ll control over you know ecam and some other controls over here that you know I've sort of spilled out and as I say it's for when I'm in a live environment where I'm going to need at least all of these buttons I want to have them to hand without having to switch between pages actually in a zoom meeting if I come into this setting then here I've got all of my zoom controls on here so these are specifically for controlling zoom and these are specifically for controlling ecam and the whole sort of presentation side of things so that means that I've got total control of my zoom meeting on these these two devices the stream deck though the stream deck excel in the middle has become useful for as you just saw you know if I press this button it'll update both of those stream decks if I press this one again it will change what is being shown on these stream decks this becomes kind of like my home profile home button home device that updates the other two and then the dials are useful if you've got a roadcaster you know the dials might seem like you know overkill or unnecessary rather but they are useful for controlling levels in ecam specifically such as you know interview guests and so on I can control those or with these individual dials and then from a productivity point of view you know using them for controlling things like brush size in photoshop and all of that kind of stuff and things where you want incremental adjustments in apps they're useful for that I also do use the stream deck pedals as well and they are really handy and you might think you know is that really necessary again none of this is technically necessary but they can be quite useful for you know easily switching between scenes maybe you're in a meeting you're taking written notes as well potentially or jotting stuff down or you're using your your apple pencil or you're using something else or you're just you know got too many other things on your your regular stream deck having things on pedals so being able to switch between different scenes with those is really useful and I use those in my live stream as well so not everything's done with the push of a button you know if I want to switch to you know the other scene up at the top I can just press this pedal down here if you want to switch back I press this other pedal so I can switch stuff you know with the the pedals as well and actually you do get really really sort of used to controlling things with your feet so again not an essential thing but definitely something to to consider if you want more control I don't use a teleprompter as you can tell with all of these scripted ums and urs but I do I do sometimes use a an actual teleprompter with text on it for example on my live streamer backstage podcast the guest intros I always have a script for that actually because I want to make sure that I yeah I get their their critical information and their intro right and so then I control the the scroll of that with the foot pedals as well so it's useful for for that purpose and then yeah really it comes down to just all of these other little things that are nice to have but I would say that first of all you know where I am now with my studio I've pretty much got everything that I need for my my regular zoom calls the only thing that I'll be upgrading imminently is actually the mac itself so uh so yeah just making some considerations there and have to put aside the actual time to do it because that will be a week in itself to just like set all of this up but that is really the only thing that I feel I need to upgrade in my studio setup at the moment just because now I you know I run into process issues from time to time and so on um and that is largely as well when I first started you know the the mac mini and one is the one that I'm using at the moment and it blew out of the water you know any previous mac that I'd owned and I thought this is you know this is amazing I always used to buy the top end max every time uh whereas this was basically like the bottom end mac and it was still you know way faster than anything that I'd had before and anything that I felt that I needed but as you start doing more things um yeah I'm probably going to read just that going forward um but now you know with doing all these other things of you know sharing multiple max multiple screens multiple cameras then yes actually more processor is is useful so the next upgrade will be to to combat all of that but in terms of the other setup though I'm pretty much there so I just wanted to say that just so that you know that the gas does have a limit there's always new things coming along but there is what is a perfect setup that you can have that will really help to elevate your online meetings and allow you to have those sort of seamless presentations that really the ultimate thing it's all about um yeah it's all about having that sort of flow in your meetings um so just a last few okay don't you Jeff Jeff just a little voice in my ear from the discord great to see you Jeff so yes just coming back to these last few questions have a similar problem with your stream deck not being recognized says no device found if that is your stream deck pedal I don't know if that is your stream deck pedal if it is your stream deck pedal then definitely get in touch with Elgato because yeah I I had it with my two stream deck pedals where the little light on the back didn't come on and as I say it was after I seemingly had like popped plugged them which I didn't think it was a thing but I wonder if that might be it that uh yeah it it didn't recognize them and there's no way to reboot them or anything like that so if it is the stream deck pedal specifically that you're having that issue with then definitely contact Elgato well even if it isn't actually contact Elgato their support there is really good and they had them replace for me very rapidly so that was all all great hey there Mr Riley great to see you thanks for watching thanks for stopping by have you been causing trouble again Rich deleting messages it's still there in the it's still there in my thing so we've still got it you might have deleted it from the actual chat but yeah it's still captured in there in the e-cam waiting for the next MacBook Pro so I was looking at the the studio and the Mac Pros and I've I might have had a change of heart about the direction of going in that but yeah so yeah I've got that as well fat finger syndrome there was a great episode of The Simpsons which was where Homer calls some weight loss clinic and it said something like you know if you want our free guide to weight loss mash the keypad now yeah all right so it's not the not the pedal that you've got the issue with it's the 15 key right definitely get in touch with them Algato then about that in any case like I say they have got really good customer support and if it's saying it's not recognized when it's plugged in do all of the things you know reboot try it plugged into a different socket and so on there is a thing in the stream deck app where you can like select to forget a device as well so that would be one thing to do and then try plugging it in just a little public surface announcement then I guess is making sure to back up profiles as well so if you come into your stream deck so in stream deck here if you go into devices this is where you've got your different devices list listed down here there is one of these which is no where is it now there's one place where you can forget device somewhere I think I definitely did that before I'm just can't find where it is now style not that one there is somewhere where you can forget the device I thought it was in there or am I going completely mad yeah maybe I'm going completely mad there I'm sure I did that somewhere how do you do that I'm not sure not sure somewhere there is a forget device button hmm okay maybe there isn't I'm sure there used to be I'm sure I did that with my pedals when they weren't working but yeah apart from that useless bit of information I can't find yeah if you if you go through and try to just restart I'm guessing you've probably done all these things anyway but yeah do the restart unplug it plug it in try it in a different socket as well so that yeah it might be just some conflict with something else that's in your dock oh yeah and the the backups by the way thanks for the message Mike I just sort of missed that but the backups if you come into your stream deck you go into profiles and then right down at the bottom there's this little arrow you can do backup all so that's the first thing is just say create backup and it will back up absolutely everything and then you can obviously you can see here restore from backup so it's worth doing that if you if you're doing work on your stream deck like building it out then you know set a recurring task in your task manager to go and just back them up so I do it like basically monthly but then you can also restore that but you can also just export a single profile as well so if you're doing something where you want to share from one thing to another you can export it there by the way if you've got multiple stream decks and you've got like the profile set up on one but you want to use it on one of the other ones you can also right click and go copy to and it brings up a list of all your other stream decks so you can send it to you know use it on that particular stream deck as well so yeah that is definitely recommended but yeah the the backing up it's it's always one of those things isn't it that we we tend to learn the hard way on these on these things let me just get this link though because there is where is it somebody's asking about video pencil I've got a link to that somewhere description so let me just pull it up here but yeah if I if I'm going to leave you with one message it is definitely don't let the gear sort of hold you back but it is also you know to have an upgrade plan to think about how you can improve and which I mean this could be live streaming or or zoom to be honest or teams or whatever but yeah having an upgrade plan in path in in having learning how to speak is also a good thing spend some time on that having an upgrade plan or an upgrade path planned out is is definitely always a good thing too to to sort of be constantly improving and constantly learning about these things too I can't just find the link to video pencil now it's definitely in the description though so if you go and look in the the actual video description it will be just down beneath the main description and then I've got something in there about the the academy and all of that kind of stuff and then I think it's just under there just above an e-cam thing I will just go and grab it though it is nearly there now it is oh actually guess what guess what I've got a stream deck button for that there it is what an idiot also if you're going to set up stream deck buttons always remember what buttons you've set up all that time people asking for the link and I've actually got a button for it over on another page there you go that's the problem with having things spread out over pages though I guess good luck with the stream x I'm sure you'll you'll love it and I'm not sure if the beta of the unify software is out yet I've got a pre-release beta and I'm not sure if the actual formal beta has been released yet but that is great for like some of this advanced routing that we are talking about thanks everyone for stopping by if you've got any questions about any of this by the way best place is to go and check out the discord community and it's free to join and don't be put off by the fact that it's discord or it's called a server it's just basically a family and it's a place to go and chat and ask any questions and it's the best place to chat just because everyone gets to benefit from the responses to the questions there and also you're not just getting my response potentially but other far more learned people than me here in there too so yeah but also if you do want any other information feel free to reach out to me as well and don't forget to check out the academy if you want to really level up your on-screen presence and impact too thanks so much for stopping by I will try to remember as soon as it's finished processing to go and leave a link to some other relevant content over here on the right hand side look forward to seeing you in another video a few more videos coming up this week about the stream deck developments and so on take care see you next time


