The Million Dollar Problem that Went Unsolved for a Century - The Poincaré Conjecture

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Topology was barely born in the late 19th century, but that didn't stop Henri Poincaré from making what is essentially the first conjecture ever in the subject. And it wasn't any ordinary conjecture - it took a hundred years of mathematical development to solve it using ideas so novel that they were worth at least a million dollars.
Here I talk about the Poincaré Conjecture by introducing fundamental topological concepts like homeomorphisms and homology, and eventually talk about its solution through Ricci Flow.
Note: At 12:25, it's 3-Manifold, not 3D Manifold. I don't know what possessed me to say 3D.
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Inspired by Kevin MacLeod
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Wind Of The Rainforest Preview by Kevin MacLeod
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Wholesome by Kevin MacLeod
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The Forest and the Trees by Kevin MacLeod
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Image Credits:
Tesseract, 24-cell and Tetrahedral Compound: Robert Webb's Stella software is the creator of this image. A link to the website: www.software3d.com/Stella.php.CC BY-SA 3.0 (creativecommons.org/licenses/..., via Wikimedia Commons
Klien Bottle Homology: Steelpillow, CC BY-SA 4.0 (creativecommons.org/licenses/..., via Wikimedia Commons
John Stallings Photograph: George M. Bergman, CC BY-SA 4.0 (creativecommons.org/licenses/..., via Wikimedia Commons
Saddle Shape: Nicoguaro, CC BY 3.0 (creativecommons.org/licenses/..., via Wikimedia Commons
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Sources and Citations:
1) people.math.osu.edu/fiedorowi...
2) www.math.unl.edu/~mbrittenham...
3) Evelyn Lamb, 'What We Talk about When We Talk about Holes,' Scientific American (blogs.scientificamerican.com/...)
4) infoshako.sk.tsukuba.ac.jp/~ha...
5) enry (math.stackexchange.com/users/..., Why is $S_1$ said to "enclose" a 1-dimensional hole if the region it encloses is a (2-dimensional) disk?, URL (version: 2017-06-06): math.stackexchange.com/q/2311769
6) Wikipedia contributors, "Homology (mathematics)," Wikipedia, The Free Encyclopedia, en.wikipedia.org/w/index.php?...
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Chapters:
00:00 - The Euler Characteristic
03:00 - Intro
03:10 - What Even is a Homeomorphism?
05:07 - The Euler Characteristic, Reprise
07:50 - Introducing Homology
09:20 - The need for a Fundamental Group
11:55 - The Conjecture
13:50 - MORE Dimensions
15:05 - Deforming Surfaces with Ricci Flow
17:02 - Ricci Flow 2.0
18:24 - The Million Dollars

Опубликовано:

2 июл 2024

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Комментарии : 110

@mlguy8376 3 года назад

I have a PhD in Theoretical physics and I enjoy your videos. Not sure what you are doing or planning on doing with your career but you have a great way of explaining things. I will be following for sure.

@evalsoftserver 2 года назад

Robert , Have you ever heard of Phase Field theory?

@jaw0449 Год назад

I’m getting my PhD in theoretical physics and I agree with this. There’s a desperate need for those who can explain concepts well

@Hermetics 3 месяца назад

Phd = Pure Heart Divided ymonkeys ;)

@dylanparker130 3 года назад

"I'll leave that one for you to figure out for yourself" You have wildly over-estimated me, sir!

@ativjoshi1049 3 года назад

Video version of "left as an exercise for the reader".

@Lufernaal 3 года назад

It's all around the tourus, as is "the outside", then you also can't do anything to change it to the other shapes, because the line will go around the insides of the tourus and come out to the same place it was before.

@dylanparker130 3 года назад

@@Lufernaal thank you!

@roman111117 2 года назад

@@Lufernaal couldn't the same be said if you put it "inside" the hole?

@l1mbo69 2 года назад

@@roman111117 they're both equivalent, I think

@chriswest8613 3 года назад

This channel deserves so many more views and subscribers! Great content!

@kinertia4238 3 года назад

Glad you think so!

