The Poincaré Conjecture

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The Poincaré Conjecture
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Special lecture on the recent spectacular developments concerning the Poincaré Conjecture.
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the 1st he was a well the next next activities afternoon is a special lecture on the contrary conjecture by some local divisive organizer to politics is always have interest in the interface
between topology another subjects algebraic John Gates very radical physics so it's no surprise that now very interested in ritzy fire and its impact on topology so as to maintain that he can say I'm produced an extensive the exposition of these ideas paramount in solving the prime conjecture that is going to offer a general view of Frank rate for all of us here so professor book auto affray thank you is a great pleasure for me the speak to you today about a stupendous achievement in mathematics theory problem is solved Richard jitters Africa the pose of this 100 EUR probe will work In my view the founding of topology as an independent discipline since its posing the problem has been studied generalized related problems have been solved it has been linked in 1 way or another to most of the progress and topology and the last 100 years now while related problems have been solved the original conjecture as posed by quite ready stood untouched resisting all attack before problems work may be viewed it as a siren song for many boats have foundered on the rocks trying to reach it there have been innumerable proposed truce innumerable proposed examples but none was scrutiny following quackery conjecture is a signal achievement for Perlman but it is also a signal achievement for mathematics
Ford is a measure of how far our understanding Of the subject of has advanced in the last 100 years Roman has seen far but to do so
he stood on the shoulders the Giants 1 giant in particular stands out Richard Hamilton who addressed the conference on Tuesday about his work In its relationship to probe over a period of 25 years Hamilton painstakingly built the solid and elaborate foundation upon which Roman constructive be edifice of his proof without Hamilton's work probably would not have been possible days I will begin by giving you a sense of the importance centrality in depth Of the prime rate conjecture that I will introduce you to surfaces and three-dimensional spaces we call amount of manifolds and tell you some of the ways apologist think about spaces next I will formulate explained in a statement of the my great conjecture that would shift subjects give you a brief introduction to different you geometry which is the field of mathematics out of which the truth of the topological problems come I'll talk about Hamilton's Richie flow equations and briefly describe Romans insights I will not be able to give you anything but a superficial metaphorical impressions of the mathematical arguments for those interested in more details I will give you links to over a thousand pages now available on the Web where you can learn more detailed my goal here to explain the significance of the conjecture its ended state let me began With the role of problems in general and for so from the beginning of the subject always been problems always played a significant role in fact during 18th and 19th centuries that petitions those each other problems often giving 6 weeks to months half a year for solutions and learned societies often pose mathematical problems to the general public and offered prizes for their solution to the nature of problems in mathematics Everly changed with Philbert Karkut International Congress of mathematics in Paris In 1900 theory proposed a list of 23 problems covering a broad range of mathematical field is problems were of a different character than the ones that have been closed In earlier times for Hill no viewed them as the simple problems of mathematics if he thought In his and he thought a study of these problems and their solutions would lead the field forward and indeed he was quite precedent for much of the progress and twenties 20th century Maddox's revolved around these questions and to solve 1 is to be recorded in medics all of them In 2000 following the tradition established by Hilbert acclaim at Medical Institute identified 7 important and simple problems in mathematics things it changed a little bit in years Reading 100 years there was no Hilbert so clay leaned on the support of a committee of leading mathematicians in order to choose these problems and were much more materialistic society so they offered a million dollars for problem for the solution but makes a good medical problems let me quote to you from what Hilbert said in his address to the congress in 1900 it comes out a little funny in English because I believe must be translation from the German but nevertheless I should still more demand for medical problem it is to be perfect for what is clear and easily comprehended tracks a complicated repelled moreover a mathematical problem should be difficult in order to entice this yet not completely in inaccessible left its mark Our efforts it should be to answer post on the tortuous