So, having seen a number of examples of CW complexes, we return to the more elaborate study of topological properties of CW complexes. To start with a CW complex X, then X power k is the kth skeleton, namely all, the union of all simplex is all cells of dimension less than or equal to k, right. So, this each skeleton Xk is a closed subset of X. This is the first statement. The second statement is each closer cell is also a closed subset of X. It is not just a closed cell, it is not I just name, but it refers to closeness of the underlying set inside X. A subset S of X is closed if it is only if S intersects each closed cell in E k, please call cell E k of X in a closed subset of E k. Topology on X is compactly generated. So, these are the four observations you want to make about the topology of a CW complex, okay. Remember a subset is closed in X if and only if it intersects each skeleton, okay, in a closed set. That is the definition of the topology on X, right. If X is actually a skeleton, then we do not need this definition because each skeleton Xk is obtained by attaching k cells to Xk minus 1. Therefore, the topology is well defined there as a quotient topology of a certain object, all right. So, this definition, this statement, statement A is needed only when X has infinitely many skeleton, the dimension of X is infinite. So, first of all, each X is a closed subset, how do you see that? Look at Xk intersect with Xl. If Xk, k is smaller than l, it is just Xk. If k is bigger than l, it is Xl. So, either it is Xl or Xk, they are closed subsets is what you have to show, right. Then only each Xk will be closed subset. If Xk is a closed subset, if Xk intersects in Xl, inside Xk is a closed subset. Now, if Xk is contained in Xl or l is bigger, then we know that Xl is obtained by attaching cells to Xk minus 1, Xk plus 1 and so on, Xl, Xl minus 1, etc. So, each Y, when you attach cells to Y to get X, we have seen that Y is a closed subset of X, okay. So, successfully Xk will be closed subset of Xk plus 1, closed in Xk plus 2, closed in Xk and so on, Xk will be closed in Xl. If l is bigger than l, so you can interchange the two roles. Therefore, intersection of Xk with every other thing is closed, therefore, Xl will be closed, okay. So, this much argument is needed here to say that Xk is closed. Next one, each closed cell is a closed subset of X. To see that, I have to show, again, take a closed cell, say let us call it as sigma, intersect it with each Xk, okay, then it must be closed inside Xk. If this cell, say cell itself is of dimension, say l, all right. If k is bigger than l, it will already contain the entire sigma, right. And each closed cell is, we have seen, when we attach a cell, they take a l, sigma is a dimension l, then it will be closed in Xl, right. Once it is closed in Xl, it will close in Xl plus 1, Xl plus 2 and so on, Xk also, all right. So, if l is smaller than k, it is fine. If l is bigger than k, what happens?  What is the intersection? Intersection is the boundary and we have seen that the boundary is closed, okay. So, this way you will see that each cell is also a closed subset of X. The next thing is, take a subset S. If it is closed, we find only if, one part is obvious. If it is closed, then the intersections are all closed because we have shown that ek's are all closed. Intersection of two closed sets is closed. Now, we have to look at the converse. The converse, S intersection ek is closed in ek, right. But we want to show that S intersection Xk is closed in Xk for every k. Does that follow from this one? If this Xk, if k and this one are same, for example, this ek is closed in Xk and so S intersection ek is closed in ek and ek is closed in Xk, so it is closed in Xk also, okay. So, showing that X intersection, since S intersection Xk is closed, will follow from, I mean, very easy for it is closed in ek because this is ek's closed, okay. So, so, so C also follows. Finally, D also will follow for the following reason. What is this? What is the meaning of compactly generated? The same thing as instead of ek, I have to put a compact set for every compact set of X, original X. Take a set S such that S intersection k is closed in k. So, this should be true for every compact. Once this is true, S must be closed. Again, we have to show only this, only this part. The other part, if S is closed, S intersection k will be closed in k. That is obvious, okay. Take any compact set. We know that it is covered by what? Finitely many closed sets. Therefore, S intersection k will be the union of X intersection some sigma. The sigmas are finitely many of them. So, union of finitely many closed sets should be closed. So, S intersection sigmas are closed inside actually in the whole of space. Therefore, they will be closed inside k. So, S intersection k which is union of finitely many of them will be closed in k. And conversely, for each, for each ek you can show that actually that is what we wanted. For each ek is compact is what we have seen already. For every compact set is closed. So, in particular for each compact set also it will be closed. So, all these things are written down carefully here. So, let us go through this one once again. So, first we want to show that this k skeletons are closed. So, we have seen that if X is obtained by attaching kth l, then why is a closed set up? From here, it will follow that X naught is closed in X 1, X 1 is closed in X 2, X 2 is closed in X k and so on. Therefore, each X i is closed in X i plus n where n is any extra positive number.  So, part b X naught being a discrete space is ausdorff. We have to take ausdorffness, the ausdorffness the ek's are not closed, need not be closed. We have you remember that by clock corollary whatever we have earlier and actively it follows that the skeleton X k is ausdorff. This was one of the exercises. If y is ausdorff by attaching a single dimension k cells, whatever you get is ausdorff. So, I am using that one here. That is why this will be these things will be drawn. So, each X k is ausdorff. Now, ek is a closed cell, then it is compact subset of it. A compact subset of a ausdorff space is closed. Since each X k is closed in X itself we are done. So, each ek is closed in X k. So, this is an easy way of saying the whole thing. But you have to use to show that a closed cell is actually closed. You have to use the ausdorffness of it. If a relative CW complex, then you should have started with y as a ausdorff space, then it is really true. The C part I will repeat this one namely I have to show that if S intersection E is closed for every E inside E then S itself is closed is what I will show. Inductively, we will show that S intersection X k is closed in X k for every k. How?  So, S intersection X naught is also closed inside. So, there is nothing to prove.  Remember, when you attach one single thing, what is the condition? It should be S intersection S, this set is closed if and only S intersection X k is closed and S intersection each k plus 1 cell must be closed and that is the condition here.  So, that is part C. Now, above compactly generatedness. So, this is the meaning of co-induced topologies compactly generated. A subset is closed in X if and only if S intersection k is closed for every compact system. If you look at each cell E, it is compact. Therefore, by this condition S intersection E is closed in E. By C, it will be closed in the whole of A. The converse is obvious. So, now I come to this, why this finite is generated? There are only finite, limited, etcetera. Why I gave this argument? That was the argument used in olden days. Not only that, this was put as a condition in the definition itself by J H C Viter to give the first definition of CW complex. So, I will elaborate on that one little bit later. Right now, once you have watched this thing, if you have worked out from, for the attaching cell, the same exercise. If Y is host or regular or normal, then X is host or regular normal. That is what I have given an exercise. Now, inductively you can do that, that if A is a host or space or regular space or whatever it is, then so is X, where X is a relative CW complex. In particular, every CW complex, E is a T-force space. So, this is left as an exercise for you now. Having solved that exercise, this will be easier. The next one is, you should pay attention to this one. It is a very, very important result on CW complexes, all the time used. So, what is said? It just says that every CW complex is a disjoint union of its open cells. Don't be under the confusion that therefore it is disconnected, you may say. These open cells are just names. They are not open subsets of X, not necessarily. Some of them may be, some of them are not. Indeed, they are open cells if they are maximal, that no additional cell of higher dimension will intersect their interior. Then only they are open cells. Open. But they are open cells, they are fine. Open cells are by definition what? It is just the interior of Dn, whatever, under the image of V, that quotient map. So, that is the open part there. It is open inside the closer cell, that is all. So, what we are saying is, take all the interiors of these cells. That will be the union of the entire CW complex. This result, we have seen it in the Simplisher complexes. It is exactly same here. In fact, much simpler. What is said? By very definition, you can write X as X naught union X1 minus X naught union X2 minus X1 and so on. So, Xk minus Xk minus 1. So, this union is patriotic union because Xk's are increasing subspaces of X. Here in this notation, I have taken X minus 1 is empty set. We start with X naught. There is nothing to subtract. X naught is the union of zero cells and zero cells are both open and closed. So, all these points belong to this disjoint union. So, we start with all the zero cells. They are open cells. Now, look at X1 minus X naught. What are they? They are open one cells and they are disjoint. Any two open one cells are always disjoint. So, Xk minus Xk minus 1 more generally is the disjoint union of open care cells. Take any K cell. Its boundary is contained inside Xk minus 1 and you are throwing away that what you left of it. So, it is interior. Okay. So, this is a totally obvious statement but this is very useful one. Now, let us see how it is useful. This lemma says take a subset that contains at most one point from each open cell. At most one point. Intersection of an open cell and S is a singleton or empty cell. So, this is the condition on S. Such a set is closed automatically and discrete subset of X. Okay. If you think a little bit, it is obvious but now let me write down the full proof of this one. Okay. There are books and books which give you long, long proofs of this one. So, even myself, I got this proof only little later and then there I found out that there are books which I have written like this also. To show that S is a closed set. What you have to do? Intercept with each Xk and show that it is closed inside Xk. Start with X0. X0 is a discrete space. Every subset is closed. That is fine. Now, assume that S intersection Xk is closed in Xk. I have to show that S intersection Xk plus 1 is closed in Xk plus 1. That is what I have to show. Then for any closed K plus 1 cell E, look at K plus 1 cell. E intersection S either is inside Xk or it is in the interior of E. If it is the interior of E, S intersection that part is at most one point. Therefore, if you look at E intersection S, it will have at most one extra point than E intersection S intersection Xk. In either case, this is a closed subset of Xk plus 1 because this already closed in Xk plus 1. What I get is one extra point. Where is the closed subset? Therefore, S intersection Xk plus 1 is closed in Xk plus 1 by our lemma because Xk plus 1 is attached is got by attaching