Poincaré Conjecture - Numberphile 2,383,997 viewsApr 23, 2014

Poincaré Conjecture - Numberphile 2,383,997 viewsApr 23, 2014 Numberphile 4.08M subscribers The famed Poincaré Conjecture - the only Millennium Problem cracked thus far. More links & stuff in full description below ↓↓↓ Ricci Flow (used to solve the problem): http://youtu.be/hwOCqA9Xw6A Riemann Hypothesis: http://youtu.be/d6c6uIyieoo Extra interview footage with Jim Isenberg: http://youtu.be/7eJleW0JcKg Grigori Perelman's paper: http://bit.ly/perelmanpaper Discussed here by Katie Steckles and James Isenberg. Support us on Patreon: http://www.patreon.com/numberphile NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com/numberphile Subscribe: http://bit.ly/Numberphile_Sub Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphile Videos by Brady Haran Brady's videos subreddit: http://www.reddit.com/r/BradyHaran/ Brady's latest videos across all channels: http://www.bradyharanblog.com/ Sign up for (occasional) emails: http://eepurl.com/YdjL9 Numberphile T-Shirts: teespring.com/stores/numberphile Other merchandise: https://store.dftba.com/collections/n... Buy Numberphile merchandise $20.84 Spring $28.99 Spring $25.99 Spring $23.14 Spring $44.14 Spring 1,311 Comments rongmaw lin Add a comment... NuncFluens NuncFluens 7 years ago An interview with Perelman would be the ultimate Numberphile video. 4.4K Misterlegoboy Misterlegoboy 6 years ago what a badass- he solves one of the 7 hardest problems in the world and then just drops the mic and leaves 6.4K The Fake Slim Shady The Fake Slim Shady 1 year ago Gregori solved this question alone in his room which contains only a desk, a bed, a lamp, and a chair. And then quit all of mathematics after this. What a guy 650 Luck Luck 5 years ago Hardness of math problems: 0. I can solve the problem 1. I can understand the solution of the problem 2. I can't understand the solution of the problem 3. I can't understand the problem 4. I can't understand why it is not obvious 2.5K JanPBtest JanPBtest 5 years ago This video never mentions one major hero of the story, the inventor of the Ricci flow himself: Richard Hamilton. This is almost like talking about relativity without ever mentioning Einstein. Perelman's achievement is undeniable, of course, but Hamilton did a lot of heavy lifting beginning with the early 1980s. In fact, one of the reasons Perelman rejected the Fields medal and other prizes was that the people who were in charge of awarding them refused (apparently) to make the prizes shared with Hamilton. 1.5K ErwinSchrodinger64 ErwinSchrodinger64 8 years ago While a very intriguing story, I'm surprised there were no reasons given why Dr. Perelman declined the money, fame, and accolades. His reasons, on some parts, I completely agree... many fields become doctrines that are devised by a very selected few. Science and mathematics are seriously suffering from a lack of openness to new ideas because of these centralized ideas (string theory, molecular quantum mechanics, and so forth). Unfortunately, not only is his intelligence and insight incredible but the his level of humbleness, unselfishness, and grace are far beyond most people. 178 Dmitriy Soloviev Dmitriy Soloviev 4 years ago Perelman refused to take any prize, thereby create Perelman Conjecture. Now mathematician trying to understand why. 2.2K TheUneuro TheUneuro 8 years ago As a mathematician, I respect so much what you do Brady ! Thank you :) 841 Jakedesnake97 Jakedesnake97 7 years ago Hey Brady, could you make a series of videos explaining every millennium problems please? 252 Bill T Bill T 8 years ago Shouldn't we stop calling it a "conjecture" now that it has been proven? 444 John Hobitakis John Hobitakis 8 years ago Perelman is a great character in the world of mathematics, thanks for this video. The millennium questions are very captivating, I wonder how long it will take for the next one to be solved. 6 Francesco Rende Francesco Rende 5 years ago do a video on every millenium prize problem 1.8K nekad2000 nekad2000 2 years ago Most brilliance languishes in obscurity. Unfortunately, this is the world we have fostered: people with few brains a tons of ambition usually take take the credit. People like Steve Jobs fit this example perfectly. 