Everybody forced tier up being in 2022 NT baseball hobby four box full case number eight random divisions Check the results anytime late sports cards calm. Let's get started everybody. Good luck Yeah, I think so too. LaMouc that game feels like a true coin flip because both of those teams are very like hot and cold, you know, I mean The Cowboys should win. They're better overall, but Dak has known to throw pick sixes and be inconsistent at times. So it do be happening like that Right, you just can't win doesn't matter what you do and the Bucks like the opposite most of the time They're pretty man. Yeah, and then every now and then the Brady's like, you know what for this quarter in this quarter only The offensive line decides the block and the receivers decide to catch everything Yeah, and then the Cowboys like opposite. They're great for three quarters and one quarter They're like, uh-oh happens usually the fourth quarter. It happens fast. It happens fast That game is gonna be pretty close though. I think either way One scored game I'm assuming the Cowboys are favored by like three or four something like that Which is crazy to be NFL playoff road favorites. It's crazy. No matter how we however you look at it Although Charger should be road favorites, right against the Jags. I would assume They are Jersey card and I was at Gallantin 99 Guys, I'm not gonna recap this one. I believe which let me check my schedule if we have time. Yeah, let me see real quick No, not gonna recap this one we're still pretty far behind everybody. So, yeah a Jumbo patch Jake Meyer 16 to 25 al West Crazy number patch right there. It's like the in take the six like that way could be that way too Patch card al central Jim Tomei 10 of 10 Jim Tomei Jersey auto al East Brendan Beck 18 of 99 Best pulled today Probably the Stargell cut auto Really Stargell cut auto. There's a cool. There's some really cool judges Jersey auto 242 is Isaac Brady's al East big poppy nickname Cal Raleigh Midnight Jersey auto Kyle Schwerber 5 out of 10 and L East Nice Schwerber that is a Washington Schwerber. That's from last year. Yeah, it's from last year Washington Schwerber So Victor, what's up, man? Speaking of Cal Raleigh, here he is al West RPA. What an insane patch 21 of 99 big dumper big patch big dumper a Josh young patch out of 11 to 25 al West prospect it there. He could surprise people this year for rookie of the year potentially American League Legendary material signatures duals Mike Piazza and L East hold one of these last cases now. There you go. Nice Piazza 25 out of 25 pizza pizza And a Shane boss RPA 56 and 99 AL East So Victor how bad are the bucks gonna lose this weekend? Skyler Thompson for the starter dude. I feel bad for that kid now Like it's your job as an echo quarterback, but you got to go against the bills for the team of destiny and just better That's tough. It would have been funny if you did sign the patch like that Matt some way we breaks is a different channel so this is just I Don't know how to explain it, but basically just not the MLB breaks Here is the Link for that stream Colossal patched car Tyrone Taylor and Al central to 25 I know I was messing with you sin.  He was like we got London on the track so every time Sean was like I'm gonna lose my mind dude. It's funny Mel Ott Jersey card Coach worn for the NL West New York Giants goes San Francisco Giants very cool die-cut Jersey auto going to the ALS Nick Selleck to 49 dual-patch auto Landon Marceau AL West to 25 NFL Mom's yeah, yeah easy just cuz Hertz miss us do games. Yeah, that's it definitive ink auto is Frank Schwindel and Al central to 99 Schwindel Schwinn that's a kind of bike I have I was a kid Schwinn an NL central RPA Lars new bar to 99 No bar dude, I can't wait for the new bar to come out Candy bar need it better sell them at every Baseball stadium and Al central again a Paul Goldschmidt legacy signatures 9 of 35 Goldie and rookie signature jumbo rookie signature jumbo material book Reese Nair NL West to 99 All right two more boxes. We got hoops after this y'all You know what's weird is I Agree with Max Keller. It was just strange. Wow, that's actually fire. I put Hertz in tier two though with those guys No, not me Okay, why okay All right, I'm done talking about that. Anyway Retro materials Jersey with Max Kellerman Cover the three Retro materials Jersey is calling McHugh to 25 for the Braves NL East Is the only doing people playing this? No All right, and Lawrence isn't up there. Come on now. So I present Alan isn't up there. He was tier two No hurts. Oh Jason Hayward Jersey guard at 25 NL Central. Where's Jalen? I appreciate you present. You're telling me. Thank you, man Currents right now that relic and I was trying to talk to each junior to 25 Brady and Jones none of them are better than Jalen hers I say I say hers is at least as good as Brady and Herbert right now Jersey auto and L West or hey Al Faro to 99. Did you see Al Faro pimped at home run and I think was Dominican We just broke.  How mad would you be if you're the picture that happens to dude? That's normal in Dominican though. Yeah, it's true dude. People were freaking online I was like that is like Latin America right there, dude. There's you're supposed to pimp it Yeah, and they'll fight after but that was just like a given Waited to the land Oh my god, he would get hit three times in the MLB if that happened there Jersey auto Jake Fraley AL West to 99. Do you guys see that? I don't know if it was Dominican League or what what league, but it's a winter league in Latin America Might have been Venezuelan league or something He hit he had some home run it's a bomb and no doubter He stands there looking at it until it lands and then throws like spikes his bat towards the dugout and then Trots around the base. Oh Man, I want to see that happen in the MLB But the Red Sox hit the home run not against the Red Sox got patch auto Andrew Vaughn to 49 AL central If you haven't seen it, just go look up Jorge Al Faro. Just Google Jorge Al Faro probably the first thing you see So if you want to see somebody pimp a home run against the Red Sox, just watch Bon Grissom's first MLB home run I gotta go see it. I have seen I gotta go watch it again RPA Mike Baumann 13 of 49 for the AL East you should pin your first one though. It was over the monster to And there's there's MLB game There's certain ones you pimp like if it's a big walk-off or a big spot I'm fine with that.  Wade Boggs auto at 25 AL East I'm cool with pimping home runs for the most part But pictures should be allowed to pin things to you know what I mean?   Yeah, you think a big walker 99 of 99 dual material signatures of wonder Franco and O'Neill cruise That is nasty That is gonna be a random I believe AL East Brian G and L central Roy and that's a random at the end Sick both on card as well fantastic card. Good luck We'll get on that as well and treasured stats Walter Johnson Washington Senators, right? So that would be what twins Senators I would assume that's Washington Senators Definitely not the national Yeah, Walter Johnson is gonna go twins. So Washington Senators became the Minnesota twins Very cool 110 shutouts. That is that is a few that is a few for sure Dude back then they were like you had like three pitchers I remember reading a stat about Who was the Giants? Pitcher like the 60s and 70s. He pitched like an 18 inning game The whole thing like 180 170 so what pitches? What was his name? He's a legend Latino I can't I can't remember it now. It's a shame.  Yep has the 25 NL Central Jumbo patch Colton Welker 4 out of 10 NL West. I don't that's a nameplate. I think oh, no, it could be the IE for Rockies Wait What's the angle of this? Is it KER for well no EL no But did you get traded you got traded no Darwin's in Who do you get ready for what? Darwin's into the Orioles for cash What are we doing? Jersey card to 49 Charlie Blackman for the NOS and I guess Darwin's in you know Hit or miss he's got crazy stuff, man But I don't know he must have been unlike 40 man and just like I don't know got guys coming up I have no idea man Jersey auto to 49 Riley Adams NL East not Jersey. Sorry batting glove auto Jersey auto steel city die cut to 99 Roanze Contreras NL central patch auto Louise Gonzalez To five for the NL West. That's pretty cool. Yeah, you don't see him very much Louise Gonzalez auto to a five Game use patch RPA ALS read that merge 62 of 99 Angels patch has been looking pretty crazy on some of those RPAs that one we pulled to 10 earlier was disgusting Legend hit here bat relic auto eight of ten Mike Schmidt NL East the best third baseman ever One of the best lefty hitters ever Allure's Montero auto 20 of 99 NL West and NL central a Harry Walker legendary Jumbo materials 32 of 49 Jumbo game-worn relic all right Harry Walker Larry Walker Big random Big random. All right. Good luck on this one AL East NL central on the dual Relic auto book Wonder Franca O'Neill cruise Good luck to those two. I haven't raised pirates as the in the random, but I'd be so good one of those divisions Five and one six times six times whoever's up top gets it Raise pirates one two three Four five and six be raised AL East AL East gets the book Thanks again everybody and he was awesome today. We're on two hoops


Hey, I'm Andrew Connell this video is an overview of one of the chapters in my course mastering the SharePoint framework that's available for purchase on my site boytanos.io. This overview video is going to give you an idea of everything that the chapter covers. You can learn more by checking out the description and the notes below the video. If you got any questions about this chapter or about the course in general, just make sure you drop a comment below in the below the video and I'll be sure to get back to you. So with that let me get out of the way enjoy the overview to this chapter. Welcome to the chapter on logging and debugging SharePoint framework solutions. While creating custom solutions developers need to have a firm grasp on different debugging options and techniques. So this chapter will walk you through different debugging options available to client-side developers and it's also going to cover some of the logging options that you can take advantage of inside of your application. You will see some things that the SharePoint framework has to offer as well as some things that you can take advantage of using common browser developer tools. Now in another chapter I'll address some advanced telemetry and usage analysis but following the approach from other chapters in this course this one is just going to focus on the fundamentals of logging and debugging. So let's get started.


