What is a Manifold? - Mikhail Gromov

so next week is Misha Gromov saw me she is known for his fundamental contribution to many areas of geometry so let's just mention selective geometry hyperbolic geometry geometric group theory and many others so in relation to this confidence just let me mention that I think probably one of the first papers by Terran man may be the first paper was to join paper boof Gromov and baraja also let me mention another fact so it's yesterday bill first mentioned that it's after some point he was 99% plus sure that Poincare conjecture is true about the other kind of as recent as 2002 I think there was a confidence somewhere and there was a conversation and Misha mentioned that from a certain point of view the probability that Poincare conjecture is true is close to zero so so actually I explained I think it's still close to zero right but this thing is right at the point about mathematics in general is completely at least logical of all Sciences all everything what we prove we are proving is just shouldn't be true right otherwise so about that you know this painting yeah and that's because I can make such beautiful painting and other people so I borrowed them from the net and you know this equation usually people ask about ourselves we ask how in but this essentially the last question it's better not to not know the answer but for many files you may ask it and now just I want brief you to show what I'll be talking about so you can read it so you had some ideas and maybe if you start asking me questions because there isn't kind of no it's not systematic talk there are kind of glimpses on different subjects which I want to make and essentially what I want to do is just change perspective because it's big event and just we want to look in the larger time scale and what is the largest time scale we can look at a tech topology right what do you think how far we can go is still see subject of topology and how many years what are you guess you go in time and still we see the Paulo G and actually we shall see what bill essentially something very similar to what bill was given state in during his lecture how high can we go 30 million years we can go 30 million years sideways not backwards not on the future but sideways how can we go 30 million sideways so let me show you yeah he is not our relative and it is around Bhutan and unlike chimpanzee is really very far from us right it's removed by 15 years by roughly 50 million years of evolution and certainly but it's by the tree is 15 million years here and 15 million there that make 30 million years and still we are now attacked and this is not this particular one but the name of which I mentioned before yeah yeah funny I can see kinky near yak is name or something yeah I have to go before he's his fellow it was recorded doing exactly the same what Bill was doing you know that so what are you doing whenever he was given a piece of a house he will make an circle right it will go through it back with the lock it up and I think it's not just funny it tells you subjects quite remarkable so two things which I gather anti-scientific but get very mathematical first it helped us very rarely right you don't see many people doing what he was doing yeah it has exceptional and also for this particular individual it was exceptional not no one of these other Apes were doing that second day for the point of view of external observer it can completely meaning his behavior right right and actually this is a psychologist who is observing that but you're saying that I'm strange doesn't fit in the idea of intelligence yeah the thought was not very intelligent but on the other hand so it cannot be accidentally complicated behavior right and it came from very different sources therefore it might be modulated by something universal so it means the apology is universal in the way we generated Universal and that's I think is fantastic proof exactly because it's so rare if it were a common thing you would say some trivial explanation because it's so rare on the other hand potentiality of this happening even reveals a much smaller it means that is something kind of fundamental behind it but I want to say this this kind of fellow justifies our interest in topology and give you external view and it's still exciting and interesting not only for us right and that's a good sign so we are not alone so so what you react what will come next so I have shown you everything here right so yes you look at this guy's aggressive pp-people whose name mentioned there they might be curious of course what I'm going to save it about the yeah so now come back and try to be quite a systematic so just yes it's at one point which is here I want to include not saying much about mathematics I just brought here the conjecture back by Whitney because it was a one of the indicators that something special was happening in small dimension special dimension for soso paradigm and of course is just something about dimension three but it's kind of an inseparable yeah small dimension are inseparable and the inside of the turn which was taken by topology was yeah because I wrote something I just expected ask question because a chemistry exactly was sittin there so and kind of about many things many things I don't know if I'm cat will be will you'll be saying there but I just want to say because just one mathematical example its theorem for each Nick it was conjecture about eating it and the interesting point but you got at this point of my personal perspective first kind of turn experience when this something strange was happening that apology because when you're learning topology we and the influence and panca read about this idea you for fate we forget everything geometric you just given this very