Roedd y cwestiynau ar gyfer oedd y ffilosofaeth yn antychwtyd yw, yw, ychydig yn mynd i'ch gydig o'r hyn o'r hyn o'r hyn o'r byw? Rwy'n aeth y cwestiynau ar gyfer o'i gweithio, drwy'r hollian yw'r hefyd, ac mae'n rhaid o wirio dwi'n gwybod. Mae'n oedd y cyfeiriadau, mae'n cyfeiriadau yn y cyflwyd ddylayr, yn gweithio, gwiaithio, yn gweithio, The other is community, that the thing that matters for human beings is relationships. What I'm going to do is take you on a kind of whistle-stop tour of Western philosophy and look at how these two core ingredients of the good life have fared in the waves of industrial development that we've been hearing about this morning. Or to put it another way, will the fourth industrial revolution really bring us the good life? So to begin with the question of agency. I'm going to look at three different ways in which different philosophers have characterised this concept. And I'm going to begin at the beginning with Plato.  That's to say you had to be master of your desires. To be ruled by your desires was to be a slave to them. It was as if you were to be ruled by a hydra. That's to say when you kill off one snake or desire for boots in my case. Another two immediately spring up in their place. You might think about jets. I don't know. I like the shoes. Another way of thinking about agency was provided by John Locke who said that in order to succeed in life, in order to successfully engage with the world, you have to understand that world. So for example if you're driving a car it's a very good thing to be able to know how to fix the engine. Which my grandfather for example used to do. I certainly can't. Or of course there are some countries in the world which don't allow people to drive. Don't allow women to drive. That's another example where it be a very good idea. Where understanding liberates us and gives us agency in the world.  Who said of course that in order to lead a fulfilling life, what you need is you need to act in the world. You need to create. You need to produce. You need to labour. But you have to have a sense of ownership over that labour. You have to identify with the work that you do. And we can I'm sure all relate to this when we think about writing or painting or climbing a mountain or skiing down it. There's very little in life that's good that doesn't come at some kind of cost of activity or creation. Activity, agency is what counts. And arguably these things are threatened by the industrial revolutions that we've been hearing about and in particular in the context of the fourth industrial revolution. Of course Marx himself gave the most piercing analysis of what happens when one is alienated from one's labour. The more you become part of a production line, the more repetitive, the more discreet, the more mindless your tasks, the less a human you are, the more of a kind of shell of a human, of a shadow of a human. The more you become like an instrument or an object until humanity is erased altogether. That's the production side. On the consumer side it looks like the fourth industrial revolution should give us a utopia. A world where all our needs are met. A world where we're completely plugged in to everything that we might possibly desire. But of course the danger here is the passivity, the opposite of the activity that our philosophers told us was crucial to the good life. But everything is given to us. We don't have to do anything anymore. A further thought is that we lose understanding. We're de-skilled. Not only can't we fix our car, we can't remember our phone numbers, which of course is very complicated when your battery runs out on your phone and you're on the top of a mountain. And also there's a danger with smart marketing and acceleration of consumerism that comes with that. That there are things we didn't even know that we wanted until they're beamed into our phone. And then suddenly we become victim again to Plato's Hydra, unable to battle off these desires that couldn't continually put in front of us. So that's agency which we ought to keep an eye on. The second thing that philosophers have said is crucial to the flourishing life of a human being is community. Again going right back to the beginning, here's Aristotle. Aristotle said, as I'm sure you all know, that man is by nature a political animal, a zoon politicon. That's to say a man who flourishes when he is in a community. And Aristotle said that community is only possible when you have ethics. That's to say when you can deal justly with other human beings, when you recognize their humanity. Simon de Beauvoir, moving forward to the 20th century, she brought together the ideas of agency and of ethics. When she said in her existentialist way that you are nothing as a human being unless you act. And moreover it's not good enough just to act in any passionate way that you want. What you have to do is you have a responsibility as a human being to act ethically to care for your fellow human beings. And of course in her mind was appeasement in the 1930s and the impetus as she saw it on France and Britain to act. Now how do we get community? How do we get ethics? Well, another way of putting that is how do you learn to care about your fellow human beings? The Scottish Enlightenment philosopher, where is he? David Hume. He said that the only way that you learn to care is by having empathy. And you don't get empathy just by understanding facts and figures.  And of course there's a danger in the context of the fourth industrial revolution that rather than increasing empathy it's going to pull us further apart. There are various motors of atomization that we have to be aware of. The first one doubtless obvious to us all is that in our increasingly virtual communication we have less and less physical contact with each other, less and less actual contact with each other. We live in increasingly siloed worlds. We have our personalised news feeds, the echo chamber of Twitter where we become less and less aware of what other people think, what other people's values are. We have the massive problem of growing inequality which comes about in part as a result of the high tech, high touch new economy that seems to be only accelerating in the context of the fourth industrial revolution. Now you might think this sounds a bit apocalyptic and depressing and so I want to end with an example, a concrete example of where I live two streets. This is two streets from where I live in Hackney in London. This shows the riots that broke out in London in 2011 when a young black man Mark Duggan was shot dead by the police and it sparked off looting and burning and it seemed at the time I myself remember trying to cycle my daughter through these riots and thinking oh god this is, this speaks to a sort of terrible dislocation, a terrible inequality. But do you know what happened the next morning? It might look like a protest but do you know what this is? This is the community spontaneously coming out on mass and cleaning up the streets. There were more people coming out to clean up than there were to riot the day before. And so the question that I want to leave you with is what kind of a world do you want to live in? Do you want to live in an atomised world with walls high both virtual and real? Or do you want to live in a community? Thank you very much.


Puffin produce was started in the mid-70s by a group of growers getting together to jointly market their Pembrokeshire Early Potatoes. We've always had sustainability as a key element of our business since the very beginning. Most of the major retailers now have got rock-solid sustainability plans and they're expecting their suppliers to work with them on those plans. We've added resource to that area of the business and we keep building what we do in that area. I'm involved in helping to plan the whole growing process from field selection, varieties of potatoes and other inputs. We gain by receiving better quality in our factory here but also our growers will produce better crops, so better returns for them. We're starting this summer to construct a £6m prep veg facility that will make what we hope is the first Welsh chip. Sustainability is going to be one of the key hillars of that expansion. We're working with three key little consultants here.  We've then got a company that we're working on offsets called Climate Partners and between all of us, we're working hard to set science-based targets for the business. We're talking about sustainable soils and we're also looking at trying to reduce fossil fuel inputs to reduce our carbon footprint. We've received a whole range of support from the Welsh Government over the last 10 to 12 years. We're also a member of the Sustainability Cluster Group. It's been excellent, there's been great information we've gained from them. The end of the day, a farming business revolves around its soil.