You know, to see this go from $37 to $38, I don't know what the net return is on that percentage-wise. We'll do a little loose math right now. That's a 2.7% move from $37 to $38. This actually broke above $2 here, the calls. Maybe hit a high of $204. So $204 minus $150 divided by $150, that's a 36% gain. So you're talking 2.7% gain versus 36% gain. Obviously, options are gonna be pretty leveraged there in that regard, but it's tough to just say that, right? You need to be pretty knowledgeable with understanding not just the core philosophies of charting, but how volume is integrated with it.


What is going on guys today? I am back with another Destiny 2 video and today I'm going to show you guys how you can solo glitch into the pipeline system within the raid. Now this can only normally be done in a full team of six because you need to work together to pull levers to open up secret doors and so forth. You can't normally do this solo but today I'm going to show you guys exactly how you can get down there and you can explore the whole entire area find chests. Obviously chests you can't open unless you've got keys but you can find the enemies down there these are red enemies they drop decent loot. If you watch my past few raid videos you'll see the amount of legendary engrams I get from these enemies so you could find them if you want to or you could just go and explore and find new secret areas and obviously a lot lot more. But before we get into the video guys I am giving away silver for Xbox and PlayStation for you to spend and tests ever versus exclusive loot to win some. Simply drop a like in the video and leave a comment down below. Also guys join my discord links within the video description. Okay so full credit to this find goes to a friend of mine and not a butter one he actually showed me this which is actually pretty smart it seriously is but once we load in I'll take it from there. Okay so we are in now once you're actually in the raid you want to make your way to the sort of like booster thing is there to sort of like fan booster things and so you shoot right away to the top because that's where we've got go we've got go right away to the very top I mean just look at this place it's so so big it's so so beautifully designed it seriously is okay so let's go let's go should have been on my hunter because I'm faster on my hunter but it doesn't really matter it doesn't really matter I think it's this next platform up we can go through yeah there we go we should be fine now if you run all the way up it takes so long there's so many ramps to go come on come on yep there we go there we go oh oh almost went down then so which one do you just want to run straight towards this door right here and straight towards it and it should open hopefully yes it does and you want to take a left here let's follow the path I am taking man the place is so big and there's so many doors everywhere you can get confused you seriously can so let's go through here and I think it's right again this is okay so you eventually come to what I call a battle arena which you basically go between every boss battle which is fine what you want to do here is make your way to that door at the back now to do this glitch you might not have to do this but this is the way I know it works I'm just gonna show you guys it this way so I make your way to this door right at the very back just keep going keep going I want you all here touch it give it a high five then run backwards go back to basically the place we just come from basically the place we just come from let's keep going and these enemies here you can farm and they're actually oh my I'm trapped I'm trapped I'm trapped whoa get out of there quick I mean they are quite tanky my time isn't weak and you can see what I've just done to me so yeah but just go to the door most basically find your way back out the way you came in and it's a left here yes it is and it's another left and then I think it's not like we're sure it's a right I was gonna say right and I think it's a left yes it is and then we're gonna go right here and you'll see by the red bleeps on your radar that this is the way you need to go and what you are here you just want to head right you want to jump up onto this and then simply to step into this water and when you respawn people you will respawn underground a place you can only get to in a full team of six now there's so many places to go here I mean underground areas of this if you watch any videos are absolutely humongous they seriously are let's just see if I can make my way to the fans I think they're down here somewhere but this chest you can find around here as well which obviously can't open unless you've got keys but there are chests there are plenty of spaces within this system what people haven't found yet I know that for sure because there's so many doors and things to activate these doors and so forth now this chest through there up to your left there's other areas through there up to your left let's try and make it I probably ain't gonna here I'm probably gonna get flown into a wall I did turn it off though but it's probably gonna spawn me back up top um yep I knew it was gonna I knew it was gonna I don't even know where I've gone now where I've spawned okay so we go we go back that way we go back that way we'll go back that way okay so that fan I turn that fan off so it shouldn't be blowing you know no it's it's just decided to come back on it just said to come back on what is wrong with this place now when I eventually get through here people I'll zoom the video from there okay so I've made my way through the first set of fans not gonna lie to you guys to talk me back 10 minutes and you can see there's a chest right there which you do need a key for a ventilator key which you will get for being a boss that's fine we're gonna carry on further into this let's see I'm just gonna hide behind these little walls it makes it uh I'm not no I'm not don't feel me oh oh that's close that was close let's go we'll turn this off anyway and then make our way back through whoa whoa whoa that almost looked not the wall um so there's a natural room oh no you open it from the other side my bad so we'll go up here come on jump there we go oh oh are you taking me I thought us okay so we go through hold up a minute no we've got through here we go through here we go down this ramp and the door to our left or open up into a room where there's watchers this is the room where you can farm enemies it can I mean it doesn't matter you want to be spotted by these watchers it doesn't make you want the enemies coming towards you and you can just come here kill these enemies farm them then simply go back out of the room let the door close and you'll be sorted well there's one watcher down that went too hard oh they're detecting me oh no oh dear come on enemies what are you gonna do I know there's a few down below as well she's fine but you can see them coming from over there and you can literally farm these if you want to whoa I'm gonna fist a panic you mate where are the other enemies who's enemies down here as well they're down here look you can see big guy big guy big guy come on come on uh where are you where are you I'm not I'm not gonna shoot a rocket at you just dodge me now if you this is the room where if you um this is actually the pipeline room so if you uh eliminate all these watchers without them spotting you which would be pretty difficult on your own it can be done but it would be pretty difficult you can actually get into that chest room right there and you get back up by going on this fan here if you want to and go up up up and away there we go uh oh well yeah there's so many like I said there's so many different places you can find things here so yes guys have an explore see what you can find do you find anything decent let me know down below in that comment section but yes guys I hope you enjoyed the video if you're new around here and enjoy daily destiny videos be sure to subscribe if you never want to miss a video upload hit that bell button next that subscribe button to receive email notifications of when I do upload but thanks as always for stopping by and hopefully people I will see you on that next one


Thank you very much, and thank you again to the organizers for having me participate in this wonderful affair. Indira was concerned that my last lectures would be dry, but that will not be the case. So, let's get to it. First, let's talk about wall spaces. My goal today is to tell you how to get from a group to a cat zero cube complex. The main theme is going to be Mikhail Sagiev's Ph.D. thesis, which has been developed upon a lot in the last 20 years or so. So, wall spaces coincidentally are a notion that arose in Orsay, I believe, because of Frédéric Palin and Frédéric Arglond. So, a wall space is a set S with a collection script W of walls. Each wall in our collection is a partition of S into two subsets. We'll assume that the subsets are disjoint from each other, and we will also assume that no two walls are the same partition for technical reasons. Although a lot of what I'm going to describe has great flexibility to it, you'll see as we proceed. So, this definition, which is the most elegant form of the notion of wall space, is suggestive of quite a lot of generalization that will become apparent as we proceed. So, we're also going to require that the number of walls separating any two points in S is finite. So, this is the number of walls separating P and Q. A wall separates P and Q if P lies in one of the half spaces of the partition, and Q lies in the other half space of the partition. If you exchange the two parts. If you exchange the two parts. Yes, is it another one? We're not going to assume that each wall is just a partition, so it's a collection of two subsets. So, it's the same wall. The order doesn't matter, even though the notation suggests that it does. The order does not matter. It's not fixed, which is left, which is right? No, it's not. It's just a convenience of notation. And now, what's known as empty? We'll assume that they're both not empty. Otherwise, it makes a little bit of a mess for certain subtle points, but don't worry about it. Okay. You missed me on one empty set, by the way, that I was wrong on last time. So, what's the picture? The set is a set, some object, maybe. Maybe it has some additional structure to it, some geometry, maybe it's a metric space, who knows? We will see.  There's a yellow wall. So, look, I'm drawing the wall as if it were a wall. But we all are thinking, oh yeah, it's dividing it into a left half and a right half, or a half and another half. Never mind the order. And then there are other walls. There's a third one. I'm not going to be so generous with the chalk in the future, just this one time. Okay, so here's a wall space. It has five walls. Okay?  The yellow things, this yellow thing, I tend to call that the wall, right? And you'll see why soon. But if you're being very careful about this definition and being faithful to the definition, you have to decide for these yellow points, are they going to be on this side or on that side? Okay, so let me choose a way. This definition is flexible in that fashion. And it's actually quite a bit more flexible than that. So let's give some examples. So some examples that you're conscious of, perhaps. I mean, that was a perfectly good example over there. But what if we took, if we're being very, very faithful to the definition, and we took the zero, the vertices of a cat zero cube complex, and the walls correspond to the pairs of half spaces associated to the hyperplanes, right? So each hyperplane in the cat zero cube complex, when you just consider the vertices of the cat zero cube complex, that hyperplane cuts the vertices into the ones that are in one of its minor half spaces and the vertices that are in the other minor half space.  