I was dancing a lot last night at karaoke, and I was snapping my fingers a lot, and I have a blister. These are some good developer hands, nice and soft. Alright, we'll get started in a couple minutes. A lot of you are here. More time for questions. Let's do this. Alright, so today we're going to talk about WebVR. What is it? Why might you do it, and how you can do it right now with a lovely little language called A-Frame. Real quickly about us, my name is Wes Ravocaba. I'm a senior front-end developer at Lullabot. I got my BFA in illustration, so I'm coming at this more of a user experience design, creative asset kind of perspective. My partner here, David. Hey, I'm David Burns. I'm a technical project manager, but also a VR enthusiast. I'll be doing the demo at the end. We're proud owners of many VR devices. We have a few out here. We both own Vibes at home, and then you have two... I don't have three. Oh, three. Yeah, no. Great. So two Samsung Gear VRs, and then some...we don't know how many Google Cardboards, and then I also own a Google Daydream. So if we talk smack about one of the other ones we don't own, you know why. So we work for Lullabot. We do strategy, design, and development. We're great. You could work with us and see this wonderful mug every morning at Scrum, so something to think about. So today we're going to go over the VR landscape, kind of what's going on right now. What's the terminology? Get our term straight, and what the user experiences are, because we're not dealing with a keyboard and mouse anymore. What's the hardware that's around? Because that really drives what you're able to do, and what people are doing with VR. Then we'll get into actual web VR and A-frame, the wonderful language you can use today. So that's what we're doing. And then a wonderful demo, man with the VR plans, doing awesome stuff. So terminology first, right? What are we talking about? So we have virtual reality. Virtual reality is when the device or the experience you're in is trying to completely envelop you as much as possible. They don't want you to know that the real world is around, right? Then we have, oh, so here's some examples of virtual reality. So this is a small girl in an HTC Vive, and she's playing with a robot dog, as one does, in the hills of somewhere in Scandinavia.  She feels like she's actually there trying to pet a robot dog, which is awesome. Here's another one. There's a guy trying to defend a castle. Again, HTC Vive, he has his head set on, and you can see, you can kind of see the controllers, but to him he's seeing a bow in this hand, and he's getting arrows, and he's trying to chase away all these bathroom icon looking people from breaking down doors. So again, nothing in there is real, right? It's a cartoony, weird universe. So then we have augmented reality. Augmented reality is when you can see the real world and there are virtual elements introduced into it. So, real good popular example, Pokemon Go.  One of my favorite implementations of augmented reality is car companies made it, I think it's Hyundai's one of the ones where if you get their app, point your device at the hood of the car, it'll actually have little tasks you can do. So change the air filter, you click on change the air filter, and there'll be a little 3D video that plays right where the air filter is to say, hey, lift this up, do this thing, the other thing, tada, air filter. So next thing you really have to think about, this isn't, although we're doing this with web technologies, the user experience and interaction types are completely different, and we don't want to make people sick and we want to offer something that's really cool and fun. So, some of the biggest things that drive that are going to be the controls and inputs you have available. So this is going to kind of depend on what kind of device you have, what class of device and what the hardware is. So, the first interaction you can have, this is kind of like having a mouse, so you have a reticle, you can see there's a circle over my face in that little image there, that'll show up when you're in the viewer, and that's kind of where your mouse point is, right? And so a gaze is kind of like a hover with a mouse, you just stare at it for a while, and some things will have an indicator where like a button will fill up with a color or something like that to kind of say, hey, you need to stare at this for a while until it activates. The other thing you can have is you can look directly at something and then press a button. Every Google Cardboard device or any other of the cooler kid devices all have some sort of click or thing like that. So this one has a little button right here which in the background is pushing a little capacitive piece of cloth onto your screen. You could also have traditional controllers, so like my daydream comes with a little media looking controller guy. There's also video game controllers are pretty common, or you might have a specialty controller talking about like a steering wheel or a flight stick or something along those lines. VR controllers, so that one's kind of a VR, that is a VR controller. It's not quite as cool as these two, but that's when you can actually see the controller in the 3D world and actually point at things.  And the goal of these things is to be as much like a hand as possible. So the Oculus ones for example, they have a ring around them so if you let go like that, the ring catches on your fingers and they know that your hand is open because on the inside of the ring there's a touch sensor. Very cool stuff. So next thing that's really important is body interaction, right? So, well you don't know that, but you're about to. So they're sitting and standing, sitting and standing when you have a device that can't track you in real space. So that means that you can do this, you can look around with your head, but if I am like this and I walk, my body says that I'm moving, but my eyes will say that I'm standing still, which is a good way to get motion sick.  The other thing that you can have is room scale tracking is what it's called right now with the HTC Vive and also Oculus can do it. It's like an extra thing you have to go through. So basically that means that they actually track you. So if I'm in a 3D world and there's a little robot dog here, I go like that and I'm now closer to the robot dog. It's getting bigger in my view. I can actually, it feels like I'm walking around in the virtual world that I'm being immersed in. So, the hardware. Low-end headsets I'm categorizing like below the $100 range. So, these guys are about 20 bucks. If you pay much more than that, they better do something really cool. Also these guys, which are a good step up, quite a good step up from those, but these are closer to the $100 range. So, they're all limited by phone hardware for the most part. This one has a little bit of extra help with the phone hardware. This device requires phones that have extra hardware in them. Yeah, so they're all sitting or standing only.  But then we have the higher-end headsets. So, this is, these are $500 and $800. And on top of that, you need to have a computer that's powering the thing that's probably going to cost $800 or more. So, these are the adoption rate for last year. This sold about 5 million units. So, that's significant. And then the Oculus has sold this year. And I don't have figures on the Vive. And the cardboard, God knows, they're everywhere. You can get free ones here somewhere. And so these all come with VR controllers now. And they provide the highest fidelity experience as far as like superior tracking, superior graphics, all that fun stuff. But, like I said, not as many of these out in the world. So, what are the applications for VR? What can you do with these? So, there's gaming and photos and video. That's kind of, it sounds boring, but actually, I take photos in Iceland while we are there. And they're 360 photos. And when you slam a little daydream on and look around and you're in Iceland all of a sudden, it's actually really cool. It's much better than just looking at a photo on the phone. You can also demonstrate macro and micro scale things. Macro being, we're in the planets and looking around at Saturn. And, you know, maybe we're as big as Saturn or not. And micro being, you're doing the magic school bus thing where you're traveling through blood vessels and you're really tiny. So, you can demonstrate things like that. You can go to our museums just in locations, event reenactment, things that happened a long time ago that, you know, we can take guesses as to what actually happened. Another big use is therapy for PTSD and phobias.  But you might think there are spiders all around you. And that is exposure therapy to kind of help you get over what you are afraid of. And then also brain injuries. They found that stroke victims they're able to focus what they're training their brain on better when they control the environment more. There's also training simulation you can think of driver's head, flight school, all these kinds of things could be really well done with VR. So, WebVR, what we're here to talk about, right? So, there's an effort that started a few years ago MOSVR by Mozilla. And what their goals are is to make a WebVR reality and to get it into the W3C specification. So, all browsers will have an API you can talk to so you can actually speak to the hardware in the device. So, you can speak directly to the controller, you can speak directly to the headset with your JavaScript. And so, the other goal of the device is not only to look forward and getting those natively in the browser, it's also to get WebVR in a state where you can use it today, which you can. So, the first big thing that came out with is WebVR boilerplate. It's just kind of a repo that has everything you need. 3JS is a big part of it and other libraries that help you do the different VR things. This is more developer focused. You want a very customized experience that's still on the web. You'll want to probably start with this. Very developer centric has, you know, your package.json file and all the shenanigans that comes along with that. A-Frame is built on top of WebVR boilerplate, but what it is has a simple looking markup language. Looks like HTML. The only difference is all the tags start with an A and they have completely different attributes. They do. But still, very approachable. So, easy to learn. You can sit down in afternoon, you can hack on it on CodePen. It can be further modified with JavaScript. So, if you want to do something that's more customized than what A-Frame handles out of the box, you can actually either extend it or do anything you would do with WebVR boilerplate since they're related. And then the project goal is to enable more creators. So, JavaScript engineers shouldn't be the only ones hacking on this. They have great ideas, but maybe the art student who kind of knows HTML also has great ideas. So, now I'll hand it off to my good friend David Burns for an A-Frame tutorial. All right. As Wes already mentioned, A-Frame is a markup language, very similar to XML or HTML, which makes it extremely user-friendly. And that's what we're going to be going over today. Good news, the documentation for A-Frame is fantastic. And I'll jump over to that when I get into the demo so you can see what I'm talking about. To build it out to get started, you just need to include one JavaScript file, which is awesome. And there's no need to compile as you make changes. You just refresh the browser and you can see what's happening. So, to get started, as I mentioned, you just add a single script, and it's called A-Frame.js.  