And speaking of games, we have Gabriel Yanagihara. He's the game guy at Eolani School, Sullivan Center. Welcome to our show. Oh, thanks for having me on this big tech. Yeah, well, we want to do it multiple times. I'm going to check in with you every now and then, see what you're doing. Oh, yeah. And what the kids are doing, actually. Oh, that's the best part. I can just show off what they've been up to. So give us an update on life in gaming and 3D reality. Yeah, so we've been expanding our computer scientist program over at Eolani School. We recently made some purchases into virtual reality. And having the students try it out, just explore with it, try it out in a bunch of our different classes, and just let the teachers kind of explore and play with it and see what they come up with. But you don't just play with it. You program it. Yeah, so for the first phase, we just want the kids to kind of get an idea of virtual reality. Right now, we're working with the Oculus Rift. Get them in there, see what the world feels like when they can't. It's not the physical table in front of them. They're in this virtual world. So letting them get their VR legs so you don't get motion sickness. You kind of get used to it. Yeah, right, right, right. So let them play around with it. And then we're going to be launching, or we're currently launching, that class right now where the students are going to be programming their own apps for virtual reality. That's fabulous. Oh, I'm so excited. Yeah, wow, wow, wow.  We had somebody in the studio. I don't think it was you. It might have been. Maybe a year ago, and you can remember things. Oh, here's a picture. There it is. That's what there it is. And she's touching things.  Yeah, this is really interesting. You build a whole universe, don't you? You can. I think when I was originally, we're starting with before VR, right? One of our most popular workshops for the students was Minecraft, right? Letting them build with, essentially, virtual Legos, right? And all the curriculum and all the lessons that you could run if you had an unlimited amount of Legos in your classroom, right? You can come up with some pretty cool classes that way. So what we were taking from there is now you're in a world where you can build and modify the entire world around you. Minecraft just launched their virtual reality version of the Minecraft for Windows 10. And with that, we're having students, so instead of just being on their iPads kind of hunched over there, they're actually up and building and moving trying out all their little creations in there. Fabulous. You know, just a footnote, I need your help on something about Microsoft 10 after you show it. Definitely. I'll take it and show you what I'm talking about. Anyway, so think about programming. So how do you program Oculus? How do you do that on a regular PC? So right now, what we're working on is we're using, so we have a Mac lab, but we're working with a free game engine called Unity. They recently released their educational licenses. So any school can apply to get free licenses as well. Indie developers, if you're on your own, if you're just to start up on your own, it's also free. There's free personal editions. And it's basically an environment that lets you start arranging and setting up your game. So in the easiest terms, it's a program that you run, and you can build games in it. And then it's very easy to publish those games on pretty much any of the platforms, Windows, OSX, iPad, Android. So you can pretty much launch it out. There's one program. There's one program. That's pretty great. And so I mean, I know we don't have it with us. We're not going to be able to show it on the screen. But conceptually, how does it work for you to program that? I mean, if you worked with any other 3D environment, I've come from a 3D modeling and animation background, so it's a 3D space or 2D if you want. And then you can just drag and drop game objects in there. Like you can drag your little Mario character. You can drag your other character and then you assign a programming script to each of those characters and have them start interacting with each other in the game. This is like animation, but on steroids or something. Yeah, and it's getting even bigger too, because a lot of the indie games that you'll see as either Steam Greenlight Projects, which is kind of like a not Kickstarter per se, but kind of a crowdsourced select which games you want to see come out option. So there's that as one option for what they can do. They can play with it. And then the other one is kind of letting them, so taking Unity, right, and then publishing their games as they want to go with it. So these kids come in, do they have game experience? So they, you know, I mean, one of my partners, his kid used to stay up till 2, 3 o'clock in the morning. He was like in the seventh grade and dad didn't know what to do because he was consuming him. They come in to you with that kind of experience? It's pretty cool. I mean, imagine coming into like a math class and then the students are the experts at math and you're kind of just like, okay, so now that you guys already know everything, they know, like they say the huge math bands, right? They know, they played all the math games. They've done every math problem. They can get their hands on, but now you're going to show them, all right, here's academically what you can do with that skill set, right? So we get a lot of kids coming in. They've played lots of video games. They're very familiar. They're digital natives, right? They're very familiar with that entire space, but they haven't quite gone to the program. Oh, here's some virtual reality. Yeah. That's the rock band. That's going to be really cool. I haven't gotten one myself, but we're going for it. How does that work? He's wearing the goggles, so you're still still playing? I think for the VR, I think the new rock band they're releasing is you put the goggles on and imagine, like you're on the front stage in front of an entire crowd of people and you can walk out, right? Because with the last rock band, like I love that game, but it'd be just my friends and I kind of just sitting in our room, right? You see, like, all our laundry on the side and all our books and computer stuff everywhere, but you put the goggles on and suddenly you're right there next to, you know, whichever celebrity will sign with that brand, right? You turn to them and see them? Yeah.  Whatever. I think it'd be really cool if they could, I mean, if they get the licensing and everything for it, reviving old bands, right? So you could play next to the Beatles or next to whichever one, whoever they sign up for it, right? That'd be super cool. I mean, my students won't really get that, right? They might be like, all right. They don't know the Beatles. We want to see, you know, chain smokers. We want to play with Daft Punk or something like that. The old guys like me, I don't care if your students understand it. You want to try it out, yeah. Blocking with the Beatles. That would be cool. Let me see if that brand, that IP, has revitalization with VR. Yeah. Well, you know, I mean, so many questions float into my mind about this, but this is moving so fast. It's amazing. When I saw you last, it wasn't like this. It's changed and it's changing too. And you got to stay up with it. Now, to stay up with it and to be relevant for them in their expectation that you're going to take them right to the edge of the envelope, yeah? How do you do that? You just go with the flow? Do you read up and study up? Well, a lot of it is. There's a lot of really great communities online. There's actually a Facebook group for educators, run by one of my colleagues, Mike Furcano. And it's a virtuality in education. So basically, it's a crowd, not just like crowdsource information feed, but basically all the teachers in the world who are kind of dabbling with virtuality. They share the content with each other, right? So he set that up. He pioneered it out here from Hawaii, right? And now all the other teachers are hopping on board and they're like, hey, I tried Google Expeditions, right? I tried this Google Street View. I tried Coast Spaces. I tried Thing Link. There's all these different startups happening. There's so much information. There's no way one person could run through it all, right? But as an education community, everyone can kind of share that a little bit. Yeah, you got to do that. So what are your favorite sources? What are your connection points? Definitely that Facebook group run by Mike Furcano. The other one is just kind of keeping up to speed. There's a bunch of YouTube channels that I just like to watch from here and there. And of course, the students, right? The students are definitely watching a lot of YouTube videos talking about Let's Plays, which is where the YouTube hosts will have his variety talk show, but he'll try on a bunch of different games that people send them. And then they'll see the good ones and they'll come to me like, oh, Mr. Y, did you hear about this game? Did you hear about this game? And then I'll be like, yeah, I totally heard about the game. Write it down in the research editor, right? I pretend like I'm kind of getting an idea for how it's going. But it's even expanding to the mobile market too, right? You can get your small phone, right? And even any phone, any Android or Windows phone or even Apple phone, right, or the iPhone, can run a virtual reality headset, right? Google is launching their DreamVR, which is their mobile headset. Oh, is that right? We work with Google a lot.  Like $1.95. If you remember back to those old days where it's the red, what is that, ViewMaster or something, where you just click the thing and then it cycles through the disk, right? It's like they rebuilt one of those digitally now. So you can like click the button and instead of being inside, just looking at one view, you can look all the way around you and you'll see whatever you're doing there. And then that same idea, right? Imagine you could get a classroom set of those, right? Give them to all your students and you can take them on a field trip in the classroom anywhere in the world. And that's been the, we've been doing a bunch of projects with that. Mike Fricano, who's also a colleague of mine, has been doing a lot of Google Cardboard expeditions with the students. I think the last big one that we did was doing the Memorial for the Pearl Harbor attacks. We had the students take virtual tours of the Arizona, of the Missouri in the classroom so they could run along. So what is it about these virtual tour, virtual reality?  