I suppose another example that the two Fred Rieks were motivated by was, I'll just say, coxsweter group or coxsweter complex, and the walls are associated. So that's going to be the set S is the coxsweter group, the elements of the coxsweter group, the walls will correspond to the reflection, the reflection walls, right? So each reflection, when you have this coxsweter complex, each reflection, or even if you imagine a coxsweter group acting in one of its usual ways, each reflection kind of give you a wall, okay? And so you can cook up a wall space from that. If we were being really, maybe I'll be particular about this. So for instance, in a dihedral group of order, so why should the, let's see, of order 3 or 6 depending on who you are, degree 3 I guess. You can see 3 walls over here, right, associated to 3 reflections, okay? And, you know, I personally, I'm not so picky about the idea that the walls should be, should be an actual partition. I'm okay with them being just some, a union that adds up, a union that is equal to S, and, you know, I allow the two half spaces to kind of intersect along what I like to think of as the wall. And that's a, that's a workable definition for what we're going to do, okay? And it's a little bit easier in a way because that way there is a thing that we actually call a wall, if you actually manage to get the two half spaces to intersect. But needless to say, over here, in the most elegant version of this definition, S would be these six points, right, the six vertices for instance, and we have our three walls. So now we talk about Sagiv's construction. It is going to be a process which, so this is going to be a construction which takes as, takes input a wall space and gives output a cat zero cube complex, which we call the dual to that wall space. So let me describe Sagiv's construction and really what I'm going to be doing is I'm going to be describing the cubes of the cube complex. So the main thing is the idea of a zero cube. That's the hardest thing over here. A zero cube in this output, in this dual cube complex, so maybe we, let me be a little formal here, a zero cube of this dual cube complex is a choice of half space for each wall. So what do I mean by choosing half space? Well, if you've got yourself a wall, then you could say I'm going to choose the left half space for the wall or you can choose, or you could say I'm going to choose the right half space for the wall. So you're going to need to choose a half space for each wall and this choice is going to have to satisfy two conditions. The most important one is that such that no two choices, no two chosen half spaces are disjoint. So what that means is that heuristically this picture is not allowed. Because I've chosen, it's kind of nice little communication device to put a little normal vector on your wall to indicate what your chosen half space is. This is going to be a choice for each of these two walls, but it's problematic because this chosen half space, the yellow chosen half space and the orange chosen half space are disjoint. And the other condition is a little bit, you can ignore it for now, but let me spell it out. All but finitely many choices, all but finitely many chosen half spaces contain some particular and hence any some fixed point little s0. And so here let's indicate the sort of behavior that we don't accept, although it can fail in other ways. So here's our point s0 in our set, the underlying set of the wall space. And then maybe there's a sequence, it doesn't have to look quite like this, but that's the easiest way to get the idea across. This is not allowed. So these are not allowed, forbidden. S0 is one of the points, so my set s probably it has a bunch of points. And you just choose some point, it doesn't matter which one, maybe this one, that's one of the points. And then we require that, well maybe there's a few of the walls that are, a few of the half spaces in our zero cube, maybe a few of the half spaces won't contain s0, but almost all of them do. But the question is quantified, do you say there exists s0 such that you also hold s0 equals? There exists for some fixed point, so you choose s0 ahead of time. Okay, and then, thank you, you choose s0 ahead of time. And then you want, and then it's also true because of this requirement over here that it doesn't depend upon your choice. Okay? Okay. Yeah, thank you. Okay, yes, please. How does two correspond to your picture? It looks like your choice of half spaces, interestingly many of them don't contain that first point. Right, so I don't like that. Okay. I like to draw the things that were not allowed. Okay. Huh? Yes, so, and they don't have to be in this kind of descending chain. They could be, you know, they could be often in all sorts of different directions. Right? They could have been a forbidden, it could have been a forbidden picture which looked like this, et cetera. And our point is over here, but there's more and more walls that are... That breaks the wall as well, right? Yes, it does. That's a great point. Who said that? So, I like this definition because I like it that a zero cube has a lot of stuff. So, I don't know, it's a lot of information. Now, we need to say what the one cubes are. We need to say what the one cubes are and it's actually pretty simple from here. I should really illustrate this. Let me say what the one cubes are and then I'm going to illustrate it, excuse me. So, there's going to be a one cube. It's going to join two zero cubes if they differ, if their associated choices differ on exactly one wall. So, I'm going to illustrate that in a second. And then, we're basically done. You then, we've built the one skeleton, add an n cube for larger n whenever you see its n minus one skeleton is there. So, you add higher cubes wherever you can, starting with the squares and then the three cubes and so forth. So, let's breathe. So, it turns out this is going to be cat zero. It's going to be simply connected and non-positively curved. Let's breathe some life into it with a few examples. So, maybe the most traditional example goes back a few hundred years, maybe 100, 150, something like that. So, this is a situation where no two of the walls cross each other, so to speak. We say the walls two walls cross if all four possible intersections of their half spaces intersect, have non-empty intersections. So, let me first draw the dual for you. Is there enough contrast between what do you guys like? Orange or red? Or orange or red? Red. I mean, I'm not talking about what you like. What do you see better? Red. Okay. So, this is, here's my wall space. My walls are the W's and my dual is this cube complex C. Okay? And let's look at one of the zero cubes and try to decide what... Let's look at one of the zero cubes and decide what choice of half space for each wall it's associated to. Okay? So, are you able to see the purple or not? Okay? So, that's a nice one. We're simply orienting everything towards this point right here. Okay? So, it's very easy now to start imagining other zero cubes. And how do you know that, how do you know the axioms are satisfied? Well, this is not a concern because any two half spaces contain my big point. Right? So, we're okay. And this one also. Let's now ask ourselves, well, what's the difference between the purple zero cube and the orange zero cube? With the orange zero cube, everything's going to be the same except for one small difference. And the difference is right here. I've taken this half space and I flipped it so that it goes in that direction. Okay? So, this one cube over here between the purple and orange zero cube is kind of associated specifically to this wall. Okay? So, now you see a one cube. Right? Well, now you know the one skeleton. It's going to get more complicated. There weren't any, there weren't any higher cubes to add. Right? Because we don't see there one skeleton. It's a tree. It's actually, it's not the most traditional usage of the word dual, but it's actually dual. Right? I think it's dual. It's a slightly more conventional dual, right? That's where we're even more familiar with the usage of the term. Okay? So, here again, these are my walls and I'm going to draw the dual and it's actually going to be even more familiar than the case of the tree. In this case, because there's a very simple, there's something very simple about this type of wall space that I've drawn. Each zero cube is actually going to correspond to a point in the complement of all of the yellow walls. And I suppose we should play, we should play the same game. Let's choose, let's focus on a particular zero cube. And it's again the case that it's associated to the choice of, it's associated to the choice where we're directing all these normal vectors towards that zero cube. That type of zero cube is called the canonical zero cube. So, and then, you know, if we flipped it over to this orange one, you know what the change is going to be. It's just that change right over there. So I'll just put the change that way. And if we flipped it to this green one, then there'd be this change here. Green is not so good. Let's see what this does. To check that those are the only zero cubes, it's, of course, it's a finite problem, but is there a way to do it, it's hard going over all the possibilities. What do you, I don't know what that means. So you define what a zero cube is in terms of the choice of half spaces. So you have some evidence, zero cubes correspond to the point you've drawn. So maybe there's a surprise zero cube that we didn't see. Yeah, we're going to do that. So let's, maybe I should move forward. So in a way, all this picture is exhibiting.  If you flip them both, you get here, right? You get to this white zero cube. If you flip, if you flip the, if you flip this one, maybe the white one's more interesting. If you flip that wall and you also flip this wall, right? And you leave all the other walls just like the purple zero cube wanted. Then you get this zero cube. Okay, so we need to do more exciting example. So I'm going to do that, right? Because otherwise this isn't so interesting because it's just like something that we all know about. We learned about when we were, you know, when we were in high school, I guess, right?  So, so what's a more interesting example? Okay, so we already know the game and we know about these canonical zero cubes, right? Where, where you just, where you just orient all the walls towards some point and that's going to satisfy the axioms. And we found seven, we found seven and maybe, maybe it's a good idea just this one time to, to do this. Okay, and what I'm going to do is let's call our walls the, the first, second, and third walls. And actually let's direct them. Let's make the, make this choice so that we're starting at this zero cube right over here. Okay, that's going to be our sort of neutral choice. And now let's, let's name all of the other zero cubes, right? Because we can make a little naming system now. I suppose maybe you'll forgive me for calling neutral one and let's, let's name, let's name them 111. That's all of these and forgive me for, for the number one being that, that just means neutral. So 111 and let me, let me proceed. 110, 100, 101, 001, 011, 010, right? So, so 010, that's this, that's this guy right over here, right? That's 010. And why is it called 010? Because we've flipped the, we've flipped the first wall. We kept the second wall in the, in its, in its original position so it hasn't changed. And we've flipped the third wall, right? So, so, right? So we, you got from 111 to 010 by flipping two walls. Now what about, so, so now we see that there's another zero cube that we left out before, right? And that's, that's 0, 0, 0. We like 0. So, so that's the, the type of zero cube, a sort of secret, secret zero cube that was hidden. And well that, that corresponds to, that corresponds to this orange cube. It, it corresponds to reversing all three of these. And you look at these three orange chosen half spaces and you see that their, their, their triple intersection is empty. So the 0, 0, 0 cube is, is not a, what I call the canonical cube, right? So, excuse me, canonical zero cube. Thank you. Okay, I, I said what a, what a canonical zero cube was. Shall I, shall I say it? Shall I say it again? I'll put it over here. But it really belongs right after we defined what a zero cube and a one cube was and we said what the cube complex was. It's an observation that zero cubes exist in the dual. Just orient all half spaces, orient all walls towards a point of s. And you'll get, let's name that point, little s, obtain a canonical zero cube, cs. What, what was your question? If s is empty, there's no such canonical. If s is empty, there's no, there's no, okay, good. Okay, you're fine. So, yeah, so the truth is, the truth is, I'll give you a private confession now. I too love the empty set and I normally don't even, I, I, it was just as a public service that I told you about the set s. Because I, I only imagine the walls, right?  