 All right, so a little past halfway through a little bit more than halfway through now, and we heard a lot of complex things Very hard Complex or very hard. I don't know what to do now. It's time to go back into the real world make your life a little bit easier And I promise today is gonna be Really nice, I'm not even gonna have groups acting on things today I'm just gonna introduce a couple a couple of geometries Extend a couple of geometries change some geometries still the rents the end But look at them in a different way than you might then we have before so today's a really easy day We're just gonna talk about All right, so All right, so let's start with the Euclidean example Euclidean is the wrong word That Yeah, yeah, okay. The Euclidean is the right word. Okay, so how do you get a one-point compactification of our end? Well, there's lots of ways. I mean a conformal compactification. I let's say let's let's make it conformal, right? Okay, one of them is to consider our N plus 1 comma 1 so it's Lorenzian, right? So Lorenzian what because Lorenzian geometry is the mother of all geometries as John already knows even in his complex world okay, and You have the standard inner product and We can write it like this too. Oh shoot. I always do that Okay, we can write it like this here. I got us. I gotta come down In front Okay, and then we'll look at the null cone right So maybe we'll draw a picture now now. I'll come go back up I'm very just like that very disturbed today So we look at the null cone Now this is where I'm going up into higher dimension. So it is Thursday. It is the right Thursday after all Okay, every other Thursday. You got to worry about which Thursday you're on Okay, and you can find the Riemann sphere the conformal compactification of our end This is by taking the projection of the normal cone Okay, all right, so These points by the way, I'm at later on I'm gonna do this We could either write them with a little Just to do the projection a projective geometry We'll either write them with the With the parentheses around a vector not too much this vector or we'll write them in homogeneous coordinates, which is just with the brackets and the colons and You notice that you can multiply by a number anytime you want Okay All right Okay, so let's embed our end into this SN that is that's an SN by the way Why is that an SN because well if it's the null cone and it's and it's all these other ones right you have V1 squared plus V2 squared plus minus v vn plus one squared you can put the vn plus one squared on the right and Multiplying by a particular number you can get it equal to one So multiplying by the number is just finding a representation for my point Okay, so this is the way I'm going to embed it first of all. I'm going to screw up my metric I'm going to not use the metric that I have been using before I'm not going to use the standard metric I'm going to use this other metric Which is ones along the diagonal and then negative one half negative one half right All right, and I will show you explicitly that that this is actually by change of variables This is actually the right. This is actually equivalent to this metric But later in a different context, but you'll see how it is, but certainly you can take Determinant and you can find the eigenvalues and there's one eigenvalue one and one eigenvalue of negative one. It's not so hard Okay, and this is the way I'm going to embed it This the x here is Is a vector this is our end right this is I'm living in a higher space I'm I'm really being weird for me because I'm a low-dimensional sort of guy Okay, I'm an X and then I'm going to take the inner product with itself and one Certainly works right you can put it in there. Is that a good embedding? Well, things should be can I use the word conformal and by the way I hate that word quasi conformal just freaks me out But conformal I don't like too much either Okay, but is it conformal is that is that obvious that it's conformal to everyone or or do we want to talk about it? Do we it's obvious? Maybe it's obvious. I Mean certainly when you're multiplying by matrices you've done that before but maybe you want to worry about Whether that's a conformal embedding or not whether well, let's take it. Let's let's worry about let's take two paths, right? through a point a plus TV What's P plus TV? You and you plus TW what are those map to? Where's that stupid thing? Okay, they're embedded into their map to You plus TV You Plus TV Dot you plus TV one The other one not so surprising is you plus TW you Plus TW Plus TW One and then let's do the multiplication because we can Because I said it would be a nice and easy day We should all have fun sit back Smoke whatever you got And I've got two two TV V You dot V and I've got plus T squared V squared and I've got the same thing down here. I don't have to do it again All right Well, how am I gonna find what what does conformal mean it means keeping angles the same I want to know what they did what what's the vector at that point and oh, no I'm gonna do something that John and I hate in this world. I'm gonna take a derivative Okay The T of this and I'm gonna evaluate it at T equals zero Okay So this vector turns out to be V Zero the only thing that survives is the two U dot V All right Down here the same thing right DDT at T dot equals zero Same one right. It's you know, it's W To T to you W Zero all right Now take the inner product Well, I'm using this inner product, right? I'm using this weird the wacky one Sorry, maybe I'm not gonna do that today I'm using this wacky inner product right here that inner product Take the inner product between This one called us a and B Right. What's a dot a dot B Well, let's use the brackets because the dots are gonna be my Euclidean one But what is on the first n variables? It's just the regular dot product, right? so it's you dot V and What happens otherwise? That's zero because it's it's it's the