Ricci Flow sounds like the name of a rapper.

@francorende4305

8 жыл бұрын

true

@mehdinadif

7 жыл бұрын

Run ABC conjecture and Ricci Flow are releasing an EP in 2017

@ck88777

6 жыл бұрын

Or a vaporwave artist

@joshuaperry4112

6 жыл бұрын

Unless a rapper changes his flow and takes it to another dimension, his circle of influence shrinks. That's deep yo.

@6grams375

6 жыл бұрын

hahaha

The fact that he's brave enough to give a 15-minute explanation to people with very little, if any, knowledge in topology, should be appreciated..

@lovaaaa2451

6 жыл бұрын

Why should it be appreciated? It clearly didn't work

@schmud68

6 жыл бұрын

i think it works ok if you've actually studied riemannian geometry before, but how can you expect to understand something like this if you haven't studied some similar material

@lovaaaa2451

6 жыл бұрын

Problem is why you would need a low level explanation of the concept if you have studied anything similar in the first place. This is very bad pedagogy since it may give people the illusion of having understood something when they have not actually done so in any real way at all. Even more so with the thing about always trying to teach topology using pictures, it makes the whole field look vague, imprecise and generally unmathematical. Point-set topology should be defined and explained before one goes into the whole deal with pictoral representations of objects in topological space at the very least, in my opinion.

@schmud68

6 жыл бұрын

yeah, i agree it's definitely not the best presentation by any means, but i find it hard to comment too much as the only familiarity i have with the content in the video is the notion of a metric tensor from the math section of my GR course. i found that the extension of the idea of a "nice" deformation of some 2D manifold embedded in 3D etc. to instead just picking the dimension of your manifold and changing the metric under similar constraints to be a rather nice generalization.

@theultimatereductionist7592

6 жыл бұрын

+Lova aaa Well said.

I now know as much about Ricci flow as I did before I watched this video.

@TheUpsidedownCheese

7 жыл бұрын

what did you expect?

@entengummitiger1576

7 жыл бұрын

That they explain Ricci flow - they explained a few things that aren't Ricci flow instead

@sherlockcipher6690

7 жыл бұрын

You realise your statement is also true if you are an expert on ricci flow.

@ganondorfchampin

7 жыл бұрын

What Ricci flow itself is is far too technical to explain in a 15 minute video for people with no background knowledge in topology.

@Ub3rSk1llz

7 жыл бұрын

no it's not, lol. wikipedia does it in about 2 sentences. if you can't explain something adequately and succinctly, you don't know as much about the subject as you think you know.

I'm glad that even really clever mathematicians are bad at drawing circles.

@user-rs5hb6gd8e

6 жыл бұрын

they are not painters)))))

@fi4re

4 жыл бұрын

One of the ideas behind the field of topology is that bad circles are still circles, as long as they aren't so bad that they cross over themselves

@rv706

4 жыл бұрын

It's a bit like saying "I'm glad even the greatest architects are bad at laying bricks" but ok... (I mean: they're architects, they're not supposed to be able to lay bricks in the first place)

@ThrillaWhale

4 жыл бұрын

Eli Jacks Lol I get you dude.

@Triantalex

9 ай бұрын

??

Apologies to Jim for name typo in video - James Isenberg is at the University of Oregon - more here: pages.uoregon.edu/isenberg/

@ffggddss

9 жыл бұрын

Son of a gun! I think this guy (Jim Isenberg) was a classmate of mine in a grad course in General Relativity at UMd/CollPk in the mid 1970's! Jim - remember MTW, in its introductory year? With Prof. C.W.M.? Fred S

@ownagebutter

9 жыл бұрын

ffggddss no i don't remember sorry

For some reason I love videos trying to understand difficult stuff, even though I don't understand it. Guess it's just entertaining to listen to someone talk about something they're very engaged in.

No need to hold on to my hat, this went so far above my head it was never in danger!

Ten minutes in - "that's still not Ricci flow"

@dancrane3807

2 жыл бұрын

I told you all that, so that I can tell you this...

This guy really knows what he's talking about. I can tell because I can't understand a word.

@niemandniemand2178

5 жыл бұрын

dumbass

@xzy7196

3 жыл бұрын

@@niemandniemand2178 dumbass

@astroboy3002

3 жыл бұрын

Its just about how much time you have to devote to math

@liviu445

2 жыл бұрын

@@astroboy3002 pretty much yeah, it's innovation which requires just below genius intelligence if not genius.