path the hidden truth ultimately rewarding pleasures of the successful solution that's what we're celebrating today pleasure for all of us Of the successful solution so I think would Hilbert was trying to say was at least these things among others betting good problem stimulate a tremendous amount of mathematical research some of its directly on the problem sometimes it simply sponsor research in related areas you never know where studying a problem will lead you do I get to start hoping it will lead you to the solution but wherever it leads you to follow probably come warned more famous they resist more and more different attempts over time at their solutions and the famous and all problems are used as standards for testing new ideas as a new idea in mathematics one-way detested power and relevant To try to use it to solve all problems and that's it toughest test passes that test it will be ensconced as important in subject so turn your particular problem migrate conjecture was formulated in 1904 by only white guy but a little too young for the made Hilbert's list for years to the problem proposed a characterization of the simplest of all three-dimensional shapes that it's a three-dimensional sphere and come back in a few minutes and tried to explain to you how to think about the 4th wall with the three-dimensional Serie A's and have a think about it it has attracted nearly 100 years by direct topological me so it was a problem in topology hypothesis to recover the hypotheses are topological the conclusion topological it's natural to try to solve it in the terms in which it posed by direct topological knees and there was no success nobody mad His managed to do that but there was a lot of effort put into stunning quackery conjecture did however lead to many advances in the study of other three-dimensional spaces by a variety of workers over a period of about 30 or 40 years we learned a lot about three-dimensional spaces but none of it was relevant to the quackery conjecture which stood there is a vegan untouched by any of the new methods in 1 of the most revolutionary ideas Our topology as far as I'm concerned Stevens mail realized in 1960 that in fact 1 to formulate the library conjecture in all dimensions wasn't too revolutionary but that in fact it was easier to solve hired tensions and he in fact went not to solve it in dimensions find buyer and for that he 1 feels level I believe that Mike erase idea was the following simple intuitive naive idea we understand surfaces two-dimensional spaces they were understood by the end of the 19th century clearly the next thing to try to do is understand three-dimensional space he wrote a long paper about them and end that paper he said well we've achieved a lot but is this leaves 1 remaining questions and then depots by firebreak conjecture surely in the intervening years people thought what if we can solve this problem in the mentioned 3 is no hope in dimension for 5 6 and aunt promises get warned more complicated we have less and less intuition about what these Sarah dimensional spaces look like and snails revolutionary idea was to the contrary it becomes much easier and In slightly earlier in 1956 Miller providing examples to a closely related problem in dimensions 7 in high he told me that he when he 1st discovered this he thought he found a counterexample to the higher dimensional Point Break conjecture but it turned out in and he had a counterexample to a closely related conjecture and for this work he also feels In about 1982 Mike Friedman an incredibly powerful toward a force took the ideas at work in higher dimensions and managed to squeeze them now so they work in dimension for at the expense of crippling the spaces ending in influences complexity but he did solve the formulation of the conjecture image for and for that you want feels muscle almost exactly the same time a young anguished apologists student of Nidal whom he later traded Michael T. for future draft choices Simon I solved a problem break closely related to what Friedman was doing but not for crinkly spaces but for nights basis basis on which you can do calculus right derivatives differential equations and contrast between but Friedman proved for crinkled spaces and went down crude for smooth bases showed us that 4 dimensions was like no other dimensions the divergent between these 2 seemingly very close notions were in fact totally different dimension for factor when it always believed until Milner's resulted there was no difference between the 2 us that's how rate of things he was lucky dimension 3 there is no moving the story along in about 19 Thurston working on hyperbolic three-dimensional space is proposed a generalization of the factory conjecture took conjecture about all three-dimensional spaces and for that I'm not for that because he did that he won feels his conjecture is different from Mike conjecture in 2 essential way 1st of all as Doherty said it covers all three-dimensional spaces where's my Caray conjectures that's true only covers 1 of the simplest and secondly the way Thurston formulated his conjecture was that the three-dimensional spaces had geometries and suggested the idea that as will see later was picked up by Richard Hamilton that 1 could study these three-dimensional questions using geometric methods rather than purely topological analytic in geometric methods and here we are today for years ago Roman solved quackery conjecture and as we now know he has a feel for when I 1st prepare this talk 2 weeks ago I had to question marks next to the feels Medal but I got a erased them for which I'm quite so what is the method of solution that I alluded to sadistic couple more words about it and we will cycle bike later told little more detailed description from a solution used differential geometry and partial differential equations to attack His problem that In its formulation is purely topological please it is an equation an evolution equation called the Ricci flow equation for ammonium nitrate now try talk both about Our remind metric is the Richie flow equations humanity is equation was introduced by Richard Hamilton who went on to develop a rich theory about the solution of this equation and program for attacking the quackery conjecture and according to Richard with crucial input from Yao Thurston's were general geometrizans conjecture Our Ramallah standard turned the topology have to tell you a little bit about the topological spaces that we're thinking about will follow and work by analogy two-dimensional spaces are easier to understand we can see them for the most part and our intuition is guided by this very concrete president representation we have a three-dimensional spaces are harder and we have argued by analogy so I will go back and forth for a few minutes between two-dimensional and three-dimensional space so start with the simplest 1 of the 2 and giving you to representations of the 2 on the right of the its mathematical equation on the left a picture of the see what I think of the tooth Fairy my wife says it looks nothing like it you can judge for yourself the 1 thing I want to emphasize this point because it often a