Let's go for a ride. Hey everybody, this is Rebels Summit Auto in Von Lack, Wisconsin, and today I'm going to show you how to take your hard top off of your Jeep Wrangler JL four door. There's a bunch of steps that you can take to ensure that you're going to be able to take it off without damaging the hard top. And to do it as quickly and as easily as possible, I'll show you all the tools that you need to use and some good tips and tricks along the way. Hope you enjoy the video. So the first thing that you want to do before you take your hard top off is to come up with a plan on where to put it. Some people have lifts for them, but if you're probably the people watching this video and this is your first time taking your hard top off and you want to just put it in your garage somewhere, you want to have a plan on where to put it before you lift that heavy hard top off. So I've got it all set up right here. We actually have four 2x4s and we have them set up. The width is going to be five feet apart and then you're going to want to have 38 inches between them.  It's really heavy and you don't want to chip the corners that you see sometimes on used Wranglers. The next thing you want to do is grab your tools. So a lot of times people come with this tool kit that's got a few Torx wrenches and a ratchet in here as well. So the one that we're going to be using mainly for this video is the Torx 50. This is also how you get your door hinges off of your ringlet right here as well. But that's a different video actually. If you want to check that video out, it's in the upper right hand corner of the screen if you click that link. Let's get to taking the hard top off.  There's a couple quick disconnects right here, right here, right here and right here and they're pretty easy to get off. The JK had these screw ins back here that are no longer there so you don't have to worry about those but they come off really easy. You just do that, that, that and then you undo that. You do the same thing to the other side and then we'll show you how to take them off. Everything unhinged underneath the hard top here, it's time to take the pieces off. You always want to start with the driver side because it's got a little lip over the passenger side one and this is the first one to come off. I'll always remember the driver's side comes off first. Now you're not very heavy but they're kind of in an awkward shape so you want to have somebody else to grab these as you take them down. For safety you always want to get a ladder or something sturdy to stand on so it's easier to get these off but they just come off nicely. Okay so we got our case that comes with your hard top. This is for your front hard top pieces.  You're going to want to put the skinny side on the driver's side on this side because that's where the lips go and then you have places for your hinges right there. So after you do that you want to make sure that's down as well. It comes with a flap that goes over like that. So when you put your other piece in it'll not rub on this piece which will keep your hard top pieces looking good. That hard top piece off you don't want to put it into the case as well. It just slides in like so. Now you always want to put this side to the back so that it doesn't scratch on any of the hinges on the other side but this is also the only way that it can go in because of how the bag is shaped so they kind of do that so you don't make any mistakes. Once you do that you can put this piece up like this, you can stand it upright so that it's easier to zip and now you can carry your hard top pieces easily. It's probably about 50 pounds between both hard top pieces in the case here but then you can put those in the back of your Wrangler so that you can drive and put them on if you need to. Now we'll take the hard top off.  So now you're going to need your tool kit so you need your ratchet and then you need your T50 that's right in there that's kind of magnetic so it sticks right in there. There's going to be three bolts on the side as you can see. One, two, three and then there's three bolts on the other side as well. We're going to take those off and then there's two quick disconnects which I'll show you how to do in just a second. Okay so we're going to take this hard top piece off. Real easy. Now when I had my Wrangler I actually had some different bolts than this. They were like ones you could hand crank off. I really like those they come in like a triangle shape but once you get all these out you just go right down the line here and then you go down that line and then there's two more on the top which I'll show you in just a second as well and then once those are off we'll show you how to get these quick disconnects off and then how to lift it off and place it in a safe area. Alright so now that you've got your three on each side bolts out there's two on the top, there's one right here and then there's one right on the other side as well on the passenger side. So you just same Torx 50 wrench and they just come out just like the other ones did and as soon as you get those out you can do your quick disconnects and then you're ready to take your hard top off and enjoy the open air feel of a Wrangler. Alright now that you have all your bolts out of your hard top and you should have eight of them and the nice thing is that the ones on the top are exactly the same as the ones on the bottom so that you don't have to worry about them getting interchanged they all are the same. So you can set those off to the side or you can put them in the holder that's right in here there's a spot to put your roof bolts underneath this case here but now that we've got those all done we're going to take these quick disconnects off so we can actually remove the hard top. So the first one you're going to want to do is the electrical so it's pretty easy just take this red tab put it down you're going to press that red tab as you're pulling down hold this piece right there and then this stores right into this spot right here so it's actually going to snap in just like it did up there and now it's locked into place the second one is your windshield washer fluid there's a silver tab on there and if you listen real closely you can hear it click so then it comes off just like so and then you can put that right there and now we are ready to take the hard top off and enjoy the open air Jeep lifestyle alright now that we have all the steps done to get the hard top off the bolts are off your quick disconnects are off you get your place where you're going to set your hard top in place the next thing you need to do is grab a friend or somebody that's pretty strong to get this hard top piece off because you cannot do it by yourself and break it back and the next step once you have your person make sure that this glass is up and that this door is closed I know it seems kind of weird because you have the door closed and you're taking the hard top off but trust me it gets in the way if you don't have it like this and it just makes it a little bit easier to grab on the back here so that you're not going to pinch your fingers and stuff when you take it off alright so I got Mason here to help me we're going to take this hard top off and we're going to take it over to our sign spot when we're storing it while we have it off and it's pretty easy you're going to want to grab on the side here so that it can make a good grip and then you're going to want to grab on the edge just to get it off and once you get it off you want to get a better grip underneath here and what you're going to do is you're going to use this on the back and then take it over to your spot and since we have everything measured up we're just going to put the quarters on each piece of the wood like so and now you have your hard top off and you're not going to damage the corners you're not going to damage the edges on your hard top and it's just right where you want it easy to store easy to grab and pick up when you want to put it back on to your regular and that's as simple as that well that's how you take your hard top off here Jeep Wrangler JL hope the video was informational give us a like, subscribe click the bell notifications for upcoming videos that we do each and every day here at Summit Auto and we're actually going to be doing a video on how to take your doors off next so stay tuned for that you can actually click that on the left hand part of the screen I'll link to subscribe to our YouTube channel in the right hand part of the screen I'll link to this video in the lower left hand part of the screen link to one of our latest YouTube videos like, subscribe and share, click the bell notifications and once again thanks for checking out the videos


I had been waiting to hear and they had said to call and I called that morning and they weren't ready, called back in an hour, not ready, this went on all day and I was trying to get the information before I was picking up my daughter and I just got on the grounds of her elementary school and my phone rang and the doctor said do you have a nice quiet place to talk and I was on the playground and I said well this is as good as it gets. He told you over the phone? Over the phone. Wow. And he said I'm sorry you have breast cancer and then I just sort of stopped listening and I remember everything about the moment, the breeze, the temperature, the birds, the kids and he started to tell me and I said you know I stopped him and I said I'm sorry am I gonna live and he goes I can't answer that question and I said okay and he started talking more and I was still standing there. Were you able to like feel that moment or were you so outside of that? I was, it was pretty outside of myself and I said wait hold on a second I need you to answer that question. Like I want you to think if I'm somebody that you love and you have my, their results in your hand and they ask you what would you tell them?


The reality of it is, is this ain't talking about who should I see like they supposed to be, they ain't talking about King Vaughn like they supposed to be, they ain't talking about pop smoke like they supposed to be, they ain't talking about Nipsey hustle like they supposed to be, because they ain't talking about X extension y'all is like they supposed to be. If your time come up, buddy, they going they going they going to represent you for a good three, four weeks, four months.  Happy birthday, bro. Miss you.  You just gonna be gone, man.