Ok, Im going to bed now

@numberphile

9 жыл бұрын

rimythemurloc good night

@giacomopamio1191

6 жыл бұрын

Are you still sleeping!?

@-Me_

4 жыл бұрын

U still sleeping?

@owicehammad6

4 жыл бұрын

Are you still sleeping?

@mueezadam8438

4 жыл бұрын

@Rimy Are you still sleeping?

I love math in general, but this topology stuff is definitely out of my league.

I think maybe a follow up video would really help explaining how Ricci Flow was applied to the Conjecture.

I come here when I think that my Calculus course is too difficult.

@shugaroony

5 жыл бұрын

If you start studying general relativity down the line, you will be studying this and using tensor calculus which can take a long time to get familiar with.

Thank you for this excellent video Brady. I realize that more advanced videos get far less views but your advanced topics videos are unrivaled. As a grad student, this is just the right mix of material I know and don't know to keep me interested.

If I understand it correctly, Ricci Flow is an analytical method to turn any arbitrary topology in any number of dimensions into a sphere or toroid in the same number of dimensions, so that the original topology can be simplified for further analysis. Correct me if I'm missing some of the nuances.

@thefloormat3297

Жыл бұрын

I would but this comment section is too small to explain all the weird little details

My brain has twisted into a singularity xD

@niemandniemand2178

5 жыл бұрын

dumbass

@primsiren1740

3 жыл бұрын

@@niemandniemand2178 bruh moment

As a software developer / math enthusiast, forget a fields medal, if I ever appear in a numberphile video I'll feel like I accomplished something in my life.

@TonyKalashnikov

3 жыл бұрын

Try computerphile

Am I correct to assume that people with normal math in their brains should have a hard time understanding this?

@AgentDexter47

10 жыл бұрын

well yes, but any human has a problem understanding aything that deals with dimensions above our 3d

@energysage9774

10 жыл бұрын

It is very different but connections between what I assume you're calling "normal" math (arithmetic, classical algebra, calculus) and this stuff do arise naturally. For example, with well-behaved functions (conservative vector fields) if you integrate from one point to another it doesn't matter what path you take. Because of this, you can just integrate along whatever path you want--whichever is easiest. An awesome example that I like is Amperes law...take a closed loop of any shape you want, add up the magnetic field at every point along the loop, and you get the total current passing through the loop. It doesn't matter if your loop is a circle, a square, an oval that stretches across the galaxy... it always works. (disclaimer-magnetic field is actually not conservative but this still works. If it were conservative the loop would integrate to zero). All of Maxwell's equations have that nice "choose your own shape" feature which is at least part of why I think many people consider them so elegant.

@infinummjb

10 жыл бұрын

Not really. As I understand it you just have a square matrix that describes your space at a particular point and values of this matrix change in accordance to some function. Those 2D/3D visualizations remind me of how gravity (variable volume) and surface tension (constant volume) behave. Combine that with some form of Riemann Zeta function as the function that drives changes in the matrix's values and you might just explain away a good portion of reality ;) ps. In real life the hour-glass blob just splits into two - the behavior of wax in lava lamp is a good 3D example :)

@RedTriangle53

10 жыл бұрын

I understand how and why it works, but putting it in general math terms just makes it less understandable. I understand it in terms of physics, like with surface tension and optimization(probabilities/tendencies), but I haven't gotten to the point where a bunch of scribbles and an X should make instant sense to me. Is a matrix similar to a tensor field? With one value for each point in the entire plane/space. I assume that the state of each point is dependent on the ones around it as well. That's at least what I've gathered, though I'm probably massively wrong.

@infinummjb

10 жыл бұрын

Every matrix is a tensor, so in a sense they are similar.

I love these videos because they always make my brain hurt, so thank you for continuing to make complex mathematics accessible to everyone.

8:46-9:00 the foreshadowing of Coronavirus brought to you by Ricci Flow

@jacobpeters5458

3 жыл бұрын

"we're not gonna work with these guys anymore, let's just try and throw it out" xD

@LunizIsGlacey

2 жыл бұрын

"Very quickly, the whole thing is not going to make any sense."

His ability to draw circles is inversely proportional to his ability to understand them.

Love videos like this that try to tackle such difficult concepts. Keep it up

Great video! This is one of the best introductions I've seen to the math behind the proof of the Poincaré conjecture.