point of confusion for 9 mathematician who we say that too here we mean the surface of the ball the ball would be the solid in a three-dimensional object but its surface which we see sitting here in space which we city over here talking about the surface itself that two-dimensional fear that is the two-dimensional sphere never other surfaces and these are understood to be the surface not with solid objects we have often called the surface of the doughnut or Taurus we have a to hold and in here have a picture I believe this is probably a 12 4 we see around the sides see 6 handles each 1 with a hole in EC 3 in front and I believe were tormented 3 more in back Back this is what all surfaces look like they look like some people Taurus I have zeroed polls the sphere here's 1 hole to hold Taurus because imagining a meeting with his 12 gold Taurus keeps going but what all these surfaces look like that's a complete list of a wire these two-dimensional they're sitting in three-dimensional space we look at them picked them up they look three-dimensional but in fact there mathematically two-dimensional because you can describe where you are on 1 of these surfaces by giving the number numbers I'm sure you're quite used to that in the case of the series to think of the spheres the surface of the earth you wanna tell somebody where you are you give latitude and longitude if you were only doughnut here on the doughnut you could give to angles again an angle around this way and an angle around that way it would take to angular numbers to specify where you are as a little harder to see how you do it on other surfaces but basically works the same way user services because it takes 2 numbers the describe where we are and something will be importer little later each of these services is our problem sets in three-dimensional space and in three-dimensional space there a boundary of the innings side so in the case of these surface of the doughnut we get had been doughnut bagel itself a solid object that would be a three-dimensional object would ban on the boundary as the surface of object call I no bodies were solid handle bodies and we refer to look to distinguish between them all we have to say is how many holes that has and the technical term for the number of holes is called the genius and we apply at both the surface and handle body so this is the surface of genus 1 bounces solid handle body emerging as 1 hears it to halt Boris surface which members of solid to hold him and so on right now we can more difficult part of the talk services are easy to see how we see the here but In fact you can see the 3 spheres I don't know anybody could see the 3 sphere the reason is it does not exist as an object in our three-dimensional space the only thing we have a direct visual visual understanding are the objects that set in three-dimensional space and we can see the three-tier doesn't sit in three-dimensional space justices To sphere doesn't sit in the planar two-dimensional space but the 2 spheres sits as an object in three-dimensional space and the 3 skiers an object in in four-dimensional but but we can't understand its properties by analogy with what's true for the 2 spheres so do we think about to see what would it take about 2 years through something called stereo graphic rejection so unimaginative this fear the Reds sphere sitting on the black plane tangent touching at 1 point which I think of the South Pole and now I imagine a light at the north Paul and I look at the shadows that like we can have a point on the sphere right here a light ray coming from the North Pole we hit that point and it would have a shadow on the line misidentified all points in the sphere except the North Pole With all the points in the place and technically the topological equivalence between we're minus 1 . and the flying maybe the picture to keep them mind this you got a ball of cheese she want a gift to somebody a present to wrap it up would you do you put on silver paper you wrap the silver paper around it entire this is a little different 1st of all our paper is no longer a finite extent its speak infinite planar goes on infinitely in all directions and secondly the way we wrap up lead any leftover tie the glory factors you go out infinity and playing you're just converging to the North Pole so going out in any direction in the plane simply converges up toward the North Pole another way we can think about the series a union of 2 disks so I started here with 2 copies of a flat this which have separated a little bit figure Boeing the upper went up a little bit and bowling the lower went down a little bit continuing the bowling until their Sony's 2 years and then simply put them together in Gulu and you have a copy of the so another way to think of the 2 you can take to desks 2 two-dimensional this glue their boundaries together entirely but keeping the interior points separate and want to end up with topologically this sphere but we do the same thing intervention 3 no problem so the freeze here can be made that taking to the solid three-dimensional balls absolutely solid objects whose boundary is this year so I have to solid three-dimensional balls and I simply glue their boundaries together entirely keeping the interior points separate dislike ended for the day easy right she can't really see the other description you can think of the 3 spheres as usual three-dimensional Euclidean space going on infinitely in all directions plus 1 more point in infinity it would be identified with indeed North Call of the three-dimensional now that the simplest of all three-dimensional spaces and the others are made many descriptions of other more