So, with that being said, let's go into the demo. In this demo we're going to just build a basic landscape and throw around some objects and animate them a bit without any JavaScript at all. All right. You guys can see my browser. So, as I mentioned before, we have a regular HTML page. We're inserting the script, which is local, and we're building out a scene. There's really not much to the scene right now. It's just going to be a white page, nothing to look at. So, first thing we're going to do is add a sky and a plane for a ground. And as Wes mentioned before, all A-Frame markup begins with A-something else. So, we're going to add a green plane. We're going to give it a width of 500 and a height of 500. And that sounds weird. And yeah, the 500 is in meters. All numbers in A-Frame are done in meters because science. Thank you. So, you ask why would you give height to a plane? Because it would be really tall. And that's true. So, one of the things we have to do to a ground plane is to rotate it on one of its axes. If you ever worked with a 3D modeler or something, you know that X axis is flat, Y is vertical, and Z is depth. So, let's go take a look and see what this looks like right now. There we go. An amazing green plane with a blue sky. Very little code. All right. But let's do something because that's kind of boring. We're going to add a basic primitive. Other primitives are like boxes or spheres, any shapes you could think of. And the thing we're going to add is a gray box. And we're going to position it about 2 meters to the left of us, 1 meter off the ground, and about 4 meters in front of us. There it is. We got a gray box. And even that's kind of boring, but didn't take a lot of code to do. So, we're going to start modifying some of the attributes about this box to make it more interesting. And this is where the documentation comes in. Aframe.io has amazing documentation for all these different components that you could add to the scene. On the left side you can see we have many components including modifying the camera, the cursor which is that red circle that Wes was talking about allows you to point at stuff. You could add basic shapes like the box, a cone, a curved image, a cylinder, so on and so forth. But if you click on one of these for example the box and you scroll down, you could see Aframe provides a whole bunch of attributes that you could modify directly in HTML or Aframe. One of the things we're going to do is add another box to the scene. This one's going to be blue. And we're going to position it a little more to the right and about the same depth in front of us. We're not going to animate it just yet, so let's go see what this looks like. There it is. We're in a scene. There's a gray box, there's a blue box. Still kind of boring. So let's start adding animations. And you can see this markup is nesting an animation within the box markup itself. If you were to tab this I'm sorry, if you didn't put the animation outside of this element, the animation wouldn't take effect at all. The first thing that we're going to do is to put a little bit of rotation on this box. You see here that we're setting the attribute rotation, tells it what we're going to change. The from is the starting position of where it is. And the to is where that animation ends. So we're going to spin it 360 degrees on its x-axis and its y-axis. And we have another attribute that says we're going to repeat this indefinitely. Let's go see what that looks like. There we go. We got a spinning box. Now this is pretty cool. You can nest these animations back to back. So let's do more than just put a rotation on it. Let's go ahead and change its position in space. And what we're going to do here is we're going to delay and wait two seconds for this animation to begin. The whole duration of the animation is taking 3,000 milliseconds. And we're starting from its original position and then we're going to throw it more to the right. Keep it one meter off the ground, but send it 10 meters back into space. One, two, boom. It's moving. This is just markup. This is pretty cool. I'm not going to bore you with many more animations, but let's just throw two more things on it and see what happens. We're going to change the color from blue to yellow and blue to yellow and back and forth. We're also going to change its opacity, just showing you how easy it is to modify an attribute of an element in A-frame. Let's see what this looks like. So there it is. It starts changing color. We send it back in space, but you notice it didn't return back to us because that's the one animation we chose not to put on a loop. Let's see what else we have to work with. So A-frame provides us the ability to pull in assets. Assets can be images, videos, even 3D models. And you declare them within A assets markup tag. You can see here we're using HTML5 tags and we're pulling in a sky image and we're giving in an ID of sky. Now that's going to be really important to remember when we apply it to the scene. So our initial sky was kind of boring because it was just blue. Let's go ahead and comment that out and do something kind of neat here. Let's go ahead and use a sky image we brought in as an asset and apply it to the scene. Oh yeah, thanks Wes. So before we do that let me show you what it looks like in a 2D space. If you all have your own camera and you download Google Cardboard camera you can take a 360 space of this and it would kind of look like this image here when you upload it to the web. What A-frame is going to do is use the metadata stored in this image to wrap it in a giant sphere around your scene. I'm going to comment out the ground too because the image looks way cooler without it. There we are. We have two boxes inside of a building. And you can see here if we had our cardboard headset on and we started looking around it feels kind of like we're here.  Alright, so I'm just going to show you a couple more of the assets that we brought in.  Let's go ahead and refresh the scene and here we go. We got our giant lullabot in the scene with us. It's also worth pointing out that Wes was saying you can be tracked in 3D space. For these you would probably want to lock down the forward and back movement just so you don't get sick. But if you have this hooked up to like an HTC Vive, the ability to move around in space is baked into A-frame and you didn't have to add any extra tags to do that. So it's really cool, really user friendly. And this is something that's super duper cool for me because I built a whole little demo that I'll show you after this. Adding video to a scene. It's so simple. There's a video tag. We're going to use this source from a mp4 video of Lullabot's MSNBC project. So I'm going to refresh. There's that. There's the video loaded. That was quick. Cool. We could get closer. Screen gets bigger obviously because we're moving closer to it in space. But it was really simple to add video to a 3D environment. So now that you have these basic knowledge of A-frame, it's just hacking it together to do something kind of cool. And when YouTube came out with its Daydream app, I looked at it and I'm like, I could probably build this. So over the course of a weekend with a lot of coffee, I built this scene that allows us to go out to YouTube and fetch the DrupalizeMe feed. It grabs thumbnails. It grabs the mp4 video. We could even navigate up and down and browse to a different video. And we have player actions of pause, play, skip video, which will probably take forever on this Wi-Fi. You could even do cool stuff like to reduce the attention on your playlist to fade the opacity after a certain amount of time. So that's pretty much it. A-frame is super easy. There's one really cool thing. So when we built the D-frame photo for Lollabot last year's Drupalcon, we had to sit there for hours and hours, a meticulously smooth place, like everybody's image. Thirty hours of my life were spent making a 3D image map. It was really sad. But as of like four months ago, they created this amazing inspector that allows you to pull up a scene and like select elements. And when you select them, it pulls up all the properties that could be modified. Living in the future. It's pretty cool. It's like having a 3D modeler in your own browser. It's pretty cool. So that's A-frame for you. I hope you go explore it a bit more. I'd love to see what... A-frame is built into the A-frame JavaScript. Yeah, it's JavaScript. So you could build, Drupal could spit out any sort of markup. And there is an A-frame module that's available. You could like upload a video or an image and say, I want this to appear in 3D space. But yeah, there's a module for it. There's always a module for it. We're gonna pass it back to Wes. So, what we've gone over. VR landscape, we went over the terminology. Augmented reality is starting to be a thing. But we didn't really have the time to work that in 25 minutes. And then, yeah, talked about the user experiences. How the kind of device you have drives the kind of experience you can offer. And, yeah, WebVR, A-frame, go check it out. It is really, really cool. And like I said, if you go to the A-frame website, when it has a try it out, it has a code pen all set up for you. And in a couple basic shapes you can mess around with and add your own. It's like a meg of JavaScript. So that's not amazing. But considering what you get, holy cow. So, yeah, that's about it. So for resources, go to A-frame.io and especially slash community. It has a lot of the stuff that's going on right now. Most VR's being so open, inclusive, and cool about all this go open source. And, yeah, please give us feedback. That's just a Google link to our session page. And you can let us know how we did. And, yeah, we have some time for questions. Let's repeat the question. Is A-frame available on all Web browsers now? So if you go to WebVR.info there's a breakdown of all the browsers that are supported. Unfortunately the one that's really not staying up to date is Safari's WebKit. And I'm going to just take a while guess it and say they didn't include it yet because they don't have hardware to sell along with it. So I think they're working on something and I expect to see it soon. It works really well on all Androids, especially in Firefox and Chrome right now. Yep. So the question was the 3D Lullabot model. What was that? How would you do that? So that was built in Blender. That's a regular DAE file. Yep.  Yeah, so there is recognition for when you're either on a phone that could support one of these devices. Oh, so the question was the goggles on bottom corner, what's up with that? So yeah, it'll detect whether or not your device has some VR capabilities, especially if it's a phone, that's what this is mostly geared for. But they've been doing amazing stuff with the Vive. If you've seen the Vive's paint, the Google painter thing, they made that A-frame. They just call it A-painter. And it works. And it works on Oculus Touch now too. So actually the WebGL stuff was in browsers about six years ago, if not more. And to do the VR stuff that's available now is a really small subset of like data that's needed. You need a position of your camera. You need stereoscopic visioning. And then any other interactions that a user would have. I'm sure you could even use the controllers without the headsets. Yeah, I don't see why not. The question was, is this maybe integrated Google Tango? So here's another cool thing about A-frame. You can make your own modules. So somebody wrote a little, I don't know if it's called module, but in our parlance modules, where you have a Vive module that supports the things. There's a lot of them. And I'm sure that somebody has been thinking on that. Yeah, especially for the hand controls, you could go in and there's a component called Vive controls. There's also Oculus Touch controls. But then they have a generic wrap around that called just A-hand controls. That'll just pick up any sort of device. So you don't have to be too specific or write special code to pick up each own hand control set. Frank? Yeah, we have this. So we almost had some AR at our booth. We had to get kind of a QR code looking thing to be our tracker. And then you could point your Android only phone and see it.  It kind of looks like a QR code. I could hold it here and you would see the 3D Lullabot model spinning in 360. And it uses all the same technology that's made available through A-frame. It uses A-frame. Yep. Any other questions? Frank? There's one where you punch these big blobs, like orange and purple blobs that are coming at you with music. One of them is called Soundbeat Audio Shield. Thank you. It gives you a work out. You can be there in 10 minutes and like sweating like crazy. Still not helping the gut yet. Yeah. I mean it kind of helps mine because I can't drink whiskey while I have the goggles on. I'd say mine's probably the bow and arrow Vive one. I'd love it. I mean it comes with the thing. But that one specifically I really love. I was trying to get it to load in stereoscopic. You need to go to the inspector and set your browser to be like Firefox Mobile. And when you hit the camera button you could see what it would look like on your phone. And you can go to vr.lullabot.com to get a link to this and the team photo thing that we've kind of shown a little bit and inspect the code. It's all really basic fun stuff. Hack it. You can open up Chrome Inspector tools the second you hack any of the properties that'll just respond immediately like it should. We have a couple minutes. I want to show you one cool thing that West did. So when we were building this experience our frame rates were dropping pretty low. And in VR you don't want to drop below 60 frames per second. So a sphere has all these different tangents and points that need to be rendered and calculated. Chris goes let's just build a giant box and a sphere. So if we move out of this scene you'll see that it's just a bunch of planes flat surfaces and it worked really well. It took our frame rates or what was it called something from like 3000 rendered surfaces down to like 1200 vertices. Man you're good. We just hack this stuff like we did with Drupal. We don't really know what we're doing at all. Personal joy. We do not get to do this for work. That was the question. But if you have an idea that you want done in VR get a hold of Lullabot and I would love to build you something. Cool. Well thank you and I invite you all to Lullabot's party tonight. Lullabot Pantheon party. We're going to go hang out at our booth right now and we can chat more. We can show you some of our toys and whatever. Feel free to come over and hang out right now and we'll do that.