I understood it until the point where he said the narrow point in the hourglass shape would close up and become infinitely narrow. Aren't narrow areas supposed to get wider and wide areas supposed to get narrower until the whole thing is uniformly round? That's what they said in the first part of the video.

@stuffandpoop

9 жыл бұрын

***** IKR! if the whole scenario is that higher curvatures move to lower curvatures, why would those parts become even more curved???

@Pining_for_the_fjords

9 жыл бұрын

stuffandpoop I don't know what IKR means, but yes, that is a better phrased version of my question.

@MattL34

9 жыл бұрын

***** The flow is determined by sort of the curvatures at each point. Intuitively, the way you described it is accurate except for in the "not nice" sort of cases when the curvature becomes "infinite." If you have, say, two cones that are lying on top of each other with their two tips touching (looks like this >

@natan9065

9 жыл бұрын

Imagine seeing the hourglass shape from the side: it looks like the tight area should go outwards. Now imagine looking from the top: it's actually a tight circle which wants to go inwards!

@glialcell6455

9 жыл бұрын

***** It's 3D. The 2D projection is pretty misleading, though. I agree.

Ricci Flow would make a sick rapper name.

@TorinCooperBennun

10 жыл бұрын

Fuck. Yes.

@magicalpencil

10 жыл бұрын

Ricci Flow: Taking it to the next dimension

8:59 sums up the video for me. "...very quickly, the whole thing is not gonna make any sense"

This guy is blisteringly smart. He explains complex things so beautifully well.

It is amazing that a concept can be so simple that a layman can understand yet so powerful that it can be used to solve a century-old problem.

It seems to me that Curve Shortening Flow would always result in a shape that is *approaching* a circle, but will become a singularity before it is a fully realized circle.

@waynebrehaut7183

8 жыл бұрын

Or, perhaps, simultaneously? Just when you think you've won you disappear!

@t8m8r

6 жыл бұрын

True

@gmcgarveyut

6 жыл бұрын

Except that you can simultaneously do a linear expansion of the entire space (and thus the entire curve) to keep the inside area constant (or the curve length constant). Then it will approach the shape of a circle.

@Rohan_Trishan

6 жыл бұрын

Can you explain this more? As the curves go inward, you have a flat plane expanding to keep same area? So while edges go in, the middle goes out? Which will keep a circle shape... am I getting it right? And that is a circle shape as in 2d? Or will it form a sphere shape as well? I feel like this dude was hinting at something more.

@beano6338

5 жыл бұрын

I don't think it will ever become a realised circle before it becomes a singularity as a singularity is just a sizeless location defined by coordinates. So the fact it has no size means that it is impossible to have a singularity before a circle. I don't think that they happen simultaneously either as this would contradict the previous point.

I really loved this video! Thanks, numberphile.

Thank you for simplifying and extremely complex matter for us mere mortals. Excellent Lecture!

When he starts talking about the hourglass object, why would that middle section shrink? Wouldn't the curvature of that section make it bloat instead? It just didn't make sense

@energysage9774

10 жыл бұрын

Think about it this way: align the hourglass in the vertical position. Then along the vertical axis that central region has negative curvature and should bloat outwards as you predicted. But along a horizontal line it has positive curvature and should shrink. Circles shrink with the mean curvature flow and a loop around that skinny section is definitely a circle. What decides whether it should expand or shrink in that region just depends on how sharp the curves on the hourglass are (i.e. for a particular hourglass shape some is the negative or positive curvature greater in the center)... but that fact that it would shrink in some of these cases is specifically the problem with relying on the mean curvature metric and is why something more advanced like the Ricci flow is needed.

@Hulks92

10 жыл бұрын

Since this example is in 3 dimensions think of that middle section as a tube where the curvature of the tube at each point on the tube's surface is 1/radius of a marble (sphere) inside the tube so as the tube is symmetric the distance between the two opposite surfaces would shrink as shown in the video.

@EmonEconomist

10 жыл бұрын

energysage Thank you so much for explaining this - I didn't understand it in the video either but it makes so much more sense now!

Love how Brady had all his Yankees gear on at the end to conflict with Jim's Red Sox hat!

I understood nine words in this video.

@colt4667

10 жыл бұрын

Braggart !!!

@jeremyj.5687

10 жыл бұрын

Wikipedia, buddy. Whenever you don´t understand something, wiki that shizzle.

@Kalumbatsch

7 жыл бұрын

That's very useful, you get a description of something you don't understand in terms of other things you don't understand.

