I’m…there have been an enormous number of wonderful really revolutionary developments in these subjects of Geometry and symbology in three dimensions over the last 30 odd [years] and even an astonishing number of new things have been learned since the Turn of the century [and] what I want to do tonight [may] be a little bit ambitious But I’m hoping to take you on a whistle-stop tour Through some of the ideas that have been involved so but in order to tell you about What’s happened in the last 15 years Really I need to start by going back to the turn [of] the previous century and We need to look at some of the things that were going on then To explain a little bit about geometry and topology as it was at that time and we’ll see how that gets carried over so the Plan of my talk, we shall first see a little bit about what is hyperbolic geometry That’s the geometry that a lot of these developments are based on. What is this word topology? Surfaces I’m sure everyone knows what a surface is We will then move on into 3D, and then we get to the sort of late 1970s a complete revolution happens with the American mathematician Bill Thurston, and so I want to give you at least try to give you some sense of the Enormous strides that Thurston introduced and then Since the Millennium there have been further Enormous strides which in some sense have completed a large part of the programs that the stand Initiated so I want to try and give a flavor of that, and then finally talk a little bit about an open problem Which I think is very fascinating and which touches on The work that I’ve [been] doing and which also relate to some of the ideas [I] shall be telling you, so that’s where we’re going to go So hyperbolic geometry if you think about geometry at the end of the 19th [century] it was a lot of discussion What is geometry? [you] could say Geometries study of Space well, we’re going to start in two dimensions space in which wherever you are in whichever Direction You look what you see is kind of the same, so whatever it is. It looks the same [everywhere] and Of course the most obvious kind is ordinary euclidean geometry But you might also be living on the surface of a sphere [and] wherever you are on the surface of a round sphere Things look the same they don’t look quite. The same as if you’re on the flat But they look the same and the third kind the [kind] that was missing Up to about 200 years ago is called hyperbolic geometry So here’s a picture that shows rather well Euclidean geometry Everybody knows the sum of Angles in a triangle is  degrees in a sphere [well], what are the straight lines on a sphere? They’re the great circles the way Aero planes would fly to go the quickest way so the great circles on the sphere And if you start playing around drawing triangles on a sphere whose sides are great circles You very quickly come to the conclusion [that] the Angles add [up] to something greater than 180 How much more than 180? depends by a very precise way on the area of the triangle and The third kind of Geometry then is the kind where? The sum [of] the Angles in a triangle is less than 180 so there’s a kind of pairing between these two things so hyperbolic geometry you can think of as Geometry on a surface which is everywhere Saddle-shaped so you see in this picture it sort of curves down that way and then over a hump in the other direction a surface of negative curvature hyperbolic Geometry was first discovered in the search for a proof of Euclid’s parallel postulate but rather than finding a proof a whole new world of Geometry was discovered and We can nowadays interpret it in this rather nice way so How are we going to model this geometry? well You can’t put the whole Sort of plane of this geometry into Ordinary flat Euclidean space because It’s too big basically, so this is one of a rather nice series of crochet Models that’s been developed by [this] person [diana] tamina There’s in fact a whole book you can buy about making crochet models of geometry there’s a whole project Which has made a coral reef out of crochet which I think is in Dublin and not quite sure anyway You can make these models and the main pointers once you [start] trying to make a surface that’s curved the right way it starts crinkling up and crinkling up and if you kept on before too long [you] would find it was just so crinkled around the edge that you just couldn’t fit it in there anymore And you would [just] have to stop because you you wouldn’t have any more space to make your stitches basically so Here’s another way to do it which perhaps some people may be familiar with so if you know anything about the theory of special relativity [you] [can] use the space which in which special relativity takes place and I’m talking about a [two-dimensional] usually one has three dimensions of space [and] one of time This we’re just going [to] have two dimensions [X&Y] space and this is time and this picture over here on the left is a rather nice model made by a colleague, so this is the show upon which Geometry takes place and let me just go [on] and It’s not so easy to understand geometry there, so rather than doing that we project it so just as you can think about Geometry on the surface of a sphere and Projected onto the flat plane So we can take geometry on this shape here with this rather curious formulas And we can project it onto this circle And this is the first time Poincaré’s name comes up his name will come up repeatedly through the talk We projected onto this circle now. I didn’t say but a Shortest distance path on this thing turns out to be the