g-conjecture - Numberphile
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- Опубліковано 7 чер 2024
- Discussing h-vectors and the g-conjecture. Featuring June Huh from the Institute for Advanced Study at Princeton University.
More links & stuff in full description below ↓↓↓
A little extra bit from this interview: • g-conjecture (extra fo...
Shapes in higher dimensions: • Perfect Shapes in High...
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Congratulations for being awarded the 2022 Fields Medal!
Damn before your comment i did not know he got that award but i was just amazed at how easily he is explaining that problem.
thx..
Really nicely explained (and edited). The modified Pascal's triangle framing is a really fun way to make these topological patterns feel like they pop out of numerical playfulness.
The remarkable substance that holds together basic number theory, graph theory and geometry is always so enjoyable to explore.
3Blue1Brown I believe I may have stumbled upon a new area of math I call complex graph theory.
It deals with operations on the graphs of functions.
Noah'sKnowledgeCenter Explain more
3 Blue 1 Brown
Hi :D
Best handwriting in all of Numberphile. June Huh has remarkable penmanship.
He resolved the Heron-Rota-Welsh conjecture on the log-concavity of the characteristic polynomial of matroids.
huh..i was gonna make the same joke so asians could be on the same level
I love his precise voice too.
Great penmanship, amazing at substraction... Gotta love this guy
Kurt Schwind because he learned it as an adult and sees it formally!!
A few years ago I attended a summer school where June Huh was one of the lecturers. It was amazing. He’s the kind of idealistic mathematician who always sees the big picture.
That’s great. We were grateful for his time.
He is Fields medalist now!!
Huh. Thats cool
Those F's are fancy as hell
Xepscern they're the queen's.
He has exceptionally neat handwriting.
Xepscern Function f's are awesome
𝑓₃-𝑓₂+𝑓₁-1=0
𝐼 𝓁𝑜𝓋𝑒 𝓈𝓊𝓅𝑒𝓇 𝓈𝑒𝒸𝓇𝑒𝓉 𝓂𝒶𝓇𝓀𝓊𝓅 𝒸𝑜𝒹𝑒𝓈. XD
I wonder if it has to do with the complexity of the characters in Korean? Maybe that translates into English penmanship with extra "flourish?"
the way he writes the letter "f" is so satisfying
But the eight though ;-;
Stop pretending to be ''autistic'' or ''OCD'' like it makes you a mathematician, mathematicians are almost never autistic.
F
The "unproven" g-conjecture was proven in a paper published in December 2018, just 6 months after this video was posted.
Nice
I'm here after June Huh's Fields Medal announcement!!! CONGRATS!!!
"Pick my favourite triangulated sphere in the 17th dimension..."
There's just so many, I can never choose just one!
I'd go for the one of which the vertices are all such points where one coordinate is +/-1 and others are 0. Edges are between each pair of points that differ in exactly 2 coordinates. The simplices of this triangulation are all such sets of vertices that no two are opposite (meaning that they have the same nonzero coordinate, one 1 and the other one -1).
This is the best penmanship I've ever seen on a Numberphile brown paper.
Same lol
Yes, except for his 8s, which he draws with two circles. And his 7s, which he does not cross.
Eoghan Connolly sheesh cut the guy a break 😂
The greatest congratulations to June Huh for having been a recipient of the Medal Field this year
Mathematicians are enormously imaginative.
Olbaid Fractalium Mathematic is about imagination.
They need to be
what is your profile image?
You can replace all Imagination by definition
JaguarFacedMan It is a fractal art of Mandelbrot Set I made. I love the Mandelbrot Set!
June Huh has a beautifully patient cadence to his presentation style.
Just seeing him write is a pleasure.
As of 2022, June Huh has been awarded a Fields Medal. Just amazing!
It’s nice knowing that, as of this filming, June Huh had a bright, bright future. Congratulations on the Fields Metal!
This is a very inconvenient time of day for me to watch a 20-minute math video, but I got the notification, so here we are.