A mathematician, a poet, an NBA player. Some people are just born gifted.

wat

How exactly do topologists arrive at these bizarre rules for how these processes work?

@DorothyTheMouse

10 жыл бұрын

It's all made up and the points don't matter

@ABooleanEarth

10 жыл бұрын

Often to solve problems like the Poincare Conjecture. You make shit up that relates to the problem you're trying to work with, and if the method works or seems promising, then you develop it further into a more generalized form.

@ganondorfchampin

7 жыл бұрын

The general rules for topology in general preserve relationships between points without taking distance into account, just how they connect. Everything is just made up witchcraft which just works.

@SilverLining1

6 жыл бұрын

For math in general, it almost always comes from trying to emulate a more natural idea. Math is all cool and dandy until you start trying to make models for events in the real world, in which you will very quickly realize how many different rules you'll need. Think about how complex high school math was with just the most simple ideas. All that time spent expanding on so few ideas. Now, take a completely different set of ideas, and of course it's just as complex. The point I want to make is that, despite appearance, there is usually a very clear progression of thought to what we want to accomplish, but the path can be quite long.

@hassanakhtar7874

4 жыл бұрын

Topology began with the 7 bridges of Königsberg problem. It gave us the epiphany that we can study how shapes are connected (topology) rather than their definite size and being (geometry).

The only numberphile video where, by the end, I haven't understood what they were talking about. This shows what a good job they do normally, and *maybe* it shows that some concepts are just difficult. It'd be interesting to see whether this final Ricci Flow step could be made more accessible - kind of like the challenge that was posed about the Higgs Boson ie who can come up with the best analogy to explain its effect?

It's amazing that it took so long to prove the conjecture. That is so simple!

one of the most hardcore videos of Numberphile

After this video, I finally get that I still have a long way to go in maths...

sooooooooooooooo i think this is the first video where i feel like i learned nothing haha

Great video. As a researcher in Control Theory I don't get a chance to hear much about ideas in mathematics (however cool they may be) unless they happen to deal with something in my field, so videos like this are much appreciated. I would have loved to hear a little more on the topic (but for all I know, that information is in the videos you linked). I really just wanted to say thank you for your awesome videos!

I am so confused right now. Am I just dumb, or?

@ProfAwesomeO

10 жыл бұрын

no it's confusing and complicated

@EntropicalNature

10 жыл бұрын

No you just missed some undergrad maths which is essential in understanding this vid...

@IdioticPlatypus

10 жыл бұрын

yes, by confusing simpletons with geometry.

@yinggling

10 жыл бұрын

Relativity and Manifold tests soon :'( Believe me, it would take a whole semester to understand these concepts...

@soundlyawake

10 жыл бұрын

Somehow my university let me graduate with only math theory under my belt >_>

Entropy was also used on solving the Poincare Conjecture. You should make a video. What is the Entropy, in statistics and probabilities and how it was used in solving the Poincare Conjecture by Perelman.

so what's ricci flow?

@Yatukih_001

5 жыл бұрын

It´s a mathematical phenomenon that results in a curvature. Ricci flows look a bit like assholes. If you´re sitting on a toilet, you get what I mean. Ricci flows thus look a bit like a fart coming out of an asshole - the diagrams used to explain them, curves in spacetime are used to help mathematicians and physicists to better understand such things as what a universe looks like. The idea is you can use Ricci flows to predict for example population growths, describe a hypothesis suggesting what the universe is shaped like and so on. If you remember the last scene in ´Men in Black´the creatures that show up look a bit like Ricci flows.

@hassanakhtar7874

4 жыл бұрын

@@Yatukih_001 ok

@rewrose2838

3 жыл бұрын

@@Yatukih_001 You really wrote your heart out with that answer

you lost me when you said that everyone can the fourth dimension...

Hats off to you , yes you who's watching this video!

yes

@sean3533

10 жыл бұрын

no

@ForestShroom

10 жыл бұрын

Sean Haggard Maybe?

@MarsyorangesGaming

10 жыл бұрын

Maybe

@TimmahDee

10 жыл бұрын

ehhh... I'm not sure about that

@abacadian

10 жыл бұрын

FreaknShrooms I don't know. Can you repeat the question?

On the positive, Brady's camerawork is getting better. Also, this is a very interesting topic. On the negative, it wasn't really explained well. I think a little preparation prior to the taping would help. Nevertheless, my interest is piqued, and I will look up more info on Ricci flow. Thanks for the video and keep bringing us more interesting nerd material. Thanks for your effort, Brady.