Okay, so actually the title is Automorphic Forms on Higher-Ranked Groups, at least the official title. Okay, well, there will obviously be some overlap with other lectures, but that's probably not a bad thing. So I guess Philipp in his lectures has focused on the group SL2R. Yeah, I think that's because it's a short one. Aha. Okay, well, never mind. In any case, SL2R is the underlying group for classical automorphic forms, and there are many ways to generalize this due to certain special isomorphisms. One can view SL2R as the connected component of SO2-1, and then the natural thing to generalize this is to look at SON1. Of course, you can generalize this even more and look at SOPQ, but let's not be too general. Another way to look at SL2R is to view it as SP2R, or depending on your taste, SP1R. And then the natural generalization is SP2NR. Or what Emmanuel suggested, you can view this as PGL2 plus R, plus means positive determinant. And then the natural generalization is PGLN. Now this has rank one, so strictly speaking, this is not a higher rank group, but in some other sense it is a higher rank group. It has rank one over R, but over QP it may have much larger rank, so perhaps it does qualify as something of higher rank. This has rank N, and this has rank N minus one. I will mostly focus on PGLN, but I will also mention some things about hyperbolic spaces and the symplectic group Siegel modular forms. So let me start with hyperbolic spaces. So we are interested in automorphic forms on SON1. And to get started, we need some coordinates. So we start with the Eva-Zava decomposition of SON1, and, well, abstractly, it's a product of three groups, N, A, and K. So I'm not quite sure how to, no, no, no, I'm just trying to find out how to organize the blackboards appropriately. Perhaps I'll just continue here. OK, so K is the maximal compact subgroup that's most easy to describe. This is one and then SON. So that's obviously isomorphic to SON. And then A determines the rank, and you can easily see it's a rank one group. It depends on one parameter, and then the rest is the identity matrix. So that's isomorphic to SON1, one, perhaps connected component of the identity. And N is a bit harder to describe. It's easier to describe the corresponding Lie algebra, and then N is just the exponential of the Lie algebra. And so it's matrices of the form identity plus a matrix N plus a matrix N squared over 2. That's the beginning of the Taylor series expansion, and turns out that the rest is 0. Where N is of the following form, N is, well, this N is not this N. So this is a matrix N. Take a different font, yeah? So it's a matrix of dimension N plus 1, and it has a vector here of dimension N minus 1, and the same vector again. And it has a vector here with a minus sign and the same vector here, and the rest is 0. So this is a vector in R to the N minus 1, and this is isomorphic as a group to R to the N minus 1. OK, so there are other ways to choose coordinates. Here I'm assuming that my quadratic form is something like minus x squared and then plus y squared plus z squared plus w squared. So it's minus 1, plus, plus, plus, plus, plus 1, and so on. Sometimes, OK, so let's write this down. Is this gone forever? So this uses the underlying quadratic form minus 1, and then 1, 1, 1, 1, 1. Or I don't know, maybe the signs are different. Maybe it's 1, and then minus 1, 1, whatever. But this is obviously a form of signature 1N. Often, the quadratic form 1, 1, and then s is used. So this has one hyperbolic plane, and then here it's the identity matrix. So this is also a form of signature 1N is used. And then everything looks a little bit different. OK, so hyperbolic space is SON1, modulo SON. And that's a natural generalization of the upper half plane. And indeed, if N equals 2, and perhaps I have to take the connected component here of the identity matrix. If N equals 2, then this is just the usual upper half plane. And if N equals 3, this is what Akshay introduced yesterday. That's hyperbolic 3 space. And again, there are several models of this hyperbolic space. There is the hyperbolic model. This is the set of all x0 up to xn in the positive reals times the reals to the power n, such that x0 square minus x1 square up minus xn square equals 1. And then you have a natural action of SON1 on this hyperboloid. Maybe more familiar to you if you have grown up with the usual upper half plane is the upper half space model. There are n coordinates in this case. Here we have n plus 1 coordinates, but one equation. Here we have n coordinates. And we just require the last coordinate to be positive. So in the case N equals 2, this is just the usual xy complex plane. And there is a third description in terms of Clifford algebras, which is sometimes quite useful. And let me spend a little bit of time defining the relevant objects. cn is the Clifford algebra, which you may or may not have heard of. So the Clifford algebra is an algebra of dimension 2 to the n. So as a vector space, it has dimension 2 to the power n. And a vector space basis is given as follows. We take the power set of some basis elements, i1 up to in. So I take n elements, and then I take the power set. And so I get obviously 2 to the n elements. And I interpret a subset. So any element in this power set is a subset of this set, i1 up to in. And I interpret it as the product. I'll give examples soon. So this is a bit up-using notation, so strictly speaking. The basis is, so it's r plus ri1 plus up to rin. And then all products are i1, no, wait, i1, i2, and ri1, i3, and so on. So it's a product and not really a subset, but it's clear how to do this. And so this defines the Clifford algebra as a vector space, which is not particularly interesting. It's an algebra, so we need some multiplication relations. And I called them i because they play a similar role as i of complex numbers. So in particular, we have ij squared is minus 1 for all j. And we have ia, ib is minus ib ia. This is what you know from Hamilton quaternions. And in fact, it turns out that the Hamilton quaternions are a special case of this. And there are some more relations. I mean, this is not enough to get all the relations, but the multiplication table is known. So as an example, c0 is just the real numbers. c1 is the complex numbers, and c2 is the Hamilton's. So c2 is r plus ri plus rj plus ij, and ij is k. OK, so inside cn minus 1, we have an important vector space, vn minus 1. And this is the vector space consisting, so generated by all basis elements of degree at most 1. So i1 plus plus up to in minus 1. So that's a vector space of dimension n inside the Clifford algebra of dimension 2 to the n minus 1. And I can view the upper half space hn, and this up there should be hn, as sitting inside this vector space. And in a very natural way, if I take the upper half space model, then I have n coordinates, the last of which is positive. And I simply map this to x0, x1 to xn minus 1. So I don't do anything in this map. OK, so you can view the upper half space as sitting inside a Clifford algebra. So in the familiar case of n equals 2, it sits inside c1, and c1 is the complex numbers. That's what you know, the upper half plane is part of the complex numbers. And as Akshay mentioned yesterday, the upper half space, n equals 3, can be viewed as sitting inside the quaternions. Now the important question is, how does the group SON1 act on this? I mean, this is obvious in the hyperbolic model. But it's not so obvious in the upper half space model, and it can be very well described in this Clifford algebra model. There exists a set which I call SV. V stands for Valen, who more than 100 years ago introduced this theory, SV of cn minus 2, which is a subset of matrices, 2 by 2 matrices, with entries in the Clifford algebra, cn minus 2, acting on Vn minus 1. Bifractional linear transformations. If I take a matrix G, a 2 by 2 matrix with not all entries, but so not all of these matrices are allowed. I have to take a certain subset. But they are 2 by 2 matrices with entries in cn minus 2. Then this is az plus b times cz plus d inverse. Now this algebra is highly non-commutative. So the order does make a difference. I cannot write az plus b over cz plus d. I'm doing all of this in the Clifford algebra, which is highly non-commutative. For G is abcd in SV cn minus 2, and Z is in Vn minus 1. And this makes sense. So abc and d are elements in cn minus 2. But of course, I can embed cn minus 2 into cn minus 1. And then I know what the product with an element in cn minus 1 is. And it turns out, which is not easy to see, you have to show it, that this is an invertible element in this algebra. Not everything in this algebra is invertible, but it turns out that this element will always be invertible. Do you want a definition of SV that is invertible? Yes, probably. So the definition is a bit more complicated, but this is certainly one of the requirements. And the definition of SV is rather complicated, but I can give you the definition in simple cases. So here are some basic examples. SV of c0 is simply sl2r. So there is no extra assumption. It's just, well, the determinant is 1. But other than that, it's just everything. The same holds for SV c1, this is just sl2c. But already SV c2 is quite complicated, or a bit more complicated. This is the set of all matrices g, abcd, 2 by 2 matrices with entries in the Hamilton's, such that the following holds. ab star minus bc star equals 1, ab star and cd star are in v2. So they have, OK, I'll explain this in a second. And what is star? Star is the involution.  So in particular, star is the identity on this vector space v2, because v2 is defined by last vanishing coordinate. And it changes the sign of the last coordinate. OK, let's make a reality check. What's the dimension of this?  