intersection of a plane through the origin with this hyperboloid. This blue Object and when you project it [down] what you see is You see a disc here, and you see these lines, so these are shortest distance Paths, Sort of Straight Lines in this geometry and what they look like in our map projection is Arcs of Circles that hit the Boundary that’s right Angles and So things are very distorted But Angles are correct, so that’s good There is a of course there are precise formulas for all this which I’m not going to try to explain tonight there are formulas and The main thing to remember that this is like a very distorted thing like when you projects onto a flat atlas Things are completely distorted up near the north pole. They’re very wrong so don’t think that this is a correct picture but nevertheless let’s carry on and Talk about tilings a lot of what I want to say is connected with tilings So everybody knows how to make tiling patterns in the ordinary Euclidean plane Kind of thing you play around with it’s cool and You can make tilings of the sphere this is another nice model made by Henry Segerman, so here is a sphere with a tiling [Pattern] on it, and he’s rather nicely projected it in this photo onto the flat plane and Of course here, you can see here’s a sphere Well here’s everything distorted so we kind of see the idea of distortion in a projection what about tiling the hyperbolic plane there is [a] picture of a tiling of the Hyperbolic plane and the thing to take in here is that although these shapes. They’re actually hexagons Although they look different actually from the Viewpoints of this geometry they’re really the same so in other words just as in the [hexagons] picture every tile is identical to every [other] one There’s a rigid motion of geometry that carries one to the other the same thing is true in this picture [one] thing about this picture is that If you count the number of tiles around here they increase Linearly as the radius of your circle if you count the number of tiles here you find they increase Exponentially fast and that’s related to this croche surface that I had that was all crumpled up There’s more and more and more of them, and there’s an infinite number of them to get from here to here It’s an infinite distance and this rather Wonderful Little video clip shows you probably better It’s going very fast, but there we [are] all of these shapes are the same and we’re zooming on Peter Leaper is another computer graphics person who’s got kind of fascinated by some of the things that can be done in Mathematics So this is a picture and as you see we go on and on and on forever And out at the Horizon it just goes [on] [and] on and it’s infinitely far away So that’s my very quick introduction to hyperbolic geometry we’re going to come back to this geometry repeatedly Let’s go on to what the word topology means So [Topology], what is it? [it’s] the study of things that remain invariant unchanged when you do deformations here is a picture of What do I mean by a continuous deformation will I take a thing and I squish it and I scrunch it and I pull it, but I don’t rip or tear it and I want to ask. What remains the same well if you like The fact that there’s a little hole [that] I could sort of go through here between her [buck] and her front It’s still here [even] though things are a bit kind of distorted well, there’s more the same in these pictures But you could distort much more than that. So that’s what’s topology is about and Well again, the name of Poincaré comes in so let me introduce Poincaré. I’m sure many people here well acquainted with Poincare a he was really one of the absolutely great mathematicians of all time [man] smart Mathematician and Mathematical Physics he contributed to enormous numbers of different areas and in particular He’s often known [as] the father of this subject of Topology although He wasn’t quite the first person to do it, but he was really the [first] person he made Introduced an enormous number of ideas that are still the fundamental tools of this subject He also of course was the person who wrote down the Lorentz transformations in their modern form which are used in the theory of relativity special relativity He was the person who probably can be credited with the invention In some sense with the invention of what we now know as Chaos theory so many many things and I rather like this quotation which I’ll leave you to read Felix Klein was one of the leading German Mathematicians at the sort of later part of the 19th century Klein actually drove himself and to have a breakdown trying to compete with Poincaré, and eventually [I] think gave up and realised and Wrote a lot about Poincaré this rather wonderful tribute so this is Poincaré as I say his name is going to keep continually cropping up throughout the so what about Surfaces then what can we say about surfaces from this Viewpoint of Topology and perhaps by the time I’ve said it you will have a bit more feeling So the idea is we take it easier to see a picture. [here’s] [a] surface now if you Take a loop a closed loop on this surface that just kind of goes round it once here And you cut it of course, it’ll come apart and what will [it] look like in Topology? Well, it would be like a cylinder, but a cylinder is really the same thing if I took a sphere and I cut two discs out of the cylinder And then I squashed it so from the Viewpoint of Topology if I make [one] cut through this thing I get a sphere with two holes and if I took this one And I cut each one of the two handle things with a cut and I opened it out I would