From these few minutes he appears to be the best advanced math teacher I have ever seen, and by the way I myself am a math teacher too.

Should it be "hold on to your hats"? Can we ask bibliophile to explain the difference between on to and onto?

@sciencefreakdog

10 жыл бұрын

"On to" is static and "onto" is combined with motion, right? (English is not my first language, so please don't scream at me if I'm wrong.)

I absolutely understood nothing in that video :(

@JrDarkPhantom

10 жыл бұрын

I'm not claiming to be a prodigy, but I can usually keep up with the Numberphile videos, but this went waaay over my head, I had little to no idea. At least I'm glad I wasn't the only one! lol :P

@Koisheep

6 жыл бұрын

I'm a mathematician, so I know what Jim was trying to get through, but I think he rushed too much and the idea is poorly explained in general. Not your fault.

I have watched different videos about Poincare Conjecture for multiples times and read some introductory books regarding this topic. My understanding is the following statement. Ricci flow grant mathematicians a tool distinguish how the ballon will shape into based on the shape it had before blowing. With the help of Ricci flow, Perelman eventually proved that every shape that does not have holes with will eventually become one sphere or multiple spheres while proving the shape of a sphere has the property that any enclosed lines on that sphere will eventually shrink into one point. Please spot the mistakes of my statement if any of the math lover find it.

@rickascii

Жыл бұрын

You have the right idea but that's not quite it. We already knew that closed loops on a sphere can shrink continuously down to a point. What Perelman showed was the converse; any surface that has the property that loops can shrink to a point has to be (basically) a sphere. In particular, it will shrink to a sphere under Ricci flow or similar. In particular he showed specifically that that this was the case for a three-dimensional sphere. That is, like what Jim was describing, an embedding of a 3D surface in 4D space (or indeed a 3D surface that needs more dimensions to be embedded). If loops can be squozen down to points, then the surface shrinks to a sphere. Others had already dealt with all other dimensions.

@tonymontana9221

Жыл бұрын

@@rickascii Wow, it was a comment I made three years ago. Thank you for your comment. You mentioned that "it will shrink under Ricci flow or similar". Assuming that there are other flows, could you offer me any examples? At the same time, you are saying that a loop on a sphere shrinks into a point and a sphere without hole shrinks into a ball has an "if and only if" logical relation, right?

@rickascii

Жыл бұрын

@@tonymontana9221 There are other flows, he mentions some in the video. Mean curvature flow for instance. Idk your background, but it's a certain class of differential equations on the metric of a Riemannian manifold. I don't know how to characterize them in an intuitive way. I don't understand Perelman's proof so I can't speak to how it was helpful in particular, but the basic idea is that as a surface follows Ricci flow, it doesn't fundamentally change the nature of the surface. Holes don't open or close. As for the Poincare conjecture, we say a surface is "simply connected" if any closed loop on the surface can be continuously deformed into a point. A sphere and a plane are simply connected, you can see this easily. The punctured plane, the plane minus one point, is not simply connected. If you draw a loop around the missing point then there's no way to shrink it down to a point without leaving the space. A more complicated example of a not-simply-connected space is a torus. You can imagine drawing a loop around or through the hole in the middle and there's no way to shrink either down into a single point. This notion generalizes to 3d space as well. Take R³ minus the x axis, then any loop going around the x axis can't be closed into a point so it's not simply connected. The Poincare conjecture is that any 3d surface that *is* simply connected (and is compact, which is another thing entirely) can itself be continuously deformed into the 3-sphere, which is the 3d surface of a 4d ball. We have similar results for all other dimensions but 3 was the hard one.

The image i had in my head about how this applies to topology is like looking at map and you want to go from point a to point b but theres a mountain between the points so you walk around the mountains base rather that climb the mountain

These videos about Real Math are amazing. I don't understand it a bit better than before, but it is fun to listen to!

How do you work out what the curvature is? I would not have thought that any 'point' would have any curvature at all (as it is a 1 dimensional description). If you want to relate a point to another point to calculate the curve, how do you get to the 'next' point (as shouldn't any two points have an infinite number of points between them)?