Today, on the breakfast, many passengers feared injured while several others kidnapped as a train-conveined passengers from Abu-Jet to Kaduna was attacked by gunmen suspected to be bandits in Kaduna state. Vice-President Jamio Shiba Joe and former President Delusio Robson Joe hopped on the need to strengthen democracy in Nigeria and Africa just what are the roles of the citizenry ahead of the 2023 elections. And as always, we're delving to the front pages of our national dailies with an analyst. Good morning. Many thanks for joining us on the breakfast and plus TV Africa. I am Justin Acadone. And I am Mesia Boko. Beautiful morning to you and thanks for joining us. Yes, it is a beautiful morning but yesterday late in the evening Nigerians were greeted with a gruesome attack along the Kaduna Abu-Jet rail track. We understand mercy that a band that attacked the train on the route of Abu-Jet and that many have been feared and kidnapped and it is really a sad situation, Mesia. Yeah, well it's very sad and that's because just yesterday we also talked about the fact that an airport was attacked in Kaduna. It's in Kaduna state. So it feels like Kaduna is, you know, in the news and a lot is actually going. But you know how the body language of the government, the state and at the federal level has not really shown any concern as regards tackling this. You could see some of the pictures, you know, from that sad incident that happened that would probably be from, you know, bullets or forceful entry to it. As we hear, this is not the first time this has happened where you have this explosive on the train track and then of course it derails the train and then they have the opportunity, you know, to go ahead and cost this to him. Very, very sad. It calls for a lot of concern and worry. And I remember the time where, you know, there was some sort of inspection in League of South last year and I happened to be around, you know, for that inspection. And I did mention, you know, to those questions that were being popped up, which, you know, I'm sure that no one paid attention to because of the imminent threats that we witness, not everybody witness. It was just a minute threat, right? It was just insignificant. I'm not sure that, you know, anyone considered that. But, you know, that's where my question came up.  I put out this question and I asked, what are the measures to, what measures have been put to ensure that, you know, the train, you know, the trainway or the railway entirely, the entire system is protected, right? Because if you find out, these trains are not, the train tracks are not in commercial areas, they're in isolated places, right? And so you don't expect, even if that might be the case, well, you can't even have it, it's not even possible. So it has to be in isolation. But the question that was put out was that, yes, as always, we would always have a question to answer, which really didn't make any sense because at the time, when we were moving, I mean, the train was moving from Lagos, you know, trying to navigate it, but even though we didn't eventually get it, but you saw young lads, children across, throwing stones at the train, throwing stones and they were throwing stones. I think what actually, it's not like you stopped them. To me, that was a threat and I'm like, why would this be happening? And so the question came, but of course, there would always be a ready and made answer for any question and not paying attention to it. So it's really sad, very sad, very unfortunate, but it just shows that we have not really taken seriously the issue of security, these bandits. Just as on the 26th, you had the attack in Keduna and now you're having this. This is not the first time this is happening. And what would have thought of putting proactive measures ever since the first time? So because if you applied through, I mean, if you've used the train, you find that that is a lonely, so we should we be expecting some patrol?  But I really don't know. It leaves me very speechless. It leaves me really sad. It leaves me. There's a lot of mixed feeling regarding this because I feel like we do not. We're not interested in this because as long as you have a wheel, there will be a way. There is no wheel here. I've not seen any wheel. There's no wheel. There's no wheel from the federal government. There's no wheel from the state government. There's no wheel. So we're all fingers across. And this is really sad because security is top on the list of every government. It is your responsibility to ensure that lives and properties are protected. People pay their taxes. Taxpayers are paying taxes. And what do they get in turn? We need to understand the origin of government. Government did not come through as all people wanted government. It's as a result of a social contract. And some people said, we will hand over our rights to you. And in turn, you will do XYZ for us. So we will obey the laws. We will pay taxes. And in turn, we expect you to do XYZ. And that's how government came through. But you see, the contract has been broken. So people are paying their taxes. But government is not keeping to the other side of the bargain. And we don't care. Of course, we're closer to the election year. What are we concerned about? Getting the political parties, I mean, trying to sort out who becomes the next governor, who becomes the next president. And these are issues that are top on the list, the issue of governance. And we're not paying attention to it. Yeah, indeed. The Cardinal State Government was also quoted as saying that they were able to put the situation, check the situation as it were. And then there was some military intervention. But my question right now will be, that was more reactionary than being proactive. In as much as the military came out to do what they could have done, I feel that is not really enough. Cardinal has been like a hotbed for lots of issues in recent times. Like you rightly said, mercy, two days, three days ago, over the weekend, the airport was attacked. One would have thought that security would have been put on high alert. The airport was attacked, now the railroad track. So what kind of signals are these bandits actually sending out there? They're trying to tell governments that your airport is not safe, your railroad tracks are not safe, your roads have never been safe over time. And they're trying to say that they are in charge. We need the Cardinal State Government to come out and give some sort of explanation to the cities. They need to really know if they can actually trust government to protect their lives and property, because this is just one attack too many. Cardinal, a railroad track has been attacked even before now. What measures were put in place to ensure that, to forestall a recurrence? So how come we are having a similar attack and not too long after? These are the questions that government need to be answering also. But let's just move away from them. But just before we move away, I mean really sorry to bring us back here. It's at the point where you have the Minister for Culture, I mean the Minister for Information and Culture, Lai Mohamed saying that Niger is safer now than ever.  We have not prioritized, it's very saddening. You know why it's saddening? Because it has not happened to you, you have not been in that situation. You know those who have lost their life in the course of all of this, those who have lost their lives, not because they would have died, but because they're living in society, they live in a country where it's not about this business. We're thinking about political parties are having their conventions and then they are selecting who becomes the next governor and president. Nobody cares about that. Nobody is saying it's not about this business. And you had talked about being proactive, but we're never proactive, we're reactive. We wait for things to happen before we send the army there. But this is not the first time this is happening. We have seen that these guys are increasing in their tactics and one would think that we would double our effort as a people, but we haven't shown all of that strength. And it's sad, very sad. And what a time to be a Nigerian. Well, I just hope that this time around we'll begin to do the right things because Nigerians have died, Nigerians have been kidnapped. We cannot continue to believe in refugees in our own land. Well, I haven't said that. I don't want to get too emotional this morning. Also making the rounds all through yesterday, Will Smith, Jada Pinkett and Chris Rock were in the news. If you just search Google, Twitter, I'm sure one of the top searches you would see will be Will Smith, Jada Pinkett, Alopecia, Hellos, Bordeness, and the Oscars and all of that. A lot of people have come out to speak for and against Will Smith over what happened at the Oscars. Some have come out to say that it is good for a man to stand up for the family, to stand up for his woman. A lot of people have said that fighting as much as he is speaking for or standing for his family, he should have kept it together somehow, should have been able to just hold it all back without having to heat and Chris Rock, who looked all short. Some people have even asked if the host of what staged. Chris Rock looked like an excellent dramatist in all of that. We thought about it and Will Smith was seeing there, seeing some expletives and how Chris Rock should have kept his wife out of the host of, as you can see, he was all tiered up when he was given his acceptance speech for the award that he won for King Richard. So, I mean, it's gotten a lot of people talking, not necessarily just Nigerians, the entire, everyone seemed to have an opinion about this particular situation. And a lot of persons have expected that you would have, Will Smith should have acted differently, but you can't actually control how people would react most times, especially when the emotions are involved at this point. It was a little bit overboard and one would expect that he should have taken it and just swallow it and probably just go behind the scenes. But until you're faced with that situation, you never can tell how much you react. On the other hand, he's apologized. I mean, he met an apology while he was accepting the speech but he's apologized again on Twitter and he's apologized to Chris for his behavior, his condemn the fact that the ultimate role is to love kindness and there's no violence. Violence is not accepted anywhere because that was a violent act and it's very condemnable, really. But you also want to look at it on the other hand, the issue of provocation. And I remember the part of Bible Word, Nigerian parents always quote that, parents, you have to be your children, you have to be your parents and all of that so you may live long, but I also forget the other part that parents don't provoke your children to hang in and all of that. That was it. Well, it can be very emotional, especially the fact that you have talked about the hashtag, alopecia and in 2018, Jedd Smith actually talked about, she came out to talk about the condition that she's faced where there's a condition that makes you begin to lose your hair and that's the reason why she took off her hair, really.  