K cells to K plus 1 cells to Xk. Intersection with say this you take this as Y. This intersection is closed and with each K plus 1 cell it is closed. Either it is empty in the interior or one extra point will come. Therefore, it is closed subset. Remember this is a hostile space. So, adding one extra point is a closed set because each point is a closed subset. Therefore, it will be closed in Xk plus 1. So, this way S intersection Xk will be closed for every K. Therefore, S is closed. So, what we have shown? S with that special property namely it contains only one extra one point or no points from each interior of interior of each cell such a property automatically is closed. Now, take any subset of S. It will have also same property. Therefore, every subset of S is closed. If every subset is closed, every subset is open also that is a discrete space. So, such a set is discrete as well as a closed set. We can use this one to prove certain properties that I was telling you that about finitely many cells and so on that comes now very easily. Take any compact set K of a CW complex. It is contained in the union of finitely many open cells. The funny thing is these open cells may not be an open cover. They do not form an open cover because they are not opens of sets not necessarily. Yet a compact set is contained in a finitely many open cells. This is an easy consequence of the previous lemma. X itself is a union of open cells. Therefore, every set inside it is contained in the union of open cells. You start with a compact set. Suppose you need infinitely many open cells to cover it. That means what? That means there will be infinitely many points, infinitely many cells which will intersect K on empty intersection. Pick up one point from each of these cells, open cells. Only one point you pick up, which is common with K. Then what will be the property of this K, this subset? That subset has the same property as previous lemma. But it is a subset of K. The previous lemma says that that subset is discrete. But it is a subset of K. K is compact. A discrete subset of a compact set is finite. So you could not have infinitely many points like that. So I repeat, if not, what happens? K will intersect infinitely many open cells, which we know are disjoint. Therefore, I can select one point in each such intersection. We will get a subset S of K, which is by lemma the discrete subset of a compact set. But by the very choice S is infinite. That is not possible. So this is the consequence closure of every cell, which we know is compact now. Meets only finitely many open cells, because every compact set intersects only finitely many open cells. This property closure meeting finitely many open cells was called closure finite property. Okay. So this is the remark I wanted to make. I am going to elaborate it on. I was all the time referring to it. The property of CW complex is known as closure finiteness. Okay. That means what? Closure of every cell is will meet only finitely many open cells. Okay. So this explains the presence of the letter C, CW, CW complex. This C was corresponding to the closure finiteness in this mysterious nomenclature of CW complex. The CW complex was the name given by the creator J. H. C. Whitehead, creator of this concept. Historically, this property was an additional part of the definition itself. J. H. C. Whitehead definition of CW complex was very, very long. He had put many of the standard properties which we are discussing now as a consequence in the definition itself. Okay. So this was one of the property closure finiteness. We have just proved that we do not need to put this extra condition in the definition. By the way, there is W. What is this W corresponding to? In the letters CW. C corresponds to closure finiteness. What is W? W corresponded to the weak topology, the compactly generated topology. Okay. That is also called weak topology. Okay. Corresponds to weak topology on it. So it might have been appropriate to call this one. There is no need to W. There is no need for C we have seen. But W is not this weak, the compactly generated or co-industrial topology that part is necessary. Therefore, you could have just awarded C and kept only W. We can say W complexes. But nobody has made that kind of suggestion. So I am alone perhaps. So let us not stick to a new, let us not create a new nomenclature here. Let us, many people just call, they have given up W also but now they are calling it a cell complexes made up of cells. So that is a good name. Cell complexes is a good name. Okay. But I would say that you better stick to CW complex because there are many notions of cell complexes. Each author has his own definition. And sometimes they just differ slightly. Like somebody doesn't put house darkness. Somebody puts something else and so on. And the structure also somewhat different. So let us not bother about new terminology. Let us call this a CW complex. Okay. So is the closure finiteness property, if you read old papers, then what you will get is this kind of definition. The closure finiteness is a consequence. All right. Of actually the fundamental thing is the quotient space structure here. Okay. Attaching cells gives you X as a quotient space. And then from X1, X2, X3, one contains other. X infinity namely X itself has the topology co-induced by these inclusion maps. You could have taken directly that compactly generated. But before defining compactly generated, you have to have topology on it. So you can't say that compactly generated. So all these problems CW complex in different DLU complex, you know, though he had the ideas, things were not available to him. So many of these things will develop much later. So it was a great invention after all. So let me go ahead with the topology in the spirit of our earlier lemma of extending the neighbourhoods so that their deformation retracts and so on. Remember that lemma. Remember that proposition. Okay. So I am going to make another such lemma here which will be useful later in the study of CW complex says. What does it say is the following. Start with the CW complex. For each end, let Vn be a neighbourhood of Xn inside Xn plus 1. Okay. Let X be a point in the interior of say some k cell. And Wk be an open neighbourhood of X inside sigma. Okay. Choose some small epsilon neighbourhood for example, W is some neighbourhood of X inside sigma. So you can by the way, here also you can say open, open everywhere, open neighbourhood, no problem. Then there exists an open neighbourhood W of X in X such that W intersection Xk is Wk and W intersection Xn is contained in Vn for all n. So this Wk which is an open subset of open neighbourhood of X inside sigma gets extended to W. Extended means what? Intersected, Xk is Wk is exactly equal to Wk. And it is controlled in the sense that intersection with Xn is contained inside Vn. It is not going, it is not becoming too big. This W is a neighbourhood of X in X itself. A neighbourhood Wk which is inside Xn plus 1, sorry, where is it? Inside Wk, inside Xk may not be open inside X. Okay. It is like you have a point on the line and a neighbourhood of that point on the line like an interval. But now interval is contained in R2 that interval is not open inside R2. But you can extend it to a square, open square. And that is precisely what we have done in the first lemma. We took any subset of the boundary and then we extend it to a neighbourhood of that inside the disk. Right. And this fundamental elementary construction is going to help us in all this. So this is the lemma that we had. Okay. Start with attaching cells. Okay. Start with a subset A, where is it? B inside Y. The B gets thickened to n epsilon B. Okay. This epsilon could be chosen at your wheel and that is a neighbourhood. That is an open subset of X. Okay. And if you intersect it with the Y, then it is just B itself. So it is extended. Q, B, P, Q is in it. It is a strong differentiator. This part is extra thing. We do not need this one right now. Okay. This is also true we can do. But all that I want is extension from not from Y to X, not from Xk to Xk plus 1, but Xk to directly to X we want now. So we have to do inductive steps. So this theorem is what we are going to use. Okay. So now I have to go back, but this is not where I am to go. So by this proposition for each k plus 1 cell tau, okay, tau intersection vk plus 1 is open subset in tau contained in the entire boundary of tau because what are v ns? v ns are neighbourhoods of x n. So they contain the boundary of any simple n plus 1 simplex. Boundary of any n plus 1 is x n. So it is contained in v n. So when you take the inverse image, it will contain the entire boundary, the boundary sphere. So I am not going above at all. I am just working on in X itself all the time. So the notations etcetera are simpler here. Okay. So for each k plus 1 cell tau, tau intersection vk plus 1 is an open subset of tau containing the entire boundary which is actually a subset of xk and contained inside vk plus 1. Therefore we can choose epsilon to epsilon tau such that once the entire sphere is there which is compact, there will be an epsilon. If it is non-compact then there would have been a problem. So a tau is equal to n epsilon of tau dot. Remember this is what is the meaning of tau dot.  This happens for every tau. So there I have to have you know sigma alpha and so on. I have reduced the notation here that is all. Okay. Therefore if you take beta which is equal to a smaller set namely n epsilon tau dot intersection wk, that will be a subset of tau intersection vk plus 1. Okay. So this is the intersection tau, this is the service in tau intersection vk plus 1. I am taking a subset here by taking the intersection of this one. Okay. This is going to be an open subset. Therefore n epsilon of that will be open subset. This is the import of this lemma. Okay. So take now, take wk plus 1 to be the union of all these beta's as done in the proposition. If you take union of all these beta's that is going to be an open neighborhood of this x because x is there in xk plus. That will be an open subset of xk plus 1. So it is an open neighborhood of xk plus 1. Okay. And its intersection with xk is precisely wk because all these neighborhoods had this one. B remember B is equal to QB is equal to what? It is intersection of n epsilon B intersection with y. So that is the meaning of wk plus 1 xk plus wk. And wk plus 1 is the union of all these beta's. It contains a vk plus 1. So inductive step is over. Okay. Now what you do? You just take w equal to, so we have got this inductive step for all of them. Now you just take w equal to union of all the wx. You start with n greater or equal to k here. You start with wk because x happens to be inside the k cell in the indeed of k cell. If x is a vertex, if x is 0 cell, you would have taken all these things n greater or equal to 0. Okay. Why this is open? Intersection with each xn is a neighborhood open subset. Wn minus. Intersection with xn is this one. Why it is contained vn plus 1? But if you have w and intersection it is precisely, so it is containing vn. That is how we have chosen. So this is controlled by these vn's but it is a neighborhood of x inside x itself. Okay. So this I call it as controlled extension. So we will use this one much later of course but we will use that. All right. So we have done quite a bit of topology but there are many more things to be done. One of the most important thing is that the cw complexes, the weak topology is very, very useful in constructing continuous functions on cw complexes. So this was the motivation for taking that or verifying, suppose you have constructed something verifying that it is a continuous defining a continuous function as well as if there is a function to verifying continuous function, the cw complex structure is very useful. Whereas if you take just an arbitrary space and so on like r infinity and so on it would be very unwieldy to do that kind of things on that. In particular we will show that there are lots of continuous functions namely what is called as partition of infinity. So that we will discuss next time. So today this is over. Thank you.