Ricci Flow is an awesome name for a rapper.

after taking a course on differential geometry last semester I think I could understand this guy, and it woult be very interesting to play with it in wolfram or any 3d simulation program

This came out when I was in middle school didnt understand a thing back then. Now I am studying math as my major and can finally understand why ricci flow is so ingenious

at about the 8th minute in the video: shouldn't the 2D hourglass flow expand at the middle, instead of shrink to a singularity? I see that the direction of the tangent at that pinch points inwards, so the whole thing should eventually become a sphere; or is it something else?

@Ares4TW

10 жыл бұрын

I'm guessing you meant 3D. If I understood this correctly, a point on a 3D surface will flow according to two curvatures/circles, on two different axes. In the case of the middle of the hourglass, you have one large circle on the outside, and one small circle on the inside. So any point close to the middle will move just a bit outwards because of the large circle, and a whole lot more inwards, because of the small circle. This results in an overall inward motion.

@user93237

10 жыл бұрын

Ares 4TW Oh, of course. Now I am curious what they did exactly to prevent these singularities from happening. From what he sketched out it looked like they remove these areas and stich the surfaces together. The result will be two spheres then?

@niemandniemand2178

5 жыл бұрын

dumbass

Props for the Red Sox Cap.

Makes me think of how a drop of a liquid is formed out of saturated vapors. Reversely could help to describe how smaller drops of a liquid could be formed out of a big drop.

Great professor taking time to explain Ricci flow I am still confused but now I understand a bit

Does Ricci flow give a solution to the traveling salesman probelm?

@Peter-dk2ov

8 жыл бұрын

I was thinking the same thing!

@jerjurko

8 жыл бұрын

+Norman Hairston From what I understand - no. But I totally get your idea, but it's the Riemann's Geometry that could be helpful to give a solution to the travelling salesman problem, but that's only when you're thinking of a 3 dimensional space - taking the curvature and altitude differences between the cities into account. Please correct me if I'm wrong.

@theflaggeddragon9472

8 жыл бұрын

+Norman Hairston The problem is that the traveling salesman exists in a discrete context while Ricci Flow can only be described with a continuous path. There is no way to define curvature on a "sharp" corner where points lines meet so there is no way to define curvature which is based on derivatives. I don't know if you're familiar with them but they require the curve to be "smooth". Clever idea though!

@Darkenedbyshadows

8 жыл бұрын

+Norman Hairston No it doesn't I tried actually tried this idea earlier today for fun but as the comment above states, it simply does not work. I was more surprised of someone trying to explain Ricci flow in 14mins and 38 seconds! :D

@michb88

6 жыл бұрын

What if you replace the 'cities' with high reward gaussian kernels and add a reward function to the ricci-flow formulation?

*nods head* .... Absolutely.

Toughest Numberphile video to date. I'm gonna have to watch that again.

Have no idea what this guy is talking about but I love his enthusiasm.

Question: After about 8:00 when they're talking about the hourglass, why is the narrow part supposed to shrink off, rather than just start getting rounder??

@ranjiniravishankar6832

9 жыл бұрын

Martin Staykov replying to get the answer..

@MartinStaykov

9 жыл бұрын

Ranjini Ravishankar I still have no idea.

@FlutterBug

9 жыл бұрын

Martin Staykov because tight curves like that get smaller faster, and since they are close together while the larger curves are farther away they hit a singularity way before the other curves do so it doesn't have a chance to become a circle.

@MartinStaykov

9 жыл бұрын

FlutterBug But why is it getting smaller? The flow says that everything should be getting rounder. At 2:35 he says that places with backward curvature should move in the opposite direction. So it should be expanding, not going towards a singularity.

@FlutterBug

9 жыл бұрын

It's increasing the curvature, and larger curvature means a tighter curve; that's why if you start with a circle the circle just gets smaller until it goes to a point.

you guys really need to put a compressor on the audio. Whenever he puts his head away from the mic, I have to turn it up double the volume, and vice versa.

@davecrupel2817

7 жыл бұрын

kiwin111 and then he starts *BLARING IN THE MIC GIVING YOU A HEART ATTACK SO YPU NEED TO DOWN THE SOUND* 2:40 suprised me that way. xD

This guy is explaining stuff at the very high level.

Ricci is when flow is outward. When it is inverted and flows in, it's a complementary version called the Poorie Flow.

what is this nice man talking about?

@Ashebrethafe

4 жыл бұрын

As I understand it, Ricci flow involves determining which way (if either) a path along the surface of a 3-dimensional shape is "curving" at each point other than its endpoints with respect to that surface (in the same sense that part of a circumference of a sphere is "straight" because it's the shortest distance between two points along the surface of the sphere), and then nudging the points toward the insides of those curves in order to gradually shorten the path until it's completely "straight".

