The Mystery of 3-Manifolds - William Thurston

okay it's it's my really great pleasure to introduce next speaker bill Thurston I was told that I have to be very brief so in brief bill introduced the all-encompassing geometrization conjecture his work on geometry zation completely revolutionized hyperbolic geometry 3-manifolds apology low dimensional topology in the course of his work on geometry zation he generated incredible number of original ideas and through that he created many different new subfields of mathematics and he revolutionized or re-energized many other subfields of mathematics all these subfields distinct from geometry zation bill also has contributed to many other parts of mathematics it's been tremendously influential outside of geometry zation in particular he's done extraordinary work in foliation rational maps contact structures and it's my pleasure to welcome bill Thurston thank you thank you this is just a hyperbolic polyhedron we're going to switch their smile you could hear me now can't get we okay I'm not used to using keynote but I'm going to try to use it play slideshow okay so I think I think a lot of what a lot of mathematics is really about how you understand things your head I mean it's people that do mathematics it's we're we're not just sort of general purpose machines were people and we we see things we feel things we think of things so I think a lot of a lot of what I did a lot of what I have done in my mathematical career has had to do with finding new ways to well build models to see things do computations anyway to get really a feel for stuff and you know it may seem unimportant but you know when I started out people drew pictures of three manifolds one way and I started drawing them a different way people do pictures of surfaces one way I started drawing them a different way there's something I mean there's something significant about how the representation in your head changes profoundly changes how you how you think now it's very hard to do a you know a brain dump it it's very hard to do that I'm but I'm still going to try to do something to give a feel for 3-manifolds you know words are one thing we can talk about geometric structures there are many precise mathematical words that can be used but they don't automatically convey the feeling for it I probably can't convey a feeling for it either but I want to try that's what I want to try okay so we now have a we do we have a very strong clear beautiful picture for three manifolds there's a every three manifold has a decomposition into canonical geometric pieces this is very well we've sort of known this was this had to be true for many years at least I felt it had to be true and other people gradually started believing in it not everybody impalement parallels proof really seals it so it's sort of established for the mathematical world it gives us a place it's but as every as many people have said it's both a stopping place it's the end of it era but it's also the beginning of something it gives us a platform for going ahead we don't have to now hedge everything we say in terms of well if the geometrization conjecture is true then ludora or every I mean they you know how it goes that when there's something not quite established the Riemann hypothesis or whatever people believe it's probably true but it's not established then everything is everything is hypothetical everything is a people talk about different kinds of things once once something is rigorously established then you go on people go on to do more things what are we going to go on to do it's very hard to predict that's one thing I've learned in mathematics I thought I would know what was coming next and it would be completely in a different direction but but still it's good to try to understand a little bit of what what it is I want to start though by just going over one little I mean so so this geometric decomposition is very clear very it's very straightforward in a sense but it's more complicated than people easily absorb in a casual encounter and well it's very easy to hear about it without understanding it so I want to I want to go over one one microcosm one one world when little mathematical world that contains all the phenomena in the in the gym association conjecture all all eight geometries and both decompositions and and it's a world that you can actually see and get a direct intuition floor without some you know complicated indirect calculation it's very important that you see it you know if you plug numbers into a computer and do a computation and it says something it gives you a very different understanding it's not the same answer as if you see something so we begin by just thinking of a sort of a magical circle a loop that doubles the world so we have we have a loop here and when you when you pass through the world suddenly you're in the United States and the coffee is Brown colored water you go you go through the loop again and we're back in Paris and you know the coffee means something and if you want to go back you can go back and we're back in the US and okay anyway so the world branches like this if you went to Jesus lecture he he what's the right word he he did this to Punk or a repeatedly he he created a doubled world where I'm Punk or a we branched copies of like punk or a in more and more complicated ways I'm going to do much the same thing but in three dimensions I didn't have the I didn't have the time and energy to put in a three-dimensional Punk or a here to to show you but you can imagine someone for yourself baby that joke got worn out last night I don't know okay so so the but you think of the basic universe is the 3-sphere the 3-sphere we think of it's all of space but we make it small by doing the one-point compactification but a single point of infinity so