cross right it's zero times two and zero times that well, that's great That's not the angle though Okay, I'm look I have to that's the dot product. That's I don't know what that is But what's the angle? I have to worry about the norm of these things So what's the norm of this vector a? Okay. Well, if you think about it, we'll just put it into your little your little thing again Okay, and it's also gonna be just V dot V. So it's okay. We're we're happier happier and large Okay, so it's V dot V also and this is W dot V. I don't know what V dot W So it actually turns out to be a conformal map, right? This is in conformal embedding and it all was just the inner product works, right? So it's a really nice way of writing these things Okay, there's a point miss though There's one point missed and that's the point which has a zero on the last on the last coordinate Right zero one zero one zero, right? We've got every other point except for that one Okay, we don't you can't have zeros but zero so We're So that's gonna be a that's our infinity, right? So this is gonna be infinity that point that we always add All right, so Maybe I won't do this next slide Because it looks just too complicated for me except to say Oh and leaves oh and plus one comma one leaves this the the the Leaves invariant than the no no cone and P. Oh and plus one is a set of conformal automorphisms of SA Okay, which we which you know in different contexts These are just nice ways of looking at the formulas of seeing how to do if I wanted to have a Euclidean similarity If I want to do since it's conformal, I should think about conformal maps, right? And all of these it's not hard to check that these formulas give you the ones that are in there They're all in oh P o n o n plus one comma one All right Okay, so They're nice little formulas do you can check it. It's nice nice exercise to check that it works All right, so Well, there's an inversion in the sphere, right? What's the inversion in the sphere? Well, we've done people have been talking about it. Everyone's been talking about inversions. I think Francoise has done some inversions. Everyone loves a good inversion in the sphere the way I'm gonna invert in the sears I'm gonna hit it with this matrix zero With the identity and then I'm gonna flip the other two right Okay, and you can see what happens right you just flip them you flip these two right? That's what the last part does and That's what is it? What is it equivalent to? Well, let's take a number and Dot dot it with this up divide by it You can see something that's within the inside the unit sphere gets mapped outside the unit sphere If it's on the unit sphere the dot products one so it stays there, right? So it's all good Everyone's happy. It's all in the same line Okay Again, where does the origin go to the origin goes to? Infinity that that that infinity that we talked about before Okay, this is all review. Everyone's everyone's happy. We're all feeling good because this is what we lived on What we grew up on when we were we lads and Lasses All right, so what should we do? We should do this same Process with the Lorentzian group with the Lorentzian with look with R2 1 or our n1 as you want because we're being very general today Okay, so We're going to embed it Into what did we what how do what did we embed it our n into our n plus 1 comma 1 so I'm gonna add 1 to my Plus plus plus pluses and 1 to my minuses Right, so I'm going to embed it into Into our n plus 1 comma 2. I even got that number writes okay, and Just like before I'm going to use a nice inner product There's the standard inner product that I can write but the nice inner product I'm going to write everything in this nice inner product With the basis with this and it's going to be it looks just like the other one. There's our plus There's our n pluses. There's our 1 minus. There's the other plus and minus are in this one Okay All right, and there when you dot product with itself when you inner product with itself I should say you get this sort of thing right v2 v3 That's what the one half because you get two copies of v2 v3 All right, and you look at the null cone just like we did before we're looking at the null cone and the project of a projectivization of the null cone is the is is What we call the Einstein universe? Okay, I By the way, this is me. This is just me crappy notation. I'll tell you no I don't think anybody else except me maybe bill occasionally, but even he's getting sick of it, too But I think it's I'm just n plus one sometimes there are just I'm n plus one just just Just to say I'm three because you are Einstein is implying the signature is n one You got one negative and n plus so The physicists will just put the note the total dimension up there certainly Okay, and we do the same darn thing. I love to do the same thing over and over again It makes me joyously happy I can I do it again. I did it once I can do it again, and I'm going to embed it this way exactly like I did before I'm going to take it. I'm going to take it I'm going to take the vector just like I did before I'm going to put the dot product and I got a one on the bottom and For the same reasons of all this it's a conformal embedding also It's not going to change anything. Nothing's going to change here So the question is what do we get? Do we get one point? Do we get more than one point? I don't know. We'll have to see. Well, maybe I do know but All right. Here are the transformations. They're they're looking much like the other transformations Okay, and the inversion in the unit sphere now. Remember, what's the unit sphere here? By the way The unit sphere here is not our it's our n plus one so the unit sphere When we're doing inversion in the unit sphere This is for Rn plus one. That's the unit sphere, right? That's when the inner product with itself is one and that's going to be exactly what you're doing What's going to be fixed by this for the same reasons