And, and then I just make sure that I don't have two walls pointing away from each other and I don't have, you know, certain, certain terrible things. Actually the second type of terrible thing doesn't bother me very much either because it's really some, it's something at infinity which I'm actually quite happy with. And points, points in s, I never give them much consideration, okay? So, oh my goodness, I'm super, super slow today. Exercises, well, started at 9.15, all right? Time flies when you're having fun. So let's, let's just call it an exercise that C is non-positively curved and C is simply connected. Both of these are actually easy. One of them I even put in the exercises even though I'm sure it's one of the ones that you're not going to get up to because I put too many. And let's move forward. Let's discuss the notion of a co-dimension one subgroup. So I guess let me sort of invite you to exercises. So one of the things that I will ask in the, in the, in our exercises is, you know, to compute the dual to a system of N families. Of parallel walls in R2, for example. So you can have some orange ones, some red ones, right? And I'm imagining that each of these families is infinite. So you get some nice wall space like this, right? And I want you to figure, figure out exactly what the dual cube complex looks like. You will be able to, okay? Let's now talk about co-dimension one subgroups. Yes?  In the first two pictures, there are no non-canonical zero cubes. So exercise when, when W does not have three pairwise crossing walls, there are no non-canonical yuck. Every zero cube is, is canonical. Thank you. Where were you when I was writing, making my exercises? Huh? That's, that's good. So, and that's very well, very well observed. And that somehow things get interesting as soon as, you know, as soon as you have a bit of a mess, right? And it's, it's exciting because there's a, we love dual because a dual means a secret, right? That there's a secret and then we feel that we're in the game and other people don't know. So if the secret gets exciting as soon as you're starting off with a wall space that seems kind of tame and you look, and there's a dual and the dual is something gargantuan, something maybe could be infinite dimensional. And, you know, with a very, very rich local, local complexity. So that's actually what happens. So let's, let's see where these come from. So I'm not talking about co-dimensional and subgroups. G be a finitely generated group with caligraph epsilon. A subgroup H of G is co-dimension 1 if there exists r more than 0 such that the caligraph minus the r neighborhood of the subgroup, right, which is just a subset of the vertices of the caligraph has at least two H orbits of components, K that are in the sense that K is not contained in any finite neighborhood of H. Okay, I need to, it's a little bit of a mouthful. So what you should be, let me spell out examples before we try to express the definition a little bit more humane form, because really we should be giving an incorrect definition which is more humane, and then we'll think about it a little more carefully. But just an example quickly to have in mind, any copy of Zn in Zn plus 1 is co-dimension 1. Any cyclic subgroup in the fundamental group of a surface of an orientable surface is co-dimension 1, right? So this is sort of suggesting the terminology. What's the picture? Well, let's first look at a very explicit example of Z inside of Z squared. So here's the caligraph of Z squared and here's a, here's a, like orange, like red. Here's a cyclic subgroup, maybe a nice diagonal cyclic subgroup. And the first part of this definition is, well, the subgroup should cut the caligraph in half, right? It should cut the caligraph. It should coarsely cut it. We want the caligraph minus the neighborhood of the subgroup should have more than one component. That's the first thing that we want, okay? So, well, I guess it already cut it right now. Maybe this wasn't an exciting example. Maybe I'll leave it as it is. I'll change it here, here. Now I chose a slightly, I chose an index 2 subgroup of the one that I had before. Okay, so now my subgroup does not quite cut the caligraph when you remove the subgroup from the caligraph. It doesn't cut it because you can sort of still travel through, right? There's a little gap over here. But you could take a neighborhood, right? And so is it a closed neighborhood or an open neighborhood? It doesn't really matter. The definitions don't really matter. You can, let's assume it's a closed neighborhood, okay? And let's take a two neighborhood. Let's take a two neighborhood of this subgroup. What does it look like? One, two, one, two, one, two. I think that's what it looks like, right? And now, as you can see, this yellow neighborhood of my subgroup. So I have the caligraph of the group. I have my subgroup. I have the, what, two neighborhood of the subgroup, okay? And it cuts, that yellow neighborhood cuts the caligraph into two, it cuts it into two components. But we want, we want those components, there's two things that we want to say about the components. One is that we want those components to be deep, right? We don't want, we want those components to be non-trivial, to be sort of impressive. So the way to state that is to say that they get very, very far from the subgroup. And that's the case over here, okay? So these components are deep. The two white components that you see are deep. They go far from the subgroup. And the other thing that you see is that, this is a little more subtle, has at least two H orbits of components, right? So actually when you look at how the subgroup is acting on this situation over here, it's quite nice. It keeps, there's actually only two H orbits, in fact, right? The top white component and the bottom white component don't get exchanged by the action of the subgroup. Now, if this had been maybe a Klein bottle, right? This could have been the caligraph of a Klein bottle, for example, of a Klein bottle group. And maybe the subgroup might have been acting by a glide reflection, then how many H orbits would there be then? Right, if H were a glide reflection, there'd only be one, and it wouldn't count as a co-dimensional one subgroup. Okay, so those are, and if you look in the literature, you'll see people make mistakes about that, myself as well, many times, especially in the early literature on this topic. I mean, so the notion of co-dimensional one subgroup is a very old notion, but it kind of became really very important after Sagiib's work. So let's see how this is now related to wall spaces. Oh, I should say this. Here's an alternate and elegant, alternate definition of co-dimension one is the following, comes from the following. So H is co-dimension one, if and only if, so this is a, G is this finitely generated group with caligraph, epsilon, H is co-dimension one, if and only if the coset diagram, the coset graph, people sometimes call it the Schreier graph, has more than one end. Okay, so if you have a graph, the number of ends of the graph, having more than one end means that you can remove a compact subgraph and get several infinite components that are, and get several, and get more than one component, which is deep in the sense that it's far from this compact subset. Okay, so there's a way of defining the number of ends. This graph will have three ends. This graph over here, a kind of train track, will have two ends, right? Because you're kind of cutting just this one point, doesn't do anything, but cutting a little more, you've got two ends over there, and so forth. On the other hand, this caligraph over here only has one end, because no matter what compact subgraph you remove, there'll only be one infinite. So if it's a locally finite graph, let's just stick to a locally finite graph, there'll only be one infinite comp-metric component when you remove a compact subset. Okay, so the number of ends is the maximal number, or the largest number allowing infinity, or the limit of the supremum of the number of infinite complementary components as you remove larger and larger compact sets. You've uncountably many ends. If you define the space of the end as the inverse limit, then if they are all finite, but in some cases you could get like a control set of ends. Well, okay, I don't want to stop and think about that. So I think I'm going to continue, and then I'll talk to you about it. I'd love to talk to you about it afterwards, okay? So to think about this lemma, what's happening actually is that the neighborhood, when you think about what's going on with that lemma, the neighborhood that you're removing, the neighborhood of the subgroup that you're removing, is going to correspond to this compact set that you're removing in the coset diagram, right? This is the graph that we're interested in. We want to say that it has more than one end. So the neighborhood that you're removing, the neighborhood of your subgroup that you're removing, is this compact set that you're removing from the coset diagram. And you want to have at least two infinite complementary components. Well, the complementary components are going to correspond to the H-orbitz. The complementary components over here in the coset diagram are going to correspond to the H-orbitz of deep components that we're interested in. That's the sort of connection to this lemma, and I think it's a much more elegant way of stating it, but maybe not always the way we think about it. So let's now conclude this by explaining the work. Let's now explain how we're going to get from a co-dimension one subgroup to a wall space. We already know how to get from a wall space to a cat zero cube complex. We've discussed the co-dimension one subgroup. Let's talk about how to get from a co-dimension one subgroup to a wall space. So what did Sageev do is he, the co-dimension one subgroup, here's his construction. Let H be co-dimension one subgroup. Choose R such that the R neighborhood of H has at least two H-orbitz of deep components. In particular, it has some deep component K, and then let W be the wall where one part is K and the other part is its complement.  Our collection of walls just be the set of all translates of these partitions. So we obtain, so this is going to give us a wall space on G. So S is going to be G. Well, I guess you can let S be the caligraph if you'd like as well. That's fine. This yields a wall space with a G action.  So this is the game. So the picture is the following, very simple picture. Here's our caligraph and here's our subgroup and it has a little neighborhood, which is cutting our caligraph up in an interesting, in an essential way. And then, well, we know that there's lurking over here, there's a way of deciding, right, just maybe choose K to be one of these, choose K to be one of the deep components. There's a lot of leeway when you're playing the game over here. K would be one of these and then the complement of K, so that's going to be left W. And then the complement of K, if you're being very particular about this, this might be right W over there, so that's the complement of K.  And then when you take that wall in the caligraph, so I'm just looking at it in a kind of more summary, heuristic fashion, and when I take that wall and I translate it around by the group action, that's that entire collection of translates of that wall, so there's going to be this one and this one and this one and this one, those are just translates. And, well, probably a translates of this wall is not going to cross itself. So when you translate this wall, well, maybe it crosses itself, so you might get that one and that one over there and this one and so forth. And now you have a whole bunch of walls, and the group is acting, permuting these walls around. And so the dual cube complex, which we now know how to define, we're already prepared for that, is likewise going to get acted on, I think I switched my conventions, but there's this dual cube complex, and it's a very concrete example of here, it looks like everything is canonical, but it's not usually so simple, okay, it isn't really such a big deal either way. Well, this cube complex C is our goal and the action of the group on the wall space gives you an action of the group on C. Okay, so G-ax on C. All right, well, of course this construction works with several co-dimension 1 subgroups and many things about the setup can be flexed a little bit. The one thing that you might consider, it's an interesting exercise, why is the number of points separated by walls finite?  Okay, so that's more or less the reason. Another thing to note, and then I'll stop there, is that as you can maybe guess from the construction, the stabilizers of hyperplanes of the dual cube complex are pretty much end up being commensurable with conjugates of our co-dimensions 1 subgroup H. Okay, so I'll stop now and I have to stop. Good.


