Nash Embedding Theorem - Numberphile
Вставка
- Опубліковано 24 лип 2024
- Featuring Edward Crane.
James Grime also discusses this: • Embedding a Torus (Joh...
Twisted Torus video: • Topology of a Twisted ...
Nash shared both the Nobel and Abel Prizes. Full details.
Nobel: www.nobelprize.org/nobel_prize...
Abel: www.abelprize.no/c63466/seksjo...
Edward Crane at the University of Bristol: www.maths.bris.ac.uk/~maetc/
Torus Balloon at Maths Gear: bit.ly/TorusBalloon
NUMBERPHILE
Website: www.numberphile.com/
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
Videos by Brady Haran
Support us on Patreon: / numberphile
Brady's videos subreddit: / bradyharan
A run-down of Brady's channels: www.bradyharan.com
Sign up for (occasional) emails: eepurl.com/YdjL9
sure I can imagine 17-dimensional space. I just imagine n-dimensional space and then substitute 17 for n .
Imagine a 16 dimensional object and place it an accelerator. You have created a 17th dimensional object. Namely the object with acceleration.
@@scin3759 Wait, so if I take a human (a 3D object) and put it in a car (an accelerator), then the human becomes 4D? Certainly not spatially 4D.
James G time is a dimension; so is acceleration. let X= object of euclidean dimension x. Let Y= object of dimension y. Let "dim" stand for dimension. Then dim (X euclidean product with Y)=dim (X)+ Dim (Y)= x+y . Acceleration can be associated with a linear continuum. The euclidean dimension of any continuous line segment is 1. It can be shown that the dimension of any circle C is 1. Since a circle C has dimensional 1. And a line segment [0, 2] has dimension 1, the Euclidean product of C with [0,2] is a two dimensional object that looks like a pipe segment of length 2 and with radius the radius of the circle C. The line segment [0,2] can be associated with all possible accelerations from 0 mph to 2 mph. So for example (C, 1) is the circle being accelerated to 1 mph. (C, 0) corresponds to the circle with acceleration 0 mph. (C,1.1111) corresponds to the circle with acceleration 1.11111 mph, and so on...
Hope this helps.
you don't know me, just as well because I couldn't trust you.
I like to think about a torus going before us but I tire easily doing so.
RIP John Nash
1928 - 2015
Someone His work inspired millions of economists and mathematicians from around the globe.
Nash's paranoias were informed by real threats to freedom, but ironically some of his theories were informed by such intense paranoia they underestimated the amount of trust people typically accord each other in everyday life. These ideas that were implemented during the Cold War crept out into policies that still affect us today. An excellent documentary on this subject is Adam Curtis' "The Trap".
Watching this video, I feel like a freshman who accidentally walked into a doctoral level math(s) course.
Rick Seiden As a maths freshman... yes.
Rick Seiden Pretty much. Topology isn't really that friendly at the start.
***** I don't doubt it. I understood it much better when Dr. James explained it in the other video. But it still went flying over my head. (As a reference, I have a minor in Mathematics, and a Masters in Mathematics Education. I've taken Master's level courses in Mathematics.)
Borikuaedu3991 I'd argue that topology is the hardest subject in the world to understand. Once you know the lingo, you have a better handle on it, but when you try to solve problems or prove conjectures, you'll need a brilliant mind just to contemplate the problem.
***** You're smarter than me, then.
17-dimensional space isn't hard to imagine. I just imagine usual n-dimensional vector space, then I make a linear map between this vector space and (n/2)-dimensional complex vector space and substitute 34 for n. Then it's easy to make a topological connection with smooth diferentiable manifold and take away complex coordinates. All you have to do is to jump with one leg around Klein bottle filled with unicorn's blood during the full moon.
Not sure if you're just being sarcastic, but I'm going to assume you are, But this isn't exactly "imagining" 17 dimensional vector space, but using the properties of Kahler Manifolds. :) which isn't something we can imagine but we derive mathematically.
You are right. But I reacted to what he said at 10:08 :)
I bet your great fun at a party Mahmood.