30 Divyanshu Pandey Divyanshu Pandey 10 months ago I am a undergrad student in Mathematics and I know this video dumbs down the conjecture so that we all can comprehend it easily but i would appreciate if someone provides a guided path to form a basic background in understanding Perelman's proof . Just some directions to navigate forward (I have basic knowledge on real analysis and calculus ) 5 Harvin Toledo Harvin Toledo 2 years ago Poincare Conjecture is a fascinating problem because we start to make many questions about what are indicated equations to solve it. It sounds simple but there are many theorems we need to know and use to get it. 3 Manisha Mohanty Manisha Mohanty 9 months ago " Some people can't be bought or bargained with, they just want to see the world learn. " 63 Emzy Emzy 2 years ago Perelman was my dad's classmate! From what I've been told, he's a super-nerd but is also a genius! 15 AlanKey86 AlanKey86 8 years ago Ooh... I just remembered an orange peeling question I had once which seems vaguely relevant to this video! Is it possible to peel an orange such that its skin comes off in a donut type shape (as defined around 1:45)? That is, one continuous loop that has a hole in it. I have an answer to this question - a most elegant proof. But this comment box is too small to contain it. 695 SystemofEleven SystemofEleven 7 years ago The only thing I know about this particular brand of math is that a donut and a coffee mug are apparently mathematically the same thing. And this is because one of my math teachers was very creative about coming up with excuses why he had food in class, heh 58 Cold Ham on Rye Cold Ham on Rye 8 years ago Thank you so much for this Brady. I love these Millenium Problems. 22 Dawn Dawn 1 year ago He has a gold heart. Humble, and sincere. 6 Kwstas Kartas Kwstas Kartas 8 years ago Your last videos are amazing Brady ! Excellent choices. 89 energysage energysage 8 years ago There's a tiny piece of misinformation in the video. When mathematicians refer to a 2-dimensional sphere they mean the same thing as the laypersons sphere which exists in 3 dimensions (because its surface is 2 dimensional, just curved). This wouldn't be an issue since it's just a matter of convention, but the distinction became important when she talked about the conjecture having been proven for 5-spheres and up. In this case she meant the mathematicians "5-sphere," (which thus exists in 6 dimensional space) which is two more dimensions than the final case which Perelman proved, and not 1 dimension higher, which is what the video implied. It's a curious fact that it was eventually easier to prove the higher dimensional versions of the conjecture (7 was proven before 5), but an intuitive way to understand that is that you have more "elbow room" so to speak. 50 Sean Haggard Sean Haggard 8 years ago I saw a video on another channel about topology showing how a sphere could be turned inside-out following those same rules. It had a really cool animation but didn't explain very well how it was done. 17 Gabsare Sarg Gabsare Sarg 1 year ago I guess if you can solve the hardest mathematical problems for humanity, you really don't think too much about money or prizes. 12 Jensen Jensen 3 years ago Came here to understand this, although an interesting video, i would have liked to have some brief explanation of what this proof actual stated and the basic logic behind what he was trying to do 15 Jaakko Oksa Jaakko Oksa 2 years ago Poincaré is also famous for figuring out the field equations of General Relativity a short time before Einstein but he did not even try to take credit for that achievement because Einstein had been working on them for a decade. 5 Max Schmidt Max Schmidt 4 years ago It amazes me that so many people watch your videos, always thought that math is not that popular. 1 Rumplestiltskin Rumplestiltskin 1 year ago I have been fascinated with his story... this man is a special soul... a sacred one... his skills and intelligence are just a testimony of what is beautiful and what is not... / he has shown a different way to live a human life... a human being is a free animal... just like other animals... 