Hello, thank you for joining. We are just going to wait a couple of minutes here for everyone to get logged on. Hello, thank you for joining. We are just going to wait a couple of minutes here for everyone to get logged on. Hi, everybody. Thank you for joining us. Welcome to the Hype Ledger in-depth webinar. We'll just get started in about a minute or two. Hi, everyone. I'll just give people another minute to join and we'll get started. Okay. Good morning, good afternoon, and good evening from wherever you are. I'd love to get started here today with our next Hype Ledger Foundation in-depth webinar with Casper Labs. We have Meta Parlikar here, CTO and co-founder of Casper Labs who's going to share with you about how they are unlocking a $3 trillion market for businesses. I'm Karen O'Toni. I'm the Senior Director of Ecosystem and Strategic Initiatives at Hype Ledger Foundation and I'll be hosting the webinar today. I have a couple of housekeeping things to go through before we dive in. One is Hype Ledger is committed to promoting diversity inclusion and making sure that we are a welcoming community. So we emphasize creating a safe and welcoming community for all and that includes how we manage ourselves in the chat and in any community meeting that we're having or event. We also go by the Linux Foundation anti-trust policy and meetings and so we have to conduct our activities and according to that, if you have any questions about the anti-trust policy, you can see it on our website or on our wiki at wiki.hypeledger.org. We are currently recording this and we are also live on YouTube. The recording will be available on the webinar page of our website after the session as well in our YouTube library of webinars and the slides will be available on the webinar page in our website as well. So don't worry, you will get them. We really encourage these sessions to be interactive. This is an opportunity for you to get a chance to speak with someone who's working on something that's of interest to you, solving a problem or challenge that you might be having as well. So please feel free to raise your hand, post questions in the Q&A box and I will be monitoring them throughout the session both in the webinar and on the YouTube live as well. So without further ado, let me welcome Meta. Please thank you for joining us and I will unshare my screen so you can get started. Terrific, Karen. Thank you so much and thanks for welcoming me here today. You're so excited to be part of the Hyperledger community and really excited to share some knowledge and hear feedback from members of the Hyperledger community about CASPER Labs and the content that we're sharing here today. So I'll just dig in. About me, I'm Meta Parlakar. I'm the Chief Technology Officer and one of the co-founders of CASPER Labs. I hail from the Web2, Cloud, SAS, whatever you want to call it space. I worked there for about 25 years as a technical leader in large organizations like Adobe, Avalara, and mp3.com. I've worked in content delivery, web analytics, fintech, and compliance, where I was responsible for product and program management and quality control. So let's dig in. So when I met Rinal, we really had a meeting of the minds. Rinal is one of my co-founders and we discovered that Enterprise needed a few things. They needed a public protocol that really solved the adoption issues that we were seeing around public blockchain infrastructure.  And they also needed a professional services company to help enterprises adopt the technology. And so from there we found at CASPER Labs in 2018. And we founded the company during a crypto winter and we built the technology during a pandemic. And so we have been very steadfast in our belief that enterprises are central to scaling the adoption of blockchain technology. And in order to do that, blockchain technology had to scale within the enterprise itself. And let's start by talking about trade-offs, namely, you know, scalability for security. And we know that blockchains, the most important property is the security of public blockchains. We also know that proof of work is very inefficient and not sustainable. And many proof of stake protocols will sacrifice security for scale, right? So they have optimistic consensus protocols or they take shortcuts in order to make the protocol go faster. And we've observed that a lot of these protocols are either not stable or it takes a very small number of participants to halt the protocol or bring the protocol down or compromise security. And this is one of the reasons why Hyperledger has broad adoption in the enterprise because it's able to scale adoption within the enterprise, right? It provides a lot of the controls that enterprises need that is not available in a public blockchain. And so what we wanted to do was kind of thread that needle and provide as much of the controls we could possibly provide to enterprise in a public manner, right? So what this means is that the blockchain itself is public versus private, but enterprises still get a lot of the control that they need in order to build a sustainable and scalable business using blockchain technology or products using blockchain technology. And so if you think about the adoption trilemma, you know, what would a chain that solve that really look like? And it was really clear that I needed to build a blockchain, public blockchain that I would use as an enterprise technology leader. So it needed to be secure and decentralized, but it needed to interoperate with my existing IT stacks. And it needed to be scalable within my organization and had to meet the needs of all of my stakeholders, right? I had to be able to use development operations. I had to be able to upgrade my contracts and introduce new code changes. I needed to have some guarantees around the controls and the security operations and best practices that my organization needed around governing that code as all code is governed within an enterprise organization. And so I really see the Casper Labs and Hyperledger Union as kind of a match made in heaven, right? So Hyperledger pioneered enterprise-grade private blockchain deployments, and it's emerged as a clear leader in the broader enterprise blockchain space. And like Hyperledger, we believe that enterprises will be central to scaling out the adoption of blockchain technology. But sometimes there is value in a public network, right? And Casper's built to provide many of these benefits that enterprise needs, but as a public network to bring the best of the two worlds together, right? And so last May, we demonstrated the first ever atomic swap between Hyperledger Fabric and Casper in tandem with our partners at IBM. And we're seeing a growing demand from businesses for this exact type of functionality, and Casper's really uniquely poised to deliver this. So enterprises really led the way from mass adoption of the internet, and they're going to do the exact same thing with blockchain, right? And we see ourselves positioned as a red hat to the open source Linux, right? Us being the red hat and then Linux, of course, being the blockchain technology that's Casper. And businesses are projected to create more than $3 trillion in value by 2030 from blockchain use cases alone, according to Gartner, right? And 93% of global executives believe that blockchain unlocks significant new revenue potential according to Deloitte. And 75% of executives believe that a failure to adopt blockchain technology in a timely manner is going to result in a loss of competitive advantages according to Deloitte. And even in an inflationary market, IT departments are continuing to invest in blockchain at higher rates year over year. And in 2021, it topped $6.5 billion alone. So there's a massive demand for a massively underserved market that we believe we're well positioned to provide support. And we've always been focused on the enterprise market, we always will be. And public blockchains have largely ignored this key market for the past four years, but not us. And we're going to continue building this over the next five to 10 years as well. So let's talk in a little bit in depth as to why Casper, right? I teased this around like upgradable smart contracts, right? And imagine somebody telling you that you can't change your code once you've installed it. And for me, that's absolutely where the story ends, right? Like I would never use a protocol platform with purely immutable code, purely immutable smart contracts. And this is where a lot of blockchains kind of end the story in the public domain. But you have to go further than just providing upgradable contracts, you need to provide strong security guarantees around the versioning of smart contracts on the blockchain. And this is something that only Casper Labs does, right? There are ways to upgrade other blockchain contracts on public networks, but you don't get any security guarantees around the semantic versioning of those contracts, which becomes massively important when you're doing any kind of transactions with value, right? You cannot afford to have any kind of security vulnerability or supply chain vulnerability within your code base. The other piece to think about is how is this blockchain going to interoperate with your enterprise infrastructure, right? And we're not talking about an integration with other blockchains. I don't really see that as a core issue. I'm talking about plugging directly into the systems that power your business, right? Like your business intelligence systems or the IDEs that your developers use or the tools that your network operations team used to manage all of your back end servers. I'm talking about those systems, right? How does blockchain become part of this larger infrastructure that is deployed in a lot of, you know, large softwares and service companies or inside the IT departments of the enterprise, right? And the CASPER virtual machine is built on Web3 W3C WebAssembly. And the CASPER Labs team is uniquely positioned to allow our customers and our community is supportive of allowing enterprise customers to drive the roadmap of the VM. And this is an extremely unique value proposition that we bring, whereas, you know, if you need a feature added to the Ethereum virtual machine, you're never going to get it. You'll just have to work around with what the EVM offers. With CASPER and CASPER Labs, we can actually drive that roadmap. And we have multiple examples where our enterprise customers have come back to us and said, hey, we see this as a really important feature for you to add to the blockchain host itself. And we've worked with our community to get those features deployed to the mainnet. And this is a unique and also a traditional way in which great products are built is through the customer, the voice of customer feedback loop, right? And this is a fundamentally different architecture, both from our business perspective, as well as our software perspective in terms of the blockchains themselves. We're also built for the other 25 million developers, right? Because you don't need to know Solidity. You don't have to have an ivory tower within your organization where you've got that one or two specialists that have this special niche capability about the public blockchain, right? The idea is, you want to, if you're going to implement a technology within your infrastructure, you want to get as much value out of that technology as possible. And blockchain, the magic happens when