Patrick Moloney i have my moments
786123-dimensional space isn't hard to imagine. I just imagine usual n-dimensional vector space, then I make a linear map between this vector space and (n/2)-dimensional complex vector space and substitute 1572246 for n. Then it's easy to make a topological connection with smooth diferentiable manifold and take away complex coordinates. All you have to do is to jump with one leg around Klein bottle filled with unicorn's blood during the full moon.
To quote Stephen Fry (I think), "Nash was a brilliant mathematician who suffered greatly from the effects of being played by Russell Crowe."
"it's easy to bend it because it's bendy"
And "squidgy". I learned another new word today.
And yet in no n-dimensional space would you say a stick is sticky.
I think I smoked weed out of a Klein bottle once ?
+wertytrewqa buhaha, funny- I think you'd need to be on the inside of the Kline bottle to do that.
Hahahahaha very nice!
+Austin Locke How can you be inside, if it only has one face? :O
+wertytrewqa there is no "out of a klein bottle" but that is something i wanna try
+wertytrewqa -- just checked my bong... no klein bottles there ;-) ... but at least the chillum penetrates the main cylinder.
I didn't think people would get anything meaningful out this vid either. Talk about not being able to communicate a single idea. They who know the topic will laugh. They who don't laugh too.
Just recently found out about Numberphile 2. I love it. Then I found about Numberphile 3. It went even deeper. Blew my mind. Then I kept digging. By the time I got to Numberphile 7, it was all just straight proofs. Too intense for me. So now I'm back here.
Did you encounter Numberphile pi? That is tough, for real. But not yet quite like Numberphile i, which deals with complex topics, I imagine.
I feel that if he explained what he was trying to explain before explaining the properties of the thing he was explaining, I would have understood what he was talking about.
you need to lay the base so that you build on top of it
Numberphile is to be praised for attempting such an ambitious topic. I'm not a mathematician, and now I have a bunch of terms to go look up, but I think I got the jist. Well done. More brave videos like this.
This video is all over the place...
My understanding: 30%
My enjoyment: 100%
Great stuff! More on differential/topological geometry in the future please, Brady. There are much more intuitive theorems to discuss for the wider Numberphile audience than Nash's Embedding Theorem, too!
Always enjoy these extra videos. It helps so much when trying to understand it :)
You could say it was a bit of a Parker Klein bottle
A video watched by many, understood by few.
Mathoma I think the problem is that the presenter was assuming a lot of implicit knowledge about the subject, for example the idea of what an abstract set/manifold is, so that if you don't know this stuff it's hard to even understand what an embedding is.
Mathoma I'm in my second year of undergrad for pure maths, and I'm currently taking a course on differential geometry, and it seems to me that this video was aimed at my level -- it's about math that is graduate level, but explained with only the basics of the subject. It would have taken more work on the part of the presenter to take it and build it up in a way understandable to a non-mathematician, so I agree.
When Russell and Whitehead published Principia Mathematica (3 vol, 1910-13) a reviewer suggested that a possible twelve people on the planet might fully understand it. In the 70s when my local bookshop got a dozen copies they sold out within 4 months. Clearly understanding is unnecessary when books can be bought for "pose value".
You can embed a flat torus in 3-space if that 3-space is a finite, repeating space (asteroids game type) rather than the infinite one we appear to live in. In the repeating 3-space it's just a straight tube spanning the full length of the space and connecting with itself.
The evaluation for n was improved few times and according to same Gromov, mentioned at the start of the video, it is n>=m^2+10m+3 - even bigger value. For 2-dimensional surface of the tire you'd have to have 27 dimension for embedding.
Ok, much more information than with James's video but also a bit all over the place which is probably why it was ostracized to Numberphile2.
Wow. For the first time I could not understand what a numberphile video is talking about.
He said that the adjacent normal vectors on a mobius strip without a fold had to proceed gradually, which I understood, but then he said that was counter intuitive. Totally lost me from that point on.
Just shows there's always so much more to learn about math I guess. I'll have to come back to this video later.
no, he said that the normal vector has to vary gradually, and that because of that you can't have a fold, because then it wouldn't be gradual.
and then he said that the fact that you can embed the torus this way is conterintuitive
+Noam Tashma I'm confused as to why that's counterintuitive, as in 3D space, the normal vectors on a torus are continuous? (I never studied pure maths so I'm not entirely sure what 'embedding' is)
yeah you need several graduate courses in mathematics before this can start to really make sense to you. Graduate analysis, graduate differential geometry, graduate PDEs, and more.