Yeah, yeah, yeah. So what is the experience that makes it so magical? So as opposed to that kind of looking at it on a TV monitor, right? So when you're looking at the monitor, it's kind of still looking at it through a picture frame. But once you put the goggles on and with the higher quality ones, like with the Oculus Rift or the Vive, where you have the audio as well, the 360 audio. When you put it on, everywhere you turn, you get the sensory input of whatever you're looking at in that position, right? So you can see here the people are kind of looking around and each of them are seeing a whole different world. So they could be one could be in China, one could be in Shanghai, one could be, you know, or in Shanghai, China, the other one could be here and in the middle of Diamond Head checking it out. And it's been launching like some really cool student projects as well, where they wanna share their spaces, right? So when they sit at the table like this, they're talking to each other. They compare notes about their own reactions to these various places. Now, where's the data coming from? Is it coming from the device itself or from a computer or a wireless? Where's it coming from? Yeah, so I mean, depending on which quality of device you're using, right? I think the best one, the Oculus Rift. So for the Oculus Rift, it's using the computer to feed a separately rendered image into each eye. So what's that doing? It's creating- Separate images for each eye. Separate images for each eye because it's like the screen inside the device is split and then it's feeding a different one into each eye. So what that allows us to see is the 3D parallaxing that happens. So like when you move your finger across your eye, your two eyes are seeing slightly different images. But now we can simulate that on the computer. So as opposed to just looking through a screen and seeing what's right there, where you can move your head around, you can walk around the room, and then you'll see all the objects from different angles. And it looks 3D to you. And somehow the machine is able to give you that 3D experience. You're among the objects, you're in the world, which is the big sell for it right now. But this is all a matter of data. Somebody has built it probably with gaming technology and all this drag and drop animation kind of graphical interface is really fabulous. But I have a question for you. My question is, well, preference to the question. We have, we just got this thing called live view. Live view is a little backpack you wear and you connect it to your video camera. And it has what they call bonding technology. And the bonding technology bonds up half a dozen cell phone accounts, modems, okay? And now it has very fast broadband. So my camera is gonna report back to my studio here, okay, with the best broadband that you can get anywhere. There are any cell phone towers. And the result is we can go live, okay? From anywhere, yeah. From anywhere, anywhere in the world, right? Yeah, that's awesome. In fact, just curious that I saw the first time I saw this was at the commemoration of the Pearl Harbor attack. Yeah. Pearl Harbor. It was a guy who had one of these things and he was sending it back live to New York, standing right next to me with a little thing. So you talked about these people who can be anywhere in the world. Yes. And see all these things on virtual reality and compare notes and have this kind of joint experience at the table with all this really fabulous input coming in. Yeah. Why can't that input be coming in from all over the world? You say, oh, they can be here, they can be there, they can be anywhere. Why can't it be live with my little backpack?  It's called the Theta S. So it's a small, it looks like the old Flip HD. It's a small rectangle like this and it launched like, I think they're on their second iteration now. The video quality isn't amazing yet. They're still working out the kinks and I think they're doing like a release stages to get better and better at it. But it also lets you live stream in 360. So what that software does and what the current workflow, I'm not super familiar with it, but the current workflow that I know of is that you can live stream from that to a Twitch account or to a live stream account and then on the opposite side, on the receiving side, well again, if we have good enough internet, you can have the goggles set up, you can have the virtual reality headset and then you can see whatever that camera is live streaming immediately. There it is. So there's some really cool tech going on with that. A regular one-dimensional camera is not gonna do it for the Oculus Rift. But one of these cameras would and if you put my live view backpack, might be called a baby, baby in the middle, then you can go to anywhere in the world and deliver Oculus Rift quality of visual experience. Wow. Yeah, the thing that's really, the one thing that's holding that back right now though is the resolution on the live streams. A lot of them are kind of standard 1920 by 1080 resolution, which is standard monitor resolution. But when you take a 360 photo, what it's doing is wrapping all the data around you and then compressing it into a flat 1920 by 1080. It's like unwrapping it onto a flat screen. So when you unwrap it, you lose a lot of resolution. So when you see the really high quality 360 videos and stuff on YouTube and stuff, there are a lot of times they're playing at 4K. So we don't, like our mobile networks and everything aren't quite ready for 4K streaming just yet. But I can definitely see that. I mean, VR has been exploding so much in the last couple of years. I can very easily see that in the pipeline. Well, I can tell you that baby could do 4K telling you that now. Yeah, 4K streaming, that would be amazing. Yeah, we take a short break so we can integrate all this because the sides of my head are hurting. We'll be right back with Gaby and Agihara. I'm Ethan Allen, host of likeable science here on Think Tech Hawaii. Every Friday afternoon at 2 p.m., you'll have a chance to come and listen and learn from scientists around the world. Scientists who talk about their work in meaningful, easy to understand ways. They'll come to appreciate science as a wonderful way of thinking, way of knowing about the world. You'll learn interesting facts, interesting ideas. You'll be stimulated to think more. Please come join us every Friday afternoon at 2 p.m. here on Think Tech Hawaii for likeable science with me, your host Ethan Allen. Hello, my name is Crystal. Let me tell you, my talk show, I'm all about health. It's healthy to talk about sex. It's healthy to talk about things that people don't talk about. It's healthy to discuss things that you think are unhealthy because you need to talk about it. So I welcome you to watch Quok Talk and engage in some provocative discussions on things that do relate to healthy issues and have a well-balanced attitude in life. Join me. I want to be there. He's already there. I've already downloaded that entire thing. It's Gabriel Vinogihara Vilani School. So you start out with gaming and you learn how to game and then you go to, it's a natural progression to visual reality, 3D, an Oculus Rift and nobody knows where that's gonna go. Yeah, it's an exciting field to be in. But you're gonna follow it. Oh, yeah, yeah. You could not follow it. At this point, I'm kind of like, I'm along for the ride. It's ups and downs and whatever happens, we're gonna be really excited to see what happens. So tell me about the Ignite program that you're doing or you're involved. Yeah, so we've been doing a lot with the robotics programs at Yolani with the app class, which is gonna be making VR projects. And we wanted to have a venue for our students and faculty who we're allowing to kind of explore with all this new cutting edge tech, see what they did, what they found is best practices and just share it out for everyone to see. So a lot of our teachers and a lot of our students are gonna be doing presentations at the Ignite Innovation Conference, which is gonna be at the Sullivan Center at Yolani School. The world begins and ends at the Sullivan Center. Yeah, it's- Everybody's a center of the university. And I get to live on the third floor. It's amazing. Wow. But no, I set up camp there and then we're gonna have the conference. I think the conference is gonna be February 18th. I wanna come. There'll be definitely, we'll love to have you. Yeah, yeah, okay. So I'm talking about the kids for a minute, okay? Kids respond to this. I mean, they come with a certain amount of experience at home and all that. And then they light up when they get into a classroom setting where you can show them things they didn't know, where you can understand things they tell you, which is important. And where you can structure their perception of these things so that now it becomes more real. It's no longer just a hobby. It has real life impact. Life and educational economic impact directly on them. If they create the next Flappy Bird game, right? They can go to any college they want and they're out of pocket, right? It's true, but not in games though because this gaming technology often leads to real technology, I shouldn't say that, but technology that is used in the marketplace for other things. Yeah, simulations. There's a lot of talk now in the medical field. How else can you use virtual reality? How else can you use gaming for either betterment of patient care, old folks' homes, right? How can you improve the lives of the people who maybe may not be as mobile, right? One of the projects that our group of students are doing now is they're doing 3D scans of the hikes here in Hawaii. And they hope, you know, hopefully once they can create their program in unity with that and create like virtual tours of Hawaii. No one is sorry for hike. There's so many side projects you can do with that. You can throw that on the Google Cardboard and give that to grandma, grandpa, people who are given to the hospital and let people explore the outside world and why they don't need to, right? Without standing out, yeah. So these kids come to you. Is it every kid comes to you or just the kids in the one? It's an elected class, the kids who want to. So I think the easiest example would be letting students, so the reason we have the intro video game design class which is seventh and eighth grade is to let students kind of get under the hood of the games that they're already excited about, right? So it's kind of like back in the day when you'd have auto shop, right? Everyone's super into cars, we love cars. All right, let's make it academic. Let's people, you know, instead of you just buying a car and then paying a couple hundred bucks every time it breaks down, let's teach you what the inside of the car is like so you can build your own version of it, right? And especially now with gaming, the gaming tools are all, you know, they're free, right? Anyone can pick up and make a game with showing them don't just be consumers of digital technology. Don't be just a consumer of the iPad where you just put the money in and then, you know, you use it. Engage with it. Get underneath it, right? Understand how it works.  We redesigned it, the whole thing. Making yourself happy. Make the next Apple. And remember me when it's time to retire. They said you would check, yeah. What that, I wanted to ask you about that game. So you take these kids who elect it because they really like this kind of information technology that's a broad term to define and you excite them and you show them things they didn't know before and you bring them into a conversation about it. You bring them into a community thought process about it. Now it sort of changes their lives and they really, they begin to see the possibilities as you do. They catch it's viral. They catch what you have. Whatever you have, they catch it. Okay. And they graduate. Yes. I thought they go to the mainland. Well, that's what's great is it's just to start, right? They've got a head start over all their peers. They're kind of, you know, one thing that I think Eulani and Dr. Cattrall has been really successful in with Eulani is pushing it, not just, you know, you don't just graduate with, well, I've memorized all the chemical formulas. I've studied for the test. I can get through that. But now it's, well, I did good on my academics, but guess what? I also have two games published. I also built my own drone. You know, I built a drone that flew over to the Allo. I put up a water sample, did bacteria testing on it. That's true. Yeah. Our robotics program has been phenomenal. Martin MD is leading our robotics program. And the kind of robotics projects that they have, seventh, eighth grade kids starting on, has been, you know, it's blown my mind. I don't know how he's been doing it. Oh, he's blowing my mind. I wasn't even there. OK, so they go to the mainland. Not only are they talented, smart, motivated, but they also have experience in the class for you with you. And the robotics, too, and all the stuff, they are poised for success in the IT world. The Lord knows where that'll take them, but it doesn't take them back to Hawaii, does it? There are no jobs here for this. Well, what's really interesting? We accept your job. Well, I work for the job. I was very lucky with getting set up where I am. But one thing that's really interested with the indie game industry and with mobile markets, right, that, right, you can be set