2 Zack Yezek Zack Yezek 4 years ago What if I was allowed N "forbidden" operations for my topological transformations (e.g. N cuts or sphere closures)? I can imagine turning donuts into spheres with a single cut in the real world, so having a generalized topology for that kind of thing seems like it would make sense. Also, it seems like you could have classes of objects where one cannot be transformed into another even with infinitely many cuts. 1 Miguel Reinozo Miguel Reinozo 3 years ago Excelente vídeo. Gracias por difundirlo. 2 Drarck Drarck 1 year ago Imagine solving a freaking millennium problem and not accepting the prize. What a legend. darthvatrayen darthvatrayen 8 years ago What about a hollowed sphere? Can that be made into a solid sphere or does that count as a "hole"? 35 Steven Zhao Steven Zhao 3 years ago Would've been nice if if the video also explained the significance of the problem. Like if you solve P vs NP there is a huge significance on what kind of problem can be solved by computers. But why do we care if you can squeeze something into sphere in four dimensional space? Not taking a shot at mathmatician just really curious. 1 james ehrhart james ehrhart 10 months ago Wow! I took one of my math classes from James Isenberg. I think it was differential geometry. Dave Robertson Dave Robertson 10 months ago So Perelman basically did the world's greatest ever hold my beer followed by an epic mic drop. Legend. 2 Chris O'Neil Chris O'Neil 8 years ago Another awesome video! Are there gonna be videos on all the millennium problems? 2 Casterly Smooth Jazz Casterly Smooth Jazz 7 years ago Have you guys thought about doing a video on Navier-Stokes? I think that's a really fascinating problem and it's seems more intuitively applicable than Poincaré. 1 Ben Toth Ben Toth 6 years ago But, according to Homer Simpson, if it's a real donut then nibbles are allowed. 262 クロノシル クロノシル 3 years ago Perelman on why he didn't accept the prize or medal: "I'm not interested in money or fame. I don't want to be put on display like an animal in the zoo." Ironically, this has made him much more famous and "on display" than if he would have accepted the awards. 1 Rational Mind Rational Mind 3 years ago this is called actual genius and not those who claim their IQ is of 100 or 200 or say they have cleared some entrance exams. 36 Andrew Andrew 2 years ago (edited) I speak Russian. From the few interviews on the web it becomes obvious that Perelman made very weighted and rational decision to refuse the money. Mostly because he thinks Hamilton made more for the solution. Actually, he hates when people think that he is some kind of a crazy genius. He is not. 20 View Inventions View Inventions 2 years ago I Love this channel with all my heart. Old Seadog Old Seadog 2 years ago If the conjecture has been proved, shouldn't it be elevated to theorem? Logan Myall Logan Myall 1 year ago Convinced Perelman has been working since he left the math community. the questions are; on what, and is it even finishable? 4 Justice Warrior Justice Warrior 1 year ago I can't even imagine how difficult this girls course was 3 Tyler Daniels Tyler Daniels 5 years ago I love how you have James wearing a Red Sox hat the whole video and Brady at the end wearing a Yankees hat. Go Blue Jays ;) 7 Roxor128 Roxor128 8 years ago I hope you're going to cover the other five problems in future videos. Might inspire some budding mathematicians to take a crack at them. 1 Jesus Christ Jesus Christ 2 years ago 0:45 There is a certain relationship between geometry and topology. Lol I always thought topology is just big daddy of geometry. 15 analógico analógico 2 months ago Katie explained it so well Gigatless Gigatless 10 days ago Anytime I feel that my coding job is too hard I go and watch some math videos and my life seems easy again 1 why not why not 2 years ago he declined the prize most probably to say " I didn't solve it for the prize ", and maybe to remind the scientific community of the real purpose of science, not money, not fame not the budget just a natural instinct to solve problems . although we have better means to solve problems, we are so far away from people like Newton, Gauss, Euler, Ibn Hayan, Ibn ElHaytam ... in a word we need more "heart". 