"imagine, if u can, this is in 5 dimensional space." haha, yeah
Beautifully presented, although the subject is intrinsically tough, and difficult. Nash's mathematics reminds me of Ramanujan;s Mock Theta Functions or his Tau Function.
Great video! Difficult for us amateurs, but within reach. Love it!
This is a perfect example of how quickly pure mathematics becomes impossible to relate to real life. It's best to take this sort of stuff on its own turf instead of trying to "make sense" of it in regular language.
isaacc7 Actually, if you try hard enough, you could probably find some application. I can give you two related topics, one more everyday and another more abstract but still kinda physical.
1. You can think about how people have difficulty on trying to make the "most accurate" map of the world and how sizes, shapes, and distances all get screwed up when you try to go from 3-D to 2-D or vise versa.
2. If you like physics, you can think about what our universe would be like going from 2 to 3 dimensions or if you like stuff like string theory, how 1 dimensional strings can make a 3-dimensional space and objects along with 11 and 12 dimensional space.
Sergio Garza Actually, the nice thing about dimensions is that it doesn't necessarily have to mean physical dimension. In statistics, very high dimensions are not uncommon due to the nature of data. We are even seeing applications of topology via homology show up in data analysis. This is done by taking a large data set, making a "point cloud" with a proximity metric, building a simplicial complex using the metric, and calculating the persistent homology of this complex.
These applications are just topology, but given the analytic nature of statistics, I wouldn't be surprised if differential geometric techniques are soon to follow. Maybe one day we'll even see Nash's Embedding Theorem lead to progress in Economics like his work in Game Theory has already done?
In fact, a quick Google search for "Differential Geometry Statistics" gives many links to books and research in this direction. From the first link, to the book "Differential Geometry and Statistics" by Murray and Rice, I see that they develop a lot of differential geometry (manifolds, differential forms, connections, curvature, Riemannian geometry, vector and fiber bundles, and tensors) with a view towards statistics, as well as the statistical tools that are built from these concepts. The point of view of the book seems to be of treating the spaces of random variables and probability measures as manifolds. This isn't exactly the same as the stuff dealing with data points above, but at least it shows that other people have probably already thought of anything I could ever think about. Maybe a way of building some sort of (statistically accurate/relevant) smooth structure from a very large data set could lead to another direction in which geometry intersects with statistics.
It's amazing what one can do by replacing one concept with another which are both essentially the same thing but surrounded by different contexts.
*****
That's fascinating! I'm definitely going to look it up today and see if I can find some books online!
Sergio Garza I agree, it is very interesting. I've never been a fan of statistics as I've been taught in the basic courses I had to take, but seeing such beautiful mathematics manifest in such unexpected, but perfectly reasonable, ways is enough to make me want to look into it, even if only as a side project. I'm not a differential geometer or a topologist so this is already testing some of the limits of my knowledge in either subject.
No no, I know that pure math is incredibly important I just think there is a limit to how much you can simplify it so that people without a mathematics background can understand it. I thought this video came up short in trying to explain Nash's ideas. I'm not sure it's even possible to do so without at least dipping into some calculus and more advanced topics.
wow! great video
tons of culture, and he explained the embedding thm rather well!
I don't think I've seen him interviewed before, you should have more of him!
He kept saying embedding and I still have no idea what he means by that. He mentioned that 2 points on the surface of the klein bottle need to be in different places to be embedded but i don't see how that applies to a doughnut shape. As far as I can see, all the points on it are in different places in 3 dimensional space.
Are there some nice simple "worst-case" scenarios where you actually need the maximum number of dimensions to get a nice and smooth embedding?
Like, what kind of curve takes at least 7 dimensions to be embedded and for what reason can a curve never require more than 8 dimensions?
Kram1032 Im more of a physicist but just thinking about a plot of a virtual particle with (terms with) those "degrees of freedom", guess it can depend on your setup, definitions and constraints (on your topology/manifold too). Or determine whether (such) values/elements belong to a set defined like that. just a thought, interesting question thank you. Orbifolds and string theory connections seem related. (For instance, from wiki: when looking for realistic 4-dimensional models with supersymmetry, the auxiliary compactified space must be a 6-dimensional Calabi-Yau manifold )
John nash is a legend .. rip
The object in the thumbnail is called a manifold and manifolds are something that look a certain way locally but are very quite different globally-- James H. Simons
Just woken up and watched this video now I have a headache! Thanks numberphile!