you know you start out going in this direct go in any direction you go far enough in you you reach the point at infinity if you can continue in a straight line you come back the other direction I don't mean the projective completion that's different this is the talking about the spherical completion the one-point compactification okay so we can we can now do this with other designs not just a round circle or if you take the Brandt if you do it if you double the steer the three sphere around this round circle you've got the space here again you can think of just I mean just as we saw within Jesus talk we can just think of a symmetry of the sphere if this is a great circle you can rotate the three sphere around the great circle and if you rotate by an angle of Pi and rotate by an angle of Pi again you come back so the rotation by angle of pi is an element of word or two in it the quotient by that symmetry gives the 3-sphere and conversely the sort of this square root process the inverse branch function branching around the axis gives the 3-sphere but but we can try it with other more complicated figures here's the here the boramae rings this is a famous link and there are two ways to think of it either think of think of creating this branch cover or just think of these links is creating a singularity in the geometry of space where a cross-section a cross-section to the length is like a is like a cone it's like a standard well it's like a cone you got by taking a piece of half a sheet of paper and take a sheet of paper and Ben bended two halves together to make it to make a cone a cone with a total hundred eighty degree cone angle so we can think of just geometrically distorting space like that if you do that you know it'll it'll metrical look kind of ugly but space has a the teacher has a beautiful metric which is distorted by the cone angle by that cone angle along these axes if you've seen the movie not not it tried to give a feel for that - just as a stepping stone - trying to get a feel for how it works you could the boy man rings can be rearranged in either of these two forms you can imagine think of well you can take three ellipses with their long axes and short axes all different and this arrangement of ellipses can be changed into the arrangement of them three circles now on this arrangement of ellipses I on in some tournaments the whole ellipses you can think of that as giving a spine a spine for the 3-sphere - what with those axes in other words everything outside those things all of space outside those axes can collapse into the it can collapse down and fill all the crevices so it's the outside of these so if you drew a ball in the 3-sphere good sort of turn inside out so so the anyway you can think of it taking a sphere and just collapsing it down to here if you if you now think of the opposite process inflate think of a paper construction made of two two phases for each of the each of the ellipses and they're joined just the way you see them the visible the visible faces are joined I don't have it as a live model here so I'll just talk about it if you inflate it can you see a cube this will inflate into a cube particularly if you think of making if you think of making those axes as I said with a 180 degree cone angle so it's like each of the axes it's like taking half of space and folding it in half so when you inflate it it just um it just opens out it makes it flat and it will become a cube here's a picture of a cube it's a little not centered but I'll go back and forth a little bit I should have put these on the same page but see just imagine imagine a paper model with a little hole where you can puff some air oops and it makes it turns it into this cube and now the cube with the cube you can this is part of a repeating crystallographic arrangement in 3-space so if you take three space with three sets of parallel axes a set of axes evenly spaced like this then an interleaving set of parallel axes like this so they they don't intersect and then that creates holes where you can make another set of axes going down as it's strong if you take a cube it's big a cube as you can it doesn't cross those axes it will it will intersect the axes just as in the picture so there's a there's a group of symmetries involved in space if you start with these axes you can think of the symmetry which is 180 degree rotation about any axis they are over here like that or well you have imaginations I would do it if I thought you didn't yeah you can so there's three there's three kinds of symmetries you can do the symmetries about any action any axis if you start composing those you get more symmetries but this will act transitively on the spaces between the axes each each cube can be sent to another cube in any other cube in a unique way so so if you take space with this crystallographic group and form the quotient by the action so you identify all points that are equivalent under the action you will get the topologically you'll get the 3-sphere with the Bornmann rings as the image of the singular act locus and if you had you know if you had some crystallized molecule with this crystallographic group you could draw a diagram if you wanted it in here with you know the bate the fundamental atoms would be aligned either on the axis or in between them and and you could you know you could figure out the structure from from this quotient picture just as you could figure out the repeating structure here okay so that's one example of a geometric structure in this microcosm world this is a little a little special case of the geometric conjecture one I mean I'm doing I'm calling them over fold / - they have only cone axes with order order - cone axes I'm always I'm only just using the underlying space is the three sphere and this is what we get this is the kind of