rarefied atmosphere of topology and was believed at least the way what I was learning from from from my professor Oakland that all topology is in low dimensions was kind with a accepting who combined it properly and just algebraic topology little jority will tell you everything but then there was this question but raised by whitney in 1940 about what possible kind of geometry of embedded surface in for space an interesting point was when towards non-orientable surface like projective plane inside of a for space like if you have a not in the three spaces might be very complicated but ever take small neighborhood or not and forget how it sees their own neighborhoods are the same product of a circle by this normal section and that's it and the same would be true for orientable service in first place yeah it will be just regardless of how to reset outside if you look on around it they're all the same but we nee realized it's not so for non-aryan table services however he conjectured but the number of this local topologies our geometry is finite and gave specific estimate and this was a conjecture and nobody could so I would end there it was solved just I think about 76 mm I see and it was using I think actually single third and the end was kind of of course the theorem already was scheduled then of course I couldn't appreciate kind of grandeur of that because I was learning it as they were scheduled learning everything else but for for who from a professor Oakland this was a really shot yeah that there was something coming from analysis and then proving something in the domain he was kind of an expert that the topologies in he broken dimension for and would never come from any anything else yeah it was first indication that geometry could change the perspective in topology in it is in dimension 4 and now of course this theory of searching for manifolds grandly developed by within the Donelson theory way and but still and this 10 for this particular you and piece of that there is no purely a kind of traditional proof right it's a very simple statement a bit it's just how a surface non-orientable surface looks inside of a for space and it's kind of looked extremely simple naive question on the like exercise look like exercising in algebraic topology but it couldn't be solved without some dramatic or analytical point of view so the word is this choice which a man this sharp chance the tap topology on the gun first it departed from geometry and it was pawn Karev actually was expressive it's very powerful very remarkable a year you can go away from that but then there was a kind of turn back and this we're kind of the last corner of the stern this theorem by Perelman now I was saying something about what yeah so jump is back so just yeah you may ask me Samsa gay because he won't be saying about anything but it certainly Ivor if you have some particular question huh who is naked yeah who is naked yeah so just when you against does my my experience just accepting this kind of coming from one career point of view that you have this necked it is the polish the negative geometry completely kind of bare and it was really painful and youth always felt cheated yeah and then you learn it and accept it and then boom you go have to go back yeah and certainly it's a so that's power in this wicked charm of this algebraic the polis is unclear how real was that yeah is it for real fabric fantastic oh it was just way to bypass kind of the real issues yeah and it's still a question I don't I don't have an answer to this question from poor point of view of this lower dimensional development well that maybe it was just naked yeah we just don't have this dressing yet this meat yet to come or maybe not at least you never know I mean well eventually may know I'm sorry but you don't know it now so I don't know who is Nate of this people for my character who's there yeah and then the word is can churn its mark first bye bye bye bye bye bye bye the work of Bill and this was just as I said the first churn coming from I think I couldn't properly appreciate it yeah because you just grow with this but we Boston and dancing the change perspective in a very radical way and so one concept coming from Beland was completely unacceptable at the time that something so special as a hyperbolic 3-manifolds make are a domain of something so huge all three manifolds absolutely kind of counter counterintuitive and on one hand we kind of knew about both domains well not me but and people knew about both domains but they look completely Calvin comparable in size and this it's kind of one point another point is that majority of manifolds in hyperbolic and then it Dwarfs completely things like pawn correct conjectures well just one tiny little thing and who cares about that yeah right because if you it was kind of shown right by by you that most manifolds are hyperbolic and white okay about one care condition but well another point is that until manipulative is many-fold you have to manipulate with metrics and many forms right and this was kind of this idea is crucial of course for the following development and another thing which was not often emphasized that you say that a different class was manifold but then how you blew them and when you glue them and it kind of potential point you think about now in dramatic tourism purely topological so in dimension 2 of course we have this circle but in dimension 3 we have two kind of symmetric gluing right it may be either torus or or sphere and certainly and your genetically glue say hyperbolic manifolds this gluing pattern is symmetric so this did and it's not canonical gluing the pelagic elites unique dramatically the common families and it's exactly those families which generate singularities in the in the flow of