Good morning, cyber traders. Welcome to Coronavirus Black Monday. Good to see you all then good to see you Jerry Dennis. Good to see you all There is no better place Than being in a cyber group right now. Okay. I You know what? You have to worry about going anywhere. You don't gotta rush and do everything to do There's no better place. There's no safer place. There's no cleaner No, you know, nothing better than the cyber group room. Would you say you guys all agree? It is just incredible All right, so listen, you know, if you hopefully some of you guys been watching and if you guys are on my Instagram account, but like every morning once a while I try to post on my way to work every morning and do a quick Post on my Instagram and I also post on YouTube and Facebook, but usually my Instagram Trying to give you guys a little heads up with the game plan for the morning is you guys all know that I do start around 730 in the morning two hours before the market opens up and I'd say I couldn't sleep all night. I mean, I woke up like four or five times Regarding about what was going on with the market. I mean yesterday. You had a Listen, I'm not here to broadcast the news through you You probably or did that already and probably been glued to the TV and you need to I mean need not need to know That they close New York City schools, which by the way is that is absolutely historic They might even and I'm gonna tell you what the whole thing behind it that which is what I think is gonna happen I want to tell you a little bit more about what is gonna be really good to trade and what's gonna be good But I'm just telling you right now before we get started There is no better place and there is no better business than being in being a trader right now.  Okay. Now. What are you supposed to do? Not do anything. They're like, I'm not prepared for this. They're trained for that. Okay, you know people are our veterans Our people on our military. Do you think they want to go to go to war? You think they prepare for they just prepare for it You know, and we just hope you know prepare for the hope for the best and prepare for the worst This is what you were trained to do you are a trader. Okay? You are now stepping into one of the things that this is this is our 9-11. This is our war This is this is what you train what to do.  Okay, period You know, this is all this is your job. You got to get ready to trade You got to put your head down you're in cash and let me tell you something and which is what a lot of traders And if you notice a lot of people that that teach or or do what we do Something that they don't do that I do. Okay, because they don't know how to do it They're not scalpers This is a scalping market. Okay? Scalping means you get in and out of trades if you don't know how to scout by the way for a student of cyber Traversity I did a scalping course.  It's in there But this is how you scalp. Okay. A lot of people are not scalpers. They're not day traders They're swing traders. That's what they are this swing traders and they literally have no idea You know because they're waiting for their indicators to kick in all that good stuff So I'm not looking to go on a rampage or you know, whatever I'm just telling you that this is a scalping market and this is what you have to do You got to think quick and you know, you got to make the money quick and just be happy you made it And that's it the only thing that separates us from them is that you got to be in cash You got to be in cash I know some of you looking at some swing trades and a lot of people still trying to pick the bottom, you know what? We saw what happened on Friday President came out market ran 1,500 points 1,500 points now if you guys watch my video actually if you go on our website We actually we showcased it when I was on Nasdaq and the bit number one thing that they were asking me It's like faster when you think the bottom is I'm gonna repeat it I'm gonna keep repeating it again. Okay, and believe me. I think we're almost there I really think we're almost there regardless what you hear about TV or everyone all the other hours I'm telling you I think we're almost there How do you know we're almost there when you got the worst of the worst of the worst of the worst news? The worst of the news and we don't go any lower. That's the bottom. Okay, so What's the worst of the worst? They closed New York City schools. They closed the restaurants and I predict probably in the next 24 to 48 hours We're gonna go to a complete lockdown in the United States to me That's the worst news and then we'll see other market, but I had a meeting with Josh this morning We always have our morning meetings to prepare you guys regarding what's going on. I could tell you this Please keep in mind. They are not shutting the stock market down. They can't we're not in New York Restaurant, okay. This is a global Financial market they will never close down the currency market. They'll never close down the futures They're gonna close the New York Stock Exchange probably the virus But they are not close to market down and then that's and there's no better place than be locked in your home in Confinement in the trading room. That's when you will have fun. Now. Let's talk about what happened on Friday I mean, I really don't even know where to start but Listen, there were so many great movers that were moving OPK, you know, I personally I could tell you the truth now You're probably gonna hear a lot of this stuff going on. I think the corona stocks are finished All the corona stocks. I mean, you might get a couple here and there that might make some pops I personally think they're done. Okay, the news is out. Remember buy-on rumors sell on news You know, they're all fighting to get this and that so be careful of trading some of these corona stocks Okay, these pharmaceutical stocks all that stuff. I would focus more on scalping and day trading and a lot of them you could see what's going on some of you guys did pretty well already some of you already did pretty well on You traded a royal Caribbean and it's funny if you look at you look at them They're starting to pop a little bit, you know, so I mean I know they locked the market to dow is halted because it broke through the 5% But if you notice here 8 o'clock this morning the stock was $27 already went to 30 scalp scalp scalp scalp That is your that is your model right now. Don't worry about picking the bottom Just scalp if you want to pick the bottom you buy a company right now I'm just telling you right now. You got a deal for the thousand points up to 2,000 points down Whatever it is, you know, there's some great great buys out there if you're looking for, you know, six months a year down the road You know what you're gonna have to deal with some of the headaches and you have to deal with it But right now as day traders, it's a scalping market. There's so many of them that are moving You know Play got crushed. I mean this one's taken a big correction like at 45 down to 10 Like I said all the All the cruise companies are like all over the place. They're taking a little bit of hip, but they're coming back a little bit Oil and transportation's all of them. So all the ETFs are all over the place. I mean look at this What's this down 70 72% I? Mean these things made these big pop that doesn't know what the hell that is But a lot of stocks got destroyed and they did come back CHK all these things where's a Gush Yet gosh, I'll tell you really really quick very interesting stock really holding pretty strong at that 50 cent level So I would keep a close eye on that looks long-term play that might be safe but listen One I know want to get right to get you guys get ready to trade because we're gonna have a really exciting market couple of Stocks I do want to kind of tell you what do I personally think is going to be a good buy well Think about this for a second What companies are actually benefiting right now during to the coronavirus other than speculation on which company's gonna come with a cure for the virus? Think about it. What who's actually making money in this market? Does anybody know think about it? Who's really doing well? Who's profiting from this? No, not hedge funds William Congratulations, hit the nail right in the head. You want a good buy food? Stanelizers keep an eye on the grocery stores Unfortunately, Perrell is not a public company. All right Toilet paper, you know can't find a damn anywhere.  Okay, look look at look at the move on Costco 160 180 to 2 to 320 I I'm telling I'm a huge fan of Costco. I max out my $500 check I get it every year So, you know how much you could actually imagine what I spend there Wish I could have bought it, you know when I when I could have but You know, I didn't because I bought something else But but here's your opportunity to buy some of these stocks. There are some great companies out there BJ's also if you look at them oops typed in a wrong window Same thing. They really haven't came down a lot Down to 21 right around here all time all time lows So if there's anything you want to look that I think you gotta come out with some huge Major great earnings announcements. I think those are probably your two best best bets out there to kind of be focused on Is Costco at 355 all day long listen, you know I'm just a trader. Okay. I'm just a trader and I know when I walked in that place. There was nothing in the store I mean you heard the president talk about on Friday people buying four to five times more food You know, I mean out there. I remember I could tell you this hurricane Sandy You don't benefit it from Hurricane Sandy. Okay, Home Depot lows The the company that the the company that makes the generators. They killed it. Okay a year down the road The thing was almost tripled in value. So listen, what did it generate right there? You go killed it killed it Okay, so I remember going to Home Depot. You couldn't even find an outlet. You know people trying to rig up their Generators and and the electric and all this is just unbelievable. So anyway So if you're looking at something that I know that we're trying to find the bottom Somebody's something he's got killed because of the the airlines and what's happening and travel and this and that but on But the food companies are killing it right now. So I'll keep an eye on that stop and shop Is that a shop is actually a stock? Yeah a shop of fly wrong one mRNA is running up. Yeah, it's got a little bit of a pop right there But guys once again if you want to keep an eye on some really good buys Keep an eye on what what's actually what's happening with this, but the food industry is gonna get killed listen Like I told you when we first started there's no better place to be than in the trading room right now You're stuck at home. I Predict within the next 24 to 48 hours the United States is probably gonna be on full lockdown Don't worry about us guys Cybertry University is all set up. Everyone's gonna be willing to be remote from home We everyone's all set up. We're prepared for this. We've been prepared Since Hurricane Sandy, so we're all set up.  I could tell you this my phone has blew up Everyone's like I have friends of mine that are brokers at some of the top broker firms in the industry and they're calling me up Because they're not traders and it's this fast. So what do you think you think is the bottom? I feel like I'm at the racetrack. Hey, what are you betting on? What are you betting on? You know, it's like trust me guys. There's no better place than being here We're gonna make a lot of fun. Just remember you remember this is what you paid for here for Cybertry University To help you what to do and that's to learn how to trade so let's go out there and work together like a team So don't forget guys What one last thing before we go big big announcement? I wanted to just remind you I am gonna be doing in a big event at 11 o'clock this morning with book map. Okay It's gonna be an exciting one. You guys want to know about level four. You know what? We're one of the most exclusive Online training schools out there that are Showcasing that platform. You do not want to miss it.  So look forward seeing you guys all there. All right guys. Good luck. Happy trading and Be safe. We'll start commentating about the next 10 minutes. All right


With the recent release of the Navy's planned 2014 budget, the C&O spoke at the House Armed Services Committee and stressed his concerns for the Navy moving forward. Two important characteristics of our naval forces describe our mandate. That we will operate forward where it matters and that we will be ready when it matters. Your Navy and Marine Corps are uniquely qualified to immediately respond to crises to assure allies, build partnerships, deter aggression and a contained conflict. Our fundamental approach to meeting this responsibility remains unchanged. We organize man, train and equip the Navy by viewing our decisions through three lens or you can call them tenants and they are war fighting first, operate forward and be ready.