up anywhere. On the beach. You can literally, like, I have two or three friends here. They hang out on the beach. They have their laptop. They go to Starbucks. They work on their app. And their app brings in a couple grand a month, and they just hang out in Hawaii, right? Not a bad deal. Not a bad deal. And then what's good is we're building up a lot of that local community here, right? There's another group, Dev League Hawaii, who are creating a lot of developers here. And there's a growing community of game designers here in Hawaii. And there's also a lot of opportunities, you know, pretty much if you decide that you want to live in Hawaii and you want to be your own boss, there's not a lot of things holding you back. Well, the thing is, if you were on a beach alone in another time, in another time, say, 20 years ago, trying to do something like this, you would be alone. You wouldn't be connected. You wouldn't be able to sell your stuff, find anybody who'd pay you for it, or, you know, help you with it, do collaborations and partnerships and the like to make yourself, to make some money. But now, on the beach, it's not necessarily the case. You can find people who will help you and mentor you and connect you and pay money to you, right? Am I right? Yes. So it's a growing community here, right? And I think that would be one of the really cool initiatives because digital IP, so digital intellectual property, digital projects like that, where you can work with people anywhere in the world, you don't need an office space. You don't need real estate, right? You don't need a whole lot of energy, right? So a lot of things that limit business here in Hawaii, you know, your IP, you have an idea that has an intellectual property based in Hawaii, right? You don't have to worry about making the kind of higher rent and stuff that you have to deal with here. Do they take older gentlemen in your class? And I sign up. We do have open nights. We do have open nights where I have open to other educators and we just share all the curriculum and all the stuff that we've been working on with anyone who's interested in growing the community. So where is it going, Gabriel? I mean, one of the things you mentioned that was touching to me before we started was the notion that one day something would happen and all of a sudden you would realize that you're not their big brother anymore, you're their uncle or your father, your grandfather, and you would not occupy the same role anymore. I know. I've been riding the cool older brother teacher for a long time. And I think I've still got a couple years left before I start getting like two gray hairs for now until I start getting a bit older. But I think for the now, I don't think I'll ever stop being engaged with the tech community. It's my final frontier. It's we've mapped the world. We've been everywhere. Technology is the last bastion of real exploration that I see that is accessible to the average person. So where is Hawaii going on this? Granted, there are some people who understand they can go to the beach, make a buck, participate in a kind of virtual community. Where is the industry going to go? Do we have it? Will we have it? Can we get together on this and make something happen? I think the biggest thing that we can do is form our closely knit and keep growing our closely knit community of game developers. We have the Global Game Jam, which is an annual event. We have a couple other meetups where all the different programmers, 3D artists, game designers, virtual reality people all get together and talk story about our industry here. It'll be really cool to see more students get involved, more educators come involved, reach out to some more of the business side of things here, and see what services can we offer here in Hawaii that are not real estate and energy based? That we can just expand and can scale anywhere. Because we can have the idea here and just pay for Amazon to have the servers in the main line in some warehouse somewhere and run the entire situation that way. So really with the internet with all that stuff, there's really not a set location that we need to be. And you can show them how. Oh, I hope so. You can give them the leverage, the understanding, the way to find a career in this at any level they want. Yes, I'm hoping so. Yeah. We got to close the show now, Gabriel. It's been really wonderful. Yeah, it's always great to come and visit. Maybe even better. I'm so excited. And one of the reasons we're closing the show is because I'm having this problem with my PC. And I need to hold on to him for a few minutes so he can help me understand what's going on. OK. Oh, I'll be glad to help you. Gabriel Yanagihara, Ilani School. Wow. I'm going to go back to school again.


Hello? Ah, yes. So let me just recap where we got to last time. We introduced this ADS spacetime, which has a neat embedding as a hyperboloid into some two-time generalization of Binkowski's space. And we saw that the isometrers, which are best seen from this embedding picture, are an SO2D. This is highly analogous to when you embed a sphere into Euclidean space, a round sphere, into Euclidean space, and obviously get an SO isometry group that has an action on the sphere. It's just got some flips of sine in the metric. So this is now a Lorentzian metric and so on. Now, I told you that there's a global chart for the metric on anti-desitter. But in fact, the chart that we're going to predominantly use is not a global one. It's one that can be thought of as coming from a sort of null wedge cut out of this space. These should meet at a point. And it's called the Poincaré chart. I gave you some explicit parameterization in terms of the hyperboloid. And this is this Minkowski metric. And in terms of a conformal diagram, a Penrose diagram, for this space, it's not a complete space. So we have the boundary, which now lives at z is 0, the conformal boundary. But we also have some Cauchy horizons associated to these null wedges, the fact that you can just continue out perfectly happily. But from the point of view of field theorists, we'll see this. We already started to see last time this is a sort of a natural coordinate system to use. We also talked about how, as many of you with a geometry background will know very well, this is a conformally compact space and has a conformal boundary, which is defined up to some conformal rescaling. And z, in its own right, here is a defining function for the boundary, the conformal boundary of this space. And when we look at what the conformal boundary is, it's just Minkowski. So in this form, the conformal boundary in these coordinates, the conformal boundary or a representative for it is just Minkowski metric. Now, you may be confused that we already said from here we know that the conformal boundary is this Einstein-Static universe time across a sphere. And now I've got a conformal boundary that's Minkowski. How did that work? And the point is that Minkowski and the Einstein-Static universe are related by a conformal transformation or a vial transformation. It's not completely obvious, but it's a simple exercise to show. Now, before I go on, I just wanted to put a caveat to what I was saying yesterday. It was a good question after for the physicists among the audience.  But please don't listen. I'm not going to explain it in terms of bundles. I have no idea how to. But for the physicists, just to avoid any confusion, this is a correlation function derived from the path integral. So of course, in Lorentzian signature, it would be time-ordered. What I've actually written here is the Euclidean continuation. If you want the Lorentzian version, that's an exercise for you, Fourier transform, use the i epsilon prescription, and so on. The point is when in all discussions of conformal field theories, it's just complicated to work in Lorentzian signature, and everything looks very pretty in Euclidean signature. And you'll see that again later in some of the other things I'm going to say. So whenever you see explicit expressions that look like this, there will be some implicit analytic continuation, as we nearly always do in field theory.  Otherwise, carry on listening. OK, so as we said last time, but I want to re-emphasize it because it's a very important point, this metric manifests certain of the isometries, certain of these isometries, very obviously. And so in particular, Lorentz and translations in the x-coordinates alone obviously leave this invariant and are part of this isometry. However, there is also. So these are holding z constant. But there's also, it's immediately clear from this form of the metric that if we were to do this diffeomorphism, again, this metric is invariant. And so this is again part of this isometry group. But it has a very interesting action on the conformal boundary. So these act on the conformal boundary as the Poincaré group. And this acts as a scale transformation. And what isn't manifest is the remaining sort of symmetry or the remaining part of the SO2D, which is this special conformal transformation. But nonetheless, it's in there. And you can show that it's action is the usual action on the conformal boundary. So what we learn is these act on the boundary as the conformal group. And by boundary, I mean the conformal boundary metric. So I'm typically going to just call the conformal boundary the boundary, but understand it's not a boundary. It's a conformal boundary, but just for simplicity. I hope that won't confuse. So the isometries of anti-disseter act on the conformal boundary as the conformal group acts. Now, that's pretty much where we got to last time. So ADS geometrically has properties that I'm sure you know. It's an Einstein space, which if I introduce explicit coordinates. And in my conventions, the Ricci tensor is given in terms of the metric in this way with what we would call in physics a negative cosmological constant. So it's a negatively curved space. And in fact, it's maximally symmetric. And the vial tensor vanishes, which means Riemann can be written in terms of Ricci in the usual way. But in particular, because of this, ADS solves the Einstein equation in the absence of matter but with a negative cosmological constant, obviously. Because this is the Einstein equation for just gravity with a negative cosmological constant. And we should interpret it as the vacuum solution for that dynamical system, for that theory. ADS is the vacuum solution or the sort of empty, the simplest solution for gravity plus a negative, let me call it lambda cosmological constant, by which I mean the theory whose Einstein equation is this. However, and so this is just an analogy with Minkowski being the usual vacuum solution for just pure gravity without a cosmological constant. It plays exactly that role. And from a physical point of view, what we're really interested in is not just the vacuum, but excitations, dynamics of this theory. So this is what we're really interested in. From the point of view of ADS CFT, we're really interested in this some gravity theory that live in spaces that are asymptotic to ADS. An asymptotic ADS spacetime is obviously one that solves some Einstein equation. Well, we will require it to solve some Einstein equation. So when I talk about asymptotic ADS spacetime, I'll be talking about it in the context of some Einstein equation with negative cosmological constant or some appropriate matter to give the right to at least look like a cosmological constant asymptotically and tends to ADS asymptotically. And I'll be more precise about that later, but let me leave that for now. So I will give you gory details in which in what way I mean, probably in the last lecture, in what way I mean that the spacetime should approach ADS. So our setting then, this is often called the bottom-up setting in ADS CFT, will consider gravity, GR, well, maybe Einstein gravity, plus some negative cosmological constant, but perhaps plus some matter in d plus 1 dimensions. This is the physical theory we're going to be interested in, such that ADS is a vacuum