Ricci Flow - Numberphile

[@taufiqutomo]

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]

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

[@schmud68]

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]

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]

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]

+Lova aaa Well said.

[@Morphimus]

Ricci Flow sounds like the name of a rapper.

[@francorende4305]

true

[@mehdinadif]

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

[@ck88777]

Or a vaporwave artist

[@joshuaperry4112]

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

[@6grams375]

hahaha

[@TheTokyoAmducias]

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

[@TheUpsidedownCheese]

what did you expect?

[@entengummitiger1576]

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

[@sherlockcipher6690]

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

[@ganondorfchampin]

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]

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.

[@numberphile]

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

[@ffggddss]

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]

ffggddss no i don't remember sorry

[@elijacks8269]

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

[@ПрикладнаЕкономіка]

they are not painters)))))

[@fi4re]

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]

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]

Eli Jacks Lol I get you dude.

[@Triantalex]

??

[@ylette]

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.

[@Stonerman023]

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

[@mxmartinelli1]

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.

[@kingofmaiars]

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

[@Fr3Eze1992]

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

[@niemandniemand2178]

dumbass

[@xzy7196]

@@niemandniemand2178 dumbass

[@astroboy3002]

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

[@liviu445]

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

[@ScottLahteine]

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]

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

[@DorothyTheMouse]

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

[@dancrane3807]

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

[@ChrisSeltzer]

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]

Try computerphile

[@rimythemurloc]

Ok, Im going to bed now

[@numberphile]

rimythemurloc good night

[@giacomopamio1191]

Are you still sleeping!?

[@-Me_]

U still sleeping?

[@owicehammad6]

Are you still sleeping?

[@mueezadam8438]

@Rimy Are you still sleeping?

[@sean3533]

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

[@labibbidabibbadum]

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

[@tonymontana9221]

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]

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]

@@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]

@@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.

[@arturslunga3415]

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

[@alexjohnson4681]

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

[@jacobpeters5458]

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

[@LunizIsGlacey]

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

[@rachell3506]

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

[@LCBrink]

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

[@wangus]

Ricci Flow would make a sick rapper name.

[@TorinCooperBennun]

Fuck. Yes.

[@magicalpencil]

Ricci Flow: Taking it to the next dimension

[@gelerobev4853]

My brain has twisted into a singularity xD

[@niemandniemand2178]

dumbass

[@primsiren1740]

@@niemandniemand2178 bruh moment

[@MercenaryGio]

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

[@shugaroony]

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.

[@EdwinSteiner]

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

[@Pining_for_the_fjords]

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]

***** 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]

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

[@MattL34]

***** 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]

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]

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

[@jimmy000]

so what's ricci flow?

[@Yatukih_001]

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.

[@rewrose2838]

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

[@Klaevin]

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

[@MadGammon]

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]

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

[@t8m8r]

True

[@gmcgarveyut]

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]

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]

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.

[@LordAugastus]

wat

[@RedTriangle53]

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

[@AgentDexter47]

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

[@energysage9774]

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]

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]

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]

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

[@johngrey5806]

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.

[@robbie4128]

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

[@have_a_nice_day399]

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.

[@FerociousKitteh]

I really loved this video! Thanks, numberphile.

[@chasefancy3092]

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

[@CreativityCurve]

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

[@swill128]

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

[@sciencefreakdog]

"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.)

[@soundlyawake]

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

[@ProfAwesomeO]

no it's confusing and complicated

[@EntropicalNature]

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

[@IdioticPlatypus]

yes, by confusing simpletons with geometry.

[@yinggling]

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

[@soundlyawake]

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

[@peterevance]

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

[@joebuckfan]

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

[@giorgospapazoglou6106]

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.

[@BiddaBiddaCherryPie]

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

[@DorothyTheMouse]

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

[@ABooleanEarth]

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]

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]

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.

[@hansschmid1298]

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.

[@ComicCulture]

I understood nine words in this video.

[@colt4667]

Braggart !!!