find I had a sphere with four holes, and if I did it on this one I would have a sphere with I think six holes so Here’s another one you might think this one is different because it’s all twisted up It’s still like this, but it’s all twisted up but it’s only twisted up from the Viewpoint of [how] it’s been put inside three dimensional space and that the Classification that we’re talking about doesn’t see how it’s sitting in three dimensional space Still if you made one cut it would still fall apart into this sort of cylinder with two ends, so it’s still a torus So the way we classify surfaces Here’s a rather beautiful. [I] Think this is a bronze cast it’s not very large I [think] the real thing Carlos [sequin] is another of these kind of computer people With a very artistic bent who makes rather wonderful representations of Topological Objects, so it’s much harder to see this has got to you can see it’s got some Holes cut in it, but it’s harder to understand. It’s got It’s genus  but let’s believe that’s what it is so that surfaces And it took I’ve told you this in lesson in a couple of minutes it took a long time to sort all this out So this is not meant to be Obvious depends what you think of surfaces to start with? Anyway, that surfaces now. We’re going to now go [into] Geometry on surfaces What can we say about Geometry on surfaces? So this is one of the very important ideas in [this] talk and one of the important ideas in the I could say really in the modern study of Topology, what’s the Geometric structure, so supposing [that] I’ve got some objects and supposing that In a little region around where I’m standing I can manage in some way to measure distance and Angles in Such a way that it looks identical with a bit of one of these basic Geometries, I’ve talked about so it might look a bit like this flat euclidean geometry it might look [like] a little piece of sphere, or it might look like a little piece of hyperbolic Geometry So [a] sphere well everywhere I stand on is fear, and I look around me I see a little piece of a sphere, so that’s not very exciting What about a torus so I’m sure that you’ve seen Pictures like this you take a rectangle of paper You glue two sides together to make our cylinder. Here’s our cylinder. You pull the two things round you stick them together and Lo and behold. You’ve got a torus I’ll tell you where this picture came from later [okay], so we can make a torus actually there’s a sort of cheat in this because of course [I] can roll up and make a cylinder if I try and to do this part actually if I try and do this in 3D Naively Plainly the things going to all crinkle and crunch all up It’s not going to work very so there is a bit of cheating here But it’s not really cheating because what I’m talking about is what you would see [if] you were [a] fly This is a kind of abstract concept if you like if you were a fly living here Well actually or if you even with a fly and you walked across this line It would be no different [from] going across here, and then coming back in here So somehow there is a meaning [to] seeing this thing is basically coming from a rectangle cut out of [Euclidian] geometry so The torus has got euclidean structure. So here’s a picture I take my square, but then if I went across this side I kind of unfold it and see another copy of the square and another copy and another copy Really I translate it, but let’s not get too detailed here, so you can so here is another picture Here’s another picture of gluing up this torus And there’s one very crucial feature in doing this that it was very important that when I glued well first of all when I glued from here to here [that] this plus that Were to 90 degrees so they made 180 degrees so they met at a straight line, and then when I put the two ends together I’ve got to lots of 180 degrees and That’s um Those add up to 360 and that makes a nice smooth flat piece it makes 360 degrees and that’s fine, so there is What we mean by saying a torus has a euclidean structure, so I’ve somehow to this surface attached a natural notion of Euclidean geometry Well, what happens if I now go to a surface which has got two holes genus 2 So here is [a] picture if you follow through this if you would to glue the opposite sides of this thing together You can go home and puzzle about this for a long time But it is true. [I] assure you that if you glue the opposite sides together you’re going to have to kind of come round yourself Maybe go through [yourself] depending How you do it? But you should in the end be able to convince yourself that if you glue it up you come out with a thing like this Okay, so let us believe that but the trouble is if you took an ordinary octagon then eight of these corners have got to meet up here and whatever the angle in an octagon is I think it’s 135 degrees if anyway if you glue aided them together There’s no way it makes 360 degrees. It’s way way more. It’s too much So you glue [em] together and the angle is much too big so you haven’t got a nice? Euclidean structure because if you try and cut it out of a piece [of] paper and glue it up At the last minute you’ll have a nasty mess in the middle where it doesn’t fit together flats So what are you going to do about it? Well you remember That in hyperbolic geometry the angles in a triangle Don’t have to add [up], or don’t add up to 180 they add [up] [to] something less [than] 180 and actually you can create triangles whose angles add [up] [to] anything you like less than 180 and here is a really beautiful tiling of the hyperbolic plane in which we’ve got a nice regular