I feel your pain
Didn't know that it was 20 mins
22, actually
you can make it 11 minutes if you watch it sped up 2x
@@peterd5843 yeah but we actually try ti understand so that wouldnt help
Dang, more videos with Dr. Huh. This was one of my favorites. He's obviously passionate about this math, and is very articulate.
Thanks. Glad you liked it.
who's here again after June Huh has won the 2022 Fields medal?
Here after June won his Fields Medal. What an amazing mathematician!
I watched this video years ago and I never would have thought that he would have won the Fields Medal.
Congratulations!!
I love his calm demeanor. What a great guy. Please do more videos with him.
We should rename maths. I suggest calling it "Euler"
Gauss would be miffed.
Maybe just the studying real/imaginary parts/calculus/number theory.
Calculus should probably be called Newton, while Geometry can be called Euclid (in a way it already is)
"what do you work in?"
"oh, you know, the Euclid-Euler-Gauss-Newton-Descartes-Hilbert-Riemann-Ramanujan-Nash-Penrose field of study"
"Ah Yes."
@@RickJaeger "Oh so you work at maths? Name all mathematicians."
JK Pls no.
@@peterdriscoll4070 Nah, Euler has probably made an "Euler's Gauss" or something like that that we can use.
교수님 축하드립니다. 찾다 보니 이 영상까지 보게 되네요. ㅎㅎㅎ
icosahedron?
You mean a pentagonal gyroelongated bipiramid?
Skeleton Rowdie you listened to Michael talk about that last night, huh
yes my man.
What about a snub disphenoid?
vsauce, michael here
Skeleton Rowdie I've seen the vid, but it will always be known as a d20 to me
"A 1 dimensional triangle is a straight line"
Cool cool
Simoneister it's true
No, its not. A one-dimensional simplex is a straight line. A two dimensional simplex is a triangle. One could say something like "the analogue of a triangle in one dimension is a line" but that is less precise in my view, as triangles could be members of other families of objects which are not lines in one dimension.
the name triangle implies three vertices
TootTootMcbumbersnazzle an infinite number.
(i) point
(ii) line
(iii) Any union of points and lines.
The first two are connected. If we restrict to connected geometric objects, then the counterexample to OP would be a sequence of geometric objects that is a point in one dimension, and a triangle in two dimensions. I leave it to you to imagine such an example.
clickaccept it's*
I love this guy. His explanations are so perfectly clear and direct.
4 years later this guy won a Fields medal!! Congratulations Mr Huh!!!
필즈상 축하드려요!
Fantastic explanation of Euler's Formula. Thinking about it as the alternating sum of the 0, 1, and 2 dimensional faces of a 3d shape really helped me understand it much better than I ever have before.
*SUBSTRACT*
I CAME TO COMMENT THIS. IT’S SO CUTE.
SMETRY
(from extra footage) *HYPOTHEETHIS*
absolutely mesmerising word; loved this presentation!
I'm pleased that this comment line is positive in nature. He was quite clear in speaking what seems to be a second language. I've had college professors who couldn't speak clearly in their native language, and I've known people who speak English as a second language who struggle quite hard. I know my own grasp of other languages is tenuous at best. Nevertheless, I did notice his interesting pronunciations.
Great introduction to h-vectors and the g-conjecture by June Huh.
You can tell he was careful to provide several examples so it would be accessible to most people.
Glad you liked it.
His explanation of 4D shapes has helped me understand them better that any of the popular animations that you may see online.
Yeees. Great video. Great mathematician. More from him, please!
A chunk in this video just helped me understand something I had been struggling with in modern GPU code. Thanks so much for your videos!
This guy is excellent, I sometimes find these videos hard to follow but his explanation is so clear!
He just won the Fields Medal!!! 🥇🏅
This is one of my favorite Numberphile videos. Always telling people that math is actually fun and to check this channel out.
Thank you.
This guys is my another numberphile favourite, such an articulated, well explained and inspiring. You can love math because of the way it is presented it to you.
oh my, he's so excited about the thing but so humble about it, I just love him. and the way he writes 8, come on
I want more of him, please!