8 truebaran truebaran 5 years ago Unfortunately there are few misleading/incorrect statements: 1. the circle is one dimensional sphere: it is one dimensional manifold which can not be embedded in R, so you need two dimension in order to see circle embedded but abstractly circle is one dimensional. The same later: the final case of Poincare conjecture was THREE dimensional, not four dimensional (four dimensional case was settled in 80' and n-dimensional with n>4 in 60') 2. as far as I know original formulation of Poincare conjecture was in three dimension. It was generalized later to arbitrary dimensions. 3. it is crucial also to understand about what kind of holes we are speaking: original formulation was in term of homology groups and in such formulation this conjecture turned out to be false (there are so called homology spheres), later reformulated using homotopy. 4. I also find very misleading this popular phrase, that in topology we don't allow tearing things apart and gluing. Topologists investigate category of topological spaces (or some special kind of spaces) with morphisms being continuous maps: many continuous operations allow gluing. In such cases the inverse operation would require tearing: but those are still legitimate morphisms. This is just the fact of life: if you have a bijection which is morphism in topological category then the inverse mapping need not to be morphism in topological category. Is it a big deal? Finally to put Poincare conjecture in some more general context: in the world of smooth manifolds (as objects) one can define three different notions of equivalence: the strongest-diffeomorphism (meaning smooth bijection with smooth inverse), homeomorphism (meaning continuous bijection with continuous inverse) and homotopy equivalence (f:M->N is homotopy equivalence if there is g:N->M such that fg and gf are homotopic to identitiy maps). Higher dimensional Poincare conjecture has the following formulation: If M is a manifold being homotopy equivalent to n-sphere, then M is automatically homeomorphic with n-sphere. And for example, there are so called exotic spheres by Milnor (starting with the n=7 dimension) which are homeomorphic to ordinary sphere but not diffeomorphic. Also there are examples of manifolds having the same homotopy type but being not homeomorphic so it seems to be very deep question: what is so special about spheres that homotopy equivalence gives automatically homeomorphism? (I find it also amazing that in dimension 3 homotopy type of sphere follows from the single assumption of having trivial fundamental group: a priori this is weaker condition but somehow when you formulate 3d Poincare conjecture in terms of fundamental group then it looks reasonable but if you formulate it in terms of homotopy type (which is a priori stronger!) then this is not at all obvious why it should be true!) 3 Ian Alvord Ian Alvord 1 year ago "Is any smooth, finite shape with no holes a sphere?" Can somebody just tell me whether it's true or false already... 3 Mayukh Mukhopadhyay Mayukh Mukhopadhyay 4 years ago Prof Grisha might have tried to calculate the tax implication from the million dollars at US and Russia and said ," man, that's complex! so where was I with P vs NP...." 1 relike868p relike868p 8 years ago Another Topology video! That's amazing... more people should be interested in these stuffs! Rosie Fay Rosie Fay 2 years ago 2:58 "three-dimensional sphere" Your words confuse me here. The thing is embedded in three-space, but it's a two-dimensional sphere, isn't it? So the three-dimensional sphere which is what Perelman characterised is not that ball of dough, or even its surface, but a three-dimensional object which needs a space of at least four dimensions to embed it in. Have I got that right? 3:50 Interesting that the conjecture about four-space wasn't settled until after the conjecture about five-space and beyond was settled. Anything particularly difficult about four-space? 1 adlsfreund adlsfreund 8 years ago How the $%&! did they manage to solve it in 5+ dimensions but NOT 4? That's crazy. 1 Ali Hamed Moosavian Ali Hamed Moosavian 7 years ago Brady, you should make a video with Perelman himself. 31 Keahn Bruzzi Keahn Bruzzi 5 years ago This is so inspiring <3 Dmitry Morozov Dmitry Morozov 7 years ago