solution when we turn off all the matter. Why is this called the bottom-up construction? In the bottom-up way of thinking in ADS CFT, we are going to think of our gravitational theory in the simplest possible terms and try and understand the general phenomena that can happen in this gravitational system, as opposed to a top-down way of thinking where you say, OK, I believe ADS CFT works in this particular case for this particular field theory being due to some particular string theory whose low energy limit is a particular supergravity with very particular fields. And I should only work in that setting, so I've got to take some very complicated gravity theory with lots of fields and so on and so forth. These are very complicated theories, but they nearly always, you can consistently truncate the dynamics away for many of the fields and in the end reduce some of the dynamics, at least to something much simpler that will always look like this. In particular, in any case where you have a conventional theory of gravity, by which I mean two derivative Einstein in however many dimensions you like, and with all sorts of complicated matter fields, fermions, bosons, all sorts of things, you can always truncate everything away to land up with just gravity and a negative cosmological constant, and any solution of that you can re-plug into your full theory. So let's look at these very simple sectors of these much more complicated theories rather than sort of saddling ourselves with a tremendous task of understanding super gravities, which are very complicated. So it's a consistent thing to do this in general, but one has to also bear in mind that at the end of the day, the physics that you look at should be embeddable in some sort of theory that you really believe exists. You can't just make up completely baroque theories of gravity that happen to be asymptotic to ADS and imagine that they will necessarily be found in some version of ADS-CFT. That's not, it's questionable whether that works. OK, so just to reiterate this point a little bit more, in the case of n equals 4 super Yang-Mills, which we discussed, there's a very specific 2B super gravity, let me call it SUGRA as usual. And in fact, it's 10 dimensional. This is a 3 plus 1 theory. And the vacuum solution here isn't just ADS. It's ADS 5 cross an S5. And that's where the 10 dimensions come from. And it's a complicated theory with lots of matter. But this S5 doesn't really play a terribly interesting role. It's just some sort of direct product, and you can, in the vacuum, and we can consistently truncate to field configurations that are constant on the S5 and remove, well, we could remove all of the matter and just be left with Einstein gravity in a negative cosmological constant in five dimensions. And it's also possible to consistently truncate and keep some matter fields. OK, let me carry on then. So for now, as I said, we'll just consider Einstein plus cosmological constant maybe plus some simple matter, maybe like a scalar. And Einstein plus lambda I would normally call the universal sector of ADS CFT because it's there in every known example where you have an Einstein-like gravity dual. You could always reduce to, if you've got a d-dimensional QFT, a d plus 1-dimensional Einstein plus cosmological constant, we'll always sit inside whatever complicated gravity theory you have, which is nice because, of course, this is just geometry. So it's a simple geometry, the sort of geometry that geometers are interested in. So you're interested in Einstein metrics. So the theory of Einstein metrics inevitably sits inside any ADS CFT correspondence. Let's have some examples of things that aren't ADS but live in this universal sector and play important physical roles. So one of them is ADS Schwarzschild. And let me call it planar ADS Schwarzschild. There is a more conventional ADS Schwarzschild, which is asymptotic whose conformal boundary is that of global ADS, meaning Einstein-static universe. But you just take the usual Schwarzschild metric in Schwarzschild coordinates and you add an r-squared, your function, your GTT function, which is normally 1 minus m over r, you add to that something going like r-squared. That will solve the Einstein equations with the negative cosmological constant. However, because we're interested in this setting where the conformal boundary is Minkowski, because from the point of view of field theory, typically you're interested in field theory but not space first, the relevant black hole is actually a different one, which is physically inequivalent. It's not isometric to the other one I mentioned. It takes a very simple form. This function F is just a function of the coordinate Z and Z naught gives the position of the horizon. Z goes to zero is again the conformal boundary and we see again Z in its own right as the defining function, the conformal boundary as Z goes to zero, F goes to one. It's again just Minkowski. So we haven't messed with the conformal boundary. So this is indeed, as we will discuss later, asymptotic to ADS in the usual sense. But it's nonetheless not ADS, it's a black hole. It's an interesting black hole. For those of you who know all about black holes in the usual setting, this is a very different black hole. One point to note is that the horizon is Euclidean space. So it's not compact. All of your lovely theorems about black holes very often use compactness of the horizon somewhere. So all sorts of interesting things happen to black holes in ADS that simply don't happen in asymptotically flat settings. For example, well, actually let me not complicate things. So that's one solution. It's often drawn if you read books or lecture notes on ADS CFT. It's often drawn like this. This is meant to depict the boundary. Z is zero. This is meant to depict the horizon. Z is Z naught. And this business is meant to depict the warping, if you like, of the spatial slices, the constant Z slices by this factor. Of course, really the warping becomes infinite as you go near the boundary. But nonetheless, that's difficult to draw. You can calculate for those who are interested the Hawking temperature of this black hole. Well, I mean in various ways, but if you like the Euclidean methods, it's very straightforward. This is, by the way, the Hawking radiation associated to the killing vector d by dt, which is the natural killing vector of the space, and also the time on the boundary. And the entropy of this diverges. But there is a sense in which you can define a density. By the entropy, I, of course, mean the area suitably divided by 4g Newton with appropriate h bars in. So if you like, the area diverges, because it's the Euclidean metric on the horizon. But nonetheless, there is some notion of a density. So because it's infinite, whenever you have infinite systems, you always have to start working with densities. So there's black hole thermodynamics and all sorts of things as usual, but defined in terms of densities rather than extrinsic quantities. Sorry, was there a hand? Another, oh, an interesting point here, just to bear in mind, as z goes to infinity keeping z fixed, this returns to the form of AdS. And one way to think about that is that the Cauchy horizon of AdS, in some sense, is like a black hole horizon whose temperature has gone to zero. So in physics language, we could also call the Cauchy horizon of Poincaré AdS an extremal horizon, meaning it's a zero temperature horizon. And there's some theory of extremal horizons that have been developed. I won't go on about that. There's another interesting metric that perhaps people are less familiar with, but certainly is very interesting. And it's called the AdS soliton. So this metric, in fact, the conformal boundary here is Minkowski, but something I can do and leave this a solution trivially is to wrap up this Euclidean space into a torus just by some identification of these coordinates. So I can wrap this into, say, a square torus just by making these space coordinates angles. And now my conformal boundary isn't Minkowski anymore. It's time cross a torus. And in fact, this is a perfectly fine thing to do. And from the point of view of AdS CFT, we'll later discuss the interpretation of this. But just from the point of GR, this is a perfectly fine thing to do. It's a perfectly good. Locally, the boundary will look like AdS, but globally, it's different to the AdS boundary. So it has this different topology. And in that sort of setting, there's actually another nice solution which is closely related to this one, which is, in fact, trivially related to this one, really. You just let me call this now some other label swivel where this theta is some circle coordinate. So maybe I've identified all of these into circles, or maybe I've only got one that's a circle and the others are extended. These are all fine from a GR point of view.  You can see there's no horizon. But instead, it's more like a space. One could draw a similar picture. And this is a space where instead of it ending at z, is z0 at a horizon. Instead, the circle, this circle on the boundary. So here, at some point, we've got a circle. What's happened is it shrunken smoothly provided, let me see, provided I identify 1 over L, provided I pick z0 appropriately, I get a smooth shrinking of this circle as I go away from the boundary in what we call the radial direction, the z direction. And this is called the ADS soliton discovered by Horowitz and Meyers. I mean, it's a very simple solution. But in terms of GR, it's actually quite interesting. If we have time later, one of the interesting points is that these have positive, supposing I wrap this up into a torus, now I can talk about the energy rather than just energy density. It's finite. And wrapped up on the torus, this has positive energy, just like Schwarzschild normally has positive energy. It's got some energy which goes to some power of z0. Unsurprisingly, in this case, this actually has a negative energy. So in some sense, this is the natural ground state of GR when the conformal boundary has circles in it. So it's quite interesting. From a GR point of view, you're thinking about proving positive energy theorems and so on. Depending on the situation on the boundary, it may be that ADS isn't the lowest energy solution that you might want to, well, yes. Already there may be lower energy solutions out there to consider. OK, anyway, those are just a couple of solutions that are not ADS but are interesting physically. This would be the vacuum solution. This is some static solution in this theory that's got some energy in it. Oh, yeah, OK, sorry. Yes, the Fs are exactly the same. I mean, if that solves the equation, this does trivially by just analytic continuation twice. I just take t to i theta. And so this t is not that t.  Yeah, well, you could identify here as well. So I mean, you see, this is an example of a case where the boundary at least has one circle. It may have more circles. But now there are multiple infilling Einstein metrics, maybe from a geometric point of view. We have the usual ADS one. Now, the usual ADS metric, if you identify a circle, actually has a cusp-like singularity. And then this is the more natural vacuum state, which is perfectly smooth and has no singularities. So the natural cusp-like singularity you get when you make this identification on the conformal boundary of ADS, it sort of resolves because there is actually a nicer and lower energy smooth solution. Anyway, let me not go on about that. There was one more question. Sorry, can you? Yes. Yes. Yes. Yeah.  