Hamilton Pearlman and then there is chopping off all implicit in this picture of 3-manifolds right and and and this what I would call entropy for the following reason so when you have mathematical difficulty a mathematical difficulty oh like in physics has to can come pouring through this it's entropy and energy and energy you kind of hear it is when you use a difficult proof and you have to go up here and you follow the author it's very hard to go and you know it's hard entropy something invisible you just have to decide where to go have many directions and help decide and one decision is made we have this picture you never noticed afterwards it was so difficult right but to find this out of the way why since those things One Direction incredibly difficult yeah it s difficult as to go uphill oh it's not the kind of difficulty and this often forgotten this little thing like this picture that she was not actual original in Hamilton presentation when habita started he was speaking about flaws and geometric context and equations and then it looks completely absurd what he was saying yeah because from topological point of view what he was saying was inconceivably impossible and it's only big past was possible when it incorporated this idea coming from from the boring geometry of afters it was one thing and Donaldson also of costly proving difficult like asana kind of high-energy theorems but there was a concept which is inconceivable before he entered because never before nonlinear differential equations the same subsequent apology just never it just never happen and so he could believe it is impossible right and it was the first instance of that and the fact it was possible it was kind of crucial and suddenly there the three components going with the pediment and one from Thurston Donaldson and from Hamilton of completely different nature because he in here we had instances of geometry truly interacting with geometry the Polish in geometry were talking to each other in a fantastic new way in Hamilton was pure geometry analysis there was no single indication that his method can give anything topological because you see financial equations the classically is people do them they have the equation and so they said coefficient and say then this interval and this interval and have escaped absurd things and in some way I think you your micro make notice that the point is to have equation the maximal generality and if you can extend them to their domain and they become topological and the real power in beauty of them is shown right this is kind of result in different words but that's a crucial point the equation become kind of powerful remarkable real part of the mathematically geometers understanding if they go to the real domain of existence and the resis boundary and then this boundary that grand things happen and this was not in Hamilton field and therefore it was I was saying I didn't believe ever in that and I never spoke to Petaluma yeah because because they had such a case once I spoke to one my friend Eddie aspect explaining him why he is method to proving that foliation exists cannot work and then be approved yeah and they have really good argument saying the session no chance like and so is here so and then yeah that's a little bit about this perspective and so what comes next right now about this numbers yeah and I think a promise to say what's special about these numbers because this was mentioned several times there were some geometric explanation but I think these were fundamental one and already a lot kabuki I keep repeating for for quite a while and this is a the same reason why this numbers a special in solving algebraic equations but now just imagine we turn kind of distort the history and suppose the first people let's develop high dimensional high degree theory equation with a grand theory and then this say there is no solutions and that's like high dimensional Poincare conjecture there is no solution except for trivial one and then after certain work they they found this formula is a fragment of this formula and now they know that look look so exciting but if you think about this incredible formula we have a simple equation and then boom this huge formula and then they would say well that's the end of the epoch and so everything is finished and there would be no galore theory in our class field theory and all England's problem no while I am nothing will stop that you had a strange attitude right but they didn't think in these terms however the point was that they didn't try to repeat this what wasn't dimensional for right they didn't try to find this formula there was something completely come even more beautiful but equally structured yeah not kind of state when something doesn't exist and that's the point this is I think in the polls we have to look for something comparable comparable interesting and now why to you this number is equal it was so special about this number what this it is not quite metaphoric but true and that for the following reason yeah there is something very special about this identity for equal to plus two and it's very different from any other identity for the following reason for equal to plus two in three different ways because you can become posed for element sets right in two in two halves and three ways if you take any other number bigger than 4 right 4 small forget about you have much more and if they compose 5 + 2 + 3 it's much more etc there is no other configuration like that so what it means yeah it means if we have 4 element set you can out of this canonical letter have 3 element set it means that permutation group of on 4 points that means they have a morphism 2 to the 1 & 3 points the group is not simple so the group of permutation 4 elements is not simple