why it was fixed before because of that formula that we had before Okay, all right, and you'll see that in our n plus one you have this nice this nice Inversion with it just flipping the two gives you the same sort of thing that we had before it's not nothing different I think I just copied and pasted all these slides The only difference here, and I have to say the difference and write the difference is that dot doesn't mean The dot doesn't mean where's the dot right there the dots there and the dots here the dot doesn't doesn't mean Regular Euclidean inner product it means my Lorentzian inner product it's the n and the one. Oh, well, let's figure that out. We're going to figure this out That's what we're going to talk about, right Yeah, we're compactifying everything everything's going to be compactified we're going to have it. It's all going to be you know because What what what's really going on here, right is that? Every point like the nice thing about the null cone is who cares about what direction every point looks the same It's a homogenous whatever beautiful, right? It's homogeneous space right no come so every point looks the same to to to this to the Lorentzian metric, right? to the Lorentzian inner product So no points different when we add a point whatever points we add they're going to be equivalent to any other point whether it's going to be Time and space like our time like or whatever. It's all the same Okay, so let's see what's happening Okay Okay, so No, that that is not what this is the Einstein universe This is the conform conformal compactification of the flat space. We'll get to anti-decider later That's the negatively curved space Anti-decider is different than than the Einstein universe There is relationships before this and we'll talk about what those relationships are But they're not the same this is this is just it and I'll show you the picture that I've seen I saw a couple times in a physics talk, so you might have seen it like a magazine everywhere as well, you know Where they talk about it so look at what happens to the origin, okay? Okay, so let's see what we now do here So we take the inversion in the origin and it flips the zero and the one to zero and that point Never gets hit by that embedding the embedding is always blah blah blah, you know x dot product One that point is not hit That point is called the improper point or I should say somebody called a proper point I'm not sure who called it the improper point, but as you'll see I think we should change the name And I'm into this people have taught me to change names now or at least let To to campaign for them. I think we should call it destiny And you'll see and maybe it'll have a child and we'll call it Beyonce, but that's a completely different thing But work I like to call it Death destiny Okay, I keep thinking about Anybody ever see What was the bad back to the future? You're my density See I it's my density. It's destiny. That's it's supposed to be destiny It's not there's my typo. Sorry about that Maybe it's not a typo Alright, so There's some other Points, so what was it? What would somebody ask me? What what happens to null points, right? There's there's a point on and now look at the remember the signature because I'm probably gonna screw this up someplace But that's a that's on the null cone of your Lorentzian space Right, so it's inner product with itself is zero So we're gonna get this right and that's a cone You know what that is that that that set of X is is that cone that we've been drawn over and over and over ad nauseam, right? Okay, and there and when you do the inversion it goes to some place that doesn't get hit You cannot hit it Okay So what does that say that somewhere out there at infinity that we didn't hit before There's another cone a crap Right, there's a cone of this is a cone, right? This is this is going to be takes one point to another point It's a one-to-one map certainly Okay And it's it's moving the cone that you're used to that your origin is looking at and saying here's my here's my light cone And it's flipping it out to to infinity Okay, there's some other points which we're not hitting Okay They're not gonna we're not going to be so easily able to find out by inversion because they're not affected by inversion And those are the points of the form where we have X Zero so it's inner product with itself is zero and zero Okay And because up there there's a one but down there. There's no one So you can multiply by X so you don't have a cone anymore. You have the projectivization of the cone What's the projectivization of cone? It's a sphere All right, you take take r2 you can see it right there. You can see that sphere, right All right, so and it's fixed point wise. We'll call that the ideal sphere Okay, and that's destiny not density. I don't even know what that word is. That's not even a real word Okay, and here's what the picture looks like Okay, let's think about this picture just for a while And I I don't like to use that after a while This dark circle these dark circles are my light. There's the origin. That's the origin right And this is the this is the cone. This is the light cone Um, I don't know why these are different color, but these are also this is my improper point That's what it's getting blue and it's it's there's also a cone here, right the red and the blue They should be the same color the red the green and the blue should be the same color and there's my sphere of infinity And what do you have in a sphere of infinity? You're you're identifying antipodal points So if you think about the the r2 one example, you're just You're turning them right so you can imagine this kind of turning so So this going around here This coming here the opposite one goes to the same point All right So that's what you're adding the all the stuff that's the infinity all the stuff that's Dashed is what you're adding. These are the points that you add It's not as