thing we get okay okay here's another example of a geometric structure a little a little bit more complicated than York Lydian space this is a picture actually actually I you know I was reading font grace works I mean I was reading I wrote about topology and he described this picture and you know he I he didn't draw the picture but but this is the picture that he he evoked and I was I was kind of amazed when I read it so this is this is called Nilda amma tree right yeah we call it nil geometry so think of so here you can think of space with an affine group of symmetry so you can it with this picture this is nil geometry or Heisenberg jungle-gym so you can go up if you go up one unit it's exactly the same if you go over a you didn't unit everything is tilted but it's tilted in a very predictable way so when when when you go from here you just sheer downwards Co downwards and similarly when you go over well I'm going over this way a shear upwards so there's is a group of transformations going right and left forward and back and up and down they're not preserving the Euclidean geometry but they're preserving some other kind of geometry this Heisenberg geometry or nil geometry this is another one of the this is another one of the eight three-dimensional geometries it doesn't actually occur very often but it occurs and it's important in understanding the whole picture it's important to understand the special cases as well as the main cases so this is one of the special cases in old geometry and this and the quotient well I wasn't I wasn't totally careful to draw the picture that corresponds to this picture but here's an example of a link in the three sphere where you if you think of this link / - with order to cone axis it has a null geometry structure and you know rather than all these things they're possible to carry out a detail but in a way the bigger point is to try to be able to split your eyes and sort of kind of get the feel for it I mean you won't get the exact feel for it just if you haven't seen it before just on one glance but I'm going to I'm going to go back and forth these these axes here these all four circles correspond to whoops sorry all four circles correspond to the vertical lines all that's all it's drawn are the vertical lines and there's a weird thing if you look at these vertical lines the the figure has it has order to you can do an order - rotation about any of the vertical lines but but now if I take if I take the four posts surrounding a square and I rotate 180 degrees about 1/100 I progress around the thing if I was doing that in Euclidian space doing those four rotations would get me back exactly how I started it would be the identity but when I do this I sort of I sort of drift upward so so the group generated by four rotations around for the Jason posts is actually has a compact quotient and the four posts in other words is a kind of warping here the poster worked with respect to each other and and you can see that here I mean I could have drawn this picture that make making it emphasizing that it would've would have embedded better rather than a flat picture at all sort of vertical picture where these four posts sort of go around and sort of twist before they come back it's the same it's the same thing but they're like four posts with a warped relation but but in fact there's a these are for hot circles they're part of the famous hop vibration to the three CEO of the 2-sphere and so you can actually there's actually a rigid Rocha motion there's an isometry the 3-sphere that sort of rotates these four circles along along themselves and rotate space around at the same time and if you if you take the quotient of the three served by that circle group action but you know the motion that rotates all those sports circles simultaneously the quotient will be well it'll be a two sphere where the four actions become for special singular points so that so this is a this is the three orbital that fibers over the two to do to two-dimensional riverboats I mean just to show I could well I'm not even sure I put all the slides in for all the geometries we can do some of these quickly here's here's a link that whose geometry gives us um two-dimensional sphere across a line if you just take two in fact that would work too if I just took two disjoint circles but it's it's sort of boring it's like we have the if you have these two gates you know this one goes from Paris to New York and back again this one goes from Paris to Moscow or st. Petersburg and back again and but then if you go you know if I go around through this lid and then come over from through this one you know know where you'll gap it might be someplace else so so you start considering all the combinations that you can end up having an infinite number of possibilities and that's like a line a line direction in this geometry I just put in this linking spear to make it look slightly athletic or to make it I don't know anyway it's a spear across behind geometry here's a here's a link which gives solved geometry I think I have a picture of this yeah here's here's another picture again punker a I learned about this geometry from clunker a once again so you can consider space with we have a group of translations going this way a group of translations going this way and now we have a strange saying that you go from this level up to the higher level and the the the grid of city blocks gets distorted so they longer this way in shorter this way anyway I think I think this is the 201 map oh no it's a Fibonacci Fibonacci map but um 0 1 1 1 anyway anyway as you go upward everything gets very elongated in this direction as you go downward everything gets very elongated in the perpendicular direction I like to think of this as a system for constructing freeways I mean not not that I