octagon and the Angles here have been manufactured each to be 45 degrees so when I stick eight of them together Lo and Behold I make 360 degrees so it’s nice and locally making the right angle that it should do and so I’ve created a surface if I could cut this piece of paper out of hyperbolic paper which has got this kind of curvature, and if I could stick them together I would have created the surface of genus 2 and Actually by just adjusting the angle using this trick. You can make a surface of any higher genus you like and this was I suppose known and understood it was being studied a lot in the 1920s and you might say well, so what [actually] [it] became very important because Anything you want to study about this surface you want to know what closed loops are on it Somehow if you look at this pattern [that’s] rather complicated But beautiful pattern you can read off all sorts of things here and one thing you can do is you can study particles? Sliding around on this surface, and it turns out that by studying the intricacies of this pattern and by realising that things Geodesics in this in this hyperbolic plane they come close together, and then they kind of diverge from each other and that’s the essence of chaotic motion, so sliding around on one of these surfaces is a Prototype for what we now call chaotic motion and this was one of the first examples. That was really studied and introduced the idea of Randomness and chaotic motion but by a perfectly well-defined formula So this was in the sort of [1920s] and probably 1930s for that So let me just recap very fast we have learned that we know how to classify Surfaces and every surface in a natural way has got a geometry on it It’s either like a sphere or it’s like a piece of Euclidean plane but in usually it’s like the hyperbolic plane right so now we are set to proceed [into] three dimensions, so [my] first problem in this is there is not a normal English language word for the 3D analog of a surface Nobody’s yet. Come up with a very good word So a thing that’s an analogue of a surface, but in more dimensions is known to mathematicians as a manifold So we took court talked about a 3-manifolds, [so] what it means is everywhere. You look in a little bit it looks like some sort of very squatchy vague version of three-dimensional space But it’s got all sorts of complicated loops and tubes and holes and goodness knows what’s in it. So some very complicated thing Well, we just seen how to make a surface by gluing up the sides of a hyperbolic octagon So supposing we took a hyperbolic polyhedron and supposing we glued up the faces [if] We were lucky we could fix the angles So it would all work and they’d fit round They’d have to fit around the edges nicely and they’d have to fit right at the corners, but if we were clever we might be able to do it and We might be able to produce one of these [3-dimensional] objects which just in the same way that we glued up in 2D and created a Surface of some genus we can create something in 3D which we can say has a hyperbolic structure and It comes from this process and now we come to some really Amazing pictures just as in 2D we could tile we could kind of unfold the thing the torus into a square tiling so in 3D we can start with for example this which is the same as the one on the previous slide and we can unfold it into a tiling of three dimensional space, so [I] don’t quite know how to make a drawing like this like it’s amazing, but anyway, so here we are we’ve got that the walls of this thing have [been] made transparent so you can see and you’re sort of seeing through to the cells [if] you like so imagine you could imagine a sort of A [cubical] tiling of ordinary Euclidean space this is an [analogue] in Hyperbolic space and Jos Leys… Jos Leys by the way has a rather nice website with all sorts of pictures in particular. There’s a whole section of Rather Wonderful tilings of Hyperbolic 3-space Okay, so hyperbolic geometry in 3D. Well it produces some pretty pictures, okay? Right okay? Before I can go and I have to give you an alternative Viewpoints if you go back to my original picture about hyperbolic Geometry sitting inside the spectrum in Minkowski space of special relativity if You look out to infinity along the light cone. You see it sort of in 2D. You see a circle in 3D You’d see a sphere of stuff. That’s way out at infinity and I Also, said we can model Two-dimensional hyperbolic [spaces] the inside of a disc so not surprisingly you can put 3D hyperbolic space inside a sphere So far so good So here is a picture another picture by Jos Leys this tiling that we saw you could stand sort of on the outside of the universe and look in and You would see something rather wonderful like this Point [quarry] I guess when he was working with these lorentz transformations and special relativity realised that the Isometries is word I haven’t used…Isometry means a rigid motion a congruence things like that are like congruence is in Euclidean Geometry So Isometry means keeping the distances the same so We have something a rigid motion inside Hyperbolic space that’s moving tiles to other tiles But actually it does something to this sphere at infinity and on this sphere at infinity You [can] calculate the formula now. It’s a formula [on] a sphere but No doubt some of you know how you can think of the sphere you can project it from the top onto the plane and You can think of the plane in terms of complex numbers? x plus iy and so if you take the plane together with one points points at infinity [you’ve] made a sphere, so you can kind of transits