If you listen to the joke by Chris Rock, now, so the G.I. Jane joke was really a joke about a movie action. So it was a movie that happened in 1997. The lead character Jordan O'Neill was the first woman to go through a Navy SEAL training, but in order for her to complete the training, I mean, you know how this military training can be very, very tough and all of that. So in order for her to go through that training, she needs to shave her hair and that when she shaved her hair, that's why she got that title G.I. Jane. And so for Chris to actually, in 2018, so if Jedd Smith had talked about, opened up and talked about her struggle, she looks very pretty with her hair. She talked about the fact that she's going through a situation, the other piece here, and this is the reason why she hasn't gone through it. It can be very painful. Some people who want to connect it to, so it's very sensitive. It's a sensitive thing. One would expect that Chris Rock should understand that. Just how far, knowing where to draw the line, just how far, what limit should you go? So one would know that this is where to draw the line. So I see a lot of reactions from Nigerians. You saw Ubi Franklin calling out, what was his name again? Was it Baskham out? I don't remember.  And when somebody says he's looking off, stop. You know, sometimes you could be joking. I could be joking when you say, hey, stop. But you don't know where to draw the line. But however you say it from both ends, I mean, no one is really right. But the most important thing is the fact that Smith has actually accepted that he's done wrong. What he did was really bad. He is embarrassed himself, but he couldn't really control his emotions, especially when his wife was being shamed about a certain illness. That's just talked about in 2018. That's a long time, the 2022. So the moral of the story for me would be that let's be very sensitive. Let's be careful. Be very sensitive. People are going through a lot. The most that we can do is let's be kind with our words and with our actions. Yes, so at least try to understand people better. You know, it might be a joke to you, but some people, these struggles are really something very deep for them, something personal. If for her to have come out to say that she has issue with him, hell, it makes her a very strong woman. Not everybody want to come out and admit to that. Some people would just hide with hair and makeup and all of that, but she actually advocated that it's something that she has had to deal with. So people who also have similar stuff can actually know how to go about it. But then again, we don't promote violence anywhere, anytime. So indeed, Will Smith has owned up and he has apologized. I think people should just let him be. And you know, there's something that really happened. There was some advice that was given to him by Dento just before, you know, somehow by how the devil attacks her. The highest point. A lot of things have actually came out of that. He's just human. I mean, it's just natural to have all of that. I think the most you forgive him. Yeah, the most important thing is, you know, both parties have... And Chris didn't take it up at all. No, he didn't take it up. I mean, because he came to him as a shock. You know, that's the thing where he should never expect it. Because he doesn't mention it just on stage, you know, empty the whole time. He's actually now behind the car. Suddenly, he just came out from nowhere and dragged out your wig.  So you'd be like, oh my God, what's going on? What did I miss? In a few seconds, you'll have to have a reboot and all of that. Hey, see, the most important thing.  I mean, people were also alluding it to the whole Bianca thing. Yeah, Nigerians are very funny. You know how we like to still stop over everything and just make phone calls of every situation. But that's it, don't. Yeah, lastly, social media was also trending. Ruga, you know, sexual assault as it is, is also still trending in various fora and people are still talking about it. The social media is a buzz about what happened during his performance. And a lot of people have blamed him that it was the way. The fact is that men also get harassed. That's one thing people should know. And for he to mention, there's a whole lot to talk about this thing. But the fact is that most people like why so much talk is just a man, you know. So why should he make so much, why should people be making so much buzz about it? But the fact is that everyone has his right and everyone, you know, should be treated equally. Be you a man or be a woman. Social media is just so funny sometimes. We just go about things. But was it his fault? Why did the woman come onto him? How do you react most really? Well, so just like you have mentioned the issue and that's why the conversation about feminism would always, when people talk about feminism and all of that, people will always say that, oh, you know, it's one sided thing. It's a gender issue. And if we talk about gender, it's not about, it's specific gender. So it cuts across, but now what happened to Ruga is what happens to a lot of men out there. And like you have mentioned, first of all, we live in a society where the culture of silence is predominant. And so whether or not you're a woman or a man, silence is always it. Because usually people don't, people are ashamed to talk about their struggles and whatever it is that they have encountered. Victims don't want to talk to them. Sometimes they feel like, oh, how would people react to that? They even shamed for it. Okay, so I feel like that video that happened, a couple of persons have constantly pointed and highlighted that that was sexual assault and harassment. But most interesting is the fact that Ruga himself hasn't really buttressed the fact that he was assaulted. He isn't saying it as an assault and harassment for whatever reason. I don't know what's going into his mind. I can't really say, but this is the point. People go through stuff, whether women or men. And mostly it might be men. And maybe because, you know, culture, I don't know where we got all of that from. And even generally with people, so you want to talk about some sort of, yeah, why you're weak, you're crying, why you're crying. People should be allowed to cry. If you want to cry, go ahead and cry. As a matter of what it is that you do cry. Talk about the struggles, talk about the pain. Because we're humans at the end of the day. So I don't know where that comes from. As long as you have blood flowing in your vein, you're a human being. And so you are subjected to all of these issues. You live in a society where you have Pevots, whether they're men or they're women. And we can't take that. But the interesting fact for me is the fact that Ruga didn't highlight the issue. But you also had another issue where he was on top of the issue. So your own issues that you're not talking about it, one would have expected that he should say it. Say that you were touched inappropriately and you didn't like it. And I mean, of course it's wrong. Someone touches you without your permission. And that is harassment. And that's all we talked about. So for the fact that he didn't talk about it is what I don't understand. Maybe he just went like, oh, it's okay to be a gentle man. Even if he necessarily doesn't mean anything to him, or he could let that slide. It would just be also a way to let people know that people need to respect people's privacy. True, I agree. I couldn't agree more. Okay, so that's a... You know, this is my prediction. I just whispered things into my voice. But I'll just let that out. No, and you saw, so there's been a lot of, we are the same. There's been a lot of comparison if you were, or the after. If it were brother or brother. If it were brother or brother. If it were brother or brother. If you have anything to say in the street, well, it would deck you. You will receive a blow from the stage. But of course that would be violence. That would be you acting differently. And but we need to say this to everyone who attends events. You also, people need to respect people's space. People know, you need to know where to draw the line. No, there's something like draw the line. You need to respect space. You can dance with your celebrity or someone, but just know that they have their previous and you should respect them. But so because we live in a society where people don't do that, have you gone to the market? I hate to go to the market.  I guess you ought to. No, when you go there, they drag you at every cost. People are dragging your hand. People are harassing you. And they don't have the right to do such things. Let's just leave it at that, Jerry. That's as much as we can take on top trending. So I mean, you know, of all the stories that's a read up and then make decision for yourself and leave better lives and know when to draw the line and know how to respect people's privacy. People have the right to their privacy. Let them believe and let's leave. That's where I'll just leave all of that. It's still the breakfast. We'll take a quick break and we'll go off the press and review the front pages of major delis in a moment to join us again.