Hi, Hossain. Thanks for joining the meeting. We'll get started at the top of the hour. How are you? I'm good. How about you? Glad to hear it. I see the ideas. Yeah. Yeah, we're a little early, but we'll have more people joining. So. Thanks for joining. Have you joined the call before? Great. I've been on some hyper leisure meeting before, but a few months. Great. Nice. Well, thanks everyone for joining. Well, we'll wait a few minutes to get started. Excuse me. Can I ask a question? Hello. Sure. Sorry, sorry. But somehow zoom says that I need a code is, is that correct? And is there a code available? Like strangely, I can join through my desktop, which I'm doing now. Oh, interesting. Are you, could it be some sort of issue with the sign in? Because there, you shouldn't need a code to enter. Okay. I'll check it out. Maybe some things wrong with the way I installed zoom or something. I'll check. Thanks. Okay. Thanks. Yeah. I hope that works out. Everybody. Hi, John. Thanks for joining. Give me one minute char and I will get, I'm going to stop the recording and then I'm going to start the stream and then restart the recording. Sounds great. I wonder. As a backup or resume recording, and I'm going to set char and Tim here. Yes. So that both of you to co-host, just in case they co-host just in case. Something happens to my. Why isn't char co-host. Something happens and I get knocked out of limousine. All right, char. Take it away. It's all yours. Sounds great. Looks like we have. Stephen current our great speaker for the day here. So I'll, I'll pass it off to Tim to walk us through the working group status updates. Yes. Good morning, everybody. Just give me one second here to share my screen so we can kind of all walk through it together. So yeah, good morning. It is June 15 today. And on our call today, we'll be reviewing some working group status updates. And then we also have a presentation from Stephen current. So thank you for joining us today, Stephen. Appreciate it. Quick reminder, we are under the hyperledger anti trust policy. So please just be aware of that. We have a few announcements. We have a few upcoming speakers. We have Nick Steele on the 29th. Stephen movie on the 13th of July and then Dimitri. Zagan doulin on the 27th. So if any of these speakers look interesting to you, please be sure to come on back in a couple weeks. We also have a hyperledger in depth with the red data technology. So that's on June 21, and anyone can register at this link right here. So jumping right in it looks like the hyperledger indie contributors working group met on the sixth. Was anyone able to attend this call. That was I can give a brief update, but I wondered if it also might be useful to see if we have any introductions that anybody wants to make on the call I think we've got, we've got a fantastic group here and probably a lot of new faces to this call. So if people want to take the mic and, and introduce yourself and say, talk about your interest in in decentralized identity. It's going to be a great time. Hi, James camp. I'm the CTO, why are green, which is a virtual power plant company and I think we might use this for by OT device identification or potentially also users. Thanks. Thanks for joining. That's great.  Thank you. Good morning everyone. My name is Michael Novak, and with the open voice network and been a longtime fan and student for digital identity. I'm very excited I come out of the IOT world. So I see this as being a key technology enabler for this, but also recently may have heard of generative AI in the news at least once or twice in the last 15 minutes. So again, verifiable credentials and digital identity are perfect fit for conversational voice, as well as taking inanimate objects. I'm really excited to check Steven's math today. And I'll run through generative AI don't worry Steven I'll make sure you're doing it correctly okay. Awesome. We're glad you're here. Any other introductions that anybody would like to make. Good morning. Steve Michael is Martin from Boeing, Boeing Vancouver. We also are working on a project to integrate the verifiable credentials and part of our authorization access and identity and access management authentication mechanism. There's a variety of applications to it that we're researching at the moment so this is a fantastic application interested in how this moves forward. So, thank you. Absolutely. Thanks for joining Steve. I'll drop the meeting page link in the chat here. Let's see. Oops. I dropped the wrong link in the chat but I will correct that right now. Let's see, so that anybody can put their name on the attendees list if they would like. So great unless anybody else has any announcements or introductions that they would like to make we could continue on with the working group updates. Sounds like that's about it. I think we left off at the Indie contributors call. Yeah, absolutely. Give us a quick summary of that. Yeah, so we talked about Indie community contribution so discussion about how they're really successful Indie networks, just few fewer code contributors and so perhaps this is because Indie just works well and deployments are pretty specific. But at the same time we have a roadmap of things we would love to add to it and and see implemented so just a discussion about that that I think was really useful and then we also spent time on the call going over open issues in Indie plenum. Steven I don't know if you have anything you wanted to add about that discussion or what we did on that call. That sounds about right. Looking forward to more discussions on, you know where he's going in and you know how we can configure on how to expand contributions. Yeah, absolutely. Awesome. Well sounds good thank you for sharing. Looks like the areas working group met just yesterday it wasn't even able to attend the most recent areas working group calls. Okay, well it looks like they're discussing some wf resolution and did peer steps and unqualified migration stuff if you want more info. These links are will take you right to their notes. The areas bifold group met on the sixth was anyone able to attend the areas bifold call. Yeah, it looks like they're working on an update and discussing some key issues. The areas cloud agent python users group man on the 13th was anyone able to attend this session. Yeah, I was there. Talked about an update on the BC gov code with us. We're working on an update at in DC or working on to update acropi to use the hyper ledger version of a non creds and so talking about the refactors got the MVP of revocation completed. Which was a big step and and the things we're looking at next are the automated registry setup, generic sizing the revocation registry registry recovery test updates cleanup as well. And the the 072 final release and and merging PR is in acropi. Stephen, I don't know if you again have have anything you'd want to add to that summary. We figure out how to unmute. Yeah, 082 is ready. We've got some final things going into that. And then we went into acropi plugins and updates and progress for applying on making stuff and we added a new maintainer project which is pretty cool. All of those are covered there. Welcome, folks to join us at the next meeting in two weeks, two weeks from this one. Sounds good. Thanks for sharing. I believe we've met. I'm not sure if this was the same day as our last meeting. But was anyone at the latest areas framework. Job script call. It looks like they're kind of discussing what the future of areas will look like and how to get started for more details and click that link. Life hyperledger and on crits looks like they met on the fifth. Did anyone attend the non crits call. We've had a couple of meetings. Yeah. Basically, we're moving forward on the stack we now have a mentor. We're also in the mentorship program working on the spec so we're moving that forward and also some super interesting discussions on the 2.0 plans and some of the substitutions of new ZKP stuff which we're going to be talking about soon on this call into an So it's the topics going on in that working group. All right. Awesome. Thanks Stephen. Looks like to IP hasn't been doing too much. Looks like the diff did come spec working group met on the fifth. Was anyone able to attend the, the did come spec working group. Looks like they're working on on compatibility for did come 2.1, and then they're working on some new marketing initiatives for did come. All right, unless I mistaken I believe that is all of our working group updates. Does anyone have any general updates or groups that we've missed. I'm just a quick call out the areas from our JavaScript recently released 0.4 point zero. The team who's worked on that including Ariel, Baron and Kareem are going to give a demo slash workshop on Wednesday June 28. The link is in the chat. It's not really going to be a hands on workshop but they want to really go in depth and all the changes and the new stuff that's in the new release so that's coming up in a little less than two weeks. All right, very cool. Thanks Sean. Yeah, I'll give another brief pause to see if there are any other updates or announcements before we hand it off to Stephen. All right Stephen the floor is yours. Great. Welcome. Let me share my screen. And I'll jump into the presentation. Close that got chat open off to the side so if anyone has comments, let me know. Let me leave that there. Well not quite there because I can't see. There we go. All right, can you see my screen. Yes, it looks great. Thank you. Okay. This is a presentation that I did at the open source summit in North America or a similar one so I'll, I'll, I'm using that I tweaked slides a bit to adjust into it but we'll share them up in the top corner. If you want to there's this bitly di dash the KPS that's the link to the slides themselves so if you want to grab those now or follow along. You should probably put that in chat but maybe someone can. So online identity with verifiable credentials and then we'll get into the meat of it which is the KP using high school math just to explain what the KP zero knowledge groups are so I'll jump in that's the agenda. So brief brief since this is the identity sake there's not a lot you need to know about online identities and then focus mostly on the zero knowledge proof section and what they are. So credentials paper credentials are what we use in the world that's what we've used for 2500 plus years. Many of them are for identity there's ones down here are things like professional attestations professional credentials like you know I'm an engineer, I'm an architect, I'm a doctor or I'm a lawyer those types of things there's supply chain there's certifications that can be for well I guess those are definitely not paper. But there's, there's lots of paper credentials in the world and the paper credential model is is one that, as they say we use a an issuer some sort of authority gives a credential to a holder that holder puts it into their wallet or puts it into their filing wallet or puts it somewhere, and sometime later in a separate transaction a verifier wants to see that piece of paper for some, some business purpose and so the