really like stereographic projection nothing more than that So you can translate the formulas into some simple formulas? with complex numbers and we can write down the effects of an isometry of [three] [dimensional] Hyperbolic space inside the ball we can write down the effects out on the boundary and This formula at the beauty of this formula is it’s just a two by two Matrix or a nice formula like that And we can just multiply two by two matrices and do all [our] calculations So instead of having to have some quite complicated 3D stuff. We can work in 2D and Hope yeah there we go Now we can see what what I’m doing? Now is I’m showing [you] the effects of a particular isometry of three dimensional Hyperbolic space But I’m just looking out at infinity and watching what it does out at infinity So this man is getting transported around And that’s why we’re watching him waiting for him. Here’s the formula so I just wants you to remember this for later on [and] so everybody will remember this a plus d is an important quantity and Why is it important well you can see that [this]? Person is going around in spirals, and he goes forward into a kind of sink, and he goes backwards into a sink So there are there’s a forward sink and a backward sink and in 3D. What’s this isometry doing? there is a hyperbolic geodesic going from a line going from one end to the other end, so it’s a sort of piece of circle or [Saag] and also my boundaries fear and This transformation is pushing along this axis and twisting as it goes so it pushes along And the twisting is you see that effect in [the] spiraling? So that’s what one of these mappings does so it’s very important when [you] look out at infinity all the facts about being rigid motions go away, but still a lot of good stuff remains and just think about this kind of motion along an axis that you translate on you rotates and That can be read off from this this a plus d number called the trace okay, so Now I need to introduce another idea that was one Isometry supposing I had two or several isometries And supposing I do a bit of one and a bit of another [and] a bit of another and a bit of another I keep Going with them, and I allowed them to go backwards as well, and I pick up the picture of this poor man And I transport him around this particular one has just got he’s got one thing That’s pushing him one way and another thing pushing him another way, and he gets twiddle around and what happens Instead of the being just two fixed points a source and a sink There are actually a whole continuum of sources [and] [sing] well a whole countable number of sources and sinks and he get you to see his images get smaller and smaller and smaller but they go all the way around the edge of this thing and this thing is called the limit set of this collection of Matrices and You can also think of it as the chaotic set when you get to that level Everything’s moving around chaotically in fact you don’t quite see it in this picture But there’s boundary here is a very complicated and rather beautiful fractal curve Now I’m not really going to go into this too much this evening, but We explore. I wrote a book some years ago with these other people and David Mumford is a very brilliant mathematician David wright was originally a student at Harvard [he’s] also a number theorist, but he’s extremely good at Computing we wrote a book [explaining] How you make pictures like this? And in particular a number of people have taken that up in particular. Jos Lays used some of our recipes, and that’s how he Created some of his pictures so let me now start I can now start telling you some more recent Mathematics and The first thing I’m going to tell you is about Hyperbolic manifolds, whose volume is finite? right well, I take a finite sort of original Tile or cell finite size, and I glow up the side so what I get is a finite size, so that’s what we’re talking about and we assume that this thing is glued up, so there’s no boundary anywhere and Now there’s something about 2D. I didn’t tell you if I make for [example] a torus by gluing up a rectangle I can make it a long thin rectangle or a short fat rectangle and I can actually skew it into a parallelogram and still glue up the sides So I can do a lot of things and it will still be euclidean, but the tiling of the plane will change And it turns out in 2D, you can always do that. There’s a whole science called tight molar theory about how you can deform Geometric structures on surfaces in [3D] a Theorem Famous Theorem called the Mostow Rigidity Theorem says in 3D. You can’t do that things have fixed. They’re rigid so what does this thing say supposing I’ve got two of these finite volume manifolds and supposing Well for the mathematicians supposing They’re fundamental groups or abstractly isomorphic But imagine not all of you were mathematicians so supposing they’re topologically the same somehow you know that you can squish one and do all kinds of mess to it forget the Geometry altogether and You can get to the other one Actually, then they’re really the same thing there isometric. So rigidly you pick one up, and it’s exactly the same I see So that’s what Masters theorem says And now I’m going to talk about something else. So that was a tile Which was finite volume I’ve been more concerned with tiles which have got infinite volume So here’s a picture in 2D So this particular yellow tile you can see it goes all the way out to infinity, and there’s a kind of Thickish piece that goes out and that means it’s got infinite size, and here’s another picture now this picture is a What you