you start using it as a proof, a centralized proof within your block, within your systems, where you can trust that data. And so if you've got like KYC information that's going on a blockchain, you want to be able to plug into that data with other teams. So it's important, even with public blockchain infrastructure, that you scale out that adoption internally so you can extract as much value as you can out of the technology, right? So all of this is available within the Casper ecosystem for you to use. We believe that the future is hybrid and we're built exactly for that reality, right? So most large firms and government organizations are never going to be able to deploy in fully decentralized environments, right? They need to retain privacy and control, but blockchain still present this tremendous opportunity to save money and open new revenue streams. And so hybrid is a great way to do this, right? You can get security guarantees from the public network without having to put sensitive data or sensitive information on the public network, and you can selectively choose how you want to interface with the public network and control on Casper uniquely, your interactions with that public network, because we provide upgradeable smart contracts. We provide a very rich and robust mechanism for you to control accounts and what accounts can do on the Casper protocol. Hi, Metta. So just going to jump in here, we got a question from someone on YouTube and maybe you'll cover this later, but they're asking if you could share some examples of why we want upgradeable smart contracts, as it may be a threat to security and more centralized devs can change the code at any point of time, perhaps exploit things. So why is that beneficial and how is it safe, I guess? Yeah, so it's absolutely beneficial for enterprises and entities that need to maintain regulatory compliance as a great example, right? So let's say there's a regulatory shift, right? And you need to now do something different, say with a set of NFTs, say with some fungible token standards, maybe you need to make a change where something that wasn't transferable before is transferable now. For whatever reason, you want to take advantage of a new market opportunity, right? You have to be able to do that in a safe manner with some guarantees. It's true that upgradeable smart contracts can present a challenge where you could have a bad actor do a bad things, but the flip side is technology doesn't solve the bad actor problem, right? You can also have a really bad vulnerability. You can have a really bad bug that you can't fix because it's immutable, right? So it cuts both ways, right? And I actually believe that trust is going to be established off-chain. I believe that today and even in the future, individuals will interact with enterprises and interact with authorities or individuals or enterprises or entities that they trust. So trust is still going to be established outside of the smart contract. So if I believe this is a good actor, I'm going to trust them, right? So it's possible to have immutable smart contracts on Casper. It's not that it's impossible. It's a contract that can never be upgraded, right? But it allows you to build your contracts incrementally. It allows you to address defects. It allows you to upgrade to maintain regulatory compliance. And if you think about it, all the software that we used today was not written in one go. Any software we've used, right? It's always been incrementally developed. And it is important to recognize that while the history of a transaction needs to be immutable, our future simply is not immutable, right? And so smart contracts need to provide that capability where the blockchain needs more of that capability that smart contracts can also adjust, evolve, adapt to what the needs of the enterprise and regulatory and customers are for the future, right? And we're seeing a lot that this is an extremely important requirement for a lot of enterprises in order to adopt blockchain technology. So in wrapping, what's our ambition is you want to onboard the next 100 million users onto blockchain technology. And they're increasingly look like cloud users today, right? AKA everyday consumers who are customers of major companies. And we believe that through enterprise adoption is how those 100 million users are going to use blockchain technology. And they're not even going to know that they're necessarily using blockchain technology, right? The blockchain is going to be a very small piece of a larger application architecture. It's going to fade to the background and users will get the benefits of the blockchain technology, be it and more guarantees, new features, and better usability for the products and services that they're used to using today. And so our mission is to handle the back end of this blockchain technology so businesses can focus on solving their core problems and delivering an exceptional customer experience. So the time is now for businesses to harness the power of blockchain and come build with us. And I'm open to hearing questions from folks that are attending the webinar. Thank you so much for your attention. Thanks, Metta. If anyone wants to come off mute, I can just raise your hand and I'm happy to give you the floor. I also just see someone, someone is just saying thanks for the information and they're a big fan of your work on YouTube. Metta, I was wondering if maybe you could talk through a little bit more in detail about the example that you did on the hybrid transaction with IBM. If you could share any more details on how that worked. And maybe were there any challenges that happened in that example that you really had to kind of work through that others might encounter when they're doing the same? Oh, yeah. So we have built a bridge. We've built a hyper ledger to a token bridge between hyper ledger and Casper. It's available in the hyper ledger repository and GitHub. And what this bridge allows folks to do is if you have tokenized assets on hyper ledger, you can bridge those assets over to Casper to gain liquidity. And this is an extremely interesting use case for particularly financial institutions that are looking to offer, for example, tokenized assets on a public network and get access to liquidity on those networks. And the way it is, it's a time locked transaction. So it's like a bid and ask if you think about it. So somebody puts up a tokenized asset on the hyper ledger side into the bridge. And somebody is willing to buy that. They exchange a shared secret. So there's a shared secret that they exchange. And when both confirm the shared secret, then the transaction takes place and the assets are basically bridged from Alice to Bob. And this is I'm kind of glossing over some of the technical details. But in essence, both parties have to acknowledge that, yes, I have exchanged the shared secret. The monies have been exchanged. And I have actually purchased this asset and now bridge it over to me on the public protocol. And so this is an extremely interesting way in which you can gain access to a larger liquidity pool using a public blockchain protocol. You can also do other interesting things. You can bridge NFT assets over. And if you want to bridge other tokenized kinds of assets over, it's a great use case for that. It's also a great use case for you to get provability that your hyper ledger instance has full immutability. Because if there isn't any decentralization, we know that security guarantees do go down. And so there may be compliance use cases and other use cases where you want to prove that, yes, the hyper ledger instance, we have everything as fully immutable. We haven't gone back and rewritten history. We haven't gone back and tampered with the global state. And increasingly, those kinds of that provability of immutability is eventually going to become more and more important as consumers and others become more aware of the capabilities of blockchain. And so tapping into a public protocol gives you great guarantees around that because the public protocol, there is no way to change anything on the Casper protocol unless the two thirds of the community agreed to do so. And that's more of like, I want to say, like an existential event, if that would ever happen. It's only happened once in Ethereum's entire history. It's never happened at Bitcoin as far as I know. And it's never happened in Casper. So I think there's a lot of value in that. And it's very nominal, right, for private blockchains to take advantage of public infrastructure this way. It's very low risk and a lot of benefit. All right. Thank you. So we have a couple questions coming in. So can you give an example of an asset that is a good use case for a public private hybrid chain? Yeah. So an example would be like, gosh, so we have a joint venture we call Nucleus. That's Nucleus Finance, right? And they are working with Nucleus Finance is a collaboration between Casper Labs and the researchers of Octus. And the Octus standard is well established in a lot of financial institutions, a lot of banking institutions. And here we are an example is we are creating tokenized assets on private blockchains. So as an example, think of leases, right, automobile leases. So you want to tokenize automobile leases, but perhaps you want to securitize some of those automobile leases by offering some kind of a security offering. So you can actually get transparency around. So you can give some guarantees that we have these tokenized assets, and you can provide guarantees to whoever's participating in the securitization of those assets that we're tracking these on a private chain, and that they are then bridged over to a public chain in the form of a security offering, where now you want to get liquidity off of it, right? So a lot of the use cases you're going to find of a tokenized asset is really going to be a case where you want to get liquidity off of the public chain and really be able to transact those those securitized offerings on the public blockchain. Okay. Let me see. Juan, and I'm sorry, I didn't offer this to you, John. I forgot. Juan, do you want to come off of mute and ask your question to Mehta? If not, I'm happy to ask for you. But just want to give you the opportunity to jump in here if you like. Hi Juan. Can you hear me? Hi, so my question is, I know that the Odra release was announced today. How does that relate to creating smart contracting work and how would that change moving forward? Yeah, absolutely. So the Odra release is actually a layer on top of the Casper-based smart contracting framework. It streamlines and simplifies some of the, think of it like a set of macros, right? It's like a set of macros and a framework that makes smart contract development on Casper easier for developers to get started with. So I don't feel like, you know, you can still write smart contracts natively in Rust, if you wish to, or you can use the Odra framework to help speed up the development of smart contracts on Casper vis-a-vis writing them from scratch using Rust. So Odra was actually developed by one of the former core developers on Casper and it's a community-driven initiative. So we're really excited about the release and we hope that it'll make things a lot easier for, you know, solidity developers to get started on Casper without having the big, you know, the big lift of Rust. I hope that answers your question. And that's going to change the WebAssembly