advocate it but it's a description of a system for freeways where if you want to go a very long distance you go up you go up till the roads are going super fast and one of the coordinates you want to traverse then you go along those roads then you come down and you go you go down below where you started in the roads go super fast in the other direction and you come up and anyway there it in this horizontal plane you can the disk within this within this graph within this picture the distance is in the horizontal plane are proportional to the log of the Euclidean distance and in fact this particular design is I mean really for that reason this particular design is very close to designs that are were used for switching networks and telephone systems and and networks of parallel computers and so forth pizza they all seem to be closely related to this this particular kind of geometry so this is just solves the AMA tree has a certain solve is somehow appropriate word I mean rigidly meant and had to be a thought it was a solvable ly group which is simpler than other kinds of groups but it's also good at solving solving problems in that um there's a theorem that if I take an arbitrary graph of finite valence I can plazi isometrically embedded quasi uniformly embedded in this with only a sort of a logarithmic expansion but core you know expanded by the log of the number of nodes anyway and here's an example of a link which when you unwrap it when you invent your unbranched the space around this u branch space around this link it goes solve geometry where you could think of them the the curve here is on one horizontal level the curve up there is on another horizontal level at maze um this the weaving back and forth it stands for the vertical lines and and there's kind of a you can kind of see if you take the loop at the bottom and push it up to the top it gets twisted around as measured by the top and when can actually continue doing that and it gets more and more twisted that's very like the solved geometry it's very like this picture here now again I don't expect you to get this in detail but it's just trying to say there's a intuitive relationship you can learn to look at this and you see it's like this I mean there's a lot a lot of techniques to help do that but here's just a random example of another one of the eight geometries this this is like like the others I've talked about so far this is based on two-dimensional hyperbolic geometry sort of extended is it based on a two-dimensional picture but extended one dimension higher it's it has to do with you can think of it as describing the set of the set of positions of a ship sailing on a hyperbolic sea or so it's the unit tangent bundle of the hyperbolic plane this is kind of geometry and these are just five hops or those five circles of the hopf fibration if I there's a kind of again there's a there's a motion a continuous motion of the 3-sphere where I can think of just continuously rotating all the circles all at the same time and it extends to the space around it they all sit on the surface of a torus the way they're drawn you can think of rotating the tourists this way and rotating the tourists this way at the same time the circles go to themselves and the the orbits face of that action the quotient space is topologically two sphere with five circles it'll actually look like two copies of a Pentagon's in some sense so in the hyperbolic plane you can take it there are right angled Pentagon's you can make if you take two right angle Pentagon's and so they're edges together it makes it you can think of like a pillowcase a pentagonal pillowcase and and this is just and you can think of the circles it's just describing of course this pentagonal pillowcase is a two-dimensional surface but if you think of the set of all all directions so your kid takes a pen and marks on it all possible ways it makes a little marks this is not just theoretical anyway anyway the the possible ways to mark there's a three-dimensional set of tangent vectors to this pillowcase and this is this is it's well this is related to its picture it's not exactly the picture that's it's good enough right now it's a very similar thing another this is this geometry is related to the hyperbolic plane cross a line okay this knot is one of the very simplest ones where which where the the orbit pole the divided by two or take or take the twofold Universal bands cover and it's hyperbolic I won't say it's a simplest because that would depend how you define simplest but but it's one of the simplest and and really it's the generic behavior all these other things I've talked about so far are very very special cases if you take a sort of generic thing a typical very tangled knot it tends to have a hyperbolic P description or depending what you define as a random not it would have a it might tend to have some small little pieces that have other kinds of geometry but the bulk of it the big component would be hyperbolic now can I show you why it's hyperbolic so I this the complements s an amazing kind of geometry I I felt inspired to try to build a spine for it by hand I don't think you can really appreciate it from these photographs but that's what it is you know you should think of picking these these are five well they're they're triangles plus half of a 10 gone and there's one more 10 gone you should really put in the middle but it doesn't show anyway here's the here's the fundamental domain the jewish-led main for this group acting in hyperbolic 3-space if I'd if I'd been more organized or had more time I would have shown it more let's see I need the this is Gian view which a lot of this software used to work better than it doesn't know using using um no I went the cameras I want to go back into the shoot the wind is smaller than I was you know