From the vastness of Africa to the bitter cold of the Arctic nature's great race is a sprint through dangerous terrain a constant search for food a struggle to find safety a battle for the right to mate and an epic journey to survive for another year


Every single one of us were written a check, okay, as Dr. King said. And we're here today to cash in on that check. We are tired of the 1% taking everything that those of us who are here have, you know, our parents have lived and died for. Those of us who are here alive and get ignored. Those of us who are denied health care, health services. Those of us who have been treated like second-class citizens our entire lives.  And I think part of that, let me tell you why it is, is because there is a moral narrative in this nation. That somehow or another doesn't really speak to poverty. Never really had, right? When you start thinking about, you know, $800 billion that are being dropped, being dropped into the military on an annual basis. $800 billion on an annual basis. What is that? And why they get away with it, is because government regulations have been relaxed or not even implemented to be able to keep them from doing so. And what happens is, is that these impact, this pollution, this toxicity impacts those folks who are poor and who are black and brown around the globe. Right? I think it's just been far too long that we remain silent about some of these things. And some of us, we're just becoming aware of some of these things. And what we've done is we've chosen to remain silent, but I came to tell you today that when Dr. King said it's time to break the silence, I think what he was saying is, he said, you know, it's time for you to stop thinking about whether you're gonna be ridiculed. It's time for you to stop thinking about whether you're going to be single out, okay? Whether you're going to, whether somebody is going to try to give you an opposing opinion or whether it make you feel bad or maybe your family is gonna disown you or all of the other things that folks wrestle with when they're trying to stand up against things like systemic racism, poverty, ecological devastation in the war economy. At some point or another, we as the people have to come together and make a conscious decision that we are not going to be silent anymore. We have to speak out on this at some point or another. All right, all right, all right. In a national call for a moral revival, people are standing up in 35 to 40 states to include the United States capitals right now, right now, with these types of signs, singing these songs that we're singing with this message on ecological devastation. We have access, same together, that we are not going to take it anymore. Keep on talking, marching on the freedom land. We shall not give up the fight we are fighting. We have only started, we shall not give up the choice. We have only started, we have only started. We have only started, we have only started. We have only started. We have only started. When a mass of people come to like this, it just drops people who are trying to hurt. It drops 10 people. We're brilliant. And all we can, and all there will have to be, we have only started. We have only fought. We have only started. We have only started. We have only started. We have only started. We have only started. Happy to the Christ. Can't you see? We need what we need. Ain't no way we're backing down what we need. Climate justice is what we need. We're backing down. Ain't no way we're backing down. Ain't no way we're backing down. What is what we need? We can fight him down. We can fight him down. Number is 802. Number is 802. 828. 828. 3322. 3322. We are the public and he's a public servant. We are the public and he's a public servant. He's supposed to be. No surprise, it's busy. No surprise, it's busy. He's reportedly at a meeting. The meeting is out here. Ready to put the message on the machine? Oh, there we go. Tomorrow. Systemic racism. Talk to him. We are new. Unsettling force for liberation. We've got nothing to lose but our chains. 200 people are home. We're working at a state minimum wage since 2017. It took 88 hours. It took 88 hours of work per week. To look forward to getting from our garbage. 117,000 workers. 117,000 workers. In Vermont. In Vermont. Make under $15 an hour. Make under $15 an hour. That is 40% of Vermont's workforce. That is 40% of Vermont's workforce. That is immoral. That is immoral. About 75,000 people. Participate in the Supplemental Nutrition Assistance Program. Participate in the Supplemental Nutrition Assistance Program. Our health bodies are for sale. We settle to change a system that has failed. Strange things have happened here. No stranger would it be. If healthcare was a human right, a right for you and me. The earth, the air, the water's not for sale. We struggle to change a system that has failed. Strange things have happened here. No stranger would it be. If we build a system that's for you and me. Our health bodies are for sale. We settle to change a system that has failed. Strange things have happened here. No stranger would it be. If we build a system that is for you and me. The original plan. The original plan for our bans. Was for the people. We go to the Governor's Office. We go to the Governor's Office. And sit there until we get arrested. And sit there until we get arrested. arrested but they locked the doors sure this campaign doesn't last a day doesn't last a week doesn't last a week not even a month not a year until there is equality for everyone this will take a long time and back again so today we go home even the strongest movements in history took decades to accomplish anything and our job is to make this movement stronger than any other movement we've ever seen stronger than any other movement we've ever seen lives depend on it thank you don't cut away next to the hunger mountain co-op the faculty of community college of Vermont are negotiating a contract we will be there to support them we will march through the streets of Montpelier through the state house at 2 30 I will see you all Monday she said


First of all, I wanted to say what a pleasure and an honour it is to speak at this conference for Alan. So, happy birthday Alan. Okay, so I just want to start off with just some motivation for what I'm going to talk about.  I don't see why he'd be friends with someone who doesn't know what a Caeligraff is. But I'll just remind you, just in case. So, we have a group G, which is finely generated by a set S. So, for the rest of this talk, the groups will be finely generated. The Caeligraff, so the vertex set of the Caeligraff is just the group G itself. And then recall that we connect to elements of the group, if we can jump from one to the other using one of the elements in our finite generating set. Okay, so this gives us a graph. And the properties of this graph mean that we really can investigate the group using this geometric object. So, okay, so this is a graph, so it's a metric space. Just using the shortest path metric. The group acts by isometries on this graph by left multiplication. I see we can use the group action as well. Okay, and so there are many connections between properties of this graph, so geometric properties of this graph, and algebraic or structural properties of the group. So, most famously, Gromov's polynomial growth theorem, which says that a group is virtually no potent, if and only if the Caeligraff has polynomial growth. Okay, so that is if we fix some element and we look at balls about that element, if the number of elements in such a ball grows polynomially with the radius, then we say that the group has polynomial growth. Okay, so that's all very nice. Oh yeah, and I should mention also, obviously if we change the generating set S, then we get a very similar object, we get something that's quasi-isometric, so something that is essentially the same as the Caeligraff with the other generating set up to finite perturbations of the metric. So, okay, and the object I'll talk about now is a kind of spin-off of this idea, so if our group has lots and lots of finite quotients, so from now on our groups will be residual finite. Okay, so recall residual finite means that for any non-trivial element in our group, we can find some finite quotients of our group in which that element remains non-trivial. Okay, so if we restrict to such groups, it would be a shame not to use the fact that we know that it has so many finite quotients in order to investigate the group, so what we can do is we can actually use the Caeligraffs of the finite quotients to make a space which kind of encodes the geometric behaviour of all of these finite quotients. Okay, so that's what's known as a box space, and so to make a box space what we do is we take some sequence, so we'll take a sequence of subgroups of G, and we'll call it a filtration if each of these is a normal subgroup of G. Sometimes we want it to be nested, so we'll maybe have that for now. It's not always necessary, but most importantly we want these to be a finite index as well. These are finite index normal subgroups, and we want them to have trivial intersection, so that when we take successive quotients by normal subgroups in the sequence, we will in a way approximate the group. We're going to get closer and closer to G. The object that we can make using such a filtration is a box space. G is residually finite, finely generated. The box space of G with respect to a filtration, and I, which we'll write in this way, is the following object. As a set, it's just the district union of these finite quotients of G by the various subgroups and I, and we want it to be a metric space. It's going to play the role of a Cayley graph for us, so we want to put a metric on it. I'm going to put the following metric, D, which there's a very natural thing to do on each finite piece in this district union. If we fix some generating set S of G, then we can induce a Cayley graph metric on each of these finite objects just via the image of S under the quotient map. D restricted to each of these pieces is just the Cayley graph metric induced by S. That's all we care about, because we want to study the geometric behaviour of all these finite quotients uniformly. A nice way to do that is to put them all into one metric space, so I'm going to define a distance between each of the pieces, but don't worry too much about that, because what's important really is that you have a space, just one space which contains all of these finite quotients. I'm just going to set this distance to be... I make some choice of this distance so that this distance is bigger than the maximum of the diameters of these finite quotients. This choice doesn't matter very much. I just want to be able to talk about all of the finite quotients together without overlapping each other. That's a box space. It turns out that this is also quite a nice object which can detect lots of properties of the group G. First of all, what are some of the properties of this object which allow us to detect these various properties? First of all, just as in the case of the Cayley graph, if we change the generating set, we get an object which is equivalent in some sense. It's invariant in a certain sense under change of the generating set, and what is the correct notion of invariance here? This is something called coarse equivalence. If we have two metric spaces x and y, then f is a coarse equivalence. If there exist some maps which bound how much f distorts the distances when we go from x to y. We have some control maps, row minus and row plus, which just go from the positive reels to the positive reels. We want these to be increasing, and we want the limit of these maps to tend to infinity. We want these maps to bound how much the metric is distorted when we pass from x to y. We want the distance in y of the image of some point x to the image of some point y to be bounded by these functions of the original distances. Not necessarily, no. This is a little bit like a quasi-asymmetric embedding at the moment. If you just replace these row plus and row minus by linear maps, then you get something stronger, a quasi-asymmetric embedding. Here we don't care about these maps too much as long as they tend to infinity. Obviously, that would be a course embedding, but we want a course equivalent. We also want some constant, such that the image of x is coosly denser in y. Such that, let's just say, from y and y, the distance between the image of x to y is bounded by c. That's the correct notion of equivalence for this talk. We have this nice invariance under the change of the generating set in particular. What I meant is that if I take the box space with respect to some metric induced by a generating set, then I'll write this for course equivalence. We'll