I liked him
That's too bad. I still do.
no way dude mark it eight
You look at objects, like women, man...
Yeah, more of this guy for sure.
Obviously you're not a golfer.
"We should start with Euler's formula"
Do you have any idea how little that narrows it down?
Please get this guy on more, he is a wonderful explainer, with great handwriting to boot!
This is my new favorite channel. I can't get enough!!!
this man is awesome!! he is so passionate and so clear :D
We think so too! :)
I noticed that with some of the shapes you get parts of Pascal's triangle when you play the subtraction triangle game with them. That's pretty cool.
Perfect explanation. Goes inexorably to the point, you have no chances other than nod and agree.
The absolute best math channel to ever exist.❤
Learned about Euler's formula in my math history class this previous semester. Didn't expect to see it used so soon.
This is level of content I like to see!
This guy's handwriting is unbelievable. Watching his hand movements while writing formulas is hypnotizing.
We need this guy again. Great mathematical insight, even better calligraphy.
Dude. Forget everything else. Can we focus on the fact that dude has PERFECT “f”s? That was amazing.
June Huh is awesome! I want to see more of him.
Excellent editing job and production values as usual, thanks!!!
Clearly in my top 5 numberphile videos ever, along with Riemann hypothesis, Glitch Primes and cyclops numbers, All the numbers, and transcendental numbers
There are so many Fields medalists that have been featured on Numberphile, it's quite boggling.
This man is so smart, he deserves the Fields Medal !
Amazing explanation, very clear, articulate and easy to understand.
Really enjoyed hearing from June Huh!
Almost halfway through the video: "And this is our starting point." Oh, ok. This on a Monday. Lol.
If anyone's interested, Michael from Vsauce did a great video on strictly-convex deltahedrons yesterday. It's a brilliant companion to this one.
You can feel how much this guy loves maths. Great vid
such fun to play with mathematics.. thank you so much for the video. Love Dr. Huh
Love this guy's handwriting.
I have never been able to understand why the Euler characteristic must flip-flop between 2 and 0. The explanation in this video is very complicated - but all you have to do is include the figure itself to get the same result: pentagon f0(5)-f1(5)+f2(1) = 1, icosahedron f0(12)-f1(30)+f2(20)-f3(1) = 1, 6-orthoplex f0(12)-f1(60)+f2(160)-f3(240)+f4(192)-f5(64)+f6(1) = 1. A pentagon contains 5 vertices, 5 edges, *and 1 pentagon*. An icosahedron is made up of 12 vertices, 30 edges, 20 triangles, *and 1 icosahedron*.
well ... it's obvious from the betti numbers. An n-sphere has betti numbers 1,0,0,0,...,0,1 (indexes 0 to n), and, since the Euler characteristic is the alternating sum of the betti numbers, you get (-1)^0 x 1+(-1)^n x 1, which is exactly 0 or 2 depending on the parity.
Thanks, your video inspired a breakthrough!!!!! Best feeling ever!!!
The best description for higher dimensions!
Congrats on his fields medal
congratulations!
I was just reading about June last week, amazing.
June Huh is actually amazing
I can't believe this was never mentioned, but I just noticed that there's a way (much easier than the pascal triangle thing) to get to 1 every single time.
The pattern is defined as follows: count the amount of objects with dimension "x" inside the solid, and take the alternating sum as x increases to d-1, where "d" is the highest dimension that the solid lives in. All you gotta do to get 1 every time (rather than oscillating between 0 and 2) is increase x to d, not d-1. Here's an example, using a 3D simplex (tetrahedron, d=3):
Vertices (x=0): 4
Edges (x=1): 6
Faces (x=2): 4
Solids (x=3): 1, because the tetrahedron contains (and is) a single 3D solid.
4-6+4-1=1.
Here's the same example with a 4D simplex (d=4):
Vertices: 5
Edges: 10
2D Faces: 10
3D Faces: 5
4D Solids: 1 (again, the entire simplex).