I can change it back to the good formal model here I mean this is this is what it looks like this is what the fundamental domain for that Universal unwrap you know for that Turks head not looks like inside hyperbolic space when you unwrap it so the so it's made of um let's see that it's actually a regular hyperbolic there's a forum which is a regular hyperbolic polyhedron I'm not sure it's still regular here because some of these operations might have distorted it a little bit there are many choices for the fundamental polyhedron but it's made of um it's a truncated dodecahedron you start with a regular 12 sided figure and then cut off the cut off the faces by cut off neighborhoods of vertices by triangles so the faces are ten Gon's in triangles to glue it up to give the turks head not you actually take each you take well they're two of the two of the ten ten gon faces like the one this red one here and this other red one here which which is joined together if you if you join if you sort of if you can think of taking the figure topologically and joining us together it'll make a solid torus and the solid torus you got oops I have to go back to the where is it keynote I have to go back to here the solid torus you got is is the complement of this figure I've drawn then all the other phases if you fold you can actually fold them up in space and it'll it'll create the spine where the cone axes create go to the red axis of the spine anyways it's just one example this is one example of fun he'll you know a nod or a link gives the surprise I mean there's a sort of canonical geometry associated with it that there's no choices event the geometry of the orbifold or manifold you got and and it's often very surprising and striking this is a slide I meant to have later so we'll come back to it um now I've talked about most of the geometries I'm not sure I mentioned all of them it does it's not important because most of them are minor geometries I just wanted to give a flavor the main geometry is hyperbolic geometry but there's also two decomposition theorems to really understand three manifolds and this microcosm of these slash two orbitals so one is the prime decomposition this is something well known and not subbing when you learn to tie a square knot you tie an overhand knot and then you tie another overhand knot that's the square knot as a sum the sum of the overhand knot plus the overhand knot to recognize when something is a is this is not prime the question is whether there's a two sphere that intersects the knot or linked in exactly well in in no points are two points if it's connected you can't have it I mean if it need are six and no points and it's trivial to sphere it doesn't matter if the link yeah anyway um so here here's an example where there's a two sphere indicated by the red line where you ended six and two points in the prime decomposition when you find one of those you then join the loose ends and you get to simpler knots and the nice theorem is that even though you might you might find different ways to cut it but when you get pieces that can't be cut anymore instead of pieces of a knot you get are canonically determined by the original law the prime some ends are you are uniquely determined and I won't go through that theory okay now there's another decomposition which John Conway pioneered the tangled decomposition of a knot so to understand the tangle decomposition you look for two spheres that intersect the knot or link in four points like like I've shown here and the theories somewhat more complicated also somewhat simpler so so to do the tangle decomposition if you find as a sphere that intersects in four points you you get four lose hands there are lots of possible ways to join up for loose ends you could join them like this you could join them like this or you could doing big twist and join them like this or you could you know you could twist this way just this way just this way they're infinitely many ways to join them up so what do you do you give up I mean you you just put them all in a point you gather them all to a point and you sort of almost inevitably led to this new concept where we're now instead of having just an odd or a link we have will also have vertices where four strands come together and we think of the vertex is just a locus that's ready to imply another tangle so you might put a little overhand knot or there's again a nice collection of ways the set of possible ways to do it or described by elements of the rational projective line or actually you know a pair of relatively prime integers up to sign is a way to see it and again again there's a canonical aspect to this if you you might find more than one different way to cut it a knot into tangles and but the answers you gather are all right the same if you define it correctly okay okay so that's the theory this is a very very powerful theory this tangle Theory I'll come back to it in a minute so there's sort of a so one of the directions I think it's just begging to be developed better is the issue of computation of three manifolds so we have some computational tools for three manifolds that are are just remarkably effective if you understand the tools and how to use them they they work very quickly I'll show you I'll show you in a minute one of one of them on the other hand there are many there many ways to ask questions about three manifolds that are sort of fitted into the well they're known to be np-complete there are they're known to be you know very hard to answer and so it depends some what what what questions you're asking what how easy it is to compute but the most important questions might be given we have some topological description for three manifold can we canonically identify it give it to three manifolds or they homomorphic and it's not known