have that these two things are coosly equivalence. I'll change my generating set to some s prime. Another nice property of these spaces is that we can recover the information that's given to us by the calligraph if we're willing to go far enough along our filtration. If we choose some radius, then if we choose an index big enough, the r-balls in the box space from that point onwards will look exactly like r-balls in the group, in the calligraph of the group. For all r, there exists some i, such that for all n bigger than or equal to i, the ball in the calligraph of g of radius r is just going to be isometric to the ball in the box space, the ball in any of the pieces of the box space from the point i onwards of radius r. We can see any finite piece of our calligraph that we wish from some point onwards in our box space. Okay, good. As I mentioned, there's lots of nice connections between geometric properties of this box space and properties of the group. Just to give you a motivating example, if we take g with property t, cash dance property t, so this is a rigidity property of actions on Hilbert space, then if we look at the pieces of the box space, then these pieces will be expanders with a uniform expansion constant. What is an expander? I'm going to talk more about this. I'll just define it quickly for you. An expander sequence is a sequence of finite graphs which are extremely well connected, but yet do not have too many edges. We want the following properties to hold. We want these graphs to get bigger and bigger. We want the degree to be bounded by some uniform constant for all of the vertices in all of the xi. We want the chiga constant, which is a measure of connectivity to be bounded from below for each of these graphs. I've messed up my board technique. She taught me how to use three boards because I came from the UK, a country where we can only afford two boards. She taught me how to use the three boards.  I'm sorry, Ella. What can I do? Let's do it like this. What's the chiga constant? I'll just quickly define it for you. If we have some finite graphics, we look at some finite subset A and X. We look at the boundary of A, which is the number of edges that we move in order to disconnect A from the rest of the graph. For example, this is my X and here's A. It's all of these edges here which are connecting A to the rest of the graph. That's the boundary. I need to divide through by the number of vertices in A and I take the infimum over this, and I guess I should take A to B at most half the number of vertices of X. This is the chiga constant. It's measuring how easy it is to disconnect X, but it's taking into account the size of the subset that we're actually disconnecting. It is a measure of connectivity. We have the definition of expander stuff. This is a result of Margulis, I should say. It was him who first realised that if you take a property T group, then you get an expander as the box base. This was actually the first explicit construction of expander graphs. These are very useful objects in computer science and network design and this gives you a systematic way to create these graphs. I think that's enough motivation for why these are nice objects. What I'd like to talk about is some questions concerning the rigidity of these objects. As for motivation, I just want to mention the following purely algebraic question, which I think is not directly related, but I think it's an algebraic version of the question that we're going to answer in a second. Imagine I give you a residual finite group and I'm going to look at the following object associated to this group. I'm going to look at the set of isomorphism classes of finite quotients of this group. It's going to be the set of finite quotients of G up to isomorphism. I give you the set of finite quotients up to isomorphism of this residual finite group and I ask you, can you tell me what the group is? How rigid is the group under this algebraic object here? In particular, a question of Remeslinikov, which is still unanswered, is as follows. Can you actually tell that suppose I give you the set and you can see it's actually the set of all finite quotients of a free group?  Anna, what is that letter? I guess it's a C. But it can be whatever you want it to be. C for class, I guess. Yeah, you're right. Why is it a C? I'm not sure. Yeah, I should have made it an A or something in honour of Allah. OK, so, yeah, if I tell you that the set of finite quotients of a group is exactly the same as that of a free group, can I actually say that my group is free? OK, and this is still unknown, this is still open, although there has been some progress on this by Brydson, Conda and Reid. If you assume some additional things about your group G, if you assume that it belongs in a certain geometric class, then you can conclude that free groups are rigid in this way, but in general it's open, which is kind of weird. G is finally generated, yeah. So, is this the equivalent sign for isomorphism? Yes, yes, this is isomorphism. Can you conclude that G really is a free group? OK, and so, with this question in mind, let's think about what this could be in the world of geometry. So, the question we're going to think about is, suppose that I have some resusally finite group G and I have a filtration, I take the box space and then I take another group H, suppose I know that the box space is geometrically similar, what can I conclude about the groups G and H? OK, so that's going to be what we'll be looking at. So, in the definition of these box spaces, you took the G over Ni with the world metric, but you made them disjoin with some fixed distances which are large enough and they were arbitrary. So, this is a choice. So, the choice actually doesn't make a difference for the course equivalence class of the box spaces. So, if I choose the distances to be growing as I did because I wanted them to be bigger than the diameters of the pieces that I'm separating, then actually this choice doesn't make any difference with respect to the course equivalence class. So, OK. So, the first result that we have in this direction is joint with Ella. So, what we were able to observe is that, if you have this course equivalence between two box spaces, then this actually implies a geometric similarity between the groups. So, here I mean the calligraphs of G and H. So, it actually implies that G and H are quasi-asymetric. And this probably isn't very surprising because as I told you, as we move along the sequence that makes up the box space, we see bigger and bigger pieces of the calligraph and it turns out that you can actually glue these things together to form a quasi-asymetry between the whole calligraphy. OK. What can we say about the opposite direction? It's not true. In fact, for the opposite direction, it's even kind of difficult to see what kind of question you might ask because something I forgot to mention is that, so if you just fix your group G but you vary the filtration, you actually get wildly differing course properties. So, you can actually have groups for which, with respect to some filtration, you actually get expanders and with respect to another filtration, you get something that embeds into Hilbert space which is kind of an opposite behaviour. So, you can really get all kinds of behaviour if you vary the filtration enough. So, it's not really clear what would be the... If there is economic filtration, you can just take all. Oh, you can take all of them. Yeah, you could. In fact, so, Thibaut Della B, who's here somewhere, he's investigated these objects as well. This is something we call full box spaces. This is also an interesting question, what happens when you take all of them. But here, in general, we take them to be nested, so we don't take all of them. It's also interesting. So, in your statement, G has property, T implies that all of these bones... Yes, exactly. There, it's all of them. For the free group, you have mixed situation. Okay, so this is kind of... Maybe not that surprising, as I was saying, but you can get some nice corollaries from this. So, in particular, one can answer the following question of Mandela Naur, who asked... Remember, we had these expanders, which are very nice course objects, which people are interested in. But how many do we actually construct, for example, via these groups of property? So, how many expanders are there, whereby this, of course, I mean up to course equivalents? Okay, so perhaps there's only really one course equivalents class of expanders that would be kind of a shame. And indeed, this result allows us to answer this question. Well, actually, sorry, I should say, of course, that Mandela Naur kind of answered their own question in a sense, so they constructed two examples of expanders which were closely different, where one had a large girth, where girth is the length of the smallest cycle, so they constructed a sequence where the girth tensed infinity, and another sequence where the girth does not, and proved that these were not coarsely equivalent. But there was no kind of scheme to show that any of the other examples are coarsely different. So, using this result with Alan, we're able to show that, for example, if you take some groups which you know are not coarsely asymmetric, so, for example, if you take various groups of property T, you can actually get infinitely many coarsely equivalence classes of expanders from property T groups. You can get continuum many, but I guess for continuum many, you kind of have to cheat a little bit, you have to vary the sequence of subgroups that you take. So, yeah, you can get continuum many. So, there are no continuum many groups of property T or two possible? Oh, I don't know, I just want some. Yes, sorry. Oh, that's a different question. Well, is it? Well, it's a really good question. Oh, there are? Yeah, then you can just play with it. Oh, then yeah, okay, then yes. Yeah, think of this as continuum in that case. So, for example, if you take various SLN, so, by a result of Eskin, if you take SLNZ and SLMZ for N not equal to N, then these are not coarsely asymmetric groups, and so the expanders that you get from these groups will not be coarsely equivalent. Okay, so maybe I should also mention some related results. So, David Hume also proved that there is a continuum of non-coarsely equivalent expanders, but using different methods which only work for expanders of large girth, so with girth tending to infinity, he used something called separation profile, which was invented by Benjamini Shraman Timar for different purposes, I think.  So, he kind of extended this result in a measure-theoretic direction. Okay, and maybe another thing I should mention is a result by Delat and Vigolo, who proved that, who did a similar kind of thing for warped cones, which are a kind of sister object of box spaces, which allow you to construct interesting examples of finer graphs using group actions, rather than quotients. Okay, so now, let's go on to something different. Okay, so, we know from calligraphs that it's very satisfying when we can assume some geometric property for the calligraph and have some kind of algebraic or analytic or structural conclusion for the group. And so, it would be nice if, with these box spaces, if we could assume some kind of geometric information about the box spaces, so if we could assume that the box spaces are coarsely equivalent, but get some kind of structural information about the groups, okay? Here, we could get some geometric information, but what about some kind of algebraic information? So, this brings me on to the next part of the talk, and this is joint work with Thibaut de Labis, who is presumably still in the same place as he was before. And Thibaut is our son with Alain, so he's a joint PhD student. And we're very proud of him. Okay, so, yeah, just some motivation for this then. So, okay, so what we'd like to do is we'd like to take this geometric equivalence between the box spaces and deduce something about, for example, the subgroups which we actually used to create these spaces. So, okay, imagine if our group was free, okay? If I look at the fundamental group, that's a really good way of detecting, so if I look at the fundamental group of a caligraph, of a quotient, of a free group by some normal subgroup, then exactly what I'll get is the subgroup N, okay? So, for the case of free groups, I have this very nice way of detecting the subgroup by which I've quotiented by looking at the fundamental group, okay? Remember, the fundamental group is the loops that you have in your space, and here, of course, we've made exactly the elements of N into loops, so we detect exactly this group, okay? In general, this may not work, okay? Well, in fact, it won't work if I look at the caligraph of some non-free group