simple as you might think Yeah, yeah, what there's a cone. It's really a cone. That's glued right so yeah, I would say rpn. Yeah Yeah, you can't I I'm not going to do that because I always forget how to do that. Um, do you remember? There are ways to do that in fact, I'll show you a little bit of how to how to do that in a second, okay Um of how to get that because I'll be able to get any point on a line and then you just rotate it around there to get a line Okay, so where I dropped things now. I've lost Oh, there it is Okay, so that's my infinity. This is my new space So I take r r r to one and I add or rn one and I add these points on And now I've got every point looks the same as homogeneous space beautiful, right? But it's But it's this weird thing right and we have this embedded Now one of these places Well, we'll talk about that that'll we'll talk about it that later So, okay, so let's first get an idea of what happens to lines. Let's start really really easy Let's start at lines through the origin Okay, okay Time like line light like lines. Okay, so we're going to take a point a point on Or a direction On The thing right and this is be the what I did to one one one one vector One zero one one zero one one Vector right Where does that go? Well, you just do like you did like you always would you take you You look at this and you can divide by You can let t go to infinity You could actually divide by t another way to write This is just divide by t and let t to go to infinity these two points go to one Because they're the same order and this one goes to zero All right, so what what type of point and and i'm going to write these in r2 one just But all of this works for any any dimension. This is just because I can only write five I just can't physically write more than that John likes to write 20 different numbers on But I can only write well five Okay, so that's an element of The ideal sphere so if you go off in a light light direction you hit the idea like we thought That's what we thought we would do like points at the end would be that's not shocking All right, so how about time like lines? through the origin and there it is The dot product with itself is Since that's the zero zero if you take the inner product with itself You get zero times zero zero times zero t times. Oh, but it's a negative. So it's negative t squared All right, so What dominates that you can divide by t squared the highest power right and then all this uh-oh What happened to my negative one? Oh, I don't care right and we're talking in projective space But maybe we should keep that in mind Maybe that we should keep in mind that we've multiplied by that that by negative one Just keep that in mind in the future okay All right, so that point. Oh, that's that's destiny. That's that's what every point that I go off I find a I find a time like Geodesic that I'm going off on and I hit my destiny All right, not improper point at the crappy name What's Ravi say? Oh, I named that You didn't name that did you Did you name that okay good that that could be me Okay, what about space like lines? What about space like lines again? I'm just going to take the really easy case right? I'm going to take I'm going to take a really easy case and I'm going to take t Zero zero one zero zero just in like a horizontal, but it won't matter everything's a bit eventually the same And again, we do this but we have a positive t squared t zero zero t squared one that also goes That really really is weird right, so You go off. Let's just think about this just for a second I go off this way and I hit the the destiny and I go out that way and I hit the destiny But when I go in between there, I hit My ideal sphere Which is really not close to destiny I mean if you remember the picture the I There was a cone of infinity that improper point or destiny was the vertex at that cone of infinity Okay, which is maybe what people call it. I think I think the now the new terminology. I've heard is covert covertx. Yeah They're they're they're they're the same. They're the same. It's just you know Because we can change there's a transformation right there is a transformation. There is an element of p o n plus 1 comma 2 Right that change can take the origin to Infinity the infinity the origin I can move I can move it to any point And then I have a different thing. I'm not going through now. It would be point lines through the origin Where is it going to? Right They are coming in pairs though. There is something to be you know There is something that well two points are related Because there are going to be two points which are which are one is a vert one's the vertex of the cone One's the covert x or destiny. It's destiny because destiny can change all the time but by changing your origin Right, it is as you change the origin. You're going to change the thing change where you where you end up All right, so let's see if I can answer friends was question here or give you kind of an answer Okay Let's look at uh slight like lines which are not through the origin Okay Okay, and then that's my I could I did this and let's hopefully I did it right I could have done it wrong. There might be a negative one here um I think I did it right Okay, well, I'll assume that I did it right No, no, no I did it right Okay, there I did it and Here's oh see as I change as I change my base point. So I'll take two lines here So if I'll take a uh, this is a light like line And if I change where it goes through I don't necessarily end up at destiny anymore I'd end up in its destiny if that was the origin that I would end up at its destiny But I'm not going to end up at the original destiny. I end up at a A point on the cone of infinity and so you can see how you can depending on where you're where which direction you're going You're going to get a whole line of these Right, you're going to get a whole line of those those points b minus c you can change the b in the c right