[@jeremyj.5687]

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

[@Kalumbatsch]

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

[@flehue]

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

[@woodyeckerslyke]

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?

[@Opfelixc]

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]

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]

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]

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!

[@Christophe_L]

Ricci Flow is an awesome name for a rapper.

[@LagartijaIncognito]

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!

[@JrDarkPhantom]

I absolutely understood nothing in that video :(

[@JrDarkPhantom]

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]

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.

[@sayunts]

Ricci flow is described by the evolutionary equation with the metric of a manifold changing over the time on the left side and the Ricci tensor on the right side, multiplied by factor 2.. Even when curvature is varying fron point to point on the maniflod (say, surface) it is not a flow yet. Flow begins when the metric tensor (g), that defines the distance between two point on the manifold, starts changing with time. You can imagine like the surface experiences deformations changing its shape from time to time. In this case the curvature, which is defined by Riccci tensor, is varying with time - increasing at one point and decreasing at another point. Physically it reminds you a flow of heat energy in the volume from place to place.

[@RokeyGames]

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

[@WeiqiSub]

Uh don’t get it. Why is the mean curvature going in in the center and reach singularity . It should bulge outward no?

[@IronSoldier]

yes

[@sean3533]

no

[@ForestShroom]

Sean Haggard Maybe?

[@MarsyorangesGaming]

Maybe

[@TimmahDee]

ehhh... I'm not sure about that

[@abacadian]

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

[@jesscarter6504]

My strict discipline is and always has been the humanities. Bs.Ed secondary education English/History...anyways..I stumbled upon this channel..the first professor I watched was Dr. Grime..so affable, cheery, and smiling..SOMEHOW..ALL of you and this channel have made me love math..I waited until student teaching to take college algebra, for goodness sake..the new conjecture is: how did I fall in love with a hard science??

[@tackyyeah8688]

JUDY CARTER - cause women love that which is hard ... for obvious reasons. Woo woo woo!

[@frenzalrhomb109]

+Tacky Yeah tacky indeed.

[@krakenmetzger]

I'm a grad student in topology, and I don't see what all the fuss is about, perfectly clear to me. I learned this stuff in class two weeks ago

[@greg55666]

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

[@GaryFerrao]

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]

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]

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]

dumbass

[@erwin.schulhoff]

one of the most hardcore videos of Numberphile

[@Sapiensiate]

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)?

[@programmer1840]

@numberfile, at 0:10 you put the name "James Isner", I couldn't find any info on him online then I see in the description this is actually James Isenberg!

[@cikif]

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

[@s091709]

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

[@simon_jakobsson]

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]

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.

[@dr.mohamedaitnouh4501]

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

[@waqar177]

Totally got it.....NOT!

[@takashikashiwase3461]

if you guys think this is the explain for five years old,you're wrong,this is essence of conception literally every topologist share. don't idealize the math,you can understand it

[@alexmcgaw]

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.

[@soxnation1000]

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.

[@MartinStaykov]

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]

Martin Staykov replying to get the answer..

[@MartinStaykov]

Ranjini Ravishankar I still have no idea.

[@FlutterBug]

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]

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]

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.

[@fordfactor]

I believe this is what he was trying to say about Ricci flow: In the first part about mean curvature, he mentioned that we have the ability to look at a surface from a higher dimension e.g. the 1D line with curvature we can view in 2D of "flat" (Euclidean) space, or the 2D curved surface of a sphere in 3 Euclidean dimensions. However, if we are confined to the surface, how can we measure its curvature if any? e.g. we can't get off the sphere's surface to look at it from the 3rd dimension. This is what Riemann considered (he mentions Riemannian surfaces). If we draw a small circle about a point on a curved surface, the area will NOT be pi r ^2 (the answer for flat 2d surface). The difference from pi r ^2 can tell us the curvature at that point in a quantity called the Ricci tensor. This is the quantity we can use to then reshape the surface, and I then imagine that this curvature "flows" across the surface as it reshapes. Ricci curvature is important General relativity i.e. the curvature of space-time that gives us gravity, as we can't go up to a 5th dimension to look at it from Euclidean flat space.