and this of course most dramatic scene in the yang-mills theory because it implies that the LI algebra so4 is not simple so it's prison - so the trivial kind of equation mean with me minimizing something like in the integral curvature splits and we have equation of the first order and because in the other heaven in Dec 30 and let you have done some third so you can say that this is kind of the proof of the dogs and theorem right like it is a kind of specific specificity of the problem or everything else huge kind of theory but Universal nothing special about city is not so clear because three it's kind of kind of simple group of course a billion simple group and then in reading the dimension three we don't have characteristic structural clarity which even dimension four we don't have it but in three we have less right and so coming back to geometry apply dimension three that special from this point of view the dimension four is clear however this identity tells you that these numbers are kind of related in an intimate way and and if we use this kind of 2,000 years ago me between you and many other things but today so what happened to my glasses now in theory but it is three equals to C say it'll have no reduction it's not less three is not smaller than three i right so you don't you don't have a muslim groups right it's still the alternating group and three elements ah yeah no not by direct an agent grupe across with voltage alternating group is too simple for three right the only case where it's not simplest for the alternating group is not simple and of course it's a billion and that of course corresponds to the abelian a class field theory i mean it i think it may it's kind of very superficial what i'm saying may seem but it may also indicate very deep very deep under undercurrent yeah this is this identity i think is one of the most fundamental mathematics so okay now my my I want to briefly indicate so where many files come from and that you don't quite don't quite have full picture of that yet because there's a rather limited number of sources on the come from and say triangulation insurgents these are kind of going to long to help combat or you'll make an easily which can be shaped can be shaped in algebraic topology however again I want to emphasize a problem which come from XP from mathematical physics which is unsolved which is in indicating that we don't understand what generation is and the problem is extremely simple you take a manifold and just ask how many triangulation it has with a given number of synthesis so it's some number depending on the strangulation so I have your manifold X and have the number of strangulation in this case implicit and you want to know what happens to it when K goes to infinity roughly that's how you take the angulation after I serve often and there immediately to alternative you can see it's bounded from below but some constant well put it this way one plus something such epsilon to the power K and it is bounded from above by K to the power K roughly that's kind of trivial just keep subdividing you have this power and if you see how many kind of automorphism you have a neck a lemon said that the question is where the truth where between these numbers you are and nothing is known just absolutely a blank right for surfaces you know it's like that and fish has kind of made computation but you can easily produce that and the point is and actually there's a manuscript these are main is given in physics when they prove this bound but as you see you mentioned the exactly make the same mistake you do they don't use the fact is the fixed manifold right the whole point manifold must be fixed right if you vary the manifold of course if you hear you can anything you would have less the question is if you fix manifold and so the subtle point is fixed pelagic Allah how this Kamil toric so this thing creation tell you something about the policy we think we understand it but when look at this problem we don't we have triangulation here abstractly topology there we just know it's the same same meaning homeomorphic your family come at a peak we drink we shower but we absolutely is not direct link between the top and well so that's a big big whack with a zero level question topology right if you start with strangulation da da da you cannot answer that and there is something behind it which possibly related phenomenon also explaining that given usual way you present the manifold for example by triangulation and on the other hand you speak about its topology on one hand 10 creation determines everything on the other hand from dimension for one I'm also confused 4 or 5 yeah it tells you nothing because you know that if you have you can you may have TV a fundamental loop and sorry tell your main for maybe two biological Ebola is fear however this equality is just that philosophical now you know it is so but you don't know in reality because the group you have the presentation of your group is this is literally not solvable which mean in practice you know to contract a loop yeah you mean just time of the universe with peanuts to that yeah it will be just in for all practical purposes will be infinite time so it's contractable in some philosophical sense in reality it's not so we don't know we cannot tell what is from the mental group of your manifold by looking the triangulation in a very deep way injustice more than any infinite you can have have any analysis yet so and and that it particularly shows that dramatic approach as we know it in low dimensions and principle cannot power I will be careful but it seems it came from principal working high dimensions and this was always also possible objection to method by Hamilton because in dimension 3 it seems it's barely work in dimension