complicated three-dimensional spaces but for me today that the representation I wanna take is the following a star with 2 solid handle bodies just a minute ago I was started with 2 and the bodies of genus Zero the ball but I could start now with for a start to solid core idea to solid to hold for life anglo their entire boundaries together keeping interior points south but he had trouble visualizing out the boundaries of tube altogether more complicated nevertheless completely accurate mathematical description of the way you can make other three-dimensional spaces and in fact all three-dimensional spaces are made this way by taking some solid moldy hole chorus of song genius to copies of it in going about together this is called a half nobody decomposition of the 3 space and its geniuses the genius of hammer but is in used in the construction so Friday indicate this was good pictures I can't go back to my to hold Taurus and nothing of this solid objects that 2 of them and I blew their boundaries together some now this time it turns out unlike the case of the steer the lots of ways to glue these boundaries together you could get lots of different three-dimensional spaces by doing it that's part of the complication of three-dimensional topology nevertheless that's how you make all three-dimensional does that said we see example the three-tier here can be obtained by gluing together to 3 balls but when and how nobody's more complicated than bolster lots of different ways to glue them together so you can get lots of different three-dimensional spaces from 1 or 2 copies of 1 solid most of the attacks the topological attacks on quackery conjecture in the final analysis involved trying to take this handle body decomposition were reviews solid or I will lots of holes and somehow simplified the decomposition in use
smaller fewer solid and bodies with fewer number of holes if we begin our way down and no holes that is willing see the manifold obtained by going to this together we would prove the point conjecture somehow had to simplify this handle body decomposition what's what proved too big intractable took as I indicated earlier where I think Caray got his motivation for this problem for his problem was he is thinking by analogy with 2 dimensions where he understood everything he was looking for some simple topological property that would characterize the 3 steered away characterize Amin of property that the 3 spearhead that no other of these three-dimensional shapes the gem a three-dimensional space and you know it has this property then you should be according to the conjecture able to conclude that it was the three-dimensional sphere tho what what property well back the surface surfaces with sought drawn here the 3 1st free surfaces in our list the sphere which ends no hold the Taurus a G equals 1 indicates that has 1 whole G forging here's a to Taurus and again demand in this list goes on and on but with the about a loop on the sphere so there's on the surface of the sphere it's pretty clear that you can take this loop and on the surface of the sphere you can drag it and shrink it down to a point when you do that for example the trace of that shrinking might look like still shrink it down over that is true for this loop it's not too hard to prove its true for 80 loop on for example but had the equator as another loop almost here I could shrink it down to the north poll over the top perjury cannot itself over the while the Taurus as honored loop it you can actually to appoint take the loop the goes around the full no matter how good on the Taurus it'll still always encompass Irby up this fall and I'll never be able to continuously triggered down 0 . Of course on the tool for a 2nd find lots of 1st like that like that you're too of when it goes around this fall when goes around that hold of course there others is an interesting goes around the middle for example there Lisa goes around the holes this way lots of other but the point is that there's at least 1 on this surface on the surface and all the other surfaces there's is at least 1 loop on the surface it cannot be continuously shrunk to appoint the series not have such a loop of this property the property that every loose on the surface shrinks to appoint characterizes the two-dimensional spheres among all surfaces now what about the 3 well the same argument that works for the 2 year works for the freeze is if you have a loop in the 3 sphere you can't shrink treated to 0 . continuously deformity to a point well when I was a young graduate student Allard every mathematical talked to have approved and here's your the we see seeing three-tier minus point bizarre Euclidean 3 space while this loop is gonna miss the point so we can think of it as a loop in three-dimensional space I want to salute the three-dimensional space it's obvious with industry could later the origin three-dimensional studies in here I'll show you have do that is a loop in three-dimensional space you just Pulitzer by straight lines down your three-dimensional space and that the shrinking of the loop to appoint and Byron vinification three-dimensional space minus point with the sphere we can use this to shrink called on the 3 figures were so the freeze here the three-dimensional specialist three-dimensional space with the same property that the to spearhead namely every loop on the 3 spheres shrink store point what about the kind that can't overstate is the point conjecture the converse says if you have a three-dimensional space and every loop on it shrinks to appoint your spaces the that's the point conjecture intuitively if you're three-dimensional space doesn't have any holes that you could wrap a loop around like the doughnut has a whole rebel you don't have any holes like them In your space must be the technically a three-dimensional space with the property that every Lupinus space shrinks to a point is topologically equivalent to the 3 it misses almost verbatim said police in the language is why should we believe it's true what