What is the dimension of this group? Well, for each entry, over the reals, you have four possibilities, because these are Hamilton's. So this is dimension 16. This means that the last coordinate vanishes, so this subtracts one dimension, another dimension. And here you have a random quaternion that has to be one, so this subtracts four more dimensions. So in total, you have 16 minus 1 minus 1 minus 4. It's dimension 10. You have 10 degrees of freedom over the reals. And the dimension of SO41 is 10. So this is good. OK, so reality check, the dimension of SV CN minus 2, sorry, C2, is 16 minus 4 minus 1 minus 1 is 10. And that's the dimension of SO41. OK, so the most interesting case is, so the most interesting case, I guess, is the case n equals 3. That's the case that Akshay mentioned yesterday. Well, perhaps the most interesting case is n equals 2. But yeah, I'm supposed to talk, not to talk on the case n equals 2. So then H3 is SL2C modulo SU2. And as we have seen in the upper half space model, typically the coordinates are x, y, and r in R3, such that r is positive. And so you can view this as a Hamilton quaternion with last vanishing coordinate. So this is sitting inside the Hamilton's with last vanishing coordinate. OK, any questions? So if you want to have further reading, literature on, especially on the hyperbolic 3 space, but also on hyperbolic n space and this valent group and Clifford algebras and so on, basically everything by Elstrott, Grunewald, and Manneke. They have several three-author papers, and you find everything in great detail in their works. What? There is a famous book. The book treats hyperbolic 3 space in complete detail. You find everything you want to know, hopefully, in this book. So that's certainly the most important reference. But this hyperbolic n space is treated in research papers. OK, so the Archimedean theory is very similar. The n case is very similar to the case n equals 2, simply because it's a rank 1 group. And in particular, so there is one Laplacian eigenvalue. There's one spectral parameter. And one has similar, for instance, one has similar bounds towards Ramanujan. There is a Kuznetsov formula, which has a very similar shape as the original Kuznetsov formula. And you can find this in many works. You can find this in work of Resnikov, Miatello Wallach. And there is also a long paper by Kokdel, Lee, Piatetsky, Shapiro, and Sarnak. So the Archimedean theory is fairly similar to the classical case. Hecker theory is quite different. Because Hecker theory, so there are Hecker operators. If you take an arithmetic subgroup, you can define Hecker operators in the usual way. But the Hecker theory is a bit different. Because over qp, SON1 may have large rank, not necessarily, but depending on p, it can have large rank. And then the theory is a little different. I mean, as Akshay said yesterday, Hecker operators, if they exist, change the picture completely. And so Hecker theory is a very important part. And the Hecker theory is more complicated. Because SON1 over qp may have rank n plus 1 over 2 Gauss bracket. So already in the case n equals 3, in the case n equals 2, you see 3 over 2, maybe 1.5. But if you take the Gauss bracket, it's still 1. But n equals 3, the rank can already be 2. And so if you have a given rank, then at least morally, and in some sense, very precisely, the Hecker algebra is generated by as many elements as the rank says. And so you see this. So if n equals 3, then the rank can be as large as 2. And you can see this in the classical picture if you view this as automorphic forms over an order in an imaginary quadratic field. There are ramified primes. Well, ramified primes are not interesting. But there are split primes and inert primes. And if you have split primes, then you get two Hecker operators for both copies of the split primes. So example, n equals 3 if p, the rational prime ideal, decomposes as p, p, bar, then one gets two Hecker operators, t, p, and t, p, bar. So this happens for half of the primes. And if p is inert, then, of course, you get only one. OK, so why is this interesting? I mean, you can define whatever you want. But does this have any arithmetic significance? Well, certainly the case n equals 3 has a lot of arithmetic significance because it's automorphic forms over an imaginary quadratic field. But what about higher rank, or higher n, so higher hyperbolic spaces? What is the arithmetic significance? So here is an arithmetic example. And that's associated with a theta series and quadratic forms. If you have a positive definite quadratic form, you can easily write down a generating series for the representation numbers, and then you get a theta series. And because the quadratic form is positive definite, the representation numbers are finite, and there are only non-negative representation numbers, so you have no problems with convergence. And you get a modular form on the upper half plane. If you have an indefinite quadratic form, then this doesn't really work because there are infinitely many, I mean, the representation numbers are infinite. And you have negative, potentially, you can represent negative numbers. So it's not really clear how to define a theta series for an indefinite quadratic form. And so Siegel developed the theory. So let q be an integral quadratic form with signature n1. And somehow we want to define a theta series attached to this quadratic form. But the naive thing of just writing down, generating series doesn't work. So Siegel introduces the following. He introduces the so-called majorant space, a majorant of q is a positive definite quadratic form. Well, it's a positive definite, say, symmetric real n by n matrix, in fact, n plus 1 by n plus 1 matrix. Matrix r satisfying, OK, I continue over here, r q inverse r equals q. And one can show that if you have one of them, you can easily write down all of them. So if you have one such matrix r, then all matrices are of the following form. These are of the form g transpose rg for g in SOq. So the special orthogonal group attached to the quadratic form q. So all of the matrices satisfying this are given by 1, and then you conjugate, well, it's not really a conjugation, but with a matrix in SOq. So that means g transpose q, g equals q. OK, and now we are ready to define the corresponding theta series, and we define the theta series attached to the matrix q as follows. It's a function of two arguments. It has an argument in the upper half plane, and it has an argument in SOq. And it's the sum over all vectors, integral vectors of dimension n plus 1, exponential. So what you would like to do is something like h transposed. This is not correct what I'm writing down, but what you would like to do is something like h transposed qh times z. This is what you would do. So then there is no g. This is what you would do if q was positive definite. But since q is not positive definite, this doesn't make sense. It doesn't converge. If you just take the x coordinate, then it's still OK, and for the y coordinate, you do something different. So you do xq, and for the y coordinate plus iy, and here you take g transpose rg for a fixed majorant. So you fix your favorite majorant, h, where z is x plus iy, z is x plus iy in the usual upper half plane, and g is in SOq. This is the so-called Siegel-Theta series, and it turns out it's a modular form in both variables. It's a modular form in z as a usual modular form on the upper half plane with respect to some congruent subgroup, which depends on q. So q has a certain level, and then you have to mod out a certain congruent subgroup. And it's also a modular form in g. So in g, in the second argument, it's a modular form for something that's isomorphic to SOn1, because q has signature n1. And in the first variable, it's an automorphic form, a usual automorphic form on the upper half plane. But if you keep the first variable fixed, then you get a nice interesting automorphic form on SOn1. So it is modular in z and g. So there is some arithmetic significance attached to these forms on hyperbolic spaces. OK, any questions? This is analogous as it starts in 0. Well, in what sense? I mean, certainly it's not a cuspital. So in this sense, it has to do with Eisenstein series. And you can probably, I mean, if you keep g fixed and view it as a function in z, then you can decompose it into Eisenstein series. Yes? In what, I'm confused in what sense it's a modular form in g.  Oh, yeah, yeah, yeah. Same with here. I mean, you have to mod out by a suitable discrete subgroup, which depends on q. So I mean, it depends on the arithmetic of q. There is some level, if you mod out by some congruent subgroup, both in z, in both variables. There was a question up there. How was the Hector operators indexed? I mean, in SL2, Hector operators are indexed by pn for all n in z. Or in SL2c, you can index Hector operator by all prime. By prime ideals. How was the Hector operator indexed in general? Good question. I don't know. I think, at least in this case, I worked it out and it turns out they are indexed by quaternions. So yeah, they are indexed by matrices. This is called the quasi-determinant, this expression here. And you can, if the quasi-determinant equals n, a real number n, then this corresponds to the nth Hector operator. Just as in the, yeah. So the point is, this has to be, I mean, a priori the quasi-determinant could be any Hamilton quaternion, but then it doesn't commute. So you have to take something from the center. And the center is just the reals in this case. And so at least in this description and for the case of C2, the Hector operators are parametrized by quasi-determinant being n. And I'm actually not sure how they are parametrized in general. And I doubt that you find this anywhere in the literature. I mean, for SN2 case, the real algebra is polynomial algebra of Tp1 and Tp1, which you can do it. So can we at least find the polynomial algebra of somewhere else?  And you can read in satake in the original paper everything.  But if you want to decompose the double cosets into single cosets, this is a complete nightmare if you want to do it in general. Yeah, anyway, I think very, very few explicit results are in the literature other than in the case n equals 2 and n equals 3, which is classical. OK, any other questions? OK, so this was just a very, very brief introduction into hyperbolic spaces just to give you an idea of some definitions so that you at least know how to start. And equally briefly, I would like to discuss the symplectic group and also give a few basic definitions. And then after that, we move on to PGLN. OK, so symplectic groups. OK, so let me first define the symplectic group. And there is great confusion. Some people call this SP2 and some people call this SPN. I call it SP2N. But if you don't like it, feel free to call it SPN. So these are all matrices in SL2NR such that m transposed j, m equals j, where j is the mother of all symplectic matrices, minus identity, identity. So this is the identity of dimension n and this is the identity of dimension n. And you can write this as explicitly as all matrices a, b, c, d in block notation where a, b, c, and d are again matrices of dimension n, such that a, d transpose minus b, c transpose equals the identity. a, b transpose is b, a transpose. And c, d transpose is d, c transpose. But in other words, these are symmetric. So this is the usual block notation that you find in most of the literature. Capital letters always denote dimension n matrices. OK, and there is an upper half space that I also call h, but it's not the h that we had for the hyperbolic spaces. hn is the set of all matrices z equals x plus i, y, n by n matrices, but now over c such that z is symmetric and y is positive definite. And you can embed this into the symplectic group as matrices i, x, i. So this is the same thing that you can know from the upper half plane and v, v inverse where v is the square root of y. So y is positive definite, so you can take a square root and this is the usual way to embed complex numbers into SL2R, where v is the unique symmetric matrix such that v transpose v equals y. And OK, so if I call this g, my group g, then this is just a quotient. g modulo, a maximal compact subgroup. OK, and g