[@dhvsheabdh]

what are happen here

[@andrejandu7181]

YES

[@picknikbasket]

Fascinating vid and surprisingly understandable. Great stuff as usual Brady

[@manheer1000]

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

[@SamFisk]

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]

+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]

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

[@joeybeauvais-feisthauer3137]

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).

[@xshortguy]

Best Numberphile Video ever.

[@Bigcubefan]

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.

[@EtzEchad]

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

[@random3767]

what is this nice man talking about?

[@Ashebrethafe]

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".

[@isabellabeckett-smith1473]

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

[@RelatedGiraffe]

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

[@ProfessorEisenoxid]

+RelatedGiraffe Really?

[@RelatedGiraffe]

ProfessorEisenoxid Yup! :)

[@phpn99]

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.

[@franks.6547]

My conclusion: Ricci flow is a specific prescription to alter a bunch of functions g_ij (the "metric") defined on a connected set of points ("manifold"). This transformation is guided by a quantity ("curvature") which can be calculated at each point from those "metric" functions g_ij. A somewhat similar prescription would be "curvature flow", here demonstrated for special cases where the point sets are bubbles in 3D Euklidian space or closed loops in 2D and "curvature" would be the visible pointiness at a given point.

[@Graviton1066]

Props for the Red Sox Cap.

[@alexanderbrowning8723]

1:28 curvature is K (kappa). circles he is drawing is osculating circles, radius is 1 / K. curvature is related to the unit tangent

[@AnindyaMahajan]

When did Robert De Niro become a mathematician?

[@swarnimvajpai6373]

I was looking for it 😂

[@JustPassingBy_]

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

[@NormanEricHairston]

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

[@Peter-dk2ov]

I was thinking the same thing!

[@jerjurko]

+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]

+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]

+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]

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

[@aunibbww]

Nice to have some more advanced topics! There was so much talk about the poincare conjecture and ricci flow after that proof so it's great to see what it's about. To those of you struggling to follow the whole video - the first part in 2d should be fairly easy to understand and the rest of it really adds no new concepts so you can wlog skip it ;-) more explicit maths wouldn't have hurt the last part though.

[@uexp4]

*nods head* .... Absolutely.

[@michaeltebo7735]

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

[@TheAlison1456]

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.

[@kiwin111]

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]

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

[@merle622]

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

[@Qman621]

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

[@JohnSmith-ut5th]

The metric tensor describes a local (usually) infinitesimal region in terms of a skewed Euclidean space. Basically, you can describe any curved surface in terms of a bunch of tiny flat surfaces. And, likewise, for higher dimensions you can describe any curved space in terms of a bunch of tiny "flat" (non-curved, i.e. skewed Euclidean) spaces. So, basically, just forget what he said at the end because it is pretty much exactly the same as the analogy he gave at the beginning, just generalized to higher dimensions.

[@acediamond5399]

You lost me when you got into the metric thing. Didn't quite understand what the different metrics represent or how they are changed or really anything about it, lol. But, one thing I don't get is, in the hourglass shape for instance, why would it ever reach a singularity in the middle of it? Wouldn't the middle part be EXPANDING?

[@acediamond5399]

I hear what you're saying but I don't get why. Am I incorrect in assuming that a three-dimensional figure would progress just the same as if you progressed every radial cross-section at the same time? Because in that case it would tend toward a sphere, right? But apparently it doesn't work that way.

[@Dominic_Muller]

Ace Diamond Because they don't use the curvature of a sphere to determine which way the surface will flow. They have to use orthogonal 2D circles (think of partial derivatives) and one of the circles is "winning" over the other. Because the circle that's "winning" is making the surface pinch, it ends up pinching and staying there since, at a radius of infinity, it stops moving.

[@potenvandebizon]

It might be because the curvature in the pinch of the hourglass is in a different direction than the bulbs on either side of it.