it Waikerie believe it's true but sigh completely clear he did believe it's true he didn't actually stated as a conjecture status question Anthony steady studying this question would take too far afield well what reasonably that's true why motivation was as we just seen analogous statement is true for 2 serious distinguishing the tooth from all of the surface as far as I'm concerned the much more important powerful reason it's true came 80 years later with Thurstan 75 years later when Thurston formulated more general conjecture about three-dimensional spaces the point Ray conjecture was a very special case of that conjecture and then he went on to establish his more general conjecture in certain special cases unfortunately not a case included the factory conjecture I think if you take a poll of match of topology when I 1st entered the subject 68 70 my sense was they were Split as many would have voted for the quackery conjecture true as would have voted for it to be false that's the same question in 1985 I think he would have got the 1 God said it was true this is a strange psychological phenomena because in the intervening 20 years 18 years there had been no advance on quackery conjecture but there had been advances on this more general conjecture putting the quackery conjecture and a more general framework saying that framework polls sometimes gave a lot of credence to the fact but it was true and three-dimensional apologists since about that I have been working on three-dimensional spaces attacking any other questions assuming that the migrate conjecture is true this does not come as a surprise the fact that somebody's proved it and the way it was proved probably comes as a surprise at the fact that it's true but shocking are right that you're lesson in topology now we switched to another field of mathematics geometry differential geometry and in particular where money and geometry this is a fun different subject so bear with me so easy three-dimensional space In fact is nothing special about 3 years based in any dimension 1 can put her imposed a notion of structural scarlet await and measure angles and length so if you have a curve in your space think of it as being parameterized by time particle moving in your space is a notion of what the velocity vector tangent vectors at each point and is a notion of how long the trajectory and you have to our needed a point is a notion Annual between them that structure cholera money in metric after remind introduced its hardly studied the problem with this idea may be problem some see it as a gift is there are huge numbers of ways to put on these structures and there is no obvious way to construct 1 that has some properties that you would like it have you decided at a time I wanted this this and this property is no obvious way to make 1 since you're working in technical Watson infinite dimensional space of possibilities finding 1 is that the the property with the property to want is gonna be easy either so now I can tell you a little bit more about Thurston more general conjecture it says it in any a three-dimensional space admits an especially nice 1 of these remind metrics now in the brackets what for
all the mathematicians and the audience that technically correct statement yet to cut the thing in pieces and some canonical way but let's not worry about that this conjecture in and of itself as I indicated earlier suggests a completely different approach the Point Break conjecture because here the conclusion is not topological anymore the conclusion is geometric the conclusion is that there's a nice Romani and mattress but we have a geometric losing conclusion instead of a topological conclusion we might try to establish that conclusion by analytic differential geometric methods and I believe that bad In the site was crucial the Hamilton as he was studying the Richie flow so Hamilton proposed to program the deal to establish these conjectures and if I can paraphrase it in this life In the following day he start with any magic remember we had no way to pick out a good when to begin with and were somebody possibilities and was gonna be reasonable to expect we could occur good when that we wanted but heated up the manifold and let it cool this is a euphemism but go on as it cools it the metric should distribute itself over the manifold homogeneously Justice Keegan a metal bar will distribute itself homogeneously as it cools Ed it could really distributors of homogeneously over the manifold Indiana we should find this Nice metric the 1 that we're looking for so in a nutshell that was Hamilton's ideas notice if it works it will establish island migrate conjecture but also Thurston's more general conjecture because it's gonna produce these nice metrics in great generality well cooling was a euphemism so let me try the sale little bit more about what really involved in Hamilton's equations evolution equation and to do that I have to to tell you about curvature and Anthony dreaded illustrator Richard Bove
following phenomena that were all familiar with it you have Top of an orange peel about that much of it and you put it down on the table Of course it will be bubbled up like this if you watch flat who will take the reason that it past terror is it is not enough of the Orange to cover the region in the play is not enough staff in New York more technically if you look at the very top of the origin you look at the amount of stock within a radius 1 within 1 unit of the top sides are happy you ask how much area do I have there and compare that with the area of the unit disk in the plane you will find you have less area and orange peel than you do in the disk in the plant so that the sense in which there isn't enough of this property that was left area in the orange peel them there is a corresponding region plane is an intrinsic property of the metric distance on the orange peel and it's a reflection of the fact that the metric has positive Kerr Richard point taxes curvature goes back to get out of the gals