So you can get any value here You can hit any points any point on that lot on that line at the cone of infinity And so if you change where you're going, yeah, right, right. Yeah, I think yes. Yes Yeah, okay. Yeah Right, right, okay I wasn't going to do planes. I mean, I just wanted to introduce things so but yeah There's all these things of what what planes do and I think yeah Okay By the way, when you take a space like Line that's not through the origin what happens to it Just because of the way it's dealt with right you can take any any point go off in the direction t But again, I don't care everything's It's symmetric with respect to t Symmetry actually it's going to be symmetric with a place where where you go Which direction you go you're always going to hit Destiny right every space like line goes off to destiny And proper point co-vertex. What do you ever you want to call it? All right, so So that's start it's and again, there's another like like friends. I always say there's another I'm going to stop kind of here pretty soon And I'm not going to go too much further on this in this path But there's a whole idea of like what the what the planes do What do half planes do what do What do crooked planes bound? Well next wednesday Virginie charrette will be talking and she'll be talking about crooked planes and and groups that you can get in the Einstein universe is her a title is to be to be announced, but I know I have an inside fact because she said what do you want me to talk about? I said, well, why don't you talk about the Einstein universe stuff She and a couple of students did some nice work and in dealing with this so And it is it's they get really weird the the crooked planes are just the when you when you When you complete them in the Einstein the Einstein universe it just gets a little wacky Okay, so Now what what do we want to do? Well? I personally don't like the idea of Going out space like directions and hitting the same point that I go when I go in that like Time like directions that seems weird So one way to get away from this is Take the double cover and and in the double cover instead of Doing projection Instead of doing the regular full projection. You might look at this Sphere of directions. I don't like that word. I'm trying to change that too. I've got I've got a lot of issues with words today Okay, I think you should but anyway instead of identifying points all that have any Any scalar multiple only take positive multiples and you'll get a double cover Okay, and what you'll get is you'll get two different Destinies right two different infinities a time like infinity Right up there and a space like infinity And I drew it like a point like a circle because it's a point think about that just for a second I drew it like a certain Okay, those two points right now are the same by the way those two points are the same And all of these points are the same Everything's gets doubled everything gets you there's always a negative and a positive right? There's always Well, right here this that you've got uh What happens is that your light cone Light cone gets the you get two different. You're you're uh Separating the future in the past at least in this one you have kind of a future in the past Although not really because you can go around because as you go the whole way up This line that goes right from the that point to that point. Well, that's just a circle So that's still a time machine and that's not well I've heard that it's not so bad, but I don't really like it either. You can take a universal cover by the way and you can do it there Okay um The area inside this triangle that I drew with like crappy draw, right? This is the worst draw I can use like some free thing That that area is a copy of r n plus one and n comma one Right that we'll call it a midkowski patch right lorenzian patch. I don't know. It's a patch. There's another patch There's another patch out here On the outside. I don't I can't draw that because in the other patch the space like Destiny the space like infinity Is the time like one for the other patch? And this one is the space like one for the previous patch right So different universes right can you get I There are these universes right that you can go between and then you can take the universal cover and just make it So that you do have some sort of time time orientation right Okay, so So the question is well my first question is what do you think the top? Well, I have a topology. How do you figure out the topology of this? I claim the topology of the double cover is s n cross s one Well, what did I tell you? I said I said we're going to use this crazy metric Don't use the crazy metric use the real metric Right the real metric you get v one square plus Plus v n squared plus v n plus one minus v n one squared minus v n n plus two squared That's that's On your null cone, right? Okay Well, let's this is one of my favorite tricks. I've used this a couple times V n squared is equal to v n plus one squared plus v n plus two squared Okay All right You're still allowed to multiply by a positive number So let's multiply by let's or divide by the norm of this the square root of this And what you'll get is you'll get okay Divide by the square root of v n Plus one squared plus v n Plus two squared And so the new Vectors these are new ones. These are v n prime plus one squared plus v n Plus two prime Squared is equal to v one prime squared Plus v Is equal to one There's your s one Right This is a positive number. Oh because it's a point, right? It's it has to be a Positive number to get this to get if that was zero then that would if these two were zero then that would that would be zero, too right So and we're we're not considering zero zero is not in our you know our space that we're considering Okay, so this here this is because after you divide by this after you do the Projectivization the only positive projectivization which I Okay, this