I mean, it's just like Schwarzschild. There's a horizon you then have to. So let me see, what does the conformal diagram look like? I think it looks like this. So this chart, as usual, is only covering the exterior of the horizon. And then there's some other charts to see the singularity. But, yeah, so maybe something to say is that this is locally asymptotic to ADS in the sense that I don't have the full infinite conformal boundary anymore in this case because of the singularity. So maybe to say this is, maybe even in this case, we should be a bit careful when we say it's asymptotic to ADS. It's actually removed a lot of the conformal boundary that ADS has. But of course, let me just say the general solution to this sort of gravity theory then looks very complicated. Let me draw it in this sort of setting. Dynamically, we're going to imagine we've got this ADS boundary or asymptotic ADS conformal boundary. And we've got, we should think about dynamics in this as requiring some asymptotics to ADS near the boundary. But also, we've got these Cauchy horizons. Well, maybe we don't even know what we have here. But certainly, we can start with some characteristic data on this null surface or Cauchy horizon. And then see what happens. Evolve the Einstein equations. Maybe we put a black hole in here.  We shoot them together.  And who knows? There's some radiation given off, bounces around.  You know, stuff hits, reaches this conformal boundary in a finite time. So this dynamics definitely knows about this boundary. It's very important. Even though it's infinitely far away from a spatial point of view, you know, radiation may go there. And you, with the boundary conditions we're going to choose, it will bounce back. And there'll be some very complicated dynamics. OK. So just because you write down only a couple of exact solutions, I mean, it's a full theory of gravity in principle. Something to note is that we mentioned this before. The time-time component, you can see from these static solutions, the time-time component, which tells us about redshift physically, in ADS, as we go away from the conformal boundary, as we go to the conformal boundary, it diverges as 1 over z squared. And as we go to the horizon, it goes to 0. Or in the black hole case, as we go by horizon, I mean, Cauchy horizon, in the case of ADS Schwarzschild, it diverges at the conformal boundary and goes to 0 at the black hole horizon. So if you think about, if you're sitting in the interior of the spacetime at some constant z, and bear in mind that that's not geodesic, so you'll have to be accelerating to do that, exactly in analogy with usual Schwarzschild to sit at constant radius, you have to accelerate. If you're accelerating sitting at constant radius z, let's say, you will see physics happening near the conformal boundary as being very high energy, very blue shifted. Whereas if, as usual, physics near the horizon will be very redshifted. So the interpretation of these sort of radial-like coordinates that take you away from the boundary is that there's also an energy. Somehow, stuff near the boundary is very energetic. Stuff far away from the boundary is low energy. And it's just in this very simple sense. So we often, in physics, which is probably confusing, would say this is the UV part of the geometry, the stuff near the boundary, and this is the IR, the infrared part of the geometry, which is far from the boundary. But just terminology in case you hear it. So now we're going to do some calculations. So it is complicated to study the dynamics of gravity, as you're all well aware. And so we're just going to study the dynamics in some detail for a scalar field in ADS. And I will need to do this so that you can see somewhat precisely how the ADS-CFT correspondence will work. And then probably in the next lecture, I will hopefully today, I'll state what the ADS-CFT, how it actually works, the correspondence. Perhaps it'll be the beginning of the next lecture. And then we'll discuss how it works in the case of gravity rather than just a scalar field in some detail, mainly because geometrically, that's probably the most interesting case, but physically not necessarily. So let's now talk about, so we're sort of moving towards ADS-CFT now. But in order to get there, we're going to consider a bulk scalar. By bulk scalar, I mean a scalar field living in ADS, it's going to obey the usual, it's a massive scalar, so to obey some wave equation. And again, we're going to take this Poincaré chart, when we think about its dynamics. We're going to think about its dynamics in this chart. And physically, imagine that you've got one of these simple, consistent truncations of some complicated theory. If you want a physical picture of this, where you had gravity, cosmological constant, and you managed to truncate everything away except one scalar field, which is entirely possible. And then you look at small excitations of the scalar. Now they will, of course, back-react on the geometry, so it won't be perfect ADS. But if the fluctuations in the scalar are small, the back-reaction will be small. OK, so this is physically relevant. It's not just some toy model. You're looking at fluctuations of a scalar about the vacuum geometry. Now it's just sort of simple calculations. Now there's an action, which you all know. There is a subtlety here, which I'll come back to later. So I probably shouldn't have mentioned it. So we have the usual action. And now we're going to consider, we're going to Fourier decompose our general solution. So our general solution will now be a function of the x. Let me suppress the indices. There will be a function of x and z. But will Fourier decompose and think of the general solution as a sum of modes, Fourier modes, in these directions, which are obviously translation. There's a natural isometry due to translations. And that naturally means we should Fourier decompose to simplify everything. So we still have non-trivial dependence on z. And then we can sum up all of these modes if we figure out for a given Fourier mode what f is. We'll have to solve some equation. We'll talk about that in a minute. We figure out for a single mode, well, we figure out for a general mode. And then we can add them all together with some coefficients to get the general solution as usual. So now let's focus on a single mode. Let's try and determine what this f is. So focus single mode. And now when we plug this into the wave equation, you'll find just a differential second order ODE for the non-trivial bit of the solution f. And if I write f as if I pull out a factor of z to the d over 2 and then write it in terms of some function of k times z, where now I should just be clear when I've written k, what I mean is this is a vector. Now if you substitute this in, I won't write it out, but h obeys a Bessel equation. So it's simple solutions. So the solutions of this are some powers of z times Bessel functions whose arguments are arguments of k and z. So you find this h of z, well, sorry, let me call it h of mu where mu is k z. H of mu is in general some sum of two Bessel functions where their parameter is given in terms of the mass of the field and the dimension. So it's a simple calculation to go through. Now we can then analyze. There are two sort of interesting regions of the spacetime. OK, yeah, go for it. This is ADS. Yes. Oh, this you mean? Yes, yes, sorry. This is the yes. Yes. Oh, yes, indeed it does. Yes, exactly. Yeah. Sorry, I'm not sure of. Sorry. Oh, thank you. Yes, OK, let me make it explicit. Yeah. In fact, I can be very, yeah, that's right. Thank you. And so by this I mean, thank you, sorry, that was too quick. Yeah. As I said, that's exactly what we're doing, yeah. So I mean you can ask just the problem of a scalar in ADS, but physically this is like small fluctuations of a scalar in a theory of gravity plus a scalar plus a cosmological constant. But the fluctuations are small enough that there is some back reaction, but it'll be quadratic in this field. So whilst it will affect the equation, it'll affect the motion of this field via the connection in here. It will only do it at some higher order in the amplitude. So we're getting the leading behavior, yeah. Yes, so that's right. So K could, depending on whether it's time-like or space-like, that's right. But if you want, let's just, we could just analytically continue time to imaginary time. This is now just the Euclidean metric, and then we don't have to worry if that's concerning you. But indeed, it could be imaginary, but not complex, either real or imaginary. OK, so let's see. So if you go through, think about the asymptotics of Bessel functions, you will find that z is 0 or mu is 0 is a regular singular point of the Bessel equation. There's some Frobenius expansion in the usual way. And putting that in here and rewriting, just defining, in fact, let me modify this just for now to make it slightly simpler. There's the leading, well, let me say, there's one power series that starts with this power and then has an expansion in powers that go up in z squared. And the other part of the power series solution has this leading term and then coefficients that go up in z squared from it. This A is this A, and this C is, in fact, some linear combination. So this is a linear combination of A and B. And just to be careful, you could have a situation for specific nu here where there's some log terms as usual for Frobenius expansions. But let's, for simplicity, consider the generic cases that there is no log case. So you can deal with the log case, but we're not going to. And then I'm just going to define this. I'm just going to redefine, instead of working with this eta, or sorry, nu rather, I'm going to define this to be d minus delta. And then this power becomes delta. So I've just defined delta to be d over 2 plus nu. And delta then, what is it? If you figure out just from here, think about what delta is. Delta solves this equation. Now, in principle, I could take any values for m here. This nu could be imaginary if I take sufficiently negative. If I took m squared to be, I'll take, if I think of m squared being real, not necessarily m. Because of course, actually, m doesn't enter anything. It's only m squared that enters the equations. I should think of m squared as being real. But it could be negative, in principle. And then if it wasn't sufficiently negative, I would get an imaginary eta, an imaginary delta. And I'm not going to allow that. I want to restrict my masses so that this is real. Sorry, that delta is real. And in fact, for simplicity, and you can do better than this, I'm going to assume, so for simplicity, I'm going to want delta to be greater than d over 2. Although, in fact, it turns out for well-poseness of the equation, you only require delta to be greater than d minus 2 over 2, greater than or equal. So that is a condition on m. So let's draw a graph of this. It has zeros at 0 and d and looks like this. This, I can depict as this curve. And so this point here is d over 2. So I'm going to assume that my mass squared is greater than this minimum value. So let me call this L squared mbf squared, where mbf squared L squared, if you work it out, is minus d squared over 4. Remember, again, just to emphasize, it's only m squared that enters the action, the equation. So it's only m squared that should be real. And I'm going to assume that delta is in this region here, which there is this restriction on the mass, but it's also a further restriction on delta, because I could have chosen delta to live. In fact, the minimum it could be is sort of here. This is this condition. And there is some bit here that one can deal with, but I'm not going to deal with it. It's more complicated. So let's just assume it lives here. That's fine for us. Note, you may be confused that the mass squared could be negative. That looks concerning. It's what we might call a tachyon in physics. Although it doesn't make the equation ill-posed, it's just an instability. But its instabilities of this nature, a negative mass squared, are instabilities on large scales, large wavelengths behave in an unstable way. Short wavelengths are still stable because they're dominated by the kinetic terms. The kinetic terms dominate this negative mass squared potential. And so because ADS acts like a box, it shelters us to some extent from long wavelength instabilities. And that's why in ADS, you're allowed to have negative mass squads as long as they're not too negative. If they go too negative, then you get complex solutions here. And then it turns