acts on this upper half space in the usual way. Group action, a matrix A, B, C, D acts on a point z, which is in fact a matrix, as A z plus B times C z plus D inverse. And again, you have to be careful with the order because matrices are not commutative anymore. OK, and as usual, we take a discrete subgroup. For instance, we can take sp2n over the integers, but we don't have to. And this comes with an inner product. There is an inner product. The inner product is just what you would guess. It's the upper half space modulo gamma. And then you take f1 of z, f2 of z bar, times an invariant measure. And the invariant measure is dx dy over determinant y to the power n plus 1. So the classical case is the case n equals 1. And then you just recover the usual thing. So these are the analog of mass forms. If you have holomorphic Siegel modular forms of a certain weight, then you have to include determinant y to some suitable power k as usual. So this looks all very similar to what you probably know from the classical case, except that all numbers are now matrices. But formally, it's very similar in many respects. So why is this interesting? Again, I mean, you can generalize as much as you wish, but why is this interesting? Here's the motivation. And the motivation comes again from quadratic forms. Motivation, why do we want to study Siegel modular forms? So assume you're given a positive definite symmetric matrix, an n by n integral matrix, symmetric positive definite, and even. By even, I mean that the diagonal elements are even. And even integral matrix is an integral matrix with even diagonal elements. And pick a positive integer, or positive less than n, or less than or equal to n, for a matrix T of dimension m with half integral entries and integral diagonal, symmetric and positive definite, study the representations of T by a. So what does this mean? Well, if T happens to be a number, so if m equals 1, then this is what you usually do. You want to know how many ways are there to write a given number as a sum of four squares. But you can just as well ask how many ways are there to write a given quadratic form, say a binary quadratic form, as a sum of four squares. So this is not representation of numbers by forms, but representation of forms by forms, but lower dimensional forms. And in the special case, m equals 1. This is just numbers. But you can, a priori, pick any m between 1 and n. So we can define the representation number. r a of T is the number of matrices g of dimension m times n. So in the classical case, m equals 1. This is a usual vector, such that 1 half g a g transpose equals T. And because a is positive definite, and T is positive, well, a is positive definite, this is a finite number. And you can encode these representation numbers into a theta series. Theta a of z is the sum over all T, r A of T, e of trace Tz. You need the trace to go back to numbers, where z lives in Hm. OK, and it turns out that this theta function has nice properties. This transforms like Cz plus d to the power n over 2. Theta a of z for gamma sum matrix with lower entries Cd in some congruent subgroup gamma of sp2mz. Is it determinant? Yes, otherwise it makes no sense. Yes. And so it turns out that theta a is a Ziegel modular form of weight n over 2 and degree or genus m. OK, so there is a natural motivation why we want to study such objects, because they are connected to representations of quadratic forms by quadratic forms. We have seen that many of the formulas look exactly the same, but there are other formulas that don't look the same. So many things become more complicated. For instance, there is typically a formula for the imaginary part of gamma z, which you can relate to the imaginary part of z. But here the formula looks much more complicated. It's Cz plus d minus transpose imaginary part of z Cz plus d bar inverse. Now you recognize, of course, if everything is numbers, then you get the usual formula imaginary part over Cz plus d squared absolute value. But here you can't do this really, because it's non-commutative. And you can imagine that explicit formulas become very ugly in this way. OK, there is a usual fundamental domain for sp2nz modulo hn, which looks very similar in some sense to the well-known case n equals 1. One needs Minkowski reduction theory. Well, yes. Well, this was. OK, so this was a motivating example. And now we continue with the usual theory, and we call the genus n. It just so happened that the genus here was m. So this is perhaps pedagogically not optimal. OK, Minkowski reduction theory tells you that there is a fundamental domain such that the coordinates of the matrix X, Xij, are bounded by 1 half. And the Yij are Minkowski reduced, which means that the off diagonal is bounded by 1 half times the diagonal. And the diagonal is non-negative. In fact, it's strictly positive. And we have 1 half square root of 3 is bounded by Y1, is bounded by Y2, bounded by Yn. And the determinant of Y is roughly the product Y1 up to Yn. In other words, the off diagonal is at least in terms of the determinant rather negligible. But I just described conditions satisfied in the fundamental domain. That's right. That's right. That's not everything is in the fundamental domain. This is a Siegel set for the fundamental domain. The exact fundamental domain has not been worked out, except for the case n equals 2. So Gotchling in his thesis 50, 60 years ago, one of the last Siegel students, wrote down, I don't know, 13 conditions or whatever, exact inequalities. 19? That's what I remember. OK, maybe it's 19. Whatever. OK, it's still a manageable number, but pretty large. So there is a finite number of conditions for SP4. But for higher genus, I don't think so no one has ever worked out exact conditions for the fundamental domain. But it's not a finite sequence? Probably is. Yeah, OK. But I think it's just in the case, as in the classical case, nobody needs that.  OK, so how can I get back this blackboard? Oh, OK. I'm supposed to stop anyway. But let me just quickly say something about the Fourier expansion, because that's something that's very important in the SL2R case. And it turns out that the Fourier expansion for Siegel modular forms is much less useful. There exists a Fourier expansion, of course, but it's much less useful. Fourier expansion, a Siegel modular form has a Fourier expansion of the following type. One sums over symmetric matrices, positive definite, or perhaps positive semi-definite and half integral. Some coefficient times E of trace Tz. And it turns out that this coefficient A of t satisfies certain symmetries. For instance, OK, so this is for a modular form of weight k. There is a determined u to the power k. A of u transpose tu for all u in gl nz. So this is invariance by units, if you want. But it's less useful for n greater than 2, or greater than or equal to 2, less useful for n. Greater than 2. In particular, A of t, the Fourier coefficient, has no direct connection to Hecker eigenvalues. No direct connection with Hecker eigenvalues. I mean, that's something that we are very much used to, that Fourier coefficients are just Hecker eigenvalues. This is not the case as soon as the degree is not 1. If the degree is 2 or more, then there is no direct connection between Fourier coefficients and Hecker eigenvalues. So the Fourier expansion is really much less useful. One has the Hecker bound. At is bounded by the determinant of t to the power k over 2, where k is the weight. And the conjecture is that this is bounded by determinant t to the power k minus k over 2 minus n plus 1 over 4 plus epsilon. So if n equals 1, you can subtract a half. If f is not a lift, and I will explain next time what I mean by lift, it's not a lift. We will do this tomorrow. But this is not known, unless in the case n equals 1 for holomorphic forms. What is known is this minus a delta. But delta is tiny. Delta is o of 1 over n. And we are actually expecting something linear in n. So yeah, there is lots of things to do. If you want to do, if you want to work on this, there are lots of open questions. OK, I guess I have to stop now. So that's all for today. Any questions or comments? Yes? For hyperbolic space, do you exactly know what is the connection for Hecker eigenvalues and the Fourier coefficients? Do we know what? The connection between Hecker eigenvalue and the Fourier coefficients for integral h and i. I don't know. Certainly in the case n equals 3, yeah, I don't know. There hasn't been much analytic number theory on these spaces. And I think it's now the time that one introduces the methods of analytic number theory for these types of automorphic forms. Yes? Would there be any consequence of analytic interest for this conjecture? Well, I mean, if n equals 2, then this is the Ramanujan conjecture, right? And so certainly there is some interest in the Ramanujan conjecture. Yeah, I mean, whenever you work with the Fourier expander and you want to have bounds, then it's certainly good to know what the best bounds are for the coefficients. And this is true on average. So in some mean square sense, this is true. But individually, it's not known. And I think there is some fundamental interest in knowing what the best possible bounds are. And one of the problems is that this is in fact wrong for certain forms that come from lower dimensional symplectic groups. But I'll discuss this tomorrow. The way to study representation of forms like forms. Yes, yes. Right, right. They go into the error term. So if you write, then you have Siegel's mass formula. So I mean, this is for cuspital forms, where the theta series are modular forms. But right, so these go into the error term, are these coefficients. And if you can bound the error term, then it's certainly a good thing. It would prove, as a principle, for more general problem than the one that I discussed, which is the representation of a quadratic lattice of rank n into a quadratic lattice of a square And so what I explained yesterday has been that there has been progress in that case, which was embedded in the case of this particular thing. So I have a comment that there's a very nice conjecture of Beuschever for SP4's that we realize it's very efficient to twist the del values of spin or a function. Yes. We certainly suggest that they are extremely irrespective, in some sense, in an even more difficult way than a cargo project in that case. Yes. Yeah, so they have certainly some intrinsic meaning. But they are not so much related to HECA, but perhaps to other arithmetic objects. Is it possible to say that the spin or L function? I will briefly mention L functions tomorrow. But my plan is not to go into most details, but just to give you an overview of the objects that we are dealing with, so that you at least get an idea how to start if you're interested in working on these things.