[@ffggddss]

At one point in this, he was explaining that for a 2D surface, there are more kinds of curvature than the single-component curvature of a 1D line in a plane. Although he didn't use the term, Gaussian curvature is one such extension; but it's total curvature that is used. At a point on a 2D surface, you can slice through it in many directions. As the slicing plane swivels around the normal to the surface, the linear curvature, K, of the slice, varies. It will be min in some direction, say, Kx; and max in another, say, Ky (at right angles to the first direction). If the surface is a saddle at that point, then Kx and Ky will have opposite signs. The product, R = Kx·Ky, is the Gaussian curvature; the sum, T = Kx+Ky, is the total curvature -- and that's the one you use for this procedure. Where the surface resembles a cylinder, one of the two, say, Kx, will =0, and R=0; but T=Ky≠0. So the flow moves the surface at that point (while if R were used, it wouldn't). Where the surface is "pinched," Kx and Ky have opposite signs, but both have large magnitudes. Their sum then has magnitude = the difference of magnitudes of Kx and Ky, but those are so large that their difference is large, too. So that makes the flow move those points very fast, pinching the neck off further, and the flow "runs away" until it pinches completely off, into a singularity. The reason it pinches more, instead of expanding, is that the K that goes around the neck, is greater than the K going across the neck, so that's the direction that dominates, and makes it want to shrink. Explaining how the metric tensor (matrix), g[ij] figures into all this, is a bit more complicated, but it so happens that g and its components, can be used to compute all the above quantities; curvatures, as well as geodesic curves in the surface. In Riemannian geometry, the metric is King!

[@acediamond5399]

ffggddss You lost me at some parts, as I don't know all the terms, but I actually understand now how it works with the 2D surface. Thanks! I should probably just study this subject someday.

[@harryaholic]

Devoured with gusto. Thank you numberphile :-)

[@NoriMori1992]

7:52 - So you tell us that this shape is going to behave opposite from how you've spent the past five minutes describing, and then don't explain why? And then base explanations of further concepts on that premise? Jeez.

[@renecabrera3515]

+NoriMori It will be quite difficult for Jim to explain the "why" in this video. I took a class on these topics, and it took us close to 2 months to formally define a Ricci curve, so that we can define a Ricci flow. And then doing the problems took several hours to fully understand these concepts. But I assure you these are some beautiful mathematical concepts.

[@NoriMori1992]

Rene Cabrera If they can't concisely explain why it wouldn't behave the way he just spent five minutes saying it would behave, then it seems pointless to do a video about it. It's one thing to talk about a complex topic, it's quite another to spend part of a video telling us it does one thing, and then spend the rest of the video expanding on ideas that rely on it doing the exact opposite.

[@renecabrera3515]

Ok, fair enough. Perhaps, Jim got excited about explaining the idea and then lost track of it--I'm sure I don't know.

[@NoriMori1992]

Rene Cabrera Fair enough. ^_^ I just found it frustrating. I guess it doesn't technically have to prevent me from enjoying the main thrust of the video - which is why I'm now considering re-watching it - but it's definitely gonna bug me. XD

[@waynebrehaut7183]

As I noted in my reply to b dzar just now, if we use a simple example of an upright, perfectly symmetrical hour-glass we need to consider the flow outward because the surface is concave in profile (split the hour-glass with a vertical plane down its main axis, say the z-axis), but also the flow inward because the cross-section there (in the x-y-plane) is a circle (view the hour-glass from the top, or split it with a horizontal plane through its center of gravity). So the result will be determined by the relative curvatures in these two planes. If we are curvy (large radius, small curvature) vertically but have a small waist (small radius so large curvature) then as we flow we'll get pinched off; if we're fairly rectangular with a small concavity (small radius so large curvature) at the waist--but a large waist--then this is unlikely to happen because the outward flow will exceed the inward flow.

[@VCT3333]

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

[@prankstersatlarge9355]

I don't understand

[@niemandniemand2178]

dumbass

[@fishyeverything8530]

Niemand Niemand you probably didn’t understand it yourself

[@danielcoulon6082]

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.