voyages cannot work if you start with subject for dimensional no geometric process billing it to something canonical because always on a very simple space you can make this kind of horrible twist and this place will look very not feel so nice however this twist is there and just you see nothing yeah you think you see you know nothing that's one of the problems and then coming back to a different presentation right so I wouldn't speak about regrouping homogeneous space everybody know that that's a basic basic point basic class of examples but I can say one word about a kind of classical way to generate many faults because by now the essential was brought in by Tom right so your start writing equations but if you write equation the Euclidian space that we intersect in general position hyper surfaces it get rather special manifolds and just Tom found a universal construction how to write equation that in order to have more radical topology and this was further refined in some work bye bye bye bye bye bye Michael idea and and then I can occur late too that when a we will be discussing episome like for some house in english' it has something to do with that yeah and some people restructure our ourselves and I don't touch it but the point of this tom presentation is that there is that will keep you as you pronounce you say some suggestion Eric right you have a map some function on a simple space so in the sphere this value I don't care I don't care where and then zero of that your manifold even business equation but with manifold only if the map is genetic and genetic is a dirty word yeah you just say this exhibit convene but you don't know what you get right and and but this is another hand the major mechanisms alleging manifolds which is by by the degeneracy and sometimes you can slightly bypass this calc item in testing situation appears even for even for hyper surfaces say you have a manifold and you want to look at hyper surfaces and the way you do is you take a function and X 0 level even is genetic the right cases you can cut it by passage analysis you have something more effective in actually this comes from geometry again if you have Romanian manifold and you can take a cycle and take minimal cycle minimal volume and you know include dimension 1 up to dimension 7 this will be non-singular which is part of remarkable 30 where the major contribution 7 car actually from Jim Simons but there is work an armor and other people and that in seven stretch number right the position and if you go to high dimensional conjectural II also seem that this kind of you know you may have singularities and but they on one hand they kind of spoil and callable may be rather disturbing on the other hand and they say become catalyst again the singularity that pretty nice the other one which IP an example they have extra symmetry so one singularity appear it may be nuisance but on the other hand it's come with extra structure and conjecture it's always the case and so then these singularities cannot exist possibly if they're not being special there are various conjectures which are on the special case approval how the singularity behave but in any case increase up to dimension several you can generate many popular generation for for dimension one at least and what happens for illustration we don't know if there is a systematic way generate manifold but some bizarre knowledge whatever without saying genetic and I think it's logically non-trivial point yeah you allow genetic we don't allow genetic for example if you take like set theory and you don't allow genetic then obviously sort of yeah the continues have words the Easter yeah but if you allow genetic obviously it is not true it's extremely very powerful that logical to and in we don't quite understand its role in this point but now yeah and another maybe case well it's about Markov partitions but probably don't have to say much this actually come from this your students from from from from from Shubin from Frank's and from Boeing that you can construct spaces and following out of the horseshoe by writing them by cattle by matrices by communicating certain kind of a pairs with Markov processes and it is similar to triangulation but not quite yeah because it may carry ax the similar to them but the problem with that when you have this presentation it's very hard to tell what topology of the space is even locally right and that I think I've checked our part you don't understand yeah because effectively like real numbers they're given by infinite fractions by infinite decimals but then you can say the certain decimals a equal right here we have nine nine nine right this is equal to 1 and this identity is again given by by kind of a Markov chain so it's pair Markov chain one give you Cantor set another tells you how to glue that they called Markov partitions and and this is a very computer convert Oriole thing you just it's different from triangulations everything in relation give you that but this more general in an awake and more fun toriel but it's rather an exploit and i'll mention where it leads yeah in one way and now I come to dancing kind of final point yeah so I also away with discussing little bit about singularities and collapse which two phenomena going together in partial differential equations and typically you perceive them as something you want to avoid yeah you lose a priori estimates you think become bad however typically when you look closer and if you're in a good situation either the two phenomena I think either collapse of singularity new structure mergers and beautiful structure make something really think much better than you expect give a pessimistic and don't like it only because we're not ready to accept this kind of new new phenomenon but it is accident