out everything is ill-posed. But you can tolerate a certain tachyonic behavior in your field. This was all explored by Breitner and Friedman many years ago, hence BF. Right, so we can think of, so for simplicity, what we're saying essentially is that delta is the larger root of this equation. And because of that, and just to emphasize again, it is possible to have logs in this expansion, but they're not generic, so we're not going to consider them. You have to do slightly more fiddly things. But because of the fact we've taken delta to be the larger of these two roots, this gives the leading behavior of the field near the conformal boundary, and this part is subleading. Supposing you sold a wave equation in a box, a real box, what could you do near the boundary of the box? You would maybe Taylor expand or power series expand your field, but it would just be a regular Taylor series-like expansion. So it would just be a simple expansion. There would be a leading piece, which is constant, and a subleading piece, which was linear in the proper distance away, the normal distance from the box. So assuming the box had this translation symmetry in the wall direction, and you decompose that way. So you could think of this as the constant piece for a normal box, and this this, so what you might call the Dirichlet, if you were fixing Dirichlet boundary conditions, that's what you would fix. And this would be if you would like the Neumann bit of the behavior, if you wanted to fix Neumann boundary conditions. It's a bit more subtle here, whether you can fix Neumann boundary conditions, so we won't go into that. But we are in the end gonna be thinking of this as the part of the data that we're gonna fix in this problem. So as we said in ADS, this boundary really acts like a physical boundary in some sense. You know, high energy modes can get near to the boundary, arbitrarily near to the boundary. We have to fix some sort of boundary conditions there. What are we gonna do in the end? We're gonna want to fix this, the leading behavior. So it's not, again, it's not necessarily the value of the field because that will typically go to zero, or may go to zero, but we're gonna fix it, the coefficient of this leading behavior. That's the analog. Now, if we think about going back to real space, and not just looking at a single mode, but looking at some general sum over modes, what this is telling us is that the structure of the solution asymptotically looks like this. Where this is associated to this constant of integration, but this was a single mode. I just had a constant of integration here. But of course, when I sum over all modes, okay, when I sum over all modes with some orbit, maybe I shouldn't have called this A, so let me call it something else. No, I've used B, C, D, E, F, no, G, no, alpha. Let me call it alpha. If I choose some arbitrary alphas, I can turn the behavior, you know, choosing appropriate arbitrary alphas, I can arrange any phi naught as a function of the X coordinates that I like. Likewise, choosing any constants here, I can arrange any phi tilde D, as I've called it here, function of the X's that I like. So these are free functions of the X's corresponding just to this bit of data and this bit of data for a single mode. And everything else is determined in terms of these, and in particular, so all of these terms are determined by phi naught, and all of these terms are determined by actually both phi naught and phi tilde D. Remember, this is the sub-leading part, so these do in the end care about what's going on here as well as what this bit of data is doing. You can just plug this into the equation and explicitly see that this form is true. It's not enormously complicated. And in fact, these have a beautiful structure, these terms, the way these are determined by this is just by derivatives. So for example, phi two is some, roughly speaking, it's the Laplacian or the wave operator, the d-dimensional wave operator acting on phi zero. Phi four here, the next term, would involve four derivatives, a local expression involving four derivatives on phi zero and so on. And these, again, are just more and more derivatives acting on phi zero and phi D. Note something really trivial. Under our trivial, yet important for what will come, under our isometry, which from the point of view of the boundary generates scale transformations, but it's just an isometry. I mean, it's just an isometry of the interior. So this, from the point of view of the interior, I can think of as diffeomorphism. This is just a scalar field, so it doesn't care what coordinates I express it with. But because of this expansion in z, if I want to preserve the form of this expansion, it will induce, if I scale z, it will induce a change in finite and it will induce a change in all of these terms in a particular way. So in particular, if I want to preserve this form, there's then this natural action here from the point of view of this reduction to d-dimensional data. So this is something we've seen earlier when we were talking about conformal field theories. I should have said, I mean, I picked this letter delta. Again, it was the scale dimension of things in the past when we were talking about conformal field theory. Here it was just some rewriting of various coefficients that came up in the scalar equation in ADS. But there will be a reason that I've called it delta. It's not entirely without reason. Now, we've discussed the behavior in the boundary and I've even said that what we'll want to do is fix this leading behavior. It's like fixing a Dirichlet condition at this boundary. But we also have to think about what to do at this past Cauchy horizon. I said, generally, we want to fix some characteristic data there. And what we're going to ask is that the field is trivial there. There's no outgoing radiation from this past horizon. You can take other boundary conditions and that's fine and you can solve what happens. But we're just interested in the simplest situation. So what we're interested in, physically, is having no scalar field radiation or energy flux through this. Instead, we're going to fix the scalar field to do something on this boundary by basically fixing this to be some non-trivial function. So we're going to excite some scalar field dynamics by making this do something. And then that will induce some response in the solution. But we're not putting energy in initially. It will come, for example, maybe we'll choose this phi 0 to be 0 here and then not 0 here. And there'll be some radiation. There'll be some complicated dynamics. It may be trivial here, but there'll be some dynamics later. So let's try and do that. And when you go through the analysis, you see that this mode function for an individual mode, as z goes to infinity. So as we approach these Cauchy horizons, there's some definite behavior that comes from these Bessel functions. But if we require that we only have outgoing radiation here, nothing incoming, what that means is we have to kill this behavior. So we have to set B as 0. So we had this second-order differential equation. There's two constants of integration for every sort of mode. And this will be one of our boundary conditions. So physically nothing's coming into this past horizon. And then the other boundary condition will be the one that fixes what we'll do on this boundary here. So now following that through back into real space, that relates this data to this data in some way that you can compute. So in particular, thinking about F again, just to be somewhat explicit, I think it's something like this. The form here is really not terribly important. I'm just trying to convince you that it's something you can calculate. So instead of having an arbitrary coefficient here, this coefficient is now determined in terms of this coefficient. That's just putting this condition into what we had earlier. And then again, going back to real space, so Fourier transforming back to real space, or inverse Fourier transforming back to real space, you have to inverse Fourier transform this. And when you do that, that gives you the relation between this and this. And it tells you that phi d, what I've called phi d tilde, let me, I put a tilde on it here. Let me not put tilde's on it. Otherwise I'll forget them everywhere. I don't know why I did now, I think about it. When you inverse Fourier transform, this data is determined by this leading data integrated against some kernel, which looks like this. Again, for the physicists in the audience, there are issues about, you have to think a little bit carefully when you're in Lorentzian space about precisely how this is defined and so on, retarded, Green's functions advanced, and so on and so forth. Anyway, let's think from a Euclidean point of view where everything is very straightforward. But for the mathematicians, don't listen to that, please. Now, what we're now going to be interested in is the action evaluated on our solution. We've now solved this problem with the physical boundary conditions we want. We're now going to solve, well, we've solved it. We are now going to be interested in the action for reasons that will be apparent when we discuss ADSEFT. So the scalar action, and let's call it onshell, by which I mean evaluated on our solution. Onshell is a sort of physics term. So let's write our action. This is now an action in the d plus 1 dimensional space. Let's integrate by parts. Let's imagine we don't take the full space. Well, we need to think about what happens at the conformal boundary, and also what happens at these null surfaces here and here. So let's imagine just taking the action evaluated in some region, M, and then we'll try and take M to be the full space. So if I integrate by parts, the action, I get this. And because we're evaluating it on a solution, this is just the equation of motion. So this whole term drops out. So the action evaluated on a solution is just given for the scalar field by boundary term. So all we have to do is now evaluate this boundary term on our solution and see what we get. So let's define i of z to be this. In fact, I think I've defined it. So explicitly, this is just an integral over the d coordinates, the x coordinates, d z, phi, where phi is this function of x and y that we've computed. And now let's think about what it does at the various boundaries. Now, the first point is that the behavior of the scalar at z goes to infinity, we see here goes like z to the d on 2 minus 1 half. So it goes like z to the d minus 1 over 2. And if you plug that in here, that should vanish as you take z to infinity. So it turns out there's no contribution from these null surfaces to the action. So the action, in fact, will just be determined by what happens near the conformal boundary. And what happens there? Well, let's plug in our solution, which is this. Let me leave that up. As z goes to zero, i of z, if you plug things in, I mean, phi is given in terms of some power series expansion. And so this i, this boundary term that's going to determine the action, or in fact, it is just the action, but it, well, it's evaluated at some z. We're looking at how it depends on z. So this is at some specific value of z provided we're near enough to the boundary. So the asymptotic expansion is good. It will take this form. And then we see it's disastrous, complete disaster. And it's complete disaster because delta is greater than d over 2. And so this is negative, this is negative power of z. And we're interested in z, trying to take z towards the boundary, z is zero. It's a complete disaster. So you get a whole load of, depending on the dimension, there'll be some number of horribly divergent terms, and then there'll be finite terms. But these are all divergent. But there's something very interesting about the structure of how awful it is, which saves us. And the point is, the divergences, it turns out, come entirely from this part of the expansion, which doesn't care or doesn't know about this data here. And it doesn't, yes, so these divergences, whilst they're awful, they depend