incurvature so that the simplest
form of curvature in higher dimensions curvature is quite a bit more complicated you have a higher dimensional space every two-dimensional direction has a curvature of the type I was just trying to to describe so you get a whole bunch of curvature is indeed point in different directions and they fit together into a complicated mathematical logic ago has the name of before but this object is intrinsically associated to the metric and color money curvature tensor after remind who introduced a late-19th-century 19th century remind went on to show that this is the only local property of your metric that's intrinsic to the metric for example remind showed if you have a small piece of an dimensional space and you can flatten it out onto a piece of the plane that it has Zero reminding incur richer and conversely there's 0 reminding curvature flat out of the so curvature captures is intrinsic fundamental notions about the way metric space metric on spaces Kerr and that's not occur richer the directly dinners and Hamilton's Richie flow equation is a related to simplify curvature call the Ricci curvature it exactly the same curricula comes up Einstein Einstein's equation may comes up for the same reason so Hamilton's cooling process call the before is really the idea of letting the metric of law by requiring it to change in time so that it's time derivative is proportional to the curvature this Ricci curvature if you write this equation out and local ordnance on your face you'll see that looks bring much like the heat equation tho it's nonlinear an incident cancer equation instead of the scale equation nonlinearity is very important because nonlinear equations as is well known in the Serie a partial differential equations often almost always developed singularity and in fact it was trying to understand these singularities and deal with them that was the hang up and Hamilton program for establishing these nice metrics to lecture metric under Richie float we think it's you hope it's Godoy a nice metric indeed Hamilton approved under certain circumstances that really happened but if you get hung up on singularities you're not going be able to prove anything but Hamilton did study this equation for 20 25 years he established a new mobile basic analytic properties of the equation solution and as I just mentioned you show that in certain circumstances you could use it to produce the nicest metric for example the metric you're looking for to produce the provide great conjecture but we do face the about singularity to really use Hamilton's program to give the topological result you have to deal with the singularities you have allies and figure out how to continue the flow in spite and that's what from so in a series of 3 PrePrint in 2002 2003 sorry misspelled his name here Grigori Perelman gave argument showing out a deal with the singularities arising from Richie float and how to continue the process something he called or is called Richie flow surgery he then uses this augmented flow this Retief surgery to prove the migrate conjecture and he indicated he indicated in the 2nd 3 PrePrint had use the same technique to prove the general 1st and Jim geometric druthers conjecture if you're interested Brazil length www . claim that or that will take you to probe the 3 pre Prince and also 3 long manuscripts the 1 that can I wrote that weighs in at 473 pages I hope you'll see that next summer on summer reading each table at your local library a article weighing in at over 300 pages by challenge to an article on the archive bike signed a lot a light 200 pages but what what problems reprints do and what these articles explained in much greater detail years a complete comprehensive correct argument the point a conjecture and is now thoroughly checked he has proved
Anthony what is exciting is the fact that the agree conjectures truth is the truth is that he did it using ideas and results from differential geometry and partial differential equations to solve this purely topological problem each have 1 sees this sort of interaction between disparate fields of mathematics firmly makes me sit up and take notice very excited whats significant well the obvious 1 it brings to a conclusion the classification a three-dimensional space now solved this 100 EUR problem and that's extremely satisfying and is a great victory for prominent and also for mathematics and as I just said reaction the interaction between the disparate fields makes that all the more interesting and important but I I think that in the significant impact in the future will be not on three-dimensional spaces but rather applications of Ricci flow in other mathematical context for example it's obvious to try to apply 4 dimensions and also to apply to technically what are known as killer among Poles roles a variety of these should be be the 1st applications to come there the closest to the three-dimensional questions but then there's singularity development and other geometric evolution equations that are similar to the Richie flow for example the mean curvature equation has many properties in common with the score so other geometric flow equations the way they develop singularities and how you treat these singularities I think will see lots of advances there and finally and probably more speculatively because I know nothing about application of mathematics the physical process cities it is nevertheless true that needy needy physical process these are described by the same kind of evolution equation technically a parabolic equation that the Richie flow ears and understanding singularity development and those equations can have tremendous applications we may look back a hundred years and see problems treatment of the singularity Richie flows as the beginning of an understanding much deeper understanding of how singularities develop equations however a handle woman freedom having untold consequences for everyday life but as I I'm not applied mathematicians take that with a grain of salt at