5-10+10-5+1 still equals 1.
As you can see, this works with all of these solids in all dimensions, assuming the oscillation between 0 and 2 in the original pattern continues indefinitely. The alternating sum happens to work out such that whenever a 2 is reached the 1 is subtracted, and whenever a 0 is reached the 1 gets added- it always ends at 1.
Side note: I totally realize that leaving out the final 1 was kinda needed for the purpose of the pascal triangle bit, I just thought that what I found was super interesting.
(btw I typed this entire comment on a crappy phone keyboard)
TL;DR What this video forgot to do was factor in the entirety of the solid along with its edges and faces, and if it did that, the pattern would be a clean string of 1s rather than an oscillation between 0 and 2.
Nice work buddy
This exactly what he explained about the last number in the row being a one.
Parrot-hD I don't understand this but I read it anyway
yes! did the same observation and don't get why to leave it out, better watch it again.
Not only that, but if you consider the null set an element and begin the alternating sum with it, you will always end up with 0. For example, an octahedron would yield 1-6+12-8+1=0. In odd dimensions the two 1s are both positive, adding 2, while in even dimensions the 1s are opposite signs, canceling each other out.
I guess you can also see it in the way that eulers formula is missing the sphere itself and that's where the 1 comes from.
Nice. That's a much simpler way of thinking about it. It's not a simplex, but it works. I wonder how much more general certain parts are, 'cause Euler's formula works not just for triangulated spheres, but any connected graph on a sphere.
And you can then also use it for connected "spheres", e.g. two triangles that share an edge: 4 - 5 + 2 = 1, or two tetrahedra that share a triangle: 5 - 9 + 7 - 2 = 1. But it becomes more complicated if there are holes, making the whole topologically equivalent to a torus. In general it depends on the genus of the whole structure.
This attracted me because it looked like a network mesh. I think the basis of the g-conjecture may be a generalization of a recurrence relation, which seems to be able to be constructed using a function that depends on recursion to instantiate itself in the lower dimensions.
June Huh is so well spoken, brilliant mind!
It’s been proven today!
really?
crazyspider17 no
Karim happens to collaborate with Huh on previous work. Interesting!
He was awarded the Fields Medal 3 days ago!!
I'm high as faq watching this and it is the most beautiful explanations ever. The thinking behind this is transcendental. I guess.
I like this guy. He has a knack for explaining things
Somehow I feel like Grasshopper absorbing the lectures of Master Po.
Master *HUH* talking about the *H* -factor and *palindromic* sequences.
This guy writes his 7's like the katakana ワ/ク, which is a great idea I wish I'd known earlier.
Reydriel In Japan (where you'd think this would be avoided because the similarities) this is common practice.
(｀・ω・´)
It used to be common in the U.S. too. I suspect it became uncommon as handwriting lessons became lax and then uncommon as well
My experience in mathematics is that the majority position (at least in the countries I've worked) is to write a 7 with a line through the middle (a bit like a backwards f). I quite like this because it clearly distinguishes it from '1'.
What a fantastic speaker. Very enjoyable video
This was an impressively clear and interesting presentation
I noticed that 1 3 3 1 was a line on Pascal’s Triangle (a+b)^3. So is 1 4 6 4 1 (a+b)^4.
Then I thought about the 1 9 9 1 one and thought that perhaps it’s because that was the next level up in complication (octahedron -> icosahedron)
And the tetrahedron was 1111 and is the simplest, so if the “complexity” was given a number like
tetrahedron: n=0
octahedron: n=1
icosahedron: n=2
then the h number would be
1^n 3^n 3^n 1^n.
I predict that the next level up in complexity would be 1 27 27 1.
The same seems to be true for the 4-dimensional objects except it’s the next level down on Pascal’s Triangle
1^n 4^n 6^n 4^n 1^n.
I’m sure the real mathematicians already know about this, though it wasn’t stated in the video.
