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".

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.

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.

And "squidgy". I learned another new word today.

And yet in no n-dimensional space would you say a stick is sticky.

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.

Did you encounter Numberphile pi? That is tough, for real. But not yet quite like Numberphile i, which deals with complex topics, I imagine.

+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.

you need to lay the base so that you build on top of it

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".

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.

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.

That's the kind of pickup line that makes folks swoon.

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 )

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

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

WAT

+Sidney Silva wat?

+Sidney Silva why are you referring to yourself in the third person?

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, :)

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).

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?

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).

Only just enough to fill the Library of Congress

Penny Lane I take it you don't believe in String Theory?

msolec2000 As long as they allow the likes of Michio Kaku as the public faces of their movement, I can't take them seriously. That would be like a pharmacy that tries to attract customers by giving away free snake oil on the street. The day I read the headline that Michio Kaku vanished into a black van and wasn't seen again, I'm ready to listen ;)

Penny Lane ew man, get out of my ears and my eyes.

Bliss Woven No need to be so rude, man. I'm just a barber showing photographs.

MichaelKingsfordGray You've lost me. What exactly are you referring to?

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!

I hope so, thanks anyway :)

Will Deary this is some pretty heavy maths, it's a theorem that won a nobel prize in the mid 1900, you know that the closer you are in time, the harder the math

Joaquin Pirotto As told in the the previous video he won the Abel Prize for this. He won his Nobel Prize for his work in game theory.

Benjamin Pedersen yes, i got mixed up, there isn't a nobel prize for maths. I watched this video first though

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?

Domosaur Sid But, does he even lift?

ReZdItalia big brains are better than muscles

vishizs Say that in a fight.

ErkaaJ the brits won ww2 because of Alan Turing, certainly not because they were outnumbered

SquidofCubes Yes, I know, I know, I am studying mathematics as of this autumn - I value knowledge above all. I just pity the people shouting "the brain beats the brawn", almost as an excuse for poor physique. None exclude the other, why not be both muscular and clever.