see [at] infinity? from a 3D tile so imagine I’ve got my Hyperbolic space inside the ball, and I’m kind of looking through from the outside in and there’s some complicated basic tile And it hits the boundary at infinity And then you can take all the copies of it and so I tile the stuff at infinity So [that’s] what you’re seeing in that picture, and you can see how the tiles get smaller and smaller and of course what do they do they pile up on this same thing the limit set of the group and If you would take this stuff out at infinity you [could] play the game [of] gluing up the sides of the tile but now We’re distorting things by these mobius maps That have forgotten how to measure distance, but they do know how to measure angles So you can think that whatever it is. I mean by Riemann surface. It’s a kind of structure That’s like geometry that only remembers angles and there was a great deal of work in the 50s and 60s Studying these things and here’s another note famous theorem. It’s a bit like him an [analogue] of the Moscow Theorem, it says that if I stick up these things then if I control what happens out it in so it’s so There’s a tile and in the middle of hyperbolic space that what we glue up the sides of the tile And we produce a solid thing that has some bit that goes off out to infinity An out at infinity, we have this floppy tile that’s glued up which only remembers angles and what Berbs theorem says is if I know what happens to the Angles that there’s stuff out at infinity, and I know that for every part of the tile that went out to infinity Then I know everything then it’s fixed so it’s sort of like an analogue of [Mostow’s] theorem so hyperbolic manifolds we can think of finite volume we can think of ones with tiles going out to infinity and Then there’s a rather subtle distinction that probably kind of lost at the moment, but anyway you can have Finitely many sides or infinitely many sides of the tile, so what do we know well there’s a tile with an infinite number of sides and We’re sitting inside a tile. It actually only takes two Transformations you wouldn’t quite believe it But because of spiraling that you don’t really see it actually only takes two isometries to glue up all of this stuff and [make] some complicated manifolds So what was known well We knew this wonderful monster rigidity theory, but we didn’t really have many examples there was some there were some nice examples, but they were sort of a bit special and Almost like curiosities. Maybe some examples to do with number theory then we had the theory Bers Theorem and There’s a picture of Bers who was I think he was originally finished, but he spent the Latter part of his life in the United States originally he worked on fluids, but he got drawn into [this] subject So well okay there was this that was pretty much not many examples, but we understood something something was understood here these were a complete mystery except Bers and The work of a few other people said there must be some funny things which don’t fall into this category So there is some that we don’t really know anything about still so that’s where we were at and then William Thurston came along, so [in] the sort of late seventies and early eighties Thurston started he came into this subject, and he started proving just incredible theorems Remembering in 2D we knew That every surface had a certain kind of geometry attached to it either Euclidean hyperbolic or spherical so the first thing that Thurston began doing Was he started taking lots of three dimensional manifolds that the topologist kind of Understood but only in terms of Topology they didn’t know anything about they didn’t have anything to do with Geometry he just took these sort of abstract constructs of Topological Objects, and he showed that actually you could impose all the in a natural way [a] hyperbolic structure you could make this Basic Crystal that you could glue up, and you could make it hyperbolic So I don’t have time to tell you what it’s very technical what classes of Manifolds, but Really very large classes, and this was completely astonishing nobody had imagined anything like this before one of the classes was if you Take 3D space, and you draw some complicated [knotted] loop and then you drill that out of space and throw it away And take everything that’s left That’s the manifold, and he showed that most of those are actually hyperbolic manifolds in a very precisely defined sense so He then went even further and he made something known as the geometrization conjecture [that] He might ask is it possible then that there’s a way of putting Geometry on every three dimensional manifold We could do it in 2D well you Nobody before him conceived anything like this But he conjectured that in view of all the things he’d already done and he’d shown were hyperbolic that actually not that there was a Particular Geometry on every Manifold, but that after you So just as if I take a curve on a surface and cut it before it opens out if I for example draw a torus Inside a manifold and cut along that I’ll get two bits of manifold the bit that’s on one side and the bit on the other and maybe the bits on one side and the bit on the other side separately have some geometry so that’s what he showed, and he also showed there were only eight types of geometry and They’re all quite simple and in particular. [we’ve] got spherical, Euclidean, and hyperbolic and Actually, so he conjectured this was true. I Didn’t prove it to just give you a sense of how Sort of what an immense conjecture this was I’m sure many people have heard