2D sphere (of a 3D ball) can be triangulated so that the h-vector is (1,n,n,1) for your chosen n >= 1. So (1,1,1,1), (1,2,2,1), (1,3,3,1), ...
3D sphere can be triangulated so that the h-vector is (1,n,m,n,1) for your chosen m and n such that 1
Yes, even I noticed that. The h numbers of the simplest sphere in a dimention are the binomial coefficients, which we can also see in the pascals' triangle as you mentioned. It is even true for the next dimention, where h numbers come as 1 5 10 10 5 1 which are the binomial coefficients of (a+b)^5.
I love the channel and videos and I have a small remark. The sound effects of the video (the ones used for counting) are too loud compared to the volume of the voice. This problem is apparent on other numberphile videos as well but this is one of the most obvious ones. It is hard to watch the video on the phone.. with love. Cheers
Halil Şen *too
Agreed.
Use the terms in the f-vector to make a polynomial e.g. 1, 8, 24, 32, 16 -> x^4 + 8x^3 +24x^2+32x + 16. Now substitute x - 1 for x and collect terms. In this case we get x^4 + 4x^3 + 6x^2 + 4x + 1 (coefficients are 1, 4, 6, 4, 1) and in general this transforms the f-vector into the h-vector.
What an incredibly likable guy.
Anyone else here cause they saw that it's now been proven and they want to understand?
Powers of 11..... 11^3=1331, 11^4=14641.... it’s hidden in Pascal’s triangle too
Cool, didn't know that
It doesn't go any higher than 11^4 tho
Ivan Myachykov that’s because the coefficients go above 10. For example one row in Pascal’s triangle reads 1 5 10 10 5 1. You’ll find that it works in that 11^5 = (1x1) + (5x10) + (10x100) + (10x1000) + (5x10,000) + (1x100,000). It’s also because the numbers in pascals triangles show up in any binomial expansion (a+b)^n.
Ohhh, of course. If it was done in a higher base, like base-16, you would see it.
Ivan Myachykov
Why not base-256?
Wow! Very clear explanation and more understandable than some of the native speakers ;-)
Very beautifully explained!
Once again Euler did find a pattern
I thought it was Terence Tao in the thumbnail.
He is so neat with that sharpie!! 10/10 would watch his penmanship again
The 1 in the problems comes from the count of the highest level object which when you have only one is 1. e.g. A tetrahedron would have f = {4, 6, 4, 1} where the 1 in this set is the tetrahedron itself. When you extend this idea to multiple objects you get higher symmetries, such as two tetrahedrons connected at a point would be different from two separate tetrahedrons and would have distinct h vectors as a result.
Every single one of the h vectors shown was a row of Pascal's triangle with elements raised to a power. Most cases that power was 1 (and the vector was the same as a row in Pascal's triangle), but in the case of [1,9,9,1] and [1,1,1,1,1], the powers would be 2 ( [1²,3²,3²,1²] ) and 0 ( [1⁰,4⁰,6⁰,4⁰,1⁰] ) respectively. Is there a counterexample to this?
Just commenting so I'm notified of any answers given
Oh man, felt like this was crying out after the first couple examples--I was hoping they might address it! But I guess there's only so much time in a video. I want to compute more examples myself now to see if there is a counter (seems unlikely...).
Very nice point! One more interesting observation is that the numbers in the h vectors always seem to add up to the exact number of n-dimensional triangles that the n-dimensional sphere was divided into. For example, 1+4+6+4+1=16, the number of cells in the hyper-octahedron
ALWAYS trivially true by considering 0 lik your last example. So no, there are no counterexamples.
like*
That f is FANCYYY!!
허준이 교수님 5년전 영상인데도 얼마전에 찍은 느낌이네요 ㅋㅋ 신기한 체험입니다.
Congratulations for the fields medal , 2022 !
Decagon infinity opens the door
Decagon infinity opens the door
Wait for answer to open the door
Decagon infinity - ah!
ayy
the gizz family grows ever stronger
neat reference