only on phi zero, not this other bit of data, phi d. So the basic idea, why not get rid of them? We're going to fix, we want to fix phi zero. It's like a Dirichlet boundary condition. We get to fix it. We know what it is. Okay, we get some divergences. But let's just subtract them off. We know before we even solve our equation what the divergences are going to look like. They don't depend on the solution, they just depend on the data. So we'll just remove them. And what we're then left with is some finite stuff, which does depend on the actual solution. And in particular, the boundary conditions we've imposed elsewhere on our equation. So let me just, in the remaining, let me have about two minutes, let me just say this and then we're pretty much done for today. So what we do, so we remove the divergences, how do we do it? We define what we call the renormalized action. The renormalized action is the limit, as epsilon goes to zero, of this action where we cut off our space at z is epsilon. So we're going to take m to be ads up to or down to, maybe I should say, z is epsilon. So we're not going to include right down to z is zero, we're just going to cut off the space. So it's got a boundary at z is epsilon. So we'll take this to be our usual action, or let's call it this i of epsilon. And then we add to that some, something we call counter terms, which are going to remove the divergences. We've got to fix our boundary condition to say what we're going to do, and we're going to fix our boundary condition in the following way. In this, in here, before we take the limit, we're going to fix our boundary condition by saying that the scalar field must behave like some fixed function times some appropriate power of epsilon. Okay, and this basically fixes in the limit, it fixes j to be phi zero. That's all we're saying. So we're basically fixing phi zero. But we don't, you know, if we want a good boundary condition, we don't want to talk about, we want to phrase it in terms of the field itself, not some asymptotic expansion. So phrase in terms of the field, we impose this in here. So this will be a functional of j, and then some unknown stuff. We evaluate this in this cutoff. And these terms here take the form of local functions. I realize, just see. I think I've, one of my pages has gone astray. The disaster, here it is. Apologies. So these are some terms with some specific coefficients. These are local functionals of, local and intrinsic functionals of the scalar field. Again, on the surface at constant z. So these will involve z derivatives. For example, they may, they may be higher. You know, there's some finite number of terms here. They may involve derivatives of z, sorry, of phi, but only in the intrinsic directions of this slice. Okay, so think of projecting the field onto the constant z slice, and then you can write down any sort of local functional of that field involving derivatives living within the z equals zero, the z equals constant slice. So this, and if you arrange these constants correctly, what they basically do is remove these terms. And you're left. There's just a finite number of them in any given dimension. You're left essentially with this bit. And, sorry, this is the very last thing I'm going to write down. And so in the end, when you, when you go through that procedure, you pick out just this one term, and the answer is then that this renormalized action, let me just call it this, doesn't really matter. Coefficient is, you could think of this as phi naught. I've just now called it J. This is the unknown bit of data. You know, basically you pick out this term. There's some minor subtlety when you actually go through the calculation, but pretty much you pick up that term. And given what we said, we can think of this. We know what this is in terms of phi naught or J equivalently, so I can write this like this. And that's where we're going to end today. And then we're going to do stuff with that tomorrow. I'm going to, I have got exactly enough to now tell you exactly how the, the ADSEFT correspondence works. How to map data from the CFT to the, to the gravity theory. But we've now, yes, so we've evaluated our scalar field. We've got a terribly divergent action, but it was divergent in a sort of trivial way that only depended on our boundary data. And there's a nice way to remove that data, just by adding some terms that live only on the boundary. So they're essentially extra boundary terms. They don't change the equations of motion. We've added some boundary terms that are constructed to remove the divergences, and we now get this finite action. That's called holographic renormalization. There are nice lectures on it you can find if you want more information. That's enough for now.


Are my stirrups even? Can you tell? Are they even? How would one even tell? My horse is standing crooked. You can't tell if stirrups are even if your horse's rib cage is off to one side or the other. If you have a long dressage whip, it's very easy to tell.  Make sure when you check if your stirrups are even, you're actually walking towards somebody because there is a swing. You can see my feet are dropping alternately as I walk, and you could even do a walk away where I walk away from that person. That would help. Another super easy way to check your stirrup leather length would be to use a rope. Any lead rope will do. You take it, you put it to the top, you bring it to the bottom, measure, keep your finger on the rope, go to the other side and check. This works for English and Western saddles, and then you will have a great idea of how far off your stirrup leathers are.


All right. So we've learned how we can construct affine sets in a vector space using a particular vector on the flat and using spanners. That is, you have some span, which is parallel to the flat in hand. But what if we don't know who the spanners are necessarily? What if we just know vectors on the plane? How can we describe the flat internally without needing these external spanners to describe it? Well, in order to do that, I'm going to introduce the notion of an affine combination, which is similar to the idea of a linear combination we talked about earlier, but it's a slightly different thing, but not by much. So by an affine combination, what mean is the following? We have a set of vectors x0, x1 up to xm, and they all belong inside of fn right here. And we say that x is an affine combination of the other x's. If x is equal to a0, x0, or a0 is a scalar plus a1, x1, which of course a1 is also a scalar, all the way up to am, xm, like so. So if I were just to stop there and we see nothing else on this page, you'd be like, well, that sounds like a linear combination to me. Well, you're right. The affine combination is actually a special type of linear combination where the coefficients involved in the linear combination have the property that they sum up to one. So a0 plus a1 plus a2 plus a3 all the way up to am always adds up to one. That's not something we require for general affine combination or for linear combinations, but it is true for affine combinations. And so then we define the affine span, the affine span, which will denote this as aff of the set of vectors. So very similar to the spans we talked about before, which we could call those linear spans to distinguish between affine spans. So the affine span will be the set of all affine combinations of the vector, much like the span, aka the linear span is the set of linear combinations of all of those vectors there. And so the affine span by definition is just the set of all affine combinations. So you take linear combinations of the vectors so that the coefficients add up to be one. So for example, we could say that a vector is one half the first vector v1 plus one half the second vector v2. That would be an affine combination. We could take like one fourth v1 plus one fourth v2 plus one half v3. That would be an example of an affine combination. But also the coefficients don't have to be positive. They just have to add up to one. So we could take, for example, two v1 minus v2. That would be an example of an affine combination because two minus one is equals one. That's what we're requiring here. So the affine span, what this has to do with our, with the flats we've been talking about, is the affine span is actually the smallest affine set. It's so in itself is an affine set, but it's the smallest affine set that'll contain those specific vectors x0, x1, x2, all the way to xm. And to try to convince you about that, consider the following. If we have an affine combination, then in all reality, we could rewrite a zero as one minus a1 minus a2 minus a3 minus a4 all the way up to am. Basically, if I know what a1 is, a2, am, if I know all the coefficients except for a0, I can actually compute what a0 is because all the other numbers have to add up to one. So if I subtract them from one, that'll give me the remainder, which is our a0 right here. So that's true for every affine combination. This is the main reason why we care about them right now. Now, hey, if I were to distribute the x0 onto all of these things, and then combine terms not by the vector, but by the scalar, this equation would transform into this one right here. x is equal to x0 plus a1, x1 minus x0, plus a2, x2 minus x0, all the way up to am times xm minus x0 right here. So this second equation right here, if we ignore the assumption on the coefficients, this right here looks like the equation of an m-flat, right? We have us the general vector x is equal to a particular vector plus a linear combination of all these other vectors right here. And so no, for the general flat, we don't require that the coefficients have to add up to one. We could relax that condition, but notice that this equation is the equation of an m-flat and this m-flat will contain each and every one of those vectors. So for example, if you set all of the a's to be zero, if you set all the a's to be zero, you'll just get x0. So this m-flat contains x0. If you set the first coefficient to be one and all the other ones to be zero, then you're going to end up with x1 minus x0. The x0's cancel, so you get x1. Similarly, I could set the first one to be zero, the second to be one, all the others to be zero. Again, the x0's will cancel and you'll be left with just an x2. So this m-flat will contain each and every one of those vectors, x0, x1, x2, all the way up to xm. So it contains all of the vectors we want. And if we then expand from affine combinations to linear combinations, we can produce a bunch of other things as well. And so basically the moral of the story here is that if we have a set of points, we can pick our favorite point to be the representative of the flat. And then if we take the difference of all the other points from our representative, if we take the difference of all of those, these right here then form the spanners. We can form the spanners of the flat by subtracting particular points on the flat there. So let's look at some examples of this. So let us find the vector equation and the associated parametric equations for a line in R3 where we have two points, 1, 2, 3, and 2, negative 2, 0. So we've often heard the phrase that two points determine a line. We are now going to actually prove that that's possible, at least in R3 right here. So these points are points on the line. So we have some x0 and we have x1. It doesn't matter whose 1 and whose 0, just make a choice. And I'm going to pick x0 to be to be the first 1, 1, 2, 3. Now to find the line, I need a single spanner. I need some spanner, which we're going to call V. V is going to be formed by taking x1 minus x0. So we're going to take 2, negative 2, 0 and subtract from it 1, 2, 3. And when we do that, we get 1, we get negative 4, and we get negative 3. This then gives us the slope vector of the line. And so therefore our equation of the line will look like x is equal to x0 plus Tv. More specifically, x looks like x1, x2, x3, because we're in R3. Plus this is equal to x0. x0, we decided it was going to