in the case of a per element this singularities are coming with extra symmetry which allows allows the process but now and this is a Momo in this thing part but what I want to say what are other out of not so traditional cases to generate many faults and when I say many folks say I mean I mean dramatically a significant space what let me start from the from the end yeah because it's pong correct so yes which we always pick on conjecture and that below concrete was another fantastic idea and which is not what written is my mathematical Magnus capelet it was not stated as a conjecture but just as the fact and the issue he was discussing his book I believe science and hypothesis was how from perception behalf of the world we deconstruct they construct the geometric structure of the world and this mathematical issue and the Poincare idealized it is a mathematical issue and then were completely forgotten and the man who brought it up in different context was Church even I was not mathematician at all but he has I may be extremely powerful mathematical mind is one of the greatest mind of the century and we don't know that yet we absolutely ignorant oh great people if they're not published in mathematical journal and he is in Genesis and he is probably the greatest Genesis classical Genesis without the of the century and I'll tell you what he's done what problems you solved yeah it was a mathematical problem but of course once you realized its mathematical how to formulate it you could have done the point was you could see mathematic behind kind of cows and pancreas mathematician and he was considering kind of easier problem so what is the problem so you look at this world so what you was in input your Assyria where your rough and I mean very simplified approximation already quite complicated it just follows OOP so you have your screen right where images appear but the it's not physical screen yeah where I kind of point there and then kind of imagine they're numbered and completely arbitrary way so eventually have a string of you know 10 to the 4 think and all images are given by according to the how they enumerated so you have sequence of blacks and you have some number of them not so many look at this world we have a billion of such pictures and then I picture the just sequences of numbers right 0 1 0 1 0 1 can you say that what you see is a public space or green space can you and punk resist if I have recess you can but you the algorithm in apparently the only one you can imagine and and I think has clear correctly a good idea what we have so np-complete problem it's np-complete it's it's well I mean it's actually exponential time if you do any naive algorithm into exponential that however we manage how we do that right and punker indicator on the solution in solution he says it's like the different worlds and what amazingly it was very much confirmed by new to see physiology of the of the recent time so what you see is not what happens to you where a light come to your to your and inside we I it's a minor issue of your vision the main issue that you record movement of your eye but you see by moving to I this what you see and there is that now very good confirmation that is completely right but for example when you sleep and your first visual visual part of your vision is paralyzed it doesn't prove it's completely dead you know this by MRI however you I move so the motto muscular system the part of the brain controlling the move crazy dreams not what the Samsung senior yeah so what we perceive and this was I do concur and here would be very nicely description with it and so this how you can stock and so so so the point is but the mathematical question which appears here is so you have this but you may experiment and something come out of this you just have numbers they have many of these sequences the question is what are the geometry behind it right if you knew it came from a screen but you don't know that and and then this was sold in one instance and I think is quite kind of fantastic thing by Stewart even who was at that time actually 18 years old and he was a terrible student working in the lab of Morgan was another Mortman it's not you John and and so what was what is done what he is shown the genome was linear that at that time they were not conceptual DNA of anything but he said it inside of genome there is a geometric structure and destruction structure is linear what was input right from where she started so what they were doing in this lab you know they were breeding flies and the penguin were particularly the fly and solve a kind of with long legs sorry with short legs and Sammy was an ADD eyes and sorry was shot eyes and the recombination say different kind of quantities and so he was collections of flies with different distribution of properties and from that we can derive there is linear structure right and its lead absolutely kind of fantastic right to think about how it can do however indeed there was a deal recombination that certain thing goes along writing certain features come together so you have a correlation matrix and this correlation matrix if you think carefully take a clogger is whatever you have a matrix which it's not exactly like he was done yet I'm simplifying it for example purpose of geometers but you think about this matrix or rather logs of this or explore this yeah as a metric and you look at this metric and you see the triangle inequality becomes a quality and so it's linear and of course it's it's not like that he you think is a metrics or cosmos but but anyway they construct they made the first genetic map of X chromosome of this drosphila so and now will be the last what I mention because there is this picture and probably a curious what your theorem but your paper