Did James get that Klein Bottle from Cliff Stoll by any chance?
Wait, is he saying a flat torus be considered homeomorphic to a regular torus? He's talking about them as if they're the same class of objects. Isn't that like saying a round sphere is homeomorphic to a disk? I mean they're homotopic equivalent but not homeomorphic. In the case of a flat torus, wouldn't the normal vector immediately flip "sides" (i.e. have an indifferentiable "edge" of sorts) as you passed over the "edge" of sorts (I forget the topological term for such)?
Is Doctor James Grimes single?
When I watch his videos, my heart beats a little bit harder and faster.
I'd like to embed him in my 3 dimensional space.
That's the kind of pickup line that makes folks swoon.
so he says that Nash embedded the torus in a way where folding isn't allowed, presumably because it creates a cusp of sorts and makes the surface non-differentiable at those points, but in the other video about Nash, it was mentioned that Nash embedded the torus by creating points where the curvature was meaningless, and it was implied that it's because he created points of non-differentiability. That seems contradictory, am I missing something?
I can't be the first one to point this out, but didn't Nash's equilibrium paper rely on Kakutani's fixed point theorem, which is a generalisation of Brouwer's? I seem to remember that from my mathematical economics class. And only because Kakutani's theorem was the whole point of the course.
Please do a video on differential forms (!)
How can the normal vector be continuous, while the shape is non-differentiable without folding? Doesn't that imply that it is folded?
Ah, now I know what Tom Lehrer was talking about with his "analytic and algebraic topology of locally Euclidean parameterisation of infinitely differentiable Riemannian manifold".
He used Kakutani’s Fixed point theorem to prove the NE.
"We've got a 5 dimensional -- if you can imagine -- 5 dimensional rubber"
Felt a little like I needed an intro to higher dimensions to grasp this one even a little bit.
thank you
somewhere in an alternate universe this video is a video about a bicycle inner-tube's properties and comment section is full of superlatives on this bicycle inner-tube and real-life stories containing a bicycle inner-tube
Awesome video!
Got to love this Vidniappe :)
What a crazy result! [Note: Those are often the best.]
When m=1, what is it that can possibly require *7 dimensions* to embed a mere 1-d manifold isometrically?
What am I missing here?
Could we get a bit more explanation why it's sometimes so difficult to preserve the intrisnic metric of an m-manifold?
Am I even asking the right question?
it said in the vid that here the 7d would only be an upper bound for number of dimensions to embed a 1d manifold isometrically and the rest i have no idea
What happens if m is another polynomial?
#call-for-experts
When does a 2nd order PDE have unique solutions? Is being parabolic or elliptic or hyperbolic enough? And where can I find a proof for that or videos for that?
relike868p what is your level of understanding currently?
Up to a first course in ODE I think...
though I know a little bit of Frobenius method in solving PDEs and Banach's fixed point theorem.
But that's it
17 clues required to solve sudoku; 17 dimensions to embed this tube. Is there a pattern emerging?
RIP John Nash
The Klien bottle looks like the torus, described by vector mathematics, thrown out of a sling shot
Such a lovely guy
As far as I remember the proof of Nash Equilibria in Game Theory exploits the Kakutani fixed point theorem not Brouwer's FPT... apart from that i like the video very much.
oh well, that was confusing and exciting at the same time
"If you can imagine five-dimensional rubber . . ."
Right.
Does this man have a word limit to reach?
I like Edward Crane. I want to see more of him.
Very interesting video!!!
well this was incomprehensibly explained
More like this!
No, I cannot imagine a 5 dimensional rubber.
I do not doubt, that he knows what he is talking about, but he certainly cannot explain it.
I just remembered why I failed A Level Maths. My mind can only work in 3 dimensions :(
I hope Brady don't take all this negative feedback about being too hard to understand too seriously. There is not much other places to put this and if it is not to be here, what shoudl he do? Create Numberphile3? Is he obligated to make every video accessible by any level on every channel of every topic?