turns into my copy of s one And this curves into my copy of sn and you can kind of see it right like I said as I said before right This is the that lot that point and that point are the same you can see the s one there It's right there. All right, and there's the sn There's a copy of s Maybe maybe that's not the right way to think about Okay, so here we have this very bizarre space of things that we're going to do All right, so let's see where are we yeah, yeah say it again. Yeah, I think it's yes No, no, no, no it's sn it's sn um That might be sn minus one. Yeah, you're right. You're right sn minus one I let me think about that for a second I'm always screwing up the numbers so but Yeah, so it has so it has to be Yeah, that's what I thought. Yeah, because this is too Oh, because I had I didn't have enough numbers up here. That's the problem. It should have been n plus three That's that's my problem. So I should have had this should be plus one two three Plus one two three two and three Uh plus one two and three. I know this is really important to get Okay, I did it right the first time. Okay Okay, so let's go back. Let's go back where this is that's kind of where I kind of want to leave Uh the Einstein universe There's lots to think about there's planes to think about Um by oh before we go any further since Um Since we're dealing with Um, this is the double cover that I did because I divided by a positive number. I didn't allow you to divide by a negative number. What do you get when you A lot when you look at the actual Einstein universe Well, you get You get the antipodal map, right? It's multiplication by negative one Right? You've got you've got the Einstein you you got this this sn cross s one Uh mod the antipodal map Well depends on whether sn is even or odd whether you get an orientable or unorientable space So in two one Einstein two one it's unorientable If I did it right, I think I did it right Okay All right, um, so let's go let's go back to the big picture of of everything, right? Let's let's let's try to I'm gonna change the change my direction here a little bit because I There's lots of like I said the Einstein universe you could spend a lot of time doing it And we could talk a lot more about it But I have a limited amount of time and I wanted to do one other thing Because we've been talking about it in several talks And we've been talking and I know that fanny's going to talk about it next week So I want to talk about something else and since ravi even asked for it whatever ravi asked for we do Okay, let's go to you didn't know you had that power so so We can think let's go let's go back to the euclidean or I don't know the anyway the different picture, right? The standard picture Okay Here was We had hyperbolic space Negative curvature Right On the boundary there was s n minus one Let's see if I can do that, right and on outside of it We can think of it either projectively or just taking a nice slice. You've got the sphere of radius one, which is a surface of constant curvature of radius one. It's not the sphere It's the quasi. I think I saw it saw last night. It's quasi sphere or something. They call it some people call it But it's a something of it's a manifold of curvature of curvature positive one Okay, and we have this division of things and the negative curvature comes from this taking taking this This negative one right This s n I'm I my numbers are way off Okay, so what are we going to do? We're going to do the same stupid thing That we did here. Oh, I guess we're not calling it stupid the same wonderful thing that we did here In the standard case and we're going to do it now for the lorenzi in case Okay the negative One sphere Right Is going to be Anti-decider space. It's the negative one constant curvature negative one everything's homogeneous. It's beautiful, right? right It's bounded This is this is now the the schematic picture is it's This is ads. This is anti-decider space. This is the negative curvature, right? This is and i'm not even going to worry about dimensions. Hopefully I got it right up there Okay, einstein space Oh the einstein space is The cone right that's what the einstein space was Right and on the outside it's decider Okay, which is the positive curvature the lorenzi in positive curvature Now one of the things that bothers me all about this is that our model spaces These are models for what whenever you take quotients of things these are model things These models aren't aren't simply connected, right? So but we can deal with it life goes on Okay, so that's the picture that we want to see we want to see this anti-decider space negative curvature inside the light cone einstein So the boundary of einstein I mean the boundary of anti-decider is einstein Which is also by the way the boundary of the double boundary of of Decider well if you take negative Okay All right, so I'm not going to worry about this too much. There's there's been some work on it People believe more in anti-decider space There's reasons for for the physics to be involved and plus we're all negative curvature people here I was Okay, I'll tell you the story the little snarky and I won't tell names At a department that I I was at An analyst who was more of a geometer said of the geometers who all did Analysis, but they also did positive curvature because positive curvature everything's a sphere, huh? He said I cut it derisively And so I was talking to one of my colleagues a lovely woman And I and I know she loves positive curvature. So I said, oh, I told her the story It she goes, yeah, isn't that great everything's a sphere And so I guess whether you like certain things really does depend on what your opinion What your viewpoints and you know, my viewpoint is everything's a sphere And her viewpoint was yeah, everything's a sphere So let's stay away from everything's a sphere, but that's a lovely place. Maybe people like that Maybe you should maybe should I don't