about Tom spaces has to do with the person's so actually there are two ways to think about many fast coming from from falafel from Archie if you want to the mill in general abstract terms and one comes from index theorem and it says that many fold is just a particular cycle indicator and the cycle is our total for every vector bundle you have to assign a number and the remarkable way to do it you assign index of certain canonical differential operator Twista to the bundle and does give you a number linear function on the space of vector bundles and you can forget about the rest of the cat world and that's a fantastic picture on manifolds which can go quite fabulous this one way but there is another way least a lot of sophisticated thing if you don't know that it looks go another look at the deceptively simple just like almost nothing to say you say aha so what is the manifold that we embedded into a trillion space and then there is this normal bundle okay so it's space and there is a bundle but so what's special what makes it manifold and the point is if you think about ambient space the sphere and you collapse boundary to the point there is a map from the sphere to the top space of this bundle so it's a bundle with a map to the sphere there and it tells the tremendous thing because it coupled with the theorem of sir which I was mentioning said there but I was not truly saying much so maybe I have to return to that because this was a kind of the highest kind of heaven of the pinnacles of this kind of soft approach to topology got a soft approach and then there was some geometric theorem behind it and still you have no picture of the geometry of that so the SAPS theorem says that if you can see that map from one sphere to another then almost always you can deform this map so that it's dilation to become bounded so for every such map there is another map you should obtain mohammed rafi variation namely and how much by haha by how much curves become longer than they were we rebounded depending on dimension except two cases it's one bad case and another is is this one when it's not right if you have usual sphere to the usual shake and change many many times around and just this number of turns you cannot make it better in this dimensional table maps but otherwise the theorem says is gamma W plus fees are finite which means we can deform every map to very simple map and if you try to deform as a geometer what is it's yes we have no idea where it is and it's not surprising because there are this numbers it cannot be universal argument so it means some algebra something must cancel there are some hidden equality there but but but use kind of the one of the central kind of absolutely incredible theorems and topology and if you apply to this situation it's essentially the theorem as I said the reason to get a finally many obstructions on the tangent bundle to be a normal bundle to be bundle to the manifold and then it was elaborated by ethereal browser you Novikov and saying that that's essentially order - yeah this up to find the many possibilities is nothing else this representation of a fundamental class or of a term space by a sphere is all the rest of the smooth structure and that's a this as your smooth manifolds so now while Lipa sonya and these kind of your life or some housing industry for some other person mmm-hmm you don't know okay Lepus I'm right lipid di slippery ice lipid eyes on my lips so and these are kind of quite fundamental for us because our cells are made resist it's been particularly retro sighs yeah and the way it's kind of the simple scatter cells and they exactly made in the same way so in this picture actually you think about your manifold not is that in this direction but it kind of is family of these kind of normal normal bundles yeah it's glued together of these normal directions and you in this kind of completely material this part but this is important yeah right so a space I built out of its normal direction and this is exactly what liposomes are they are built of this molecules yeah absolutely fit of a lipid monocle and these things just may flow in solution yes the way the one do this we shall see make sure them but you should cooling them they conglomerate and form particular surfaces yeah but structures built out on them yeah and that's fantastically complicated process it's complicated in this detail but we it's complicated because we have no idea which kind of a mathematical behind it yeah there are kind of even the most elementary questions we cannot answer so just one thing of course when you come to this kind of low temperature limit we have the services faster than differential equation the variation equations already this equation out of our scope yeah we don't know about them most elementary question which I will just believe they know the answer what we don't you know our later side is like that yeah and this spherical exactly symmetric and the model which you have for editor side the built out on these things and they stick together and they completely flexible in this direction I know the reason of friction in the molecular world but they don't want to learn Bend therefore this bending energy which essentially squared curvature in integral sweat curvature that so ritter sight per given area and volume at bound has minimal bending energy assuming y solution is symmetric yeah even semantics you know it has a shape yeah that's this apparently what this but I always whoever the camera down the equation but what is symmetric there cause my understanding they take it for granted multi symmetric but why it symmetrical will channel and this is a pin kind of little beginning or web is apparent very big subject which we don't understand at all and the most familiar most