All right. So we're here with the E6. Hi. Hello. So please introduce yourself. Yeah. I'm Kinseng Chen. I'm the co-founder of E6. Now I'm doing the CO. All right. And this device, I saw it at the other booth over there. It's so many antennas. What's happened? Yeah. Actually, this is a dual 4G mode router. So many antennas because we want them to be more stable because it's for industrial purpose. So where do you put the dual 4G? There's two modems in there? Yeah. There's an inside here. Two SIM cards. So SIM card and two modem SIM card. And so it could be like three antennas for one, three for the other, or something like that? Oh, no. These will be have a two for one and two for one. And another two is for Wi-Fi. All right. And you bond. It's bonding. Yeah. It's bonding. Yeah. Why we need to have a two 4G, the two purpose? The first purpose is because of bonding. We want sometimes that they need speed. So the 4G is widely available in the world. They can use the 4G to bonding faster speed. Second is maybe retentance because of some of the places they have one mobile router operator is very stable. Another may be stable. They have two operator work together. So maybe as a single line. It's like redundancy and being backup solution and seeing to get the optimal bandwidth all the time. Right. 4G. Right. And you can do even more. I saw that you had four in there. And the other one? Oh, yeah. We got some, we deployed some of them on the 5G. You know, 5G is now starting available in the world. For example, I'm in Hong Kong. Hong Kong 90% of the curve is by 5G. So we can use the 5G now. All right. So which one? Yeah, this one. This one there. So there's a lot of antenna connectors right there. How many, how many modems? Yeah, this one is also two modems, two 5G modems. Each 5G modem requires 8 antennas. So you see a lot of antennas here. All right. Lots of antennas. These are the antenna ports. And you put the two SIM cards in somewhere in there? Yeah. For this one is very different from the previous one. This is called the power is waterproof. IP67 means that you can drop into water without damage. Of course, radio waves cannot work inside the water. But also you can drop into water with a heavy rain, a powerful water on it. It's okay. Maybe it's on the boat? Yeah, it's on the boat. Yeah, sometimes the wave, big wave on it. It's okay. It could be container ships. Yeah, container ships, regular boats, anything. You see the, where is the SIM card? SIM card is inside. So you really need the screwdriver to open it. And then put the SIM card in and put it back. So in order to protect it from the water. And at the other booth over there, it seems there was four SIM card or two is maximum you do? Yeah, actually this one you can put four SIM cards. Four. Yeah. But the two are active. So you can switch. Yeah. Yeah. And then the two will be called standby. Standby. Yes. So if there's a bad signal on one of them, it can switch to another one to see if it's better. Yeah. You have provided deployed in top 10 ports, three of them. One is in Tianjin, another in Shenzhen, another in Hong Kong. You know in the port requirement is very high. For example in Tianjin, in summer it's 40 degrees, 50 degrees. But in winter, minus 30 degrees, we still need to continue working 24 hours a day. Because whenever you miss some of this port, for example, miss 40 or 24 hours, they will lose 150,000 US dollars. So reliability is very important for them. So they can spend $150,000 on the modem instead? Right. Of course. Is it expensive? It's not very expensive. That's why they love us very much. Really? Compared to the 150,000 US dollars, right? Nice. Right over there at your partner's booth, I saw if you can have a look here. Okay. Is that a different model or is it the same you showed me before? What is this? Oh, okay. Yeah. Okay. This one we call satellite max. You see a lot of internal. It's exactly the same as a power, but it's not waterproof. It's four? Yes. This four is for one 5G, another four for second 5G, and this for Wi-Fi. Wow. Anything of them or make it duplicated. The purpose is because it's industry grade. We put in some kinds of smart lamp posts and also in lots of factories. For example, in Toyota factory in Tianjin city. Also in a multi-factory in China. Nice. What have you seen in the front? Just a bunch of Ethernet? Yeah.  The Ethernet itself is what we call PoE. It could provide power. If some of the equipment, they connect to it, they do not require external power. They just use it as a power and network. Wow. Cool. Let's go back and check at your booth. There was some other device right here. What is this one there? Yeah. This one we call satellite ST2. Yeah. It's a very tiny one. It's a single modem. The main is a 5G modem. There's a 5G antenna on here. Okay. The main proposal for this one is for lights. For example, in some of the vehicles, we call it AGV. In Toyota factory, they put a lot of them on there. Instead of using the Wi-Fi, Wi-Fi is noisy. Sometimes it's not unstable. But for 5G it's much more stable. They put it on the AGV and moving it controlled by the 5G. Wi-Fi was designed for micro ovens. Yeah. It's not really optimal for data. Right. It doesn't go very far. For home use, it's okay. But for industrial use, sometimes it's really dangerous.  And you lose $150,000. Right. Again. Yeah. Was this the same or? This is the same. It's the same. All right. This is also illustrating a little bit. Since when have you company do this kind of technology? I'm sorry? How long have you been doing this technology? Oh, it's a long time. We started our R&D in 2017. All right. Yeah. So in 2019, we deployed the first case in Tianjin port. So it's been very busy the last three years during the corona, just growing business? Oh, it's mainly busy on the R&D and the product development. Actually, whenever we go into the port, our engineer cannot go out. So they are very busy inside. You know, it's because of COVID. So they brought the whole port. Anybody inside cannot go out. People outside cannot go in. So did the engineer live there for three years? Oh, no. They lived there for a few months.  It's been very busy time. Yeah. And then how does it work? Because I really like the bonding technologies. Yeah. Bonding network is really cool. Yeah. How stable is it? Let's say you have two networks. One of them starts failing for some reason. For sure the other one will not get confused. It will just continue overtake. Yeah. And it will be very, very stable to continue. Yeah. It's very stable to continue. Think about it. When you use a fixed line, the fixed line drops. Broken. It's gone. But when we use a two line, even four line, each line only 19.9 percent. Oh, sorry. If the issue of the line is a 0.1 percent failure rate and four line together, the failure rate is very, very slow. This is what we do for that. But you said there was two standby and two active. Yeah. You also have four active or you don't have four active device? We currently have two standby, two active. But wherever there's one fail, the standby active one, they just swap each other. Instantly. Instantly. Yeah. This is pretty fast. And also if you really want to have a four active, even eight active, we can use this one. We can use a cascade mode. Oh. So when they cascade together, this line connects to each other and then they can have a maximum 16 of them. Cascade means load balancing. Load balancing, they connect together, make it a single router. 16. 16. But there's no space for 16. One, two, three, four. Ah, you just stack them. Stack them together. Custom means stacking. All right. We can stack four of them together. And then another four, and then make it 16. Is there something about interference from one modem to the other? How do you design interference-free design? You're very testing everything inside? Yeah. Interference is a problem on the radio part. But we choose a very good brand name from a market and one of the good factory here and they make it to avoid the interference. This is pretty good. Just like when we two together, sitting together, have a handset talking on the phone. We do interference in each other. We've got this technology already. So no issue. There's no issue at all. All right. Doesn't like Wi-Fi. Wi-Fi will have an issue, but 5G is okay. It works really well.  Yeah. In terms of bandwidth and reliability, interference and everything. Yeah. There's something we can see. For example, bandwidth. The bandwidth is 10 to 100 times. It depends on the location. It's much bigger. And another one is a very critical industry. It's response time. The response time in 5G is much, much faster than 4G. That means that you can have instant control for all these kinds of facilities, machines, et cetera, even the cranes. How about the signal strength, the signal quality or the interference or that kind of stuff. Did it improve a lot with 5G or a similar problem like before? For the signal strength, the strength itself is the distance lower. However, there are a lot of the number of devices you can connect. So there's a call and back. But on 5G, because there are special frequency called 700 megahertz. On that frequency, we test it in China on the coast. You can have a length of 90 kilometers long or this is really incredible. When we test it, we also supplied. We thought it was 40 kilometers, but the customer told us 90 kilometers. It's all the way to Taiwan? Not yet. It's a shift for a lot of boats. They are close to the coast. For example, they are shipping around the coast, along the coast, within 50 kilometers or 30 kilometers. It works well. You do not require a satellite. You just use 5G. You save a lot of money and also increase a lot of bandwidth. Would you just say it makes Elon Musk nervous? Because you sell the satellite? Exactly. Don't tell him. I'm joking. Do you also work with the satellite connection somehow? When we work with the satellite, we use the satellite as a background network or some of the test message. Some important message you cannot lose must now use the satellite. For a lot of video, camera or photo message, they use it, use 5G. Could be Starlink? Yeah. Could be Starlink, too. So you can connect the Starlink to there? Yeah. One of our customers doing the mobile operation, they told us they want to try it with Starlink. So we gave them some samples. They tried it by themselves. Maybe they can have the Starlink when they are more than 90 kilometers out of the sea. And when they come closer, it switches to 5G. Yeah. Exactly. This is the way they want to do. They want to have a container ship across the continents. So they use the Starlink and 5G combined together. That's amazing. Yeah. It's awesome. Think about when the container ship, when they are shipping from one country to another country, during the shipping, there are lots of information they want to see to tell the destination port, video, photo, different kind of message. If they use the satellite to send you out, it's called maybe 100 megabytes or maybe gigabytes. It costs a lot of money for that. But when they are approaching the ports, they can start to use the 5G to send out. When they are approaching the port, take them about two hours, one hour to ready parking. At that time, they don't waste. They use 5G. I think every boat should have this switchover from satellite to a 5G automatic system. And I think boats should have like, I don't know, a fishing boat, for example, should have some kind of technology with the cameras and everything. The fish. And then as they approach the harbor, all the data is sent over. But that's a different market, maybe. Right. This is a really big market on that. It's called the maritime market. Actually, we are trying to test it. And also, some of the boats are still in the store. As you mentioned, shipping boats. They try it. Right. And you're based in Hong Kong? Yeah, I'm based in Hong Kong. Also, we have a business in mainland China, Hong Kong, Southeast Asia, et cetera. In Europe, in USA? No, yeah. I want to find some of the partners in Europe and then we can work together to expand the business there. Because your technology is ready to export everywhere? Yeah, it's ready for everywhere. You already have customers around the world? Yeah, I can say in Asia. Not America, not Europe yet. All right. What's your big competitor? We don't have a very solid competitor. For one big competitor. We've got a lot of competitors for different setters. For example, in ports, we want Huawei. Huawei is our competitor. We want them several times. That's why Yangshan port tried to find us again. But in the commercial world, for example, we sell to 7-Eleven. There are lots of home user routers. It will be our competitor. Nice. But there's not a single competitor that can compete with us for the whole market. Can you talk about what is the modem chipset that you use? Yeah, the modem chipset we use is the Qualcomm chipset. So it's still the best. So Qualcomm has all the solutions for 4G and 5G? Yeah. They got the solution starting from 2G, after now it's still 5G. So they are still leading. We can see the performance is very well on using that. So the whole set of our router, we use the Qualcomm. Because some countries have some kind of discussions about what chipset they use and maybe the Americans are pushing their solutions and maybe that opens up different markets. But Qualcomm is welcomed by all the world. All right. Cool. All right. Thanks a lot. Thank you.