architecture, right? Not at all. It's going to be using WebAssembly. Right, 100%. Yes, 100%. And what's interesting about the Casper virtual machine, I didn't really dig into this that much, is the way the Casper architecture works is transactions are executed after consensus. And so, you know, this is actually very different than anywhere else in the space. And what makes it really unique is that you don't need to know anything about consensus or the underlying node semantics for to build and run a smart contract. And this enables Casper contracts to actually be built and run in an IDE. The Casper virtual machine can be run on a build server and Casper contracts can be built natively using continuous integration and continuous deployment. And the reason for this is we keep WebAssembly very, very pure. You know, if you work on some of the other public blockchains, their implementations of WebAssembly basically yank a lot of like they add a lot of, you know, complication to the WebAssembly layer. So now you need to understand how Solana works, you need to understand how Nier works, you have to understand how Polkadot works in order to actually write the smart contracts effectively. And now you can no longer run them using CI CD, because you need those full nodes, you need the full semantic of the blockchain in order to run the smart contracts. Right. So with Casper, you don't need that. We kept the WebAssembly piece very, very, very pure. There's additional capability within Rust on the host that the host enforces, but like wasm is very, very pure. So you can actually send any kind of wasm in any language that compiles to wasm and it will run on Casper. That's super exciting. Thank you for that update, Rick. Thanks. Thank you, Juan, for the question. And John, if you have any follow-up questions, feel free to jump in as well. There was a couple of questions here on our YouTube attendees from Agata. What does Casper Labs take on consolidation versus decentralization? I believe that the underlying blockchain infrastructure absolutely has to remain decentralized as decentralized as possible. The Casper Association and the Validator set work really, really hard to make sure that our Nakamoto coefficient is always on the rise. Right. So we are looking to drive decentralization. So there's more and more participants in the network, more and more spread in terms of how the stake is distributed in the network. So it's more and more decentralized. Now, with respects to what's built on chain, right, like we believe that enterprises need to be responsible and accountable to regulatory bodies. They're responsible and accountable to boards of directors, to customers. And so whatever's built on chain is going to be governed. It'll need to be governed by those enterprises. And so they need to control what's built on chain. So I believe that the underlying blockchain has to provide tremendous security. And in order to do that, it has to be very, very decentralized. But I also believe that anything built on top of the blockchain, like all software is going to be governed, right? And the protocol should provide the necessary tools and capabilities for participants to govern their on-chain contracts in the manner that they see fit. So what does that mean? Well, it means like, if an account controls a contract, I should be able to set up multi-signature for upgrades and deployments of that contract. Well, the host provides that capability, right? So if I want to build a decentralized, autonomous organization, and I deem that there's 50 participants that decide whether there's a change made to how the DAO contracts function, the CASPA protocol will support that on the host, right? And we do that through our very rich account model. We provide capability to do smart contract upgrades. Then we leave it up to those participants on how they want to govern their contracts, right? Rather than me dictating that or us dictating that. And so it's all about providing those capabilities within the host so that people that want to use the infrastructure, whomever they may be, can decide how they want to use it and how they want to govern it. That's awesome. Another question here is how does CASPA labs ensure that node power will not be consolidated? So with our CASPA 2.0 release that's planned next year, we're going to be opening it up the validator set even more. Right now we're limited to 100 validators or 100 nodes that can participate in the network. With the ZUG consensus protocol release, which is slated for 2023, we should be able to open that up well over 250. And based on our research, there's no network that really has more than 250 validators in the system. And we feel that this will actually help the network decentralize even more. So we are well aware of this risk of node consolidation. And our goal is to make it more robust over time. And we work, the community and the validator set and the association works to ensure that there is not node consolidation. We work with even the staking providers. Some of the larger staking providers have been pushing forward to actually provision more node infrastructure and select more staking providers to help the network remain decentralized. So it's definitely a focus for us. And will there be any utility of the Casper token for private companies using the blockchain Casper? Will there be utility for private companies of the CSPR token using. So if companies choose to use a hybrid model, right, if they're going to use the hybrid blockchain, they're going to be using the Casper public network to get those security guarantees on those private chains. So that will be the way enterprises will use the public protocol is really to secure those private chains, right? So then they can prove to auditors or customers or whomever they need to prove to that our blockchain is in fact immutable, right? And we think that's an extremely important value proposition that the public network brings. Great. And let me know if you have any other questions, please feel free to post in the Q&A or in the chat. You know, we had our Hyperledger member summit in Dublin, Ireland, Metta, you were there with us. It was our first time back in person, which was really exciting. And you know, one of the sessions we had was on hybrid blockchains. And it's something that I think our community is going to be more and more engaged in, especially with what you're all doing and offering. And so I kind of just wanted to zoom out a little bit and get your perspective on what you think is going to be what you think 2023 is really going to be about in terms of how much we're going to see exploration, maybe even production examples of hybrid blockchain networks and how the people in the audience can get involved in exploring that with us. Yeah. So it was really terrific to be there in Dublin with everybody. It was such a wonderful conference and I learned a ton. Firstly, I thought it was one of the few conferences where I went to actually attend all these sessions and each one of them was super informative. So usually when I go to a conference, I'm not as drawn to the sessions as I was the Hyperledger conference. So it was just really, really tremendous. I'm super excited. So in January, IPV who happens to be Hyperledger customers actually going live with their hybrid blockchain implementation on Casper, they've already started minting tokens on the Casper mainnet and our goal is to mint a million tokens on Casper by January. And we're really, really excited about this. I think it's going to be a fantastic demonstration of the way hybrid blockchains can work. So the patents are secured with the infrastructure of a public network. It's public data that is publicly available and the provenance and the ownership of that patent is public information. But the terms of sale and the negotiation and how the buy and sell and the leasing of those patents is done on the private blockchain. So this is an excellent example of where the terms are negotiated using a private Hyperledger instance. And then once the sale is complete, the patent is transferred or minted on Casper on the public network. And it provides really best of both worlds because it's very clear in terms of who owns a patent and you can observe the patent as well publicly because that provenance privacy was not known. It was not known who owned a patent. It was only known that once the USPTO issued the patent, that was the initial owner. But then after that, if the patent was sold, transferred leased, there was no information around what the current state of the ownership of that patent was. So this is happening in 2023. I'm really excited about that. And we have some other initiatives which I can't go into a lot of detail with unfortunately due to NDAs. But there's multiple hybrid blockchain implementations we are looking to deploy some for many household names. So we're really excited about this work. These are private public Casper blockchains. We have a few private Hyperledger blockchains that are also interested in a hybrid deployment with Casper. So we're super excited about that as well. So there's lots of work ahead. Yeah, tons of work ahead. More than I it's pretty daunting, but it's very, very exciting. John, you raised your hand. Yes, thank you. So thanks again for doing the presentation. One of the things that brought me to Hyperledger was really the cost control of having the blockchain history, but still keeping it private. And I'm very interested in the hybrid model, but I'm curious with Casper tokens. What's the risk of it? So we do very, very small transaction sizes and it's prohibitive to do, you know, Ethereum gas fees, for instance, are just prohibitive. So what's the cost model slash what's the risk of having a public token in the change of costs for an enterprise? Yep. So we recognize that enterprise infrastructure needs to be budgeted for and Casper has, we have a unique model. So presently in mainnet right now, the way Casper mainnet functions, it functions very similar to Ethereum. There is gas fees. Of course, the Casper chain isn't nearly as busy as Ethereum. So presently, it's not as expensive, but we want to be future proof and we're always looking towards the future. So with Casper 2.0, we have a revolutionary model that we have, we have actually already completed the research for this. We know what the implementation is, but it will completely stabilize gas fees. And it will stabilize them in a manner to where enterprises will be able to treat the gas fees, the utilization of public network, much, much more as a capex versus an apex. You'll be able to hold the asset on the books and it will be something that you will essentially just recycle and reuse without going into too much details about the implementation yet as it's not live. But I can tell you that in 2023, we will be introducing a revolutionary new way to stabilize gas fees for the enterprise. It's been a promise that we've wanted to deliver on from the day we've scoped out, the day I scoped out the blockchain, that's to stabilize gas fees are absolutely essential for the enterprise. And we think we found a really great solution for that. Agreed. Fantastic. Thank you. I'll leave with one note. I assume when we're finished, you guys will leave some sort of who we can follow up with to get more information knowing that we like what we've heard. Thank you. Juan, you had a question. Do you want to come off mute? Yeah, hopefully, yeah, hear