about the Poincaré conjecture which I will come on to Let me not say what the Poincaré conjecture is right now people may have heard of it But the Poincaré conjecture was a problem that throughout the 20th century was basically driving Mathematicians nuts and Says he not to me is not only that he proved these Wonderful theorems, but he introduced all sorts of wonderful Concepts he was a very visual hands-on Mathematician Sadly he died quite young and we’re not young, but I mean he was not old you see a couple of years ago But he was always tremendously keen on getting students and schoolchildren a [saw] and doing things by hand and making shapes and playing with shapes and so on So he He left an enormous legacy of of ideas and he had so let me Show my next slide so by the time Thurston Had Been working with [a] large group of students who were attracted to work with him what has happened in Finite volume closed Manifolds well an Enormous amount from being a few [isolated] examples with a good theorem muster rigidity It became a huge big subject huge big area Large large large classes of Manifolds are actually hyperbolic and A great deal of effort went into writing up sort of nice versions of his proofs and so on here’s a picture of this and more reason there was a big conference on his 60th birthday and All of these students of his have them in themselves and of themselves made large contributions to Mathematics so this case well let me just say it was recast from A Subject that was mainly about mobius maps out of infinity into something to do with Topology and Geometry and 3D and This rather mysterious thing that must exist well Thurston did not Solve the problem, but he began to tell us what to do So what has happened since the year? 2000 well these two conjectures [which] I only had told you what the poincare conjecture is but the poincare conjecture and this Geometrization conjecture, which is much bigger and includes the poincare conjecture Then there are three things whose names Won’t mean anything, but three other very important things which Thurston and some of his school put together that between them if you understood this you would have understood this mysterious exotic case and People really didn’t know how to do this stuff Maybe they had a little bit of idea about how to do this as I’ll show you but really Beyond that there was a lot of unknown stuff and What have we done used to show you mathematicians been quite busy? so famously With an enormous amount of press coverage Grigori Perelman Proved the poincare conjecture [right] that I mean depends when you think a theorem was proved that from the time the person first Announces the proof to the time when it gets checked and possibly published can be really quite long in mathematics So dates are a little bit so Perelman proved this Poincaré conjecture whatever that says and then based on these methods this Perelman went further and he explained her to use his ideas which were completely coming from a completely different direction to [prove] geometrization he left a little some gaps that have been filled in Then these three things [all] of these were large Pieces of work, so we’ve kind of done it all you know everything Right [so] let me tell you in a little bit more detail. What’s the Poincaré conjecture? the Poincaré conjecture in 2D if I draw a loop on a sphere I can shrink it down to a point. So I could ask the opposite question if I had a surface such at any time. I drew a loop on [it] I could shrink it down to a point on the surface Would it be a sphere and that’s part of how you classify? surfaces so it’s true that if I gave you a surface and Every loop that. You could draw around the surface So here’s a torus [I] can draw a loop so [this] orange thing rounds here And no matter how I move it around and stretch and twist, [I] will never get rid [of] it I can’t shrink it down to nothing So I can detect if I was a fly I could detect whether I was on a sphere or on the torus By looking at these loops and could I shrink them [down] or not? So an innocuous sounding thing supposing I went up to 3D Three dimensional thing if I was in a three dimensional sphere I could do the same thing could shrink everything to a point What happens the other way [around] if I could shrink everything would it be a sphere? Well you might think that wouldn’t be so hard But actually it’s incredibly hard. I’m really incredibly hard and all sorts of people tried and There were many Sort of incorrect proofs that were announced and then people worked hard and found the mistake in them and this went on and on Another slightly strange thing in mathematics you might not be able to do something in three dimensions So let’s try and go to higher dimensions. Maybe it’s easier And it’s kind [of] easier because there’s more space So it’s kind of it’s [harder] to think about it. We don’t think about it [you] can just move things so you can kind of move things out [of] the way and move them around each other and so It could be proved in Dimensions greater than or equals five and then it was proved in Dimension 4 and it was still sitting there unsolved in Dimension 3 and Some big prizes the claimed Millennium prizes were set up Just before the year 2000 of some outstanding Problems of mathematics there was going to be a million-dollar prize for the solution of any of these the Poincaré conjecture was one of them and Well just about the time Grigori Perelman you see here Proved it and he used. I’m not an expert on this at all, but his methods were completely different He started with not a