but it appears to be that there probably you know quite efficient algorithms that work but I mean in practice they're very efficient algorithms that work but but why and how do they work and do they always work those are the those are questions that are not adequately I've never adequately been addressed I mean it's easy to prove they're algorithms that do work but the only proofs I know about involve these sort of doubly exponential things or they quote the you know the solvability of you know it's known that the elementary theory of real numbers is solvable like tarski proved this amazing theorem and but the algorithms are very bad but for three manifolds the algorithms seem to be very work good so you know the I think the you know the central focus of this is emanated from snap P which is a program Jeff weeks began as a PhD student he was sort of kind of thinking of dropping out because he didn't think he was set to be a research mathematician but he went off to teach at a at a college a teaching college for a year but you know he but but I thought snappy was worthy with Wells a central point of a thesis so so anyway that he wrote a thesis surrounding that and then he's developed it a lot ever since and then a lot of other people too have helped and written other programs and etc etc but a lot of it comes back to snappy let's see if I have the yeah so but unfortunately the oh shoot this is this is distorted here um you know I'm going to try something I'm going to quit this I'm going to restart it and see if it gets a better a better perspective because this is a run okay this is an emulator of a very old fashioned Macintosh I see it's it's distorted in this it doesn't understand the display so we'll just deal with it so this is an example of a program that it was faster 20 years ago than it is now but someday I mean there are other modern versions that are quite fast but they don't have the same nice user interface so let me see if I can do this I wasn't expecting that come on do we have new link projection okay we'll try this anyway under under duress I could go to the I could go to a modern version of snap T but um so basically I can draw a knot or a link I'll just draw something I'll just put in a few circles come on close it up I'll draw another one I want it to be not too complicated because because the the program is inefficient running in this this old this Motorola I mean this emulator for an old chip it's not even that anyway I'm going to make it alternate now I can so that here's an example of a not a Lang not terribly complicated but not terribly simpler simple now I'll ask it to find a hydraulic structure it did it's volume is twenty nine point one five eight five nine five eight two two eight two eight four as you can read it's possible of course to compute this as accurately as you like so that that's a unique topological invariant of the not but but volume volume of the link compliment is not a does not uniquely characterize the link give given a certain volume gets it give it a certain value for volume there's only a finite number of possible different links with that volume but we don't know what they are we don't know what the values of volume are and we don't know what the set of volumes are I think tomorrow we'll hear from Dave Gavai about that we now know what the lowest volume for hyperbolic figure out what it is but in general it's a very difficult question however there are combinatorial there is combinatorial information to do it I mean what I wanted to show the this is this is a different metaphor this is a different world now in the in the notation where I was talking about knots divided by 2 2 before he writes and this is a not divided by infinity so it's computed a hyperbolic structure where you can think of making it the not into it order three cone axis in order for cone axes order a million cone axis and there's the limit as n goes to it as a million goes to infinity where the knot itself is infinitely far away so there's a metric in space there's a metric on the 3-sphere which is hyperbolic it's a metric on the complement of the not everything but the knot the knot is a singularity like a it's infinitely far away but but the neighborhoods of it get really really thin they sort of decrease they're like Tauruses going around they're not shrink exponentially fast as you go toward the knot in the metric so so this metric has a finite body complete metric of finite volume with with very standard costs one invariant one invariant for the knot is is the shape of the costs unfortunately these are going to be distorted by an affine transformation but there they are oh you can't you can't see it very well but no these Tauruses these Tauruses are just all similar Tauruses but they shrink so the shape is well defined this the the size is not and these are the shapes they're kind of long and skinny that's typical for alternating knots like this is I mean there's there's a lot of theory about what the shape can be now that's again a topological invariant of the link compliment I'm gonna I'm gonna see if I can um I'm gonna see if it's this emulator is good enough to compute the custom neighborhoods this used to be you know quick or on a modern there it is okay it's going to do it I'm going to I'll show you another view that gives you an idea of why there's a canonical combinatorial picture you have to imagine that these may look like circles here but they're really I mean they may look like ellipses but they're actually circles from my point of view they look like circles actually so this picture is a canonical again a canonical function of the topology of that link complement there's a there's some combinatorial information though sorry there's some combinatorial information that goes with it namely the forward domain here