@xavierg733 3 года назад

I am currently a high school senior taking taking AP Calc BC. I plan on being a math major and it’s videos like yours that motivate me every day

@catcatcatcatcatcatcatcatcatca 3 года назад

There is something deeply beautiful in topology, where the shapes defining feature is not as much the area shaping the hole, or any such property that can be expressed as positively being and thus causing the hole to appear. That all can be abstracted away, morphed and scaled, but the hole itself and crucially only the hole survives as defining feature that gives raise to unique characteristics. A negative being, lack of being, is what defines the object or objects that fall under the category. The fact that we humans feel often so keen and connected to something we project as being and notice the lack off, or absence, and that it makes deep and complex sense mathematically, is to me beautiful. After all that is required to even notice a hole: you project something that is not there, and give particular meaning to the lack of it. Else it would be a tunnel; maybe set of low ceiling, walls and floor. There is no definite need to notify the absence or mark it with meaning: all things lack almost everything. Yet we gravitate towards negations and not only give them meaning but define and assign meaning to whole things based on what negations they include.

@austin2150 Год назад

What a beautiful comment

@Mr0rris0 Год назад

You make me sick Where do you hide your red sashes Ringo Detestable thuggie strawdog pontoon

@lyrimetacurl0 11 месяцев назад

I just realised why odd dimensional shapes have Euler characteristic 2 and even dimensional shapes have Euler characteristic 0. Because each shape also has 1 of itself and 1 null set (negative 1 dimensional thing). This makes all Euler characteristics 0 (without holes).

@mr22guy 2 года назад

To shut yourself away and focus on something until you solve it, that's dedication.

@richardslater677 2 года назад

Just sen this channel. That was excellent. That was by miles the best explanation of a very difficult topic that I have ever seen. Excellent delivery too. I don’t understand the maths involved in topology but now I at least understand what the Poincaré conjecture was all about. Brilliant thanks.

@Simson616 3 года назад

That's the first time I've been told, why he rejected the price. Up until now I liked to believe that he valued math so much higher than anything else that he could simply not see the necessity of having a million dollars...

@5wplush243 3 года назад

Excellent vid! Beautifully well done explanations and top-notch quality!

@The-KP 10 месяцев назад

One of the best, clearest videos on any topic, let alone maths or topology. So nice I watched it twice!

@marklama6435 3 года назад

This is great! Thank you for giving me at least an intuitive notion of what Ricci flow is!

@shreyasjv4877 3 года назад

This was really well done! Kudos to you man!

@MadScientyst 2 года назад

By far the BEST exposition I've yet seen on The P.C., Topology & Ricci Flow! I've 2 Math Degrees & was able to follow quite easily & enjoyably...LOL U deserve more views & subscribers my young, versatle friend! Keep up the excellent work indeed...

@jinjunliu2401 3 года назад

The quality of your videos are amazing man!

@kinertia4238 3 года назад

I appreciate that!

@TheCanon66 2 года назад

I am surprised this doesn't have more views. Great video!

@back2d_lobby Год назад

The best video I’ve seen on this abstruse proof, well done. You are talented at this!

@nyckhusan2634 9 месяцев назад

In 2010 Gregory Perelman proved that we live in 4-D Universe ( X^2+Y^2+Z^2+t^2=1). Now he is working on the problem of " Holes ". Possibly, will prove how this Universe was born.

@143TYAGI 3 года назад

Excellent job buddy. I love the way you explained. The no. of views on your videos shows that very few viewers are interested in such specialized topics. I wish more people get motivated to take up advanced mathematics.

@tim40gabby25 11 месяцев назад

I disagree with your conclusion. The number of viewers is primarily determined by the algorithmic shuffling of these videos, not the number who would enjoy if such videos were presented in their intray, as happened to me. Just saying.

@mediamannaman 2 года назад

I never made it past high school Trig, and I am able to follow along with you. You are an excellent communicator.

@kishorechhetri4018 3 года назад

Brilliant content with smooth explanation ! Please keep it up.

@kinertia4238 3 года назад

Thanks a lot!

@maypiatt3766 3 года назад

inb4 10k subs for the people who are coming from 1 mil! This channel is going places fs

@stanislavmatusevschi6142 2 года назад

Great piece of work, nice job

@RahulKumar-cl6xq 3 года назад

well i am a biology student and came here just out of curiosity and i was awestruck by the way you explained it .. SUBSCRIBED

@cycklist 3 года назад

Wonderful visuals. What a superb video.

@cara-seyun Год назад

The third path on the torus is where you draw it on top and encircling the hole, right?

@harransingh8398 3 года назад

Great to see more videos, keep it up!

@SanjaySingh-oh7hv 7 месяцев назад

Let me echo what others have said. This video is so very excellent for explaining and making accessible a complex and difficult and abstract topic. The creator and host of this video is a gifted teacher and orator. So glad I found this video. Adding it to my playlist for future reference!