Have you ever wondered why a jet of water breaks up into water drops? Ricci flow explains it.

@ProfessorEisenoxid

7 жыл бұрын

+RelatedGiraffe Really?

@RelatedGiraffe

7 жыл бұрын

ProfessorEisenoxid Yup! :)

@phpn99

7 жыл бұрын

Nope. It doesn't. At best, Ricci flow would approximate the behaviour of the surface, but the explanation is rooted in physics and chemistry; math models are not 'explanations' of natural phenomena, but representations - analogues.

Devoured with gusto. Thank you numberphile :-)

im so excited to be able to understand this one day!

what are happen here

@andrejandu7181

4 жыл бұрын

YES

why does curvature move inward instead of outward at 8:13

@SamFisk

7 жыл бұрын

I would like to know this. I do have an idea: imagine the slopes coming into the chokepoint, they would expand to form part of the 2 spheres, whilst the bridge between gets stretched between those spheres. In theory (say with someone giving an example of water droplet formation above) a much smaller droplet would form out of the bridge. Can't explain why this wouldn't happen in 2d as well, though, so my speculation might be totally off.

@panonymous9659

7 жыл бұрын

+Sam Fisk Remember the circles, small circles move faster inwards than bigger circles. The point that seems to move in the wrong direction is a saddle point. It has different curvatures in opposite directions. The first one is the one we see in the drawn sideview, moving outwards, but if we would turn it 90 degrees (so we would look from the top or the bottom of the hourglass), there is a circle (through which the sand would fall). Thit circle obviously wants to move inwards. I'm not sure, but I assume the net speed is both of these speeds added. So if it is very narrow, it will move inwards.

@TheRMeerkerk

7 жыл бұрын

This bugged me a bit. I'm quite certain the curvature is negative and should therefore go outward instead of inward.

@joeybeauvais-feisthauer3137

7 жыл бұрын

Maybe you're thinking about Gaussian curvature, where saddle points are negatively curved and cups are positively curved. Then you're right, the curvature there is negative. However this flow deals with mean curvature. Roughly speaking, at any point on a 2d surface there are two directions (called principal directions) in which the curvature is maximal (the principal curvatures). For instance, at the saddle point we're interested in, going around the neck would have a curvature that is maximal and inwards, while going away from the neck up or down would have a curvature that is maximal and outwards. The Gaussian curvature is the product of these two, and the mean curvature is the sum. So while the space is very negatively curved in the Gaussian sense there, it might still happen that the inwards principal curvature is bigger than the outwards one and so the mean curvature is still inwards. The flow then makes the problem even worse with time, until we reach infinite curvature in a finite time and we have to apply surgery (surgery).

One of the best maths videos I’ve ever seen :)

Jeebus. I'm astounded by the brilliance of some of our fellow humans like this man. The potential of the human mind is awe-inspiring.

so 15 minutes for not knowing what ricci flow is... cool

@hakkbak

10 жыл бұрын

Lol, he says he can visualise 4 dimensions, but struggles to explain what Ricci flow is in 15 minutes...

@unvergebeneid

10 жыл бұрын

hakkihan tunbak Given that it _seems_ to be related to dynamically changing the space a manifold lives in itself, I'm not _so_ surprised that it's easier to picture a 4D object for example as a sequence of 3D objects than it is to explain Ricci flow in 15 minutes ;)

@hakkbak

10 жыл бұрын

Penny Lane Yeah, but for someone who worked extensively with the problem, its quite disappointing that after 12 minutes of saying "how can I explain this, huff puff" he just throws matrices and equations in without any explanation.

@unvergebeneid

10 жыл бұрын

hakkihan tunbak I totally agree.

@unvergebeneid

9 жыл бұрын

Edeinawc It's probably just a language thing but every time someone calls a total stranger "my friend" on the internet, it sounds _incredibly_ condescending to me. Let alone "my dear friend."

When did Robert De Niro become a mathematician?

@swarnimvajpai6373

2 жыл бұрын

I was looking for it 😂

Best Numberphile Video ever.

sounds like something they would expect me to fully grasp and have a working program to demonstrate by the next morning in school.

Is this what the smoothing function in Maya does?

