two disconnected manifolds touch, does one need to use the time reversal of the surgery step before the spheres meet at the point of intersection?
Did he solved N=3 or N=4? I’m confused.
N=3
That's a very simple and cute video for beginners. I hope you will get more attentions.
Aleph 0 is a great channel name btw.
I thought his name was Grigori Perelman and I think it was more recent than 2002?
So.. is liquid Terminator an example of Ricci flow?
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?
Well done! Nice channel!
7:38 why do we see necks forming? this part seems like the crux of how he solved the problem and is somewhat rushed in the video :/
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 :)
great explanation, thank you. But I an confused about something, if the singularities are manually removed, then we are leaving the part of the set out of the proof, how is that addressed?
When we "hit the rewind button" and go back to look for necks- those necks are what we had right before manually removing them and replacing them with caps. So, without the surgery the neck would stretch out until we had a singularity but we performed surgery on the neck to get two balls- by rewinding back to when the necks existed we've rewound to before we removed parts of the set.
Does that make sense?
@@austinalderete2730 I see. I looked up surgery theory and it makes perfect sense. Do you know if the choice of the manifold to replaces the necks is unique? like instead of unconnected cups, he could have used a single cylinder like manifold - what's the problem with that? Also, seems like the surgery manifold must be of positive curvature, or is that not necessary condition?
@@austinalderete2730 that clarified a lot for me, thanks! It's often a question of semantics..
When he said "surgery" in the video, I thought to myself "what are the rules of that game, what he is even talking about, surgery?". Turns out it's a whole theory!
Man that was awesome
Simplicity at its core 💯💯
i've got an astrophysics degree and worked as a seismic processor for a decade...i find this analogous to deconvolution...
they set off a charge of dynamite down a hole and a whole array of detectors record the sound waves
and by modeling the original explosion, they can deconvolute the signal to produce a picture of the earth below...
in this case, the sphere is the final product and you're deconvoluting it from the initial condition to define the characteristics of the vector space?
Can anybody help explain to me why it's not just completely arbitrary and kind of just cheating to chop it in two and add a couple of half spheres? And how do the two separated manifolds just magically join back together? Wouldn't the definition of the 'surgery' tactic already presuppose success because half spheres were chosen and not, say, the top of a pyramid? This is hard for my brain
Wouldn't removing the problem-areas manually change the underlying structure of the manifold, just like a ricci flow would with singularities? Why is it allowed to just substitute problem-areas with spheres' sections?
As explained in the video, you're right, it does change the structure of the manifold. In fact it disconnects the manifold into a number of parts, where on each part Ricci Flow can be done properly. The last step of the proof, then, is to show that when you put these parts back together in the same way you took them apart, you still end up with a sphere.
@@avz1865 ah, I think I grasp that concept now, thank you. I still feel the need to look deeper into the mathematics of those last two steps, but I'm afraid that, even with my engineering degree, the proof would be way out of my reach ':)
Amazing explanation!
Is that the same Ricci thats found in Einstein's field equations?
Yes
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!
@John Cavanaugh Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
There is a 4th law of thermodynamics based upon the fact that your mind is predicting reality into existence, entangled entropy or mutual information is correlated information or syntropy.
Predictions are used to track targets -- teleology.
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"Through imagination and reason we turn experience into foresight (prediction)" -- Spinoza describing syntropy.
The word "spammy" is incorrect according to Spinoza, he would have disagreed with you.
@Electro_blob That is because it is correct.
Mind (intangible) is dual to matter (tangible) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality.
Active matter (life) is dual to passive matter (atoms, forces) -- matter duality.
Mind duality is dual to matter duality.
There is a second level of duality beyond Descartes according to Kant, Aquinas and physics.
Energy is duality, duality is energy, an apple converts potential energy into kinetic energy when falling in a gravitational field or duality is being conserved.
Electro is dual to magnetic, light or photons are dual -- Maxwell's equations.
Positive charge is dual to negative charge -- Electric fields.
North poles are dual to south poles -- Magnetic fields.
Electro-magnetic energy is therefore dual.
Everything in physics is made out of energy or duality!
Duality creates reality, the big bang is a Janus hole/point, all holes in topology are dual -- the 1st homology group of the real projective plane is a finite group with two elements [0,1].
The bad news is that "philosophy is dead" -- Stephen Hawking.
If you think that your mind/soul is dual (Kant, Aquinas) then Stephen Hawking is wrong!
Wow great channel .. Nice explanation
I hope to become such a fantastic mathematician in the future!
Thank u for this one...💯❤
I love your explanation and understanding madness genius of Grigori Perelman
What is the justification for surgery (cutting parts out and glueing semispheres to cover them)?
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.
hey! can you make videos on all the millenium problems?
That's the plan!