holder pulls it out of their wallet or takes it down to the office of the verifier and shows them some sort of paper and the piece of paper in theory. And I put quotes around proves who issued the credential so there's some sort of marker on credential that shows who issued it who holds the credential there's some sort of binding on the credential between the person presenting it and the credential itself. And some sort of verification that the claims are unchanged and proves is done because the big thing here is is concerned with forgeries and things like that that the holder somehow manipulated the document either created it themselves or altered it in some way. Trust is largely this between the holder and the verifier but there's also the trust between the verifier and the issuer the verifier chooses what issuers, the credentials of what issuers they're willing to correct to accept. So, in when I talked about that there's both the technology and a governance aspect to it does the technology does it look like it's on the right paper does it look like what that that organization that issuer organization produces. Does it look like there's, you know, ink marks on it where I changed my date of birth on my driver's license so that I could use it for other purposes. And then the governance is things about what's the source of the of the authority of the issuer. Is it a trustworthy organization what where the where does their authority come from when they issue a piece of paper. What are the processes they use for that so those are all the things we talked about in identity paper identity paper credentials online are basically done these days by taking a picture of men scanning them and that's, and that's the where we are today generally with with the use of credentials digitally. What we want is a verifiable credential model. Again, very, I think everyone here should be familiar with this. The issuer provides an issuance of a credential and is got some cryptographic backing to it they hand it to the holder. The holder at some later time that the holder puts it in their wallet holds on to it their digital wallet holds on to it. Some later time they present it to the verifier. And there's a verifiable data registry is is a place where cryptographic material goes such that when the verifier gets the credential from the holder. They are able to verify it, not by contacting issuer and finding out whether it's about it's it's accurate whether it's it can be verified but rather by going to some independent place to get information such as public keys and so on to verify the cryptography. We use this list 1234 in fact verifiable credentials with capital V capital C as in defined by the W through C only talks about the first two which is who issued the credential and the claims are unchanged so there is a path to find out who issued the credential via the information in the presentation provided from the holder to the verifier and there is a signature on it cryptographic signature to verify the claims are unchanged. In the non creds world and places we work. There is a formal way of defining who holds the credential of binding between the person presenting the credential and the credential itself how they are associated. In fact, in, in a non creds is formally defined as part of the cryptography in other in W3C data data model standard that's outside of the stack and has to be determined in some other way so something like there's a picture of the person in it and that's the binding, or there's a did, and the person proves control over that did some sort of mechanism to find it. And then as well there's a fourth item which is available in some types of based on the issuers use case and and how the issuer handles it which is the claims have not been revoked. Those are the proofs that come about. Way less concerned about whether the holder forced it is almost it's pretty much impossible to force those types of things much more on, do you trust the issuer. And so that's a big piece. There's also concerns about the software that goes along so do you trust the issuer software the trust the holder software and so on. This is different from open ID connect and log in by Facebook so I did want to underline that when I talked about this for for those new to the topic. Again I think everyone knows that here and the and the big issue is that the issuer is involved in every interaction when you're using open ID connect that the, there is only a single process and in that process the user sort of consents to both the issuer and the private party and the issuer delivers the data directly and of course in a verifiable credential model on presentation the issuer is kept out of the picture and the interaction is only between the issuer and the relying party. Okay, that's the background on verifiable credentials and what we're using them for hyper ledger and non credits is a an instance of a way to use verifiable credentials or verifiable credential type. In the project in the hyper hyper ledger foundation. There's a complete open source implementation of it in rust. That is based on the non credit specification that is also being built and created in the hyper ledger foundation. This this implementation has a long history. For Indy came out about seven years ago in the self service self sovereign identity stack and non credits has been pulled out and revamped from that Indy implementation that Indy implementation itself derived from a from an IBM implementation so there's a long history of this. The big change that was implemented in pulling an on credits out of Indy is verifiable data registry and agnosticism ledger agnostic which means you do not have to use an indie ledger to store the objects. It's not necessary to have the non credits interactions they can be published in a variety of places, and people have already published such objects in a number of places outside of Indy Indy still the most, you know, common place you'll see them but it's no longer a requirement and so that's a big push that we're trying to do in the, in the non credits community. So that's what we've added to the picture which is privacy and that privacy comes in privacy preserving elements and that comes in for or flavors one is selective disclosure so that when you have a potential that you've been issued, and you present it. You can present the entire document so unlike a paper document where you hand over the paper document to be looked at. You can actually redact if you will some of the fields and just present the things necessary for the business transaction you're conducting. And so the verifier can still see who issued it can still verify that it's the various aspects of it but they don't see all of the raw data of the attributes within it. So predicate proofs are where a. This is the most obvious zero knowledge proof where you prove that you are, for example, older than a certain age, based on a date of birth in the credential without sharing the date of birth itself so you're you're proving something in in the credential but you're not actually sharing the data for it, and, and by prove you're not claiming or suggesting self attesting you're actually proving it cryptographically. And the big one that of why non creds is is really important is unlinkable identifiers so in in pretty much every other verifiable credential model and approach when you share a presentation you're sharing unique identifiers either for yourself or for the credential itself if you are given a verifiable credential and the way of presenting it is simply to show the other party the credential itself that the signature on it is a unique identifier it's very much unique and and so you're actually sharing a unique identifiers for it and so what an on credits does is goes highly very far out of its way to make sure that there is no linkable identifiers simply by presenting a verifiable credential. That is a key place where where zkps are used their knowledge proofs which can both to get into we're getting there, which is that you can prove that the signature, for example, is valid on a verifiable credential without sharing the signature itself. And again, prove being not really there. And finally, multi multi credential presentation so inherent in an on credits is a that you can present multiple credentials at the same time, and prove that they're tied together and do that all with selective disclosure and again that allows for a data and minimization if you need to prove that you're a lawyer and you know who you are as a as a as a resident of say British Columbia and prove that you're a lawyer you can present those two credentials, minimize the data share, and, and still prove those things and prove that they were both an issue to you, or to your wallet. So that's the key features that are added. I should throw that I do throw in that I do a lot of my work with the digital identity team in the government of British Columbia. This slide sort of highlights why government of British Columbia is so engaged in this. BC and every other jurisdiction puts a ton of focus on physical identity cards and, and the importance they provided in underpinning the economy and, and life in in a jurisdiction. The world is moving online. BC therefore is investing in figuring out the best ways to provide those same services to make it safe for citizens to operate online for residents to operate online and I highlighted duty to protect data privacy and security and that's in particular why BC is so interested in non credits. And that the organization wants to, to keep trying to make it that the approach use for verifiable credentials is as private and secure as possible. With that we move on to the fun part. The high school math edition, zero knowledge proof so we're going to talk about we're going to jump back to your, your high school math and talk about how graphic graphic proofs work with zero knowledge. I'm going to go back to Professor Kazusako, who did the preliminary, you did early versions of this the first time I saw this type of thing, Mike lotter from sovereign and now it in other organizations but very involved in the and on credit community did a bunch of these and actually it was my daughter that did a lot of the slides and presentation and math parts of these that you're going to see so high kudos to those so what is a zero knowledge proof. Here's the quote, you know, a method, one party can prove to another party that they know a value x and we're going to talk about x a lot in this without conveying any information apart from the fact that they know that value. This, as mentioned it's the proof it's the core of an on credits, and that example that I give, you know, I'm older than 19, based on my date of birth. And, but without sharing my date of birth so one of the approaches used to do, for instance, age verification and this is proposed in the ISO MDL model and, and some of the things I've seen in other places is, oh well let's put in, you know, a field that says older than 19 older than 21 older than 25 and so that's another way to get around that particular use case and it is a super important use case. With an on credits