@balciusfreefall6403

7 жыл бұрын

SuperStingray - No by all means. They say that if you iterate enough in hourglass-like shapes, you get singularities. The smoothing tool doesn't behave like that. Most smoothing functions work through averaging, and similar moving matrix operations.

Totally got it.....NOT!

beautifully explained

There was a paper recently claiming to have solved the Navier-Stokea equations. Any chance of a video about that? Probably the easiest of the Millienium problems to explain as it is just a PDE. Really enjpy your videos. Simple enough for an aeronautical engineering student to follow, yet still enough to break my mind as most pure maths does!!

I'm not sure I liked this video. I know you don't care much for comments but I thought I'd offer my two cents on this one. Fortunately, I completed a course in Riemannian geometry last year, in the penultimate year of my degree, so I knew what he was talking about when setting up the Riemannian metric, but I think this is a very demanding topic to just shoe horn into the last few minutes of a video. Not only that, but I'm not even sure we got any closer to explaining the key ideas - what is Ricci flow again? I don't feel as enlightened about the topic as I would have liked to have been/as I usually am after watching your videos. I think the presenter could have paced himself a little better to make things a bit clearer too, and the editing could have been a bit less choppy. It may have been better to actually outline all the ideas which need to be discussed BEFORE shooting so that there wasn't an awkward moment of "Okay, I'd better explain Riemann geometry now - how do I do this?" For example, Simon Pampena's latest video on circle inversion was perfect - well structured, enthusiastic, well paced and he delivered a satisfying conclusion to the question posed at the start of the video.

I'm not sure if it was a good idea to make a video on this topic. Since Ricci Flow is too difficult for non-mathematicians to understand, might as well just skip it, instead of making a video which leaves everyone just with their brain hurt and totally clueless.

Great video but I'm left with the powerful feeling that there is so much more to learn about this topic.

Whenever my smol brain gets confused, I visit the comments section to feel normal again.

@TheAlison1456

3 жыл бұрын

I'm kind of the same. I visit the comment section to look for normal comments. That is, people that aren't trying to be funny, attention-seeking, or adulating. Just people asking questions. It's surprisingly hard to find those comments.

I was feeling super smart because I understood the mean flow, and I thought it was the ricci flow. But nope, lost me again. Haha, even so, I really like these videos! Good stuff!

@element4element4

7 жыл бұрын

If you got the intuition behind the mean flow, you got the essential aspects of ricci flow. The idea is that, if you put (technically: embed) a surface (sphere, torus etc) in 3D space, you can start talking about "curvature". Mean flow is about studying this geometry, by dynamically changing this curvature over time. Now, this curvature is called "extrinsic curvature" since it depends on how you embed the surface in the surrounding 3D space. There is a more abstract way to define "curvature", which is "intrinsic" and independent on the "surrounding space". This is what he very briefly mentions as metric, ricci curvature etc. If you now play the same game, but with the abstract ricci curvature rather than the "extrinsic curvature", you get ricci flow. It's just an abstraction/generalization of what you already understood.

You're just wearing that hat at the end to get on James nerves aren't you...

That auditorium is so nice; those chairs...

Well! That certainly clears things up.

I think my ears are bleeding. is there something i can search on google that can help me visualize higher dimensions "spheres" i got lost at the matrix part, something to do with Neo I think. Numberphile

@allyourcode

10 жыл бұрын

Most people (mathematicians included) aren't able to visualize in higher dimensions. Therefore, if you want a way to visualize what's going on, you have to resort analogs in lower dimensions. That was the purpose of presenting the curve shortening flow, and the mean curvature flow. Those are things that can be visualized, and you can use them to get a sense of what's going on in higher dimensions. One of the things to keep in mind is that analogies often break down, so don't count on getting a perfect picture by thinking about lower dimensional analogs. Those are just to give a bit of intuition. For example, the curve shortening flow where area is preserved does NOT result in collapse. But in the mean curvature flow, collapse is possible, as in the case of an hourglass shape.

@dammitdanFTW

10 жыл бұрын

Look for a video here on youtube called, how to imagine the 10th dimension. Or something close to that. I had to watch it about 3 times but it was pretty awesome

I don't understand

@niemandniemand2178

5 жыл бұрын

dumbass

@fishyeverything8530

5 жыл бұрын

Niemand Niemand you probably didn’t understand it yourself

Love this video!!

so this is the "pure math" behind general relativity right? I already knew about the Ricci tensor described in general relativity so that got me interested.