@prasoongupta12 3 года назад

Can I request a video on P vs Np particularly if P = NP with a fairly practical algorithm to solve NP problem in practical times?

@yogeshshahi 2 года назад

Brilliant mann just keep going RU-vid takes some time to reach you to the people who actually loves it... Don't stop keep going ❤️

@AbhishekTiwari-nw5sq 3 года назад

3:00 what's the tune name, btw liked the video 👍, make one on yang miller's and reimann too

@netomdam 3 года назад

I really like your videos, keep it up

@TheArkadyuti 3 года назад

Awesome video, but little bit more info on Perelman solution would have been great 🙏 One of the finest and apt explanation on the problem statement BTW.

@kinertia4238 3 года назад

Noted. Thanks!

@bazboy24 Год назад

You are a very gifted teacher indeed

@PierreFT 3 года назад

Your animations are so beautiful ! What softwares do you use to make them ?

@kinertia4238 3 года назад

Normally Adobe After Effects, but there's a couple of 3D effects in this video which I used C4D to make.

@analogico3615 2 года назад

Oh my god you explained it so well, thank you!

@Lufernaal 3 года назад

The other way is all around the tourus in a horizontal line. The first shape is a square, right? The second is a "ring" that circles the tourus like a ring circles a finger in a sort of vertical way and it can move around horizontally. The third is a straight line going all around the tourus, in a sort of horizontal way, and connecting it around it. You can then go around vertically, reaching the inside of the tourus. You can never make that line become the other two shapes.

@mattthecat5036 Год назад

Rad! Best explanation on RU-vid bar-none.

@jabunapg1387 2 года назад

Great explanation

@MrController12345 3 года назад

great work....

@Leo-if5tn 2 года назад

Great video

@HadiM-rb7yo 3 года назад

Your videos are amazing 👍👍

@kinertia4238 3 года назад

Glad you like them!

@prasoongupta12 3 года назад

Bhai bhai! People like you make indians proud!

@pimthephysicsguy1056 3 года назад

Wonderful!

@kinertia4238 3 года назад

Glad you think so!

@user-nx5ob7ny4l Год назад

my guess for the 3rd unique type of loop on the surfuse of a turso is a circle around the girth of the shape.

@itsRAWRtime007 Месяц назад

great analogy

@MarkSummersCAD 3 года назад

Very nice job my friend...

@kinertia4238 3 года назад

Glad you liked it!

@omargaber3122 3 года назад

You are so genius💖

@williamzame3708 9 месяцев назад

Smale came *before* Stallings, not after - but their work was independent of each other.

@jake_runs_the_world 3 года назад

Really good

@mathisfunbybikisharma2579 3 года назад

Great content .. What are u doing brother ???

@marcocecchi9853 3 года назад

Is a manifold the set of points (x,y) €R^n*R such that y=f(x) with f being differentiable and with a non zero jacobian? Great job btw, keep it up!

@MG-wj5bn 2 года назад

What’s the third way to draw the line around the thing???? I’m going insane I can’t figure it out

@gekkkoincroe 2 года назад

Thank you

@lebusiness2847 3 года назад

Hey, i love the content of your videos! You deserve much more attention. Only thing that makes it sometimes hard to unterstand and fallow is your accent. I know that is hard to change. I am from Germany so my accent isnt perfect neither. You can use an app called Elsa Speak to exercise your accent, it uses AI and is very good. I know this may sound stupid to you because you are probably a native speaker but it could help grow your channel a lot.

@kinertia4238 3 года назад

Thanks for the tips! This is not my normal accent, it's one that I'm putting on by popular demand. Hopefully as I get more used to speaking in a more Americanized accent (I'm also getting a new mic, so the entirety of the audio will eventually improve) the issue gets sorted. Meanwhile I've put subtitles on the video.

@migzleon4047 2 года назад

Good stuff...

@ManishKumar-ix5jj 3 года назад

Waooo keep it going 🙌

@ExhaustedPenguin 3 года назад

Second video, aaand subbed.

@ctai010 Год назад

Your explanation is very good but background music is very distracting

@calvinjackson8110 2 года назад

I still did not get why the man rejected the prize? What was his reason?

@satyajitsaha29ss 2 года назад

I am a student in this subject and my research interest is in low dimension topology. You made it good but one thing I want to say that if you were bit slow it would be better..