quotiented by some subgroup, okay? This will not be equal to N, okay? And why is that? Well, if I think about the loops that I've got in this space, I'm not just detecting the elements that are in N, I'm also detecting the elements which are already loops in G, okay? So, the... If she's pi1, that's a caligraph of the free group quotiented by some normal subgroup. So, with respect to the free generating set of Fn. But I'm confused. If pi1 for graph, isn't that always a free group? Yeah, exactly. But N is a free group. Everything's fine. In the end, everything's fine. Yeah, so, yeah, exactly. Exactly. So, yeah, the pi1 of a graph is always a free group, so that's a reason in some sense why this doesn't work for a general group G, right? Because, you know, if you think about what this quotient will look like, your... as loops in the space, okay, you've got the elements that are now trivial, the elements of N, but you also had already loops in the caligraph of G, so you're also going to be detecting those. So, that's no good. So, what you'd like to do is you'd like to be able to mimic this behaviour in some sense for a non-free group, okay? And the way to do that is via these course fundamental groups. So, we need a course version of pi1, and, in fact, there already existed one, which was created by Barsalo, Cremor, Laubenbacher and Weaver, okay? Okay, so, let me talk you through what this course version is. So, we're going to define what our paths are, first of all. So, we're going to fix some parameter R for example, R equals 60. I really tried to get 60 into my talk, but there's nothing, there's no numbers. So, yeah, if you fix some kind of parameter R, we're going to look at the fundamental group on this scale, okay? So, what I'm going to define, first of all, are our paths. Okay, and our paths are just going to be maps from a discrete set of integers up to, say, the length of this path, let's call this map P, that's going to be a path. So, in the normal fundamental group case, paths are continuous maps from the interval to your space. Here, we've got this map, which is going from a discrete set of cardinality length of this path to our space X, so, yeah, let's fix some metric space X. For us, it's just going to be a caligraph, so, let's fix some base points as well. And we want this map to be our Lipschitz. Okay, that's going to be our replacement of continuity in this discrete case. So, and of course, now we want to define what the correct notion of homotopy will be for these discrete paths. And in order to define that, I'm just going to, first of all, define another notion called our closeness. So, I'm going to say that two paths P and Q are going to be our close if one of the following two conditions holds. So, one of them is quite sort of obvious. Okay, if I have some path, okay, maybe this is my R path in my graph X. Okay, I'm going to say that this other path is R close to it if it has the same length so if L of P and L of Q are equal and each of these distances is bounded by R. Okay, so that's a very natural notion of equivalence between these paths. And the other version just takes it into account what happens when P and Q have different lengths. So, I'm going to also allow them to be R close if so one path does something. Okay, maybe the other path is longer than it, so what it does is it just does exactly the same thing as this path but then it just stays here. Okay, so it just waits for the other path at the end. Okay, so this doesn't involve R but R is involved there. Okay, and now I can define my notion of our homotopy. This is just called the end or the middle or? Just at the end, yeah. It's just allowed to wait for it at the end. Is this what I would call fellow traffic? Ah, yeah, possibly. Yeah, I guess it is, yeah. Does fellow travelling have a parameter associated to it? I guess if you compose these two moves then it's going to end up being fellow travelling actually probably, yeah. So what we're going to do now is we're going to be able to use both of these moves to move from one path to another path. Except I guess what we're going to lose is the control on the parameter at each, oh yeah, I'm just going to write it down then. So P and Q are our homotopic. If there is some sequence of paths Pi such that we start off at P we move up to some path Pn, which is going to be Q and each pair of paths, each pair of consecutive paths are going to be our close. So we can make these moves via these two relations of our closeness from P to Q. But I guess what we do is, we don't care about how big N gets. So we might be making a lot of these moves so that we kind of lose control on this parameter R. We can no longer say that the paths P and Q are very close but we can get from one to the other by some number of paths which are close to each other. And the endpoints are the same or they can change also, thank you. Yeah, the endpoints can change because here they just have to be close. Okay, so that's a discrete version of homotopy and now we can define this discrete fundamental group for coarse fundamental group in the obvious way. So this coarse fundamental group at scale R of X with base point E is just going to be the set of loops based at E, so well let's say R loops based at E up to R homotopy. Okay, and so why is this a good idea? What can we do with this? Well, the main thing is that now using these moves of our closeness, we can actually, via our homotopy, jump over holes of size around 2R or R if you want. So now if we have some path, if we have some loop that does something like this, if this length is less than 2R or R, something in terms of R, then this we can actually homotop down to just the trivial loop in our space. Okay, so we can ignore short loops. And what this allows us to do is the following thing. So now if we take a group in which we can actually control how big, remember the problem here, why didn't pi1 work is because in this pi1 we got the loops of G mixed up among the loops from N. So if the loops coming from N and the loops coming from G are far enough apart in length, then we can actually pick a parameter R, take the fundamental group at scale R and it's going to kill off all the groups that are coming from G and it's going to preserve all the ones that are coming from N. So that's the idea here. So what kind of group can we do that in? Well, it's a finely presented group, right? So if we have G a finely presented group and I'm going to call some things something, or maybe not, maybe. So maybe I'll just take some normal subgroup N and G. Now if my parameter R is sufficiently far between the longest loop in G, so the longest length of a relator in G but it's smaller than the shortest non-trivial element in N. Yeah, the length coming from the word length on G. Yeah. Yes, oh yes, oh sorry, I see what you mean. Yeah, and here obviously it's the length in the free group on the generating set of G. So if I take this parameter sufficiently far between these two things, then actually if I look at, if I look at pi 1 at the scale of my caligraph of this quotient based at the identity, then it will actually detect the normal subgroup by which I've quotiented. Okay. So. Is N simply a finite index or? No, no. So the notion of homotopy is that the two loops are homotopic if they are homotopic through loops based on SE because otherwise like in topology you can have free homotopy of loops which gives something else. Yeah, yeah, so yeah we base them at base at E, yeah. I guess this is kind of the same thing as if you, you know, you take your caligraph of G modern and then you just glue in some cells of, for cells of size less than 2R. So you kind of kill off the, the relators that are coming from G in this way. Okay, and so why is, what is the, a good context in which we can apply this? Well, box spaces are a very good context. So now we can answer our question in a more structural algebraic way. So now if we assume that G is finally presented and residual finite, then if we have some course equivalence between two box spaces, well this in particular also implies that H is finally presented because the groups will be quasi-asymetric and so what this results on the course fundamental group gives is simply, it's simply telling us that actually there's an isomorphism between pretty much all of these subgroups. So I'm going to write this like this. So this is isomorphism, whenever I write two lines here, it's an isomorphism. So I write this in quotation marks just because there could be some small perturbations for finally many of these guys where maybe they're not isomorphic but in the long run they will just be isomorphic and maybe up to some finite permutations. Essentially you're getting this. So in particular this implies that the groups are commensurable which is a very strong structural conclusion. So commensurable just means that they share isomorphic finite index subgroups. But yeah of course saying this is stronger. So just some comments about the proof of this. So it's not absolutely immediate but this actually uses the result with Alain. So in the result with Alain what ends up happening is that each of the pieces that are making up the box spaces you actually get a uniform quasi-osometry between these pieces and then you can use that in order to apply this to the course fundamental group. So and how would you apply that? Well actually what you end up getting is that if you have some let's say calligraph of some group A and you have a quasi-osometry of that with some calligraph of a group B then what you end up getting is that there's a map, there's a subjective map from the course fundamental group at scale r of A onto the course fundamental group of B except it's not with the same parameter you actually have to change the parameter a little bit according to the quasi-osometry constant C. So whatever quasi-osometry constant C you have here controlling your quasi-osometry you actually get a subjective map from this course fundamental group onto a course fundamental group of your other group but at a slight cost you have to change the parameter slightly. But what you can do in box spaces because you have this limiting behaviour that the NI are getting smaller and smaller it's a filtration, they have trivial intersection so what you can do is actually you can go back the other way okay and so you can go back the other way via a suggestion of the cost again of changing the parameter here but if your NI is sufficiently far along in your sequence then actually this will just stay the same. So what you end up getting is a suggestion of this group onto itself it's isomorphic to the subgroup by which we quotiented in the case of our box spaces and because we're taking residually finite groups these groups are actually hopfian. So it actually uses hopficity here in quite a funny way so we're actually using the hopficity of the subgroups in the filtration. So okay great I'm almost out of time so just some small remarks about this result and some consequences so of course just as in the result with Alain you can also conclude that some objects which are constructed in this way are coarsely different just as we did with expanders showing that there are infinitely many cost-equivalence class of expanders so you can do such things for example you can prove that there's infinitely many cost-equivalence classes containing Ramanujan graphs I mean being Ramanujan isn't a course invariant but you have infinitely many cost-equivalence classes containing Ramanujan graphs but a more interesting kind of strange consequence of this is that you can have the following situation because of the very strong rigidity of these objects so we're used to with Cayley graphs if we have groups which are the same up to finite index we're used to the Cayley graphs just being quasi-asymetric that's something we've all kind of internalised in the case of book space you can have some weird situations so I have the following corollary so there exists a group G with filtrations Mi and Ni such that there is some uniform bound let's say D such that the index of Ni is always bounded by D in Mi so these subgroups are very, very close to each other in a uniform way all the time but such that the box spaces with respect to these subgroups are actually different in the same group G nested sorry, yeah, thank you indeed, yeah so we always have this and the index of Ni is always bounded in Mi and yet these box spaces that we're getting are not going to be costly equivalent so if you think about it this condition means that the G mod Ni are just bounded covers of these G mod Mi with this bounded index here but yet these objects have very different course behaviour which is kind of strange and just to finish I just wanted to mention another result by Vigolo so Vigolo exploited this kind of