you actually put the date of birth in but the holder does not share the date of birth they just share a proof that they are older than a given age requested by the verifier. So that's what we're after. This is the interaction that happens we've got a holder prover that knows some piece of information and wants to prove it, without revealing the value. Likewise, the verifier does not know x wants to know, wants to verify that the prover knows x, without learning about x itself so both parties have a want to participate in this. So let's start with a nursery school edition so this is an example of, you know, really getting simple with it so you recall those who grew up in the age of where's Waldo, or had kids that did relished in the school that they knew where Waldo was on any particular page in the book, but they never wanted to let their friends know where they were because then their friends could claim they found it themselves so. So what is so how can you do that prove that you know where Waldo is but not share where Waldo actually is so the way you can do that is make a sheet of paper that's four times the size of the page in the Waldo book, put a little piece of paper behind it, and then move the page, the Waldo page around behind it, such that Waldo appears inside that little hole. The person looking at it can see Waldo, they know that you know where Waldo is but they can't see where on the page the person where Waldo is. The simplest the nursery school edition. Three requirements of ZKP completeness if the statement is true the honest verifier will be convinced that it's in fact, it is known by the honest prover. If the statement is false no cheating prover can convince the honest verifier that this is true, except with some small probability and we're going to get to that in a bit probability involved in in ZKPs. And finally the zero knowledge component if the statement is true the verifier learns nothing other than the fact that the statement is true they don't actually learn about the date of birth the value underlying. Keep those in mind complete completeness soundness and zero knowledge. Attributes mentioned this little earlier. ZKPs are actually probabilistic not deterministic. You are not going to get 100% knowledge there you are going to get a probabilistic, but we're getting pretty darn close and you'll see that. There's an element of randomness always in it which plays into how the zero knowledge proof is is provided. And then we're going to talk about the different forms of ZKPs notably interactive ZKPs and not interactive ZKPs foreshadowing a bit non interactive is better. We'll see why that is. Okay, high school math. Here's where we get to the refresher for high school math. We need to cover functions and inverse functions so we'll talk about functions we'll talk about exponents, and some of the rules of exponents because they come into play. Very clearly in this the module operator and prime numbers. And basically these components. Literally you covered probably in what what we in North America have is great 10 math very 11 math are are all you need to know RSA the Diffie-Helman. Diffie-Helman algorithm shot 256 cash all of these cryptographic things are all based on these core components of the matter. A function is equation for which any x can be plugged in and exactly one y comes out of the crazy one one result comes out of the equation so simple one there. f of x equals x plus two. So if I put 25 in f of x is 27. If I put two in its for and so on. All of these are examples of functions. And basically you have in these case one variable that you insert you do the calculation and you get your result out so easy stuff you know that stuff. The inverse of function is where you reverse it so you, given the output how do you figure out what the input is. So you do the manipulations you probably remember doing those oh I can take the two over the other side by by converting the plus sign into a minus sign so we remember that so x equals y minus two. And we get the inverse function so we've got our example of our original function we can calculate the inverse function. So these ones in these examples and all of these ones, what you'll figure out is it's pretty easy to go from the original function to the inverse and back. Those are all easy. What we want for ZKPs is a function that is essentially impossible to invert. So what we want is something that we cannot do the inverse for. And that is a core feature a core requirement. And in fact what a lot of the working cryptography is defined. We'll see, not just the, not just the functions, but the attribute or the, the, the numbers the types of numbers that contribute to making it impossible to enter, invert those functions, inverse functions, exponents. So exponent refers to the number of times is a number is multiplied by itself so two times two times two is two to the exponent three. So again, we remember that x to the fifth. We got that so exponents pretty easy. You've seen those in regular life, laws of exponents non exhaustive but we've got a few that play are really important here x to the zero is one. x to the one is x itself you just drop the exponent off. x times a gives you x to the a plus B. This is we're going to use a bunch actually we're going to use all of these but so again the example expands out why that is true so you can see that two to the third times two to the second is actually two to the fifth so adding the two together. And finally, x to the a to the B is x a times B. So again, you can do the same sort of expansion out and see that that's true. So, good, we got exponents covered. This is faster than you did it in high school I suspect. This is a modular operator. This is modular operator gives the remainder after division so it's done the same way as division but the answer rather than being you know how many times this this five go into 17. Rather, we care about the remainder. And so that's what you see here, 17 mod five is two. Those of you with a calculator handy or quick with math 321 mod 17 equals 15. A lot of people like to think of this as the clock ticks. I, I'm not so good on this one but I put it up there because many people relate to it but basically you count the ticks around, and what you stop at is the modulo 12 of a number so then you go around twice to 12 you come back all the way to the three that that's the modulo of it so there you go. That's a way to think of the modulo prime numbers. Last one, pretty easy, a number divisible only by itself and one. So, infinite number of these. The prime numbers are pretty important in cryptography and again that that's comes back to the need for these things to, to make that inversion of the function. So, we're going to come back to it but we're going to start again with another example that's commonly used and quite a good one. This is Alibaba's cave, and we're going to show that how a ZKP is interactive and probabilistic. So that's where these two concepts coming to play with with this. So, Alibaba's cave, Bob's the verifier Alice is the prover because of course we can't have anything in this community without Alice and Bob being involved in the cave. There are paths through the cave, a and B and there's a magic door between them. So, Alice is claiming to Bob that she knows the code to open the magic door, and she's going to prove to Bob that she knows that but she doesn't want Bob to know that code she just wants to have it her as her own secret. She's not allowed to tell Bob that. Okay, Bob and Alice figure out to determine whether she knows it is Bob stands outside the cave Alice goes in and then as she goes in she picks either a or B to go down. So in this case she picked a and then Bob, Bob does not know which path Alice took Bob stands there and says hey Alice come out one of the sides hey come out a. Alice goes out a and that was easy because she didn't even have to use her code she just came out a because she picked the same one Bob picked. So, Bob now has some evidence that Alice knows the code because if she went in B, she would have had to use the code but you know she could have gone in a so Bob really doesn't believe Alice yet. So, let's do it again. Alice goes in again Bob goes in again. This time, Bob says, Oh, come out B thinking Alice is going to pick the same way in. Alice can, of course go in a since Alice knows the code. She uses the code goes through the magic door comes outside B and Bob goes twice that worked. Now, the way you get it is the interactive part. We've got probabilistic. We've got randomness going in Alice randomly picks a or B we've got randomness from Bob, Bob's randomly picking a or B to come out. We've got interaction. We're having it repeated over and over. And every time Alice is coming out the wrong side, the right side. And Bob now thinks well there's no way Alice can be reading my mind and know which way I'm going to guess. So, I'm getting pretty convinced as I do this, you know, 1020 30 times that probably Alice knows it. This is the probabilistic nature. Bob doesn't know absolutely deterministically for sure Alice knows the code. It's just extremely unlikely that she would have guessed the same thing that he that he suggested every time. So, I like Bob is a completeness if Alice honestly knows the secret code Bob will eventually be convinced she knows the code. And that's done through repetition interaction. If Alice did not know the secret code is highly unlikely through repetition that she would be able to convince Bob she knows it. If ever the chance came that she went the wrong one for what Bob convinced she can't come out the the correct side of the cave, you know, and of course zero knowledge Bob didn't learn the secret code. So now we switch to math. Now we go over to to using those four elements of high school math and and we figure out Ali Baba's cave. So the first thing we do is we need a one way function, one where the inverse is essentially impossible so coming back to that you know x it's easy to find f of x if you know f of x, it's pretty much impossible to go backwards and find x. And this function right here is the one that's commonly that is used for zero knowledge groups. So, G, G is some public and known value G is known by Alison Bob. X of course is the number we're trying to figure out and then we do modulo P on it where again P is public and known value and it's a prime. So, Alison Bob share G and P only Alice knows X, and we're going to use this for Bob to know that Alice knows that that Bob can determine that Alice really knows X. So summary of steps Bob and Alice agree on G and P. Alice knows X. And so Alice tells Bob f of x and again confident that knowing f of x does not allow Bob to determine X. Alice generates a random number are so Alice picks one this is the equivalent to Alice entering the cave and randomly choosing a or B out Alice generates a random number and calculates f of our and shares it with Bob. So, Bob randomly sends Alice at a, a constant C, which is either zero or one in this first case. And again, this is the equivalent of Ali Bob is paid that's Bob saying come out a or come out me. Bob finds a new variable are plus X times C so Alice knows all of these items. So Alice knows be because she knows all of these things. And then she shares the of F. V, sorry, F of V with Bob. Again, Bob can't determine V. Bob verifies the results by checking that F of R, given by Alice F of X, given by Alice to the C, which Bob chose himself equals the F of V that Alice share. And if the two sides match Alice passes. So, this is the key slide that says how it's done and this is the, the manipulation that goes on with the exponents basically, F of R times F of X times C equals F of F of V that's what we said we needed to check so let's go down this F of R expands to this F of X to the C expands to this. Then we use our rules of an X exponentiation G of X to the C is G to the X times C, and then multiplying that by G to the R is adding to it.  