be 1, 2, 3. But we could have interchanged the roles. And this could have been x0 if we wanted to. It's just a selection there. And then plus T times our directional spanner here, we got 1, negative 4, negative 3. And so this then gives us the vector equation. If we want to switch this over to the parametric equation, so we get an equation for each coordinate. I'm writing them as x1, x2, x3, but you could call them x1 and z. That's more typical in a calculus class. x1 equals 1 plus T. x2 equals 2 minus 4T. And then finally, x3 equals 3 minus 3T. And so that's all there is to it. If you give me the points on the line, I can find the spanners and then I can use my vector equation to come up with the parametric equations if I want them. Let's look at another example of such a thing right here. So in this example, what we're going to do is we're going to find an equation of a plane inside of R4. And to determine a plane, we don't need two, but we need three points. We need one more point than the dimension, the flat. You might have noticed when we talked about those affine combinations, how the vectors were 0, x1, all the way up to xm, and these were useful to find an m-flat. In order to find an m-flat, you need one more point than the flat itself. So we need m plus 1. That's why you started counting at 0. So it doesn't matter who is who. So we'll just go in the order that they provided to us. This will be our x0, this will be our x1, and this will be our x2. The preference, the ordering doesn't really matter too much, but then we have to create our spanners by subtracting these things. So our first spanner we'll create by taking x1 minus x0. So that will look like negative 13, 3, 25, and negative 2. Take away negative 17, 6, 29, and 0. And so simplifying that actually gives us the spanner of 4, negative 3, negative 4, and negative 2. So that's going to be our first spanner. The second spanner we're going to form by taking x2, take away x0, like so. So x2 was listed above. We get negative 15, 6, 25, and negative 1. We're then going to subtract from it x0, which is still negative 17, negative 17, 6, 29, and 0. And then taking the difference of those things right there, negative 15, take away negative 17, so positive 2. You probably don't need me to narrate the entire arithmetic here. We're going to get 2, 0, negative 4, and negative 1. And so those are going to be our spanners. Therefore, using these spanners right here, our vector equation in general will look like x equals x0 plus su plus tv. And so if we plug in specifically what those things mean, we're talking about a plane in r4. So our x is going to look like x1, x2, x3, x4, like so. x0, as we've been using it here, we had a negative 17, 6, 29, and 0. For our first spanner here, we can't see it on the screen anymore, but we had 4, negative 3, negative 4, and negative 2. And then for the second spanner, we can still see it 2, 0, negative 4, and negative 1 right there. That's our spanner. And this gives us the vector equation. Well, this vector equation is a system of linear equations. We actually, be careful though, when you talk about the system of equation in this situation, I want you to be aware it's actually a system of 6 variables for equations because you get an equation for each of the coordinates x1, x2, x3, x4. Those are variables because they could be different numbers. Now, by construction, we're going to have four dependent variables, x1, x2, x3, x4, will then be dependent on the three variables, s and t, because x1 is equal to negative 17 plus 4s plus 2t. x2 is equal to, x2 is going to equal 6 minus 3s plus 0t, so that disappears. x3 is going to equal 29 minus 4s minus 4t. And then lastly, x4 is going to equal 0. We don't need to include that negative 2s minus t, like so. And so this gives us the parametric equations for this plane. If we don't have the spanners for a flat, like a line, a plane, a hyperplane, whatever, we can construct them by taking the difference of points on the flat, and those will then produce the spanners because of what we were talking about earlier with those affine combinations.


When India chose the Republican form of government and universal franchise the prophets of doom described it as The biggest gamble in history. The finishing line is yet to come But at this moment we can surely pause to smell the roses. We can take a step back and say Looks like the gamble paid off What a humbling sight the privileged and the marginalized Standing in the same line waiting to cast their vote Candidates in India span a wide spectrum of backgrounds Castes religions and status the world calls it a miracle But India knows better it has been a tough and tumultuous journey riddled with Unimaginable complexities and challenges that may seem bewildering to the modern mind Multi-hued and lively in more ways than one Indian elections have had many amazing moments and well-defined milestones Indian elections entail organizing and supervising about 700 million voters in around 800,000 polling stations spread across widely varying geographic and climatic zones Polling stations are located in the towering Snow-clad Himalayan peaks the deserts of Rajasthan the vast Gangetic Plains the coastal regions further south and in the sparsely populated islands in the Indian Ocean The constitution makers of India had dared to dream They were visionaries who knew they had to create a workable mechanism that would reconcile the myriad and mind-boggling contradictions of the nation to create an awe-inspiring democracy Intense deliberation and the measured stoke of a pen created a neutral and independent body to conduct Manage and oversee the election process The Election Commission of India has an extraordinary international reputation In fact, I remember talking to the former Algerian Foreign Minister Laktab Rahimi when we were setting up the mechanisms in Iraq for the United Nations And his first thing is why are we going to any Western countries? The best election experts in the world are the Election Commission of India Let's go there That's the sort of reputation that the Indian Election Commission has acquired over the years and it's richly deserved The nation accords great respect to each vote and the Indian Election Commission goes to unimaginable lengths to record it in a unique experiment the Election Commission Introduced six mobile booths in the inaccessible areas of Bar Meir and Jaisalmeir in Rajasthan This was the first time that such booths would be criss-crossing the far-flung areas of the district The mobile booths were a huge success. In fact, in keeping with times Indian elections have gone completely high-tech 2004 saw the creative and pioneering use of GIS technology for electoral management in the troubled Naxalite area of Orissa. In the 2004 General Elections, there were 1351 candidates from six national parties 801 candidates from 36 state parties 898 candidates from officially registered parties and 2385 independent candidates The 2004 General Elections stand out in the history of election management in India, if not the whole world Indians saw the country-wide use of electronic voting machines for the first time 1.07 million EVMs were used in 700,000 polling stations Most election results were announced in less than a day The media are encouraged and provided with facilities to cover the election In fact, the media is the biggest watchdog and ally that the election commission has The history of India shows ample evidence of a deeply rooted democracy Today that route has blossomed and is here to spread its fragrance The mainstay of this democratic journey from fledgling to skilled lies in India's undying commitment to the electoral process


Well in Minnesota, 21 blood drives were canceled or shortened in March for bad weather and now the Red Cross is in urgent need of donations. Our A.J. Feldman was in Pequot Lakes today as people rolled up their sleeves and donated. Pequot Lakes held a blood drive today as members of the community gave back by giving their pint. Pequot Lakes has been doing the blood drives for quite a long time now and it's always the Pequot Lakes student council that's in charge setting it up and finding a location. Winter weather hitting the entire country has canceled numerous blood drives making now an important time to donate blood. And then the other thing that we have to look at too is in the early like the February's and January's or the winter months is the flu's and the cold seasons and the flu's taken quite a get this year too so that also does not help our supply as far as getting caught up. Over 50 people gave blood today in Pequot Lakes as a steady flow of people filled the trail side center all day long. There's just kind of a big need for it. We have a pretty common blood type so they kind of need as much they can get. Not only is it a good cause it can also provide for some healthy competition. We race to see who's the fastest blood letter. I won. She did win today but I won in the past. People gave blood for numerous reasons whether they feel it's just the right thing to do or they have a personal reason to donate. I like to give blood because it does help a lot of people and it really hits home for me. My father has leukemia and is getting blood transfusions quite often and so I'd like to replenish that system you know of blood for other people because it benefits my dad quite a bit. Simpson says she's donated throughout her life and now it's just that much more important for her to give back. This has just really kind of hit home for me. I've been donating blood since I was 16. It's something that I've done for a long time because I'm a universal donor. When I was in college I was in a sorority and we worked for the Red Cross doing running blood donations just like this and so I've just done this all my life. The Red Cross appreciates any donation and encourages people to give blood whenever they can. It's a good cause to come out and give. It doesn't cost you anything other than a little bit of time and we're always very appreciative of it. Reporting in Pequot Lakes, A.J. Feldman, Lakeland News. Blood drives are being held throughout the month of April and you can find your nearest location at redcrossblood.org.


A cold Melbourne morning wasn't enough to stop more than two and a half thousand people from becoming Australian citizens at Victoria's largest ever citizenship ceremony at the Melbourne Cricket Ground. It took 60 DIAC staff more than six months to plan and prepare for the mega ceremony. Forty staff were present on the day to ensure proceedings ran as smoothly as possible. We have broken up the event today in two stages. Stage one is the registration which started from about eight thirty in the morning. Our staff were here from seven o'clock setting up. Then at two o'clock we expect around two thousand six hundred and forty eight conferees to come through plus an additional six thousand guests that will come at the same time. Among the thousands of people celebrating was English born Dr Paul Slade. After moving to Melbourne in 2006 Dr Slade is delighted to share the experience of receiving Australian citizenship with his family and friends. Today's really important and exciting day for myself, for my wife, for the daughters. After a couple of friends we've got who are getting citizenship on the same day. It's a world famous stadium. It's been fantastic to watch sport here but actually to have such an important event in our lives taking place at the MCG is just a fantastic experience. Raul Espina his wife Gina and their three children also came to Australia in 2006. Their family is looking forward to creating opportunities for others now they've become Australians officially. I want to have my own business so I can help other migrants to be employed here in Australia. Congratulations, you are now Australian citizens. Nick Hatzeglu said he was thrilled the AFL was able to take part in such a significant event. I'd like to thank the Department of Immigration and Citizenship for partnering up with us.