course fundamental group framework in order to prove that actually for these warped cones so these objects which are created using group actions on manifolds there actually are some examples of expanders which are coarsely simply connected so which have trivial course fundamental group and that proves that they're completely different from any of these expanders constructed using this box space structure so any of these expanders coming from property T groups or others so they're these warped space expanders with trivial course fundamental group really a very new example of expanders that we have okay, I'll finish there, thank you so if you apply the last theorem you mentioned to try to understand the remeslinic of questions so why you can't solve this question I guess the remeslinic of question is more about the kind of the algebraic nature of the quotions so a lot of these results in that direction they use things like the profinant completion which is like an algebraic way of encoding these quotions whereas we assume something a priori stronger well something different something geometric about these quotions in fact you see these box spaces are much finer objects than profinant completions because if you just to give an example so if you take a profinant completion with respect to some sequence and with respect to an interlicking nested sequence of subgroups then the profinant completion you end up getting is the same and a lot of these results which explore this object use profinant completions whereas box spaces you can have these kind of interlinking subgroups but which give you again box spaces with very different course properties so box spaces are finer objects and thus assuming some kind of equivalence between them it's a stronger statement so we're assuming something stronger so unfortunately it can't seem to shed any light on this kind of thing so if you choose the subgroup to produce box spaces which are equivalent then you deduce that the two groups are commensurable and what is missing is that you want an isomorph is there something that you didn't expect? When you assume that the box spaces are closely equivalent that's a strong geometric assumption whereas here what you would be assuming is that really the set of isomorphism classes of all the finite quotions is just the same but you don't know anything about how the quotions will behave geometrically so if you fix just because you know what the set of finite quotions might be you don't know that there will be the same set but it won't be the same set geometrically possibly so you need to fix generators for H when they behave in the same way as the generators Exactly, and you can't do that Yeah, it's to do with choosing the generating set for those that it works Is the theorem okay when G is just profanically profanically presented? Ah So what does it mean profanically profanically presented? Let's say if profanically profanically presented is profanically presented as profanically presented possibly it might work I'm not 100% sure as it stands but it might also work I know it doesn't work for finally generated groups in general you can have some examples very simple examples using Wreath products for which the groups are the groups are not commensurable but the box spaces are even just isometrically box spaces you can get with finally generated groups but yeah I would have to check, that's a nice question


So, today we will be looking at expanding on the simple structures we saw we have been seeing both positive and negative feedback systems. So, start with let us consider this scenario as shown in the diagram. So, we have population is affected only by the birth rate or net birth rate that as you can see is a positive feedback system. So, we are going to introduce a new variable called as a carrying capacity which is going to define how much the population can grow, nothing can grow forever there has to be some limiting constraint some limiting resource which finally slows down the growth. So, in this simple example let us assume it is called as carrying capacity. Now as and as you can see carrying capacity is external to the system now in the first order system that we saw the simple positive feedback system we have seen that in population affects net birth rate, net birth rate affects population and that is net birth rate is defined by something called as fractional birth rate. Now by introducing this carrying capacity what you are saying is will as the population grows towards the carrying capacity it is going to affect the effective fractional birth rate which in turn will affect the axial birth rate. So, we have positive feedback system here followed with a and a negative feedback system right here. So, as the population comes closer to the carrying capacity we would like the effective birth rate to fall down when population is far less than the carrying capacity then there is no real restriction on the birth rate it can be very high as it as population approaches carrying capacity we expect the birth rate to fall or when population is exceeds the carrying capacity we can even expect the birth rate to be negative. So, this is a very simple scenario before we start simulating these things let us try to see what will the expected behavioural system be we will try to plot the expected rate level graph level is another name for stock and how we can expect the fractional birth rate to change with population divided by capacity ratio. So, let us look at those three things right now convenience have reproduced this the same causal loop diagram here. So, here we have a positive feedback system and here is a negative feedback system. Let us take this scenario the questions from let us say bottom up kind of approach let us mark on the y axis effective birth rate and on x axis here let us define it as p divided by c. So, let us denote carrying capacity as c so, the population is nothing but the ratios nothing but p by c ok. So, on the x axis we have ratio p by c and y axis we want to plot what is going to be the effective birth rate. So, that we call here so, we can place it in context we have population we have births and we have. So, this is a simple positive feedback system that we had seen and we had modeled it like a stock and flow like this and here we put a growth factor g. So, this is what we had done when we studied positive feedback systems where we defined as a stock and then we told it is going to increase where rate is equal to g into p and it became a positive feedback system. So, this is a model that has been expanded on the left side where this fractional birth rate now g does not directly affect g affect the rate that is right here. So, I am here using this is birth rate we are talking about I put a small b it of g equivalent. So, we are saying that that is will change based on the carrying capacity. So, that is the additional loop that has come into play right here. So, now, how would we like this fractional birth rate to change? So, for convenience you can think you know as this p by c ratio changes what will happen to this value of g or this value of b what we expected to happen.  So, now, if population is very low initially carrying capacity is very large then there is really no restriction on the birth rate right we can allow the birth rate to be as high as it can be and. So, let us assume that birth rate starts here somewhere and for some time we can even expect in the birth rate will be kind of it is not at all affected by the carrying capacity right. Even suppose we have an excess of 10,000 suppose is the carrying capacity and your current population is just 10, we can expect the birth rate to be constant, but again the current population is say 100 still 10,000 is quite far off or population is 500 or whatever number. So, up to some extent we can expect that the effective birth rate can continue to remain constant nothing really affects it to change its direction right. So, this can continue up to some point. Another way to think about it is how much you are going to consume for example whatever you can take let us say the food you eat suppose there is let us say you are hungry and there is say 10 pizzas available. So, as per normal rate whatever you can your hunger is satisfied when there are when you eat say one full pizza then hunger is satisfied. So, there are 10 pizzas still you eat only hunger is satisfied. But even if there is 5 pizzas still you eat one and hunger is still satisfied. So, when there is excess capacity your consumption continues to remain unaffected right up to a point. So, as the population comes closer to the carrying capacity or then I will start to say ration the amount of intake. So, we can expect that this slope you know can slowly reduce over time and as we approach 1 we continue to reduce and as P and C starts to go very very large at some point the growth get get halted. The population is way way high as I told there is ok. So, give an example like as you told as you started with there is a 100 pizza and 10 guys see each of you have one it is fine then there is 50 pizzas 10 guys no problem or 50 guys 100 pizzas still no problem then suddenly it is 100 guys 100 pizzas still may not be any problem imagine there are 10,000 guys 100 pizzas probably you can just get a small piece of it 1 million guys and there is 10 100 pizzas probably nobody will eat will all be busy fighting you may not get chance to eat. So, that means your consumption effectively has dropped to 0 as P by C close to large right. Your pizzas you are carrying capacity and population is a number of people who want to have a slice of the pizza or slice of the resource here we are denoting simply as the carrying capacity. This is the effective birth rate that we can expect. So, given this fractional rate value for G if G is constant what will be the behavior? So, when G is constant we can expect a exponential growth now here in this graph it shows that this G is falling down then what is the behavior we can expect as G goes down we can expect an asymptotic convergence to the goal or a goal seeking behavior. This now if we model the population P over time then we can expect a kind of a behavior like this where initially there is a exponential growth and here it became let us say a goal seeking or asymptotic growth and this point we call it as the inflection point. The point where the loop dominance changes at inflection point what happens at inflection point loop dominance changes from positive loop to negative goal seeking. So, after the inflection point this loop is active after the inflection point I mean this loop is dominant rather after inflection point this loop can be expected to be dominant given this it has to eventually saturate in this kind of system. So, let us expand into one more type of graph there is a rate level graph how will this graph be how can we expect this graph to be. So, initially remember I need to get this growth behavior. So, if you recall in a pure positive feedback system when there is a population of the stock value those exponential growth then rate level graph your rate level graph had a line like this and we saw that whatever the positive slope it will show this exponential growth behavior and if it is goal seeking then we told that the line has to be like this. So, if I am going to get a figure like this then I need to get a triangular or a hump shape graph like this if you get something like this this is your rate or the net rate there this net rate is what is to occur here. So, just to you confirm to look at this graph we have shown three different graphs with three different x and y axis. The first one we told was first effective birth rate or how we can expect this to change with respect to this ratio get just hypothesized on that and based on that it will ok here it is constant. So, we can expect exponential growth where we did value of population over time graph and as this continues to fall and reaches 0 then it has to the addition to the stock will incrementally reduce till successfully reduce until it reaches the maximum value. The third graph we plotted over net rate versus level as we told level large stock in our case it is the population p and this is the net birth rate that will be kind of hump shape.  See this curve here as you can see is non-linear right. Now, trying to fit an equation for it whereas, p by c changes in value I need to come up with a curve like this it is kind of little more difficult. Now, I need to come up with some if there are conditions or I need to draw it as a graph and then say if the value of p by c is say between 0 to 0.5 take this value if it is between this take this value etcetera that is one way or if I know the equation for this curve then I write how this birth rate can change as a function of p by c.