And that's our results on this side. F of V is G to the V mod P. And recall that V was calculated by Alice is R plus X times C. And here, here we get these matching the mod P is is a factor that just moves out it's a common factor therefore we can move it out and and have it separated from the rest of the calculations of G. And as a result of that, we get this way that Alice only Alice knows X out only Alice knows are. And yet, Bob can be confident that Alice knows is accurately representing those values. So here's some numbers on it we're going to use really small numbers. X is for in this case, don't tell anyone only Alice knows that G we're going to use this five this is public 17 is the prime number so G is our constant P is our prime number. And F of X therefore is 13 you can do the math on that are this random number that Alice picks is seven against only Alice knows that not Bob F of R is 10. Bob then declares either a zero or a one randomly he chooses it tells that to Alice. And then V is calculated. Only Alice knows it again because it's V is dependent on X and R and only she knows it. She does that calculation and then sends does that calculation and so she can then do F of V. We get the actual math for it, we get case one of C equals zero, and we get V equals seven, and we get G to the V mod P is 10. Bob verifies that Bob knows whoops, F of R, F of X to the C, well if C is a zero, we know that anything to the zero is one. So we wind up this being 10 to the mod 17 which is of course 10. And we get verification that these two Alice past few case one case two here's here's where we use one, the only difference here being 13 to the one is 13 so this is 10 times 13 mod 17. So if you do the math you'll find that's 11 Alice passes in either case. Now we have to repeat this process where we use a different are a different C over and over and over again interactively. On the first pass, there's a probability of a half of giving the correct value because she knows Bob's going to send either a C zero or a C one. So she can figure out what it is. If they do this 20 times, it's a one in a million chance that Alice does not know X. Alice kept guessing the right thing that Bob was going to send a C, a zero or one and then, and so there's a one in a million chance that that was sent up that was determined. That's pretty close to being accurate and only 20 times going through this. Generalizing this, instead of C being zero or one, we can see to choose a C in the range of zero to P minus one remember P, P is our prime number that we're using. So this is the equivalent of adding many paths to Alibaba's cave. Alice has to choose which one of many Alice or Bob says come back. One of many, and this reduces the number of iterations necessary to prove it so basically if you have from Professor Sako's presentation, she basically explained it as basically each bit in C is an instance of a zero or one iteration. In other words, if we can get C to be 20 bits of information, we've got it down to a one in a million chance that Alice guessed correctly and produce the right zero knowledge proof, or the right value to send to to Bob. But even if we get it down to one back and forth it's still an interactive process Alice and Bob are still going back and forth. We want to get down to a simple request response process. Bob makes a request Alice sends a proof Bob verifies it so how do we eliminate that extra step where Alice, Bob has to send that extra value C. And that's one more piece of it, which is figuring out how to do non non interactive ZKP so we've got interactive. We don't want to use repetition to reduce probability we want to get down to a single back and forth. What's done is a hash function H again a hash function in a non invertible one with a random number I Alice's use that to define C as f of R, comma I, and using that function that Bob, Bob shared so Bob and Alice both know H the function works. Bob provides as part of his request I, and Alice knows makes up our Alice just creates it because R is a random number that Alice chooses. Bob still know see Bob can calculate see once, once f of R is known. And so it's a shared secret between them a shared value between them Bob and Alice both know it, and it is sufficiently random. R is used as to prevent replay attacks so those familiar with was cryptography and zero knowledge proofs and verify the credentials know about replay attacks with basically Alice and Bob Alice requests a proof from a bug request approved from Alice. Alice prepares and sends that proof along the way some outsider Mallory we often talk about Mallory malicious Mallory Mallory listens in and records the proof that Alice sent to Bob. Later, Bob asked Mallory for a proof and Mallory replays the recorded proof from Alice and claims it to be their own, and Bob can verify the proof so doesn't know. By using a different eye on every time a request is sent out. The proof that is received is different every time. And as a result, even if Mallory hears Alice's proof Mallory can't play it back and pretend it's their own because the eyes, the eye the random factor that what's called the nonce is different. And that prevents a replay attack. So almost at the end. Oops, there we go in real life. This is what number P looks like in a an on friends situation. He is a little larger than the 17 that we chose in our example. It's quite a large number and that is a decimal number. C is between one and P minus one so given P from the previous slide C is between one and this number remember that when we talked about C. C is a each bit of C represents a piece of entropy. So we said that, you know, 20 bits of of it would allow a one in a million chance that Alice luckily selected well there's a whole lot more than 20 bits of information in this large decimal number. So a whole lot more likelihood that C is is going to give enough that that probable probability is extraordinarily low that Alice could pick it. And then in real life this is what G looks like again another big and big, big huge number. So that's that completes the coverage of the high school math part. And I'm just at the end of time so that works of this session so works out well. ZKPs, as I mentioned there's, there's basically four commonly used. I go over a couple of them here. Blinding and identifier of the holder. This is the holder binding that ability that I talked about that is sort of outside of the W3C spec of how that gets done, but in an on creds it's very formally defined and it's always the same. Basically, the holder has a link secret. A big number of blinded version is put into the verifiable credential the holder proves the verifier they know the link secret, but they don't actually reveal it. And then secondly they prove that the same link secret is used with all the presented credentials. So the value of values for selective disclosure is the same sort of thing in this case the issuer signs encoded versions of the data and I don't know where I mentioned this but notice that everything I did x is a number x is always a number in zero knowledge So as a result if you want to do something with data involved in zero knowledge proofs you have to encode that data as a number. And so an on credits has a encoding scheme that converts all of the attributes all of the data elements into numbers. So it's actually in an on credit actually it is the encoded value that gets signed, not the actual data. So issuer signs the encoded versions of the data the numbers representing the data, the holder blinds the signatures in present in presenting those. So it removes to the verifier it knows the signature without revealing the signatures, and then the holder reveals the raw data values of the attributes and the verifier verifies they could inform, they correspond to the signed values. So that's how selective disclosure works, and a little bit on how these signatures are blinded in an on credits. There's similar capabilities and predicates similar in revocation but I just didn't think it was worth going through all the details of those. A pile of references for you. This was based on, you know, I went to this IW 26 presentation by Professor Sacco and it was outstanding. This is a little more formal than she did there's notes on the on on her presentation so you actually got asked to do a second instance of it so there's notes but not kind of the presentation and the math so that's this is helpful. So, like lotter did a presentation, when he's with the sovereign foundation. That's linked here which is about 160 odd slides that includes detailed math of this a little yet okay a lot more advanced than the high school math. But if you're interested in seeing all of the steps involved in this. That's a good presentation for that. And then some other posts about CL signatures, which is the the academic paper upon which all of this is involved. And I did that. And then some other interesting papers, oh, David Chum 1998 paper on blinded signatures just to show you that this is not brand new stuff. This was the sort of basis upon which CK Z cash came about and a bunch of the papers on blinded signatures so lots of lots of things to look at there absolutely I'll be sharing the presentation. Why do they matter CK P's no shared new identifier to present for governments this is huge, not sharing, creating the SSN or the social insurance number the social. Those identifiers are subject to legislation and so on creating new ones is difficult. Unlink ability, minimize sharing, and again unlink ability fighting back against online tracking at my time so I better stop some other things in here, but, and a call to action get involved in this. Feel free to reach out to me if you're interested and want to get involved, want to learn more. Welcome to do so. And with that I'll stop sharing and turn it over because we're at time. So that was a fantastic presentation super interesting. We covered a lot. So thank you. I think one of the questions is, if, if the if people can access the slides. After. Yeah. Great. Sounds good I can post those as well on the meeting page. So, if you go to this page. In a moment, I will upload the slides there. So, great. Yeah. Thank you so much Steven and thanks everyone for joining in with your working group updates and we'll see you all in two weeks. Thanks. Thank you.