want to say anything you should feel free But I think in this room, I think we're Going to the to the not everything's a sphere Okay, so anti to sitter space I'm usually really bad with things to sitter was I don't know who he was What I forget his nationality and I what Dutch that does sound right because it's But also I just well, it's just so sad that the thing that we study more often is anti instead of the real thing, right? I just feel it's weird. All right, so We're going to talk about the hyperboloid model Of anti to sitter space and I put should put quotes around the hyperboloid because I don't know what type of hyperboloid it is anymore And we're going to look at what all the vectors in this big space in r22, right? You think about what dimension that does that? That moves the the the negative thing down by one could go one two But we always put the positive more positive than negative And so we'll take the inner product with itself with negative one Okay um You're also going to double that of course because you you you just the same thing we did we do The before you it's one sheet. I think it's one sheet okay There's the project of model two Right There's we could take all these vectors we could look at their inner product Just say their inner product and we'll just write everything as as a As this as we're not going to use that too much what we're going to use and you know I used to look down I used to look down and I spit on the client model and I have to admit I'm getting used to it. I like it a little bit more Okay lines or lines straight lines or straight lines Geodesics appear as straight lines in the client model because you're projecting onto it. You're projecting onto a A plane isn't it a wonderful thing instead of those which circles are where? So I like the client model and it's great here. We're going to use the client model The picture that we all talk about that everyone says. Oh, here's anti-dissidious space. It's the client model It's taking this space and projecting at the fourth the fourth Fourth thing onto v v four is equal to one Okay, you are actually missing a point some points. You're actually missing some I'm not going to worry about this. There's actually some identification like there was before in the inside manifold So anti-dissidious space is the stuff inside, right? It's less than zero, right? We put one there, but we're going to put it less than zero and it's in the stuff inside anti-dissidious space Is the stuff inside here? Straight lines are straight lines Right and there's also there's an identification the top gets mapped to the bottom Okay, we're happy Okay, well that's enough to do. Oh, well everyone keeps saying these things Everyone keeps saying well, you know, it's psl2r. What's I don't really get that Okay. Well, let's I guess I guess that we're going to have to worry about what psl2r is Everyone says to me psl2r is is the anti-dissidious space. Well, how do I know? What's the map? How do I get from one to the other? I don't really see it Okay As I said before the 2 2 the 2 2, uh, there's our There's our signature, right? You there's our inner it's it's it's not really written as an inner product because this is like the norm, right? This would be a half It would be like two of those matrices that I had on the bottom before right a half 0 1 1 0 1 1 right Okay And I'll do change of variables Okay, and I'm doing it a little quick. I haven't written things as the inner product, but that's okay. We'll we'll deal with it Okay, and I'm going to do this change of variables. I don't know if that's the right one or well It's it worked for me. So I'm okay with it And all I do is I substitute a million and I look at it and I multiply everything out and I get And because it's equal to one, but I put it on the I multiply both sides by negative one So I get x squared plus y squared minus z squared is minus w squared is equal to is equal to negative one So that matrix just because you could have done it by eigenvalues two and and look at everything But you can see by a change of variables. You can see that that Interproduct is just a different way of writing this The norm that comes from a different inner product The plus plus minus minus inner product Okay, I will say I'm not going to say too much about this but The isometry group Is you want to stay you want to what can you multiply a matrix by? A linear you want it to be a linear thing, right? What can you multiply a matrix by? An element of psl2r to make it stay in psl2r. Well, since it's a group you can multiply by an element of psl2r You can multiply on the right Or you can multiply on the left Okay, and you do this nice little thing to make sure your numbers turn out, right? It's a better way to write it a the element a b of psl2 cross ps2 to r is just a x b inverse Okay And they preserve the quadric so you're okay All right, so We can see that psl2r is anti-decider space Okay Oh, let's have some fun. This will be my last thing I do My bad. Oh, I thought I was going to be late. So now I'm going to be early again all right I'm going to write I'm going to map my element of a b c d psl2r well s l2r. I'm not going to worry about the plus or negative And I'm going to it's mapped to a point in that model x squared plus y squared Minus z squared is less than one It's going to map down to that model by this is you just solve it You know ask your favorite idiot to do it. Maybe you should do it by hand. It would be good for you Oh pepe do it by hand. Don't don't let don't let mathematics. I could do it for you I'd be ashamed of you Okay All right, so here's my question So let's do something crazy. Let's let's look at all the points That chorus let's take a geodesic Let's take a geodesic in h2 because psl2r acts on h2, right? So let's take a geodesic on h2 And let's look at all the points where there's a fixed point on the geodesic Okay, and in fact, I'm going to want and if it has two fixed