I'm not a mathematician and i'm not into differential geometry. I could say I understood roughly 70% of the video, but it was a worthy 70% and I could not find detailed information like this on a Numberphile video for a long time.
I havent understand anything. I watched russina document about Perelman and Michail Gromov was part of that document. I later watched videos about Ricci flow or some lectures by Gromov and again I didnt understand a word. This is so strange to me.
"It's impossible to embed a Klein bottle in 3D space because 3D space is a topological illusion of a material, actually a condensed substance of timing from quantum duality-multiplicity, ie 3D is a kind of antilog "3-ness" of dominant probability in "textured" timespace, 1-2-3D time-timing history in 1-0D of eternity now.
This must be one of the hardest thing's to explain
Sometimes when I can't get to sleep I find it helps to imagine a Klein bottle (a normal one embedded in 3D space) and an ant crawling over its surface and making its way back and forth from one part to another via the tubular "neck".
Continuosly, but not differentially? So it suddenly speeds up and slows down all the time on some surfaces?
Continuously so that it doesn't suddenly break apart/not exist somewhere, deferentially such that you can draw a tangent line on it. A continuous non-differentiable shape is one where you can draw it without lifting up a pencil, but you can argue on which way the tangent line goes on some or all points.
Josh McGillivray I know what it means... I had Higher Mathematics course in the University. I was just surprised that it had that kind of property.
ah kk, :)
Numberphile2 “If you can't explain it to a six year old, you don't understand it yourself.” - Albert Einstein.
And thank you brady for asking all the "right" questions.
mos ab Interesting, then, that decades later most college graduates still do not understand Special Relativity. Perhaps Einstein was a bit optimistic - or perhaps he was referring to 6 year old Einsteins, :D
I think you mean "they don't understand general relativity."
mos ab No, that is not what I meant - but most college graduates do not understand that either.
Heather Spoonheim Most college graduates don't understand any kind of maths; that doesn't mean you have to be a genius to do so, just that most of them switch off.
mos ab Are you sure you mean Einstein? I know that Feynman said that if he couldn't explain some piece of physics in a way a freshman could understand then that meant he didn't understand it himself. But that's not Einstein and it's not six year olds. Could you provide a reference?
A Brazilian, discovered the "PI" the absolute number as it is done, the formula that was used, the means to reach this conclusion, as some thinkers of the time reported that the figure was "unchangeable", was a number "Irrational" , a number that accepted not be done in fractions, for being "irrational", it is infinite, and could not be a rational number, is that Sidney Silva managed to unravel this mystery of this giant number; which to date had never been studied to reach such a conclusion; it proves and drop the whole "theory", "theorem" and the "thesis" of the time, which stated with complete truthfulness that he is "changeable" therefore accepts changes it is "rational", is compatible will a fraction (2205 / 700), (3.15), it was researched and investigated to be 100% accurate for calculations in mathematics, it is finite, as it is an accurate and consistent number will a fraction; throw this challenge to academics (as) students (as), Amigos (as), and colleagues Known (as) and to all who want to bring down the "thesis" of Sidney Silva, on this great discovery of the number of "PI".
WAT
+Sidney Silva wat?
+Sidney Silva why are you referring to yourself in the third person?
Ok flew right over my head.
What exactly does "embed" mean?
uuurgaah A space has a notion of just existing by itself (see tomahwak thehawker's post and discussion). For instance, the real line (R^1) can be thought of as a line that just is a line. When we embed the real line into some space (like R^3) we are taking a continuous function from the line into R^3. You are basically drawing the line continuously in R^3 (though it may be continuous... like no tearing or cutting your space up into pieces... you can bend and crease it as much as you like since the embedding does not have to be smooth). You can basically think of embedding as taking a space and just putting it inside another space.
I think the embedding is also one-to-one, meaning that when I draw my line in R^3 I am required to make sure that I do not allow twp points to be drawn at the same place (no crossings).
I actually understood everything... FEELING GOOD.
aha all of this went completely over my head :)
Grady hits the snooze button at 9:52
While moving the with constant speed the acceleration is well definded. It is 0.
But I do think I understand what he is trying to say. Not sure how good that analogy really is...