remarkable as main equation threat data but what I think most remarkable abstractly mathematically which you don't understand when you heat it up they all disappear however they all come with same Gibson sample this is just one parameter and therefore all our differential equation all equational Bachna formulas identities to attack here they cannot die I adjusted analytically continues but continues to what right so there is something in this ensemble of thus particles when they heated sure that you knows what they come back and become this particular surface and this instance the simplest instance of this equation and actually something like that mentioned in the back by param area in the first people I know his okay of his article he speaks about entropy and he gives got a geometric interpretation of of some version of the energy function for for Ricci flow and explained it's kind of similar to Gibson sample right I don't do the same what's it need areas I know the words I never penetrated that this picture is more pictures because incomplete space and that's an abstract kind of equation but this abstract equation also may have this this interpretation is kind of converged through something much more flexible by still remembering basic symmetries the basic equations and the other thing is a as fun to pursue yeah okay so any questions or comments in this last day if there's some energy could be due to some stability material no well at the end I mean it justit is it takes a variational problem but the question is what happens to this equation of this radiation problem when temperature goes up it just beasts kind of this is just in a Gibson sample it's very simple function you look at one particular value classroom values at this point if you look at the paper card if you try to imagine perturbation theory for that yeah so you look expansion you immediately run out of the Naxalite differential equation immediately sitting completely this continues you just try to be close to physical model it will be perturbation mathematically you know I know nothing I think nobody knows anything yet we just all differential equations become kind of kind of babies yeah compared to this problem but this on the other hand is we see we live in this area will be dead without that in instantaneously and it's some kind of mathematics these say well just low limit at the tatata and they don't concerned but I know some other examples besides so first it fits you know I think not quite superficially this picture manifold yeah yes it featured this picture know some other mathematical instance of that when this same picture emerges also in topology but in statistical mechanics nobody looked at this from this angle so the point is what if you take a beautiful mathematical equation like that or Perelman or whatever and think about this is a limit or some statistical thing the composer than two infinitesimal elements but in more intelligent way and break the sediments but keep something so they will know how to come back can you do that like for for Hodge equations for any equation this is a meme essentially minimal surface equation you can do that but we can yeah you know nature does this we don't know how to do it if you look at the what goes actually it's not so simple so that the particles stick together there is not sticking energy between them not at all yeah it's much more subtle why they come together why they come together it's entropic phenomenon so that energy we see it as an energy but it's interpret so it's not something years you know you know you don't know this I believe well maybe some people people who do experiments probably know something but Tara tations I doubt at least I never tried as some people nobody has any idea of that so sorry for the question about the asymptotics easy easy a to the K asymptotic is not known even for the case of a three sphere manifold apathy spheres are not it's unknown it's a no yeah it's which may now in view of the parallel mind is developing hyperbolic manifolds it's possible it will be exponential possible yeah but it certainly has none no it's very well beyond what we can be done oh but for instance isn't there a two sphere that one can find directly in the triangulation which an equatorial like no sir okay so what you find so what okay well it's absolute that's not it's perfectly the question is you may have very strange turns and the strangulation okay you cannot have the hidden tricks Turing machine this for sure which a permit may be relevant for them but it's not clear why the two things already well there are questions about the historical National is the whole literature in physical ease about that and there are many version of this question can formulate in a very elementary commentary or terms about graphs or about matrices and as many reformulation but they're all looks extremely difficult because we have object built out of local data and then you have this global and variance even very simple and then you realize this link is kind of philosophical yeah when you make some step yes you kind of logical you make it but when you look at this issue nothing and that's instance it shows you how cool you understand it yeah it's very clear-cut problem and measure of our ignorance thank you okay more questions so if not then you

Channel: PoincareDuality

Views: 221,232

Rating: 4.8911176 out of 5

Keywords: Clay, Mathematics, Institute, Millennium, Prize, Problems, Stephen, Smale, Perelman, Poincare, Conjecture, Topology, Commutative, Algebra, Michael, Atiyah, Geometry, Dimension, Poincaré Conjecture, John Morgan, Math, Lectures, Topological Manifold, Diffeomorphism, Vector, Field, Tangent, Bundle, Atlas

Id: u5DLpAqX4YA

Length: 53min 55sec (3235 seconds)

Published: Sat Nov 26 2011