it shows it you know I'll show the parallelogram too so you can think of I have to shrink it so we can see it all in one picture okay there's a that you can see I don't know if you can make it out but there's a parallelogram in the picture the parallelogram is meant to be identified to at or this Taurus stands for the a Taurus going around the the blue link component in the original picture there it is so if you if you make it redisplay is painful anyway if you make a little tourist going around the blue link component and a little tourist going around the red link component and you think of inflating them blow into this one blow air into this one until they touch they they will they will they will touch on I'm sorry they will touch on those parallelograms I'm going to take away the four balls just so you could see it there's a combinatorial pattern where they touch I haven't shown the red one that they touch in a same way so again this combinatorial gluing pattern is a is a canonical function it's a canonical combinatorial function of the knot any other if you if you if you distorted this knot drew it any other way or embedded in his face in any other way this will snap he would find exactly the same picture and you can recognize and totally identify the the manifold by that pattern and this is quite general okay so I said some of this already the way the way these algorithms actually work is by starting with the not finding a common oil decomposition into what we call ideal tetrahedra they're like tetrahedra with blind with all their vertices missing their vertices are kind of spiraling towards the knot and then it turns out the shape of an ideal tetrahedron is very simple to describe you can determine it by a single complex number associated with any particular edge it's like an angle it's very much like an angle generalization of the angle of a triangle and the three there's an easy formula to get from one pair of opposite edges to the other three pairs of opposite edges and then there's simple gluing equations that the product of these numbers going around any edge has to equal one because it has to match up and and then snap P and similar programs work just by numerically solving for when the gluing equations are true and what seems miraculous is that these that they almost always converge I mean why should they these are equations with I mean as soon as they get a complicated knot you know it's crossing kind of multiplies the degree of the if you've wrote it as a single polynomial you know multiplies the degree by two or something like that so do 100 crossings it would be degree two to the hundredth it's an algebraic equation it's you know certainly algorithmically solvable we're looking for one particular solution but we zip right to it with these programs how many just a few minutes okay yeah okay so let me give the rest of my talk so they're good serial numbers I think there has to be something more going on and somebody needs to figure out what it is so so in practice we have really good ways to identify three manifolds um there are a lot of other mysteries remaining so I just want to touch on them quickly and I should have my time here but um I don't actually have a watch so one of them is another there are many structures associated with three manifolds one of them is the fields the algebraic and the information of the number theory so any hyperbolic stream of oil is associated with a algebraic number field canonically associated with that field and i mean there's more algebraic data as well but let's just start with the field and there's just a basic question that what fields arise in this way there's a easy statement that the field has to have at least one complex place so it and then anything that has exactly one complex place occurs as an arithmetic 300 but there are lots of others that that are not like that and I kind of suspected it may be that every there's a hyperbolic you and flaccidity with every number field that's not not totally real but it could be right or wrong but it's a very fascinating and mysterious question I wish I knew how much time I actually had I have to look at my iPhone what a minute okay okay another interesting structure is to try to understand another thing I think we'll be trying to understand for a long time is how to understand the set of all three manifolds so just because we know how to identify a single one you know have we have a one credit card that's valid it doesn't mean we know what are all possible valid credit card numbers we take a random number you can check if it's valid but doesn't mean you can just write if you give a random number to the hotel and have it work so not that I try but so so if it there's a there are some very nice structures that encompass all three manifolds one of these is the Dane filling space and there's it's a kind of metric complex where you can measure it has these sheets where you can measure the distance from one to another along a sheet we given two manifolds we can try to navigate within this space to get from one to another if you generalize it in the right way it's actually not dissimilar from the kind of space of metrics as Perelman might have looked at I mean there's some things that are seem similar analogous to the Ricci flow but haven't made it been made to work using these combinatorial geometry um Oh skip that they'll skip this to that well let's let's leave give people some breathing room I can stop okay so there's time for- two questions you

Channel: PoincareDuality

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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, Manifold, Diffeomorphism, Tangent, Field, Bundle, Differential, Atlas, Topological Manifold

Id: 4jdmkUQDWtQ

Length: 58min 40sec (3520 seconds)

Published: Sun Nov 27 2011