Anytime you play on the road, it's a difficult task to do to get the boys ready. You know, in the background you can see we're working hard. We've had a good week of practice so far, so hopefully we can come out and have a competitive game. Very good ball club as far as running the full ball.  And that's one of the big tasks that we have is tackling. And we're just hoping that we can come out there and compete a little bit with them. Freshman, it's always hard. It's not middle school ball anymore, so he's coming up. Had a pretty good scrimmage last week against three very good teams. You know, he's got a strong arm, good kid. That's what I like right there, is he's very coachable. So we'll see what the future holds for him. We've got to protect the football.  And on the flip side of that, we have to create some turnovers, make sure we secure the tackle and just play for four quarters. You know, we lost a few ball games last year because we didn't finish a game and hopefully we don't do that again this year.


Everybody forced here ripping one box of 2022 pristine baseball hobby for Travis K You'd love Travis See what you got man Go pack by pack on these Start with the refractors here. You got Eduardo Escobar and Emmanuel Rivera Sario Hernandez Pure Power Brice Harper insert first in case is a Teosca Hernandez not numbered on that refractor Chris Bryant and a purple rookie Max Crannick to 99 Shroud face Pedro on the insert and a Matt Olson non numbered as well Yeah, most of the cards of pristine look pretty sharp Got Salvador Perez and Hermon Marquez Miguel Cabrera inserts Got Greg Deekman rookie blue is 75 Cubs color match Kyle Lewis and an orange Manny Machado to 25 Nice Machado Bray you insert. I hope you like Machado to orange 25 because here's an orange Machado to 25 Still looking for autos, right? I think it's been one Yanni Hernandez rookie and a Marcus Simeon a Cormac base rookie Wander Franco rookie insert there The first auto is a Shane Bieber Paco demand auto That is non numbered there'd be two autos here and it looks like there is Ken Lario with a Gold Jersey auto JT Romuto number to 50 Solid hit there Clayton Kershaw and on the back is a Jared Kelnick Non numbered auto Jared Kelnick That's gonna do it. Thanks again Travis. We'll get those right out to you


Charley Company, one of the 137th Aviation Regiment, took part in Tobruk Legacy 2016 in Slovakia from the 19th to the 30th of September. Slovakia and the Indiana National Guard have been working together under the State Partnership Program since 1993, and this exercise brought together 1,200 troops from eight NATO countries to foster interoperability with ground-based air defense and joint operational procedures such as medical and casualty evacuations. Our mission here has been to partner with the Slovaks and the other NATO nations, introduce them to the different techniques of flying and missions that we do, Medevac, Kazavac, Air Assault. For the very first time, we tried to get a lot of nations, different nations together at this single place here, got the opportunity to teach them the NATO procedures. We brought in the Indiana National Guard to provide Medevac support with four UH-60s. Without the National Guard support in this exercise, we've really been missing lots of gaps within our capabilities to bring to the exercise. So it was very exciting to see all the assets come together from deployment from state side to the marshaling and RSO and I within Europe, and then moving out here into the exercise and integrating all the capabilities into such a robust exercise. It was really great. Excellent cooperation. Really from pilots we got a lot of cooperation like to fly our missions. The medic on board was excellent. The objective of medics were true response to incidents, medical response to incidents, then medical training, medical evacuation, especially with air assets, then control and command in medical field. Toburg Legacy is a testament to the multinational task force and the continued partnership of the Indiana National Guard and Slovakian Armed Forces. Reporting for the Indiana National Guard, I'm First Lieutenant Brandon Rex.


Everybody forced here ripping 20 21 select football hobby six box number 24 pick your team style We've got Adam be with the Jags Anthony P. Bucks Brian C. Seahawks Chris B. Panthers Colin M. Texans Dan M. Ravens Broncos Saints and bangles George why Cardinals Viking Steelers E and W Packers Jeffrey H. Niners Jets Jeffrey C. Titans John P. Bears Kerry G. Chiefs Landon G with the Giants Matt W. Eagles and Falcons Matt L. Raiders Michael R. Bills Peter M. Browns Richard S. Dolphins Ronnie B. Cowboys Lions Ryan B. Has the Chargers Scott F. Patriots Sean T. Washington Stephen W. Rams and Will T. With the Colts good stuff. Good luck everybody. Let's get started There's a big Joey B in here. You shall see You guys just the Jags and Pats up next still a few teams left on the others after that as well Let's get these breaks filled up to base cards Devontae Smith 53 and 99 rookie Jersey card for the Eagles got premier level silver Packers Brett Barf Ricky Williams Taylor waddle club level rookie Dolphins premier level tricolor Dolphins Devontae Parker 189 of 199 and silver club level Raiders Alex Leatherwood Got waved man. That's crazy Chris Carter 46 and 49 Jersey auto for the Vikings and I said Vikings Parker Like someone's got to be up if they're waving you after one season Silver snapshots Chargers Justin Herbert Concourse silver Buck Stalen Darden Like you have to have like a bad attitude on top of not playing well or something like that to get to get Waved after one year as a first-rounder Concourse tricolor Giants Cadarius Tony Number 214 to 249 and a cold level silver rookie Zach Wilson for the Jets Waddle premier level Silverfield level Falcons Deion Sanders Yeah, I guess sometimes that happens Eland Magie Harris purple concourse Mack Jones. Heck yeah patriots Number 52 of 75 Good stuff Concourse purple man, that looks great Patriots, it was Scott, right? Patriots. Yep, Scott F. Congrats again Scott Big hits last week too Cowboys rookie Otto Semi Fajoco number 60 of 60 vikings of guitars. He has some of my stuff Silver phenomenon Brady for the Bucks silver concourse terrace Marshall jr. Panthers rookie Mond got a tie-dye coming up now. There's a mills Trayvon Moorig Tie-dye club level Colts Darius Leonard Very nice. So you go Colts 8 to 25 great linebacker premier level silver pat crime with rookie Steelers Diggs maroon concourse buccaneers Jalen Darden Number 86 of 149 premier level silver James Robbins and Jags Tony tie-dye no tie-dye XRC Whoo tie-dye XRC Jets Garrett Wilson kaboom Boom Big XRC nine out of 25 congrats jets Yeah Incredible first box here This one's going to Jeffrey H. Congrats Jeffrey That's the thing of beauty right there nine out of 25 Garrett Wilson concourse silver matt stafford rams field double rookie fajoco for the cowboys Silver concourse kylan hill packers rookie And a seguan field level box two Got white premier level seguan barkley.  I love to see it man 180 to 199 again silver premier level st. brown for the lions Brady Concourse silver rashan slater rookie for the chargers Rogers field level base Jalen waddle dolphins gold patch four of ten There you go fins Silver premier level cardinals rondale moore Anthony Schwartz Good break so far guys Great break so far Purple darby premier level rookie falcons. You could mag the waddle 68 of 75 Club level silver rookie jets michael carter Hoco Hey, what's up dean Long gain while silver field level nico collins texans rookie Got seguan We're so club level rookie tricolor bills number to 99 181 no, sorry 148 of 199 or 149 And gold rookie hot hojags travis etn jr. Some last spot mojo for you adam nine out of 10 Last spot mojo Whoo Break is scorching every finkel rookie Now he was a rookie in what late late late eighties Silver phenomenon cardinals kyler murray laces out concourse silver uh texans david johnson That would be cool if they could get um I know shan johnson the actress played like the other version of rickville. Who was like the guy in the photo? They need to get the refingal auto silver snapshots josh allen bills Concourse silver juju for the Steelers Sean johnson wasn't in a ton of movie. She was in bladerunner though. Nice matura. There's chase base Tricolor jets elijah moore rookie There you go jets again 131 of 249 Premier level silver jonathan taylor colts xrc again val kins desmond ridder val kins congrats on that one Concourse silver vitzpatch for the titans in a michael carter field level rookie jets All right box three now Yeah, maybe paul the market the market. She's gone up since he came out But that's what happened last year too 2020 select is around 2k a box now 2020 select hobby It's craziness Justin fields 38 of 99 rookie jersey card bears suburb premier level falcons matt ryan Premier level rookie nausea here of steelers And i mean 2021 prism is out what like 1900 a box so Select will approach that territory. I would think Premier