homogeneous kind of metric at all, but a way of measuring distance That was just sort of anything and as a clever way of kind of smoothing it out. Smoothing it out smoothing it out and you come to Sort of singularities, and you have to get round and cleverly but eventually you proved that this thing that you Didn’t know was a sphere has to be a sphere and then he went on in outlines how to prove geometrization and using similar methods and as I say the word there was some some gaps, but they’ve been for Okay, and then we have the other three conjectures which is more the direction I’ve been involved in so here’s a picture one of the things this and did he didn’t have time to write up his stuff nicely just throwing out ideas [everyday], but there is a huge volume known as Thurston’s notes which with lectures he gave and were typed up and the sort of beginning part has now been Very beautifully [edited] and filled in and produces a book the latter part is still sitting there You can find it online of course here’s a picture about So we’ve got this manifold with infinite volume and the thing is What he said, it looks like you don’t really don’t know what happens as [it] goes off to [infinity] but what he said it might look like is this you might have [a] Surface and then just take copies of the surface all the way going up and create a three-dimensional object So just like you can take a circle And then a line and make a cylinder you can take a surface of some kind genus 2 in this picture and go up And you make a 3D thing and the key point really was that as you go out along here in These weird examples these exotic examples there would be Geodesic loops that wouldn’t be getting very long all the way out along this thing forever and ever so the on the face of it this End of this manifold is infinite volume bit We couldn’t understand might have done anything it might have been twisted in all kinds of terrible Wild ways but actually he sort of tamed it and This timeless conjecture says that well maybe it always looks like this so Thurston showed in certain cases it looks like this and then Thurston also showed that Well, how would you recognise which type it was? It turns out these curves that go sort of lie on these surfaces these geodesic loops that go out and out and out They you can sort of think of them like rational numbers And they have a limit which is some kind of irrational number an object thirst and introduced called a geodesic Lamination and what he conjectured was if you knew what the Irrational numbers as it were connected to each of these ends were then you’d know everything well that seems pretty remarkable But actually turns out. That’s true Not only is it true If you know what happens on the ends here actually you can more or less Reconstruct everything that happens in the middle and this was not thirst and this was largely yair, Minsky working with various other people introducing some really beautiful mathematics and understanding the kind of course Geometry of the whole Hyperbolic manifold [just] from a very small [amount] [of] Data about [the] ends And then finally well could there be any of these things kind of sitting out there? We don’t know where they are Well the answer is no they couldn’t They’d always have to be limits the ones that we knew about and it turned out that once We kind of got this far. This bit was Fairly Routine and quite routine, but it was more obvious to see how to go about proving that so that was also proved right well now I don’t know if I have time, but let me try to go very quickly through so You I’d say we’ve solved all the problems. So we one thing I didn’t say was These [three-dimensional] things you can put geometry on them. We know a great [deal] now about the geometry examples It actually turns out that if you in many senses pick a manifold at random. It’s going to be hyperbolic Just as most surfaces are hyperbolic so in three dimensions by far the most common thing to happen is for your manifold to be hyperbolic, so hyperbolic has gone from being a kind of slight Sideshow to something really fundamental, and I would say we’re at a point now. Where we kind of understand But I said that surfaces were used for example to develop things to do with Chaos theory People are playing now, but I think the facts We now know about these [three-dimensional] things are surely going to come into play [in] the future in ways that we haven’t as yet Understood ok so well very fast okay, so discreteness so discrete means That you can move things away Frigid motions of some tiling you can move the tile away from itself and it never comes back and hits itself On the other hand if you start [rotating] thing and you go rounds an irrational angle an angle that’s that’s not commensurable with [2PI] then you can go round and You’ll come back, but you won’t come [back] exactly on top of yourself And you’ll keep going round and round and [round] and you’ll end up producing a picture like this and the longer you go the more It’s the [more] you’ll overlap yourself So being discrete is kind of nice let me just skip here and Then let me just show you my last slide, so I don’t have time to tell you that but let me just leave you with this last picture, so Just a picture of what it looks like to fly around in three dimensional hyperbolic space [this] picture was made by this Geometry center which was part of the offshoots of all the flurry of activity round cisterns work, so I’ll leave you that thank you [very] much What would a galaxy your solar system look like in the universe with hyperbolic geometry?