Faxter313 I guess your comment was meant for the video with James. He tried to explain the concept of differentiability in one dimension (A graph is a two-dimensional representation of a one-dimensional function). Curvature in one dimension is the gradient of the gradient (acceleration is the curvature of the distance), so in order to have curvature in a point the function needs to be a least two times differentiable. In higher dimensions the notion of curvature is not as simple as that, but it still relates closely to the second derivative (gradient of the gradient).
More Famous Grouse please!
Makes sense to me!
Is there any literature on this topic?
Only just enough to fill the Library of Congress
i wonder who russ la cro is
Oops! 3:21 Once he has found a hole he is instinctively inserting his middle finger in it.
I think I understood it but it's making me question my dream of being a topology professor. Then again, I'm only just now about to enter college so... I'll give it time :P
What is he saying???
Do the extra dimention cosmology people know about this?
7:10 he said "valve". Half-Life 3 confirmed.
..and place is price,
paper is piper.
Sweet bong.
I don't know if he is not as good at explaining or if i just can't quite comprehend this subject lol
I’m not sure my man knows what Nash was talking about either 😂😂😂😂😂
He uses way to many mathematical terms most people won't know about and sometimes he doesn't actually need those terms to get the point across. At other times it would be better if he explained some things a bit more in depth like what a Riemanian metric is for example. I do know (approximately) what a metric is but I have no idea when a metric is Riemanian.
String theorists would love this.
I had somewhat of a difficult time understanding the video and the way things were explained.
Ah, well. I guess I'll just move on. If it becomes important for me to know the concepts in this video later on, I'll check back in.
What do mathematicians define as space?
tomahwak thehawker Very shortly, it is a set of vectors. For more info, check Wikipedia's article about Vector space, the english version is rather simple I think. (I used to read the french version which is very academic and hard to understand for people that didn't study linear algebra.)
tomahwak thehawker From Wikipedia "a space is a set (sometimes called a universe) with some added structure." The key point to understand here is that sets don't intrinsically have any of the structures needed to do geometry (distance) or algebra (operations). For example the set of real numbers (R) is simply a set, the usual operations (+ and *) on R aren't intrinsic to R, they are functions which take two real numbers and output one other real numbers. This example also displays that not all "sets with some added structure" are typically called spaces, the set of real numbers with + and * is an example of what is called a field rather than "+ and * space" (R can form a vector space over the field R but the point remains). The main types of mathematical objects I've heard of which are commonly called spaces are topological, metric, and vector spaces. A writeup which does these concepts justice would be horribly long but I point anyone interested to Analysis I by Amann and Escher. It covers metric and vector spaces quite well though topological spaces are only treated through metric spaces and not in full generality.
tomahwak thehawker When a mathematician says "a space," they generally mean "a topological space." This is just a set with certain subsets being labeled "open." It's a very general idea on which all geometric ideas/definitions are founded. Check wikipedia for more info.
Oftentimes something being a topological space isn't enough data. Maybe we want to know how far apart two points in our space are. If we can define this, then we have made our space into a "metric space." Maybe we want our space to locally resemble euclidean space. Such a space is called a "topological manifold." Maybe we also want to know how to differentiate functions defined on our space. If we can do this then our space is a "smooth manifold." The list goes on....
The kind of space that they were talking about in this video is a "Riemannian manifold." That is, it's a smooth manifold which also has a sense of distance between points. Think of it like a smooth manifold which is also a metric space.
These are just the first ideas that one learns about when studying differential geometry.
mnkyman66332 ^ What he said because that was pretty damn spot-on.
mnkyman66332 This is the best explanation!
James Blunt? I prefer James Grime lol
Trippy.
Sooo. Asteroids is played on a two dimentional surface in 17 dimentional space. Wow.
Think this wouldve been on the main one if it was 2020 i think weve got a lot nerdier
In order to explain Nash embedding theorem, he starts to explain topology in a few minutes. He fails.
will anybody tell me why numberphile 2?
Nasrin Akter So they could bring in more mathematicians without denying those in the original channel their fair share of being the star for the day.
Tex Rittenberg thanks
So as to share extra material for those who are specially interested in a topic without saturating the original channel and without posting overly complicated content that may put off newcomers.
morgengabe1 Well said....ahhhh, say what?
clever guy