me. I'm going to hotel with the Wi-Fi and not share the speakers out on this laptop. But yeah, I know that you're really big on security and that you guys were researching quantum brews with the University of Kent. Can you speak on any progress or how that's going and how you guys plan to basically remain future proof? Yeah. So, you know, quantum is a big area of interest in the public blockchain space and everybody's, you know, people are really concerned around, you know, what we can do when quantum computing becomes a real risk to particularly asymmetric key cryptography, right? There's a lot of other things to consider around quantum, right? So absolutely, we're doing research on the quantum side. I don't have a current status update on how that research is going, but we are keenly interested in quantum research, also ZK proof research. We're kicking off initiative at that with the association around ZK snarks and the value that they bring from privacy and being able to conduct private transactions even on a public blockchain because we see privacy is a really, really big important use case requirement for enterprises. But on quantum, I can comment in this way. The Casper Protocol is one of the few protocols that supports multiple encryption schemes for public keys. So you're not going to find any other blockchain protocol that supports multiple key types. We support ED25519 and we support SECP256K1. And the reason we chose to do this is because we wanted to provide a seamless user experience for anybody that's transacting with Bitcoin or Ethereum. So you can use the same private key infrastructure from a user experience perspective to transact across both blockchains. And so we're, again, ready for hybrid technology from the very, very beginning. This was designed three years ago when we initially built the blockchain. And the reason we built that architecture is I want to be able to drop in a new key type at any point in time with absolutely no changes to global state and no changes to the blockchain. Very minimal changes. So we can introduce quantum key support with virtually no changes to the blockchain. It would be a very nominal wallet change where you would then associate a quantum key with your account, revoke the main key that's associated with your account, and now your account is quantum ready. That's literally all that needs to be done on Casper because Casper does support multiple key types. So for us to add quantum keys to Casper is very, very straightforward. And that's because of the underlying architecture. The Casper blockchain does not store public keys in the global state. It stores an account hash. And it doesn't really care what kind of encryption scheme you use for signing transactions on the state. So it's not like you would have to go through this massive migration of all the existing accounts. All the other blockchains use public keys as the accounts on the blockchain. Casper is one of the only ones that doesn't, if not the only one. And so from a quantum readiness perspective, we're ready for what comes in the future. So if I'm understanding you correctly, basically the Casper's modular architecture makes it easy for this plug-and-play key for you to drop in the future versus other layer ones. Yeah, that's exactly right. Yeah, you can basically drop in a brand new encryption scheme and then it's very simple, user transaction to basically migrate. And you don't have to create new accounts. You can just add a quantum key to your account. And now your account is quantum resistant. You add one key, revoke the other, and you're done. That's right. That's awesome. Thank you, meta. This is the way. Another question here on YouTube is, do we have a date or a month for Casper 2.0? We do not have a date or a month for Casper 2.0. Well, I would say it's probably going to be Q3. The reason being is we're going to be bringing NFTs host side. So whenever you bring, again, one of the value propositions with Casper is we can make changes to the virtual machine to meet the needs of Enterprise. And what we're learning from Enterprise is they're not just interested in collections of NFTs that are 150 NFTs. They want collections of NFTs are 300,000, 1 million, 5 million, 10 million. This is what Enterprise grade scale is really about. And so our recent changes to our NFT contract will now allow you to create collections as big as 300,000 NFTs with stable gas fees for minting these NFTs. And this is something that's really unique on Casper. And we can bring it to the host. We can actually modify the blockchain, bring NFTs onto the host. And we know that the community will be hugely in favor of this, because then I can actually, we can actually say that NFT mint costs one CSPR or half a CSPR. And we can do 100 times more NFTs in every block than we can today, just like we do that with CSPR transactions. And this is something we're going to add to Casper 2.0 because of the overwhelming adoption of NFTs and everything we're hearing from enterprises. So we're going to increase the scope of Casper 2.0. And that will push out the date a little bit, but we believe this is a massively valuable feature that enterprises and our communities is going to be really excited about. That's exciting. Is there anything else that's part of Casper 2.0 besides the NFT features? Yeah. So there's several things. So I hinted about the gas stabilization feature. The other feature we're going to have is contracts will be able to pay for themselves. So when I call a contract, it'll be able to pay for the execution on its own.  It's you want to have guarantees that I signed the transaction with my key, right? A lot of systems, what they do today with Web 3 is I pay a subscription fee and then I transact using the interface, but in order for me to sign the transaction, the subscription provider needs to fund my account because there's no way for me to sign otherwise or they transact on my behalf. And really the benefit of blockchain is that you want guarantees that I actually signed something like I did it, right? Well, the only way to do that is if I hold the native token to sign that transaction. So what Casper Labs is building for the Casper protocol is we're going to unify the notion of accounts and contracts. And so what that means now is accounts can sign transactions, but contracts can pay for the transaction. And so what that means is I signed the transaction, but the contract pays for it. And this will represent a dramatic streamlining of the user experience overall for Web 3 because now I no longer need to hold token, right? In order to sign a transaction and the provider can now get guarantees that, wait, Meta, you signed this transaction. I didn't sign it on your behalf. I don't have to look at some Web 2 database to know that I did a transaction. I can actually see on the blockchain that this private key, this public key signed this transaction. And that's like the most important part of blockchain, right? And so when you have to short circuit that, because it's simply too hard and too complicated to get the token in order to transact, that becomes a real problem. So with 2.0, we're going to unify this notion of accounts and contracts. Contracts will be able to pay for transactions directly, even though a user signs it. This is a really, really important value feature, right? And you can also have multi-signature associated with contract upgrades, right?  Even an account is a contract. And it simplifies and streamlines a lot of things. There's a new consensus protocol. We're going to be one of the only blockchains in the space that will upgrade their consensus protocol within 2 years of launch, right? And the reason we can do that is we have a very modular architecture. So we have a version of CASPER in test right now that runs both protocols. It goes back and forth. It'll run the production protocol and then it'll run the new protocol. It'll run the production protocol and it switches every two eras. And this is massively important from a testing perspective to understand, even from a deployment perspective, if we want to run it in testnet in this hybrid model, where we go back and forth between two consensus protocols, we can actually do that. And super exciting work. I'm really excited. But so those are like the big, the big features. And then of course, NFTs on the host. That's awesome. And I'm sorry, did you say Q2? Q3. Q3. Q3. Okay. So we can we'll wait until summer for all of that exciting stuff that you're all working on. Let me see. Any other questions from those of us on the webinar or on YouTube? I'll give you a minute to jump in. And if not, we will wrap it up. And did you have a way for people to reach out to you, Meta, if you wanted to share that? Oh, yes. You can absolutely email me at meta at casperlabs.io. If you want to reach me directly. Very transparent about my email address. Feel free to reach out. And you can also contact us at hello at casperlabs.io if you're looking for more information about us and our offering. Happy to help out. Great. Well, I don't see any other questions. So I'm just going to pull up. If you could stop sharing your slides, Meta. And I will just share my screen real quick and wrap this up. Thank you, everybody, for all the engagement and questions. This is exactly what we love to see on these webinars is you getting your questions answered about how you can leverage hyperledger technologies and the things that our members are building on top of it and with it. Another way to stay engaged is also to join our hyperledger discord. This is where our community meets to just get questions answered in real time. Each of our projects has a channel as do our different special interest groups as well. So please get on there and join the conversation there as well. And then I did see one last question here. Actually, let me just address it. Is it possible to buy Casper and stake on Casper in the U.S.? You can purchase Casper and stake it using uphold in the United States. There we go. Thank you for that last question. So, yeah, come join us on discord and you can engage in our community there. We do do these member webinars regularly. We have one more before the end of the year. We'll have that on our events page on hyperledger.org pretty soon. So keep an eye out for that or when we post it on our Twitter or LinkedIn. This one just came in so we don't have the sign up ready yet. But this will be with IDS. Dave built this blockchain network in India. It's one of the largest academic consortia networks. A really exciting use case to cover as well and that'll be on December 14th. We also have a very vibrant meetup community all over the world. So if you go to just Google hyperledger meetups, you can find probably a meetup in where you're located. We do also tend to stream these live as well. So here's an upcoming meetup that I thought might be relevant for the audience here to attend. It's going to be on December 8th and it's going to be about some challenges in consortia blockchain using hyperledger Flabberg. Thank you everybody again for your engagement. Thank you for all the questions. Feel free to reach out to Meta with the information she shared or us here at Hyperledger Foundation. We're always happy to help you get engaged, involved, and contribute and we look forward to seeing you on a future webinar, future event, whether it be virtual or in person. Thank you all so much. Have a great day.