level tricolor Ravens tylen wallis 191 of 199 silver club level derrick gorge chiefs rookie darby Guys i'm gonna check the next few spots here. Got trevor lorenz mark is about the scaling 81 and 99 jersey auto packers harry mcguire This is this is this is american football Also harry mcguire Not many fans of harry mcguire these days silver phenomenon titans derrick henry silver concourse rookie colts quitty pay subject We've been on tiktok. Um probably a year now. We've only started streaming just recently tie-dye after the mills club level Is baker mayfield browns concourse Browns baker for a 25 on the tie-dye got premier level silver teres marshal jr panthers jc jackson Is that another xrc? Yes, it is It is a rookie lions Aiden hutchinson xrc very nice If you go lions one of the two that you could get Aiden hutchinson congrats on that one concourse silver rookie niners trey lance tom bradyfield level all right guys next up six box number 25 Jags and pats There's a link in chat There was a shield last night. No one sent me a picture dude concourse red davante smith eagles 90 of 99 for the rookie there Silver club level michael strahan giants silver field level etn for the jags robinson wilson base my homes Uh, tony field level tricolor for the giants. This one's numbered 59 of 75 A lechon mccoy tie-dye auto. That's a nice header for the eagles four of 25 shady auto Is that pulled here? Get a uh premier level fields On the base there for the bears Yes, this has been a really good break so far Silver snapshots dolphins to a tango by loa silver concourse cardinals carlin murray Mitchell Zebra habib last night. That's awesome Big stuff man davante smith God javante williams premier level blue maroncos 1444 of 149 Premier level silver nuwang wu for the vikings nico collins sensation zack wilson concourse silver atwell rams rookie And lorenz field level rookie base trevor lorenz All right box four chase base Trey lance 49ers rookie jumbo jersey auto 42 of 99 Congrats sand fran a lot of big teams with big hits so far Some teams are straight up crushing a man jeffrey h also has the niners jeffrey went big and is paying off for him A tyson cambell Hey, thanks for the fire Have to keep it going now silver phenomenon derrick henry silver concourse josh allen for the bills Got a very good six boxes here silver snapshots to a for the dolphins silver concourse to a for the dolphins and tonio brown I put them on my team Tri-color matt stafford rams 242 of 249 silver premier level falcons kyle pits rookie mondry stevensson The link is in the bio to join group breaks silver concourse in book rookie for the saints and nazi hairs field level rookie for the stealers jk dobbins 1 of 25 patch card tie-dye Ravens silver premier level patent turner saints rookie marquise brown purple concourse kylan granson colt's rookie 40 of 75 called all the silver mooney for the bears Sensation zack wilson silver field level rookie titans deserts patrick tommy tremble tie-dye club level here Looks like saints colors Yep saints mark ingram club level tie-dye 23 of 25 club level silver bears justin fields And a jamar chase premier level base orange quinn Williams jets concourse Auto coming up next 37 to 49 And bangles auto is sam hovered maroon 56 of 75 talon smoochy wallace I'm not sure what that is.  That's a pretty nasty card right there What a break concourse silver mon for the vikings and a cd field level man. I'm gonna We're gonna enjoy this recap. That's for sure Two more boxes and a recap you guys Let's keep it rolling Let's keep it rolling waddle allen blue premier level stevenson for the pats got a die cut after that three of 149 die cut rod smith tie die auto 14 to 25 broncos nice hit denver this legend hit there Sensation silver trail ants niners concourse silver montana for the niners Book Yeah, firestorms are cool. They're shorter printed as is and they have lower numbered parallels Premier level dolphin strike color ricky williams 170 of 199 silver premier level falcons matt ryan Um, I think there can be some ricky inserts dean. I think I've seen a zack wilson, but you can find them on the checklist phenomenon It's a redemption after the waddle and it is debonte smith jumbo rookie signature swatches prism eagles debonte smith There you go silver selections elijah more jets Silver tirey kill concourse for the chiefs On those are rike. Yeah, I'm just not sure which rookies, but I have seen zack wilson personally On that insert there's a mac jones club level gain while light blue premier level eagles 97 and 99 silver premier level dawn for the rams agi harris Silver concourse rookie patriots from andre stevenson chris carter at club level jail and waddle 16 of 49 nice one dolphins Silver club level damien harris for the pats Silver concourse for the dolphins hunter long the rookie I'll larry roundtree field level rookie Sponsored to parker jersey card dolphins 39 to 49 Silver premier level panthers terrace marshal jr. kellen mon premier level Silver field level chase claypool steward gary's tony premier level White club level tray lands the 49ers again jeffree h four of 35 big card Big hits for jeffree dude. Are you kidding me? He bought some big teams bought some big teams That is freaking sweet. Yep jeffree h the niners four out of 35 white club level This has been a Pretty pretty bonkers break here club level silver chargers josh palmer. All right guys Let me see if we still have those two teams left on the next one There was jags and pats Let's see Still jags and pats Moving right along I don't know on the top of my head how much the niners were trailing sensations kylan hill one of one nfl shield laundry tag rookie auto packers Are you serious? two one of ones And a ton of monster hits On top of those in this six box break Goodness gracious. Congrats packers now enw This is pretty nuts This is pretty crazy That's the fun digs This is wild silver nazi harris selection silver concourse mohomes for the chiefs water premier level Got mac jones club level tricolor nazi harris rookie for the stealers now 121 49 silver premier level jj watt with the cardinals 7 of 15 tie-dye auto Going to the titans the oilers go to the titans Nice warren moon Love his signature lean signature on the moon Silver phenomenon bucks tom brady silver concourse derrick gore rookie for the chiefs I wish every break was like this then again. I mean And the hits would be a special right? Every break was like this tricolor stealers pat fry booth 216 to 249 premier level silver 22 outlaw rams the unfortunate nature There'll be big hits. There have to be some that are not so big fields nice oven cook snapshots tie-dye for the Vikings 21 to 25 nice one Vikings silver concourse eskridge for the seahawks And a bar more rookie field level whoo cadarius tony on the zebra field level giants rookie now This is wild This is getting crazy silver club level sainte's marking. I mean it's been crazy. Just hadn't stopped Another tie-dye what what the heck man? And it's davis mills concourse tie-dye rookie texans Six of 25 I don't know I don't know what more we can pull man This is insane collin m Yeah, dude, this half case is loaded Silver field level brandon cooks texans Oh my god There's another zebra Oh goodness. What is this now? Let's see. It's phenomenon insert It's bills. It's josh allen Okay, then All right. Well There you go bills. We'll get that magged for you concourse silver sack wilson jets Chase field level rookie Holy shnikes, dude Ronald jones 17 to 75 rookie jersey or not rookie, but uh jersey for the bucks premier level silver lance for the niners Naji Harris and a penny sewell blue field level lions color match there 13 of 49 silver club level javante williams That is an insane amount of mags guards for six boxes Wait till you guys see this recap if you missed any part of that break just get ready just get ready That's six boxes worth of mags right there Okay, start with our relics buccaneers ronald jones Uh dolphins devante parker ravens tie-dye jk dobbins bears justin fields Eagles devante smith autos now vikings chris carder jersey auto We got tie-dye oilers go to the titans on the warren moon auto Got eagles devante smith rick jumbo signature swatches prism on the redemption tie-dye rod smith auto for the broncos Bengal sam Hubbard tie-dye shady for the eagles jersey auto mvs packers a j brown titans simifahoko cowboys Everything else is in a bag This is a six box break We got tie-dye concourse davis mills for the texans Xrc lions aiden hutchinson Xrc falcons desmond ridder gold rookie patch dolphins jalen waddle tie-dye xrc garret wilson for the jets blue concourse jets zack wilson It's 199 zebra phenomenon josh allen bills Concourse purple mac jones patriots rookie 52 of 75 gold rookie auto jags travis etn jr 9 out of 10 club level white for the niners tray lance Number four of 35 jumbo jersey rookie auto niners tray lance 42 of 99 A one-of-one rookie laundry tag shield auto kylan hill for the packers And a one-of-one firestorm raiders derrick car.  We got six more coming up