points. I want one of them to be attracting Okay, pretty pretty nice idea right pretty standard thing you might think Francoise back there Gritting his teeth how I do I'm doing this but anyway, you'll like it a bit. I'm sure Okay, so that's what I want to look at and I'm going to call that That object c. I don't know why I'm calling it c, but I'm going to call it c Okay, the first thing let's let's oh, let's before we get it up. Let's put it up so I can I can use this What's the map? Let's do the further map right here because this is I mean I've written it in homogeneous coordinates, but I changed it to The one here. So that's really just the x y z coordinates x squared plus y squared plus z minus z squared is less than one right? So let's put this map in a b c maps to b Plus c over a plus d nice. Oh trace. I like that and of course I'm going to miss some points. It's okay I'm not going to worry about that a minus d a plus d b minus c In fact, let's just write this as as points b minus c over a plus d Okay, the denominator is a plus d the rest of them are b plus c a minus d and b minus c Okay, you just multiply it You're getting excited because something's going to happen and it's going to be good Okay Okay, so the first thing is The identity that's good to know it goes to the origin. Oh, isn't that beautiful? Okay That's the origin that goes to 000. Okay all right, so Okay, I'm going to fix the in fact. I'm going to be nice. I'm not going to take any Geodesic any old geodesic. I'm going to pick a nice one that I have some feel for Right, I'm going to pick this is psl2. This is the hyperbolic plane and I'm going to pick The geodesic here Okay, and I'm going to look at parabolic elements which fix infinity and zero Okay, all right And I do that. I I do this embedding right? I do this map not embedding. It's a map It should be minus t I apologize out another typo I got some small ones There were other ones that but that one I felt like that that that I deserved The density was not that was just mean Okay, there should be one so there's actually it goes to two lines Now I haven't told you because it's a little bit a little bit tricky to do this, but those two lines Not only are there two lines. They are like like lines okay so in our in our Universe Right Okay, there are The y is constant it's x and z it goes like That one And that Okay, and all I did was take the parabolic the parabolic elements which fix those I've seen those crossing lines before Crossing lines must be What you've seen the c before too I should call it d g k, but that's that's too many letters Okay, so let's do Let's do a hyperbolic That Who's attracting fixed point is infinity Something like this Right You're not going to get anything for that miss in zero. I don't care what anybody says There's uh, that's that's a hyperbolic s is greater than one you want s greater than one so that it you're attracting right And and you can't have some other parameter t there any parameter t will do Okay Okay, and I put it into my little machinery And I notice that it's that It's just like this except it's parallel to what I had before Except it's in the x direction. It's coming Which way is it going it's going to the positive x direction Okay So or positive y direction positive y direction sorry So it's going uh, that was wrong That was wrong of me It doesn't hit the boundary here. So it goes like this And the downstairs goes like this until it hits Until it hits the boundary of anti-deciders space. It doesn't go off forever and ever it hits the boundary of anti-deciders space And of course, what are you going to have on the other side? It's the same thing different one one minus s over s Where s is greater than one because we have it so that it's attracting zero is that plane I'm not even going to Bother you with the next one the next slide to go through it But what you see is that you have see something in between these two Like this and it's following this line And it's following that line and this is not my construction. Of course it is Danziger garret garoto and casel What they recognized is that a crooked plane The circle will not be unbroken right where we've come back to the the crooked plane the crooked plane and anti-deciders space are these are are things Isometries that deal with this line. They have a fixed point on this line And they're able to get whether a lot whether I mean this is going to be further than I want to go to they're able to understand how these things intersect with How two lines are moved away from each other. It's really really Shocking right and they get all kind of results Not looking at the That my geometry my favorite geometry the three-dimensional geometry But looking at something that happens on two dimensions because it's all about what is happening on two dimensions It was really really nice okay there's a Danziger when he's his in his phd thesis and and since then he was One of the ways he was thinking about One of the things he was doing was he was understanding how Manifolds changed as you as you change like you could look at negative negative one curvature Negative a half curvature negative and see how it changes. And so what they're seeing is that the The flat case is really a limit of these more More more fertile negative curvature case. So it's really a limit of this Okay, and here's my favorite thing Which really tells you how cool all of this is is that At the end of my first lecture, we're talking about crooked planes, right? And what did what did they prove they proved that every freedist great group is bounded by Two n plus two n crooked disjoint crooked planes It is not true in anti-deceptive space There's a weird thing going on okay so The future This is the future All right, and I think I will stop there and I'm not I don't even want to hear what What uh, france. Wilde wants how many how many times he wants to correct me on what I said, but anyway Anyway, that's it