Just a couple of days ago, Google claimed that it has reached quantum supremacy. How do you think this will affect mining and the cryptocurrency world in general? I love the way that they named their success in quantum computing. Quantum supremacy, it sounds so amazing, but in fact, it's not really that amazing. Some great marketing there caused a whole ripple of fear all across the technology industry, as people misunderstood what quantum supremacy means. Up to now, quantum computers could solve problems that classical computers could also solve. They might solve them a bit faster, but it was really difficult to see if, in fact, the quantum effects were giving a significant advantage to the quantum computer, to the point where a classical system is simply unable to solve that in any meaningful amount of time. Quantum supremacy, as Google calls it, is simply the first time they demonstrated a solution to a problem that was solved with a quantum computer in a reasonable amount of time, and with a classical computer that would take tens of thousands of years. Therefore, that shows that the quantum effects being operated in this quantum computer really confer a significant advantage in computation. It's no longer a question of, was this really a quantum result, or was it just a slight optimization on a classical result? This is a whole other realm of computing, because of course now you can do something with a quantum computer that a classical computer cannot do in any practical sense because of the time difference. That's what they mean by quantum supremacy. However, we're still talking about 100-something qubits without error correction, and this is a really, really important point. In fact, doing quantum computing with error correction is a much higher bar, difficulty if you like, than doing quantum computing without error correction. For cryptography, you actually need error corrected bits, and you need thousands of them. You need thousands of qubits to be doing quantum corrected calculations in order to break something like ECDSA, to be able to reverse a public key to a private key on the elliptic curve, for example. What is the effect on mining in the cryptocurrency world in general? Zip, buccas, nada, nothing really happens. Quantum supremacy, what Google described, is demonstrating the practical applicability of quantum computers to certain classes of problems. Those classes of problems are not the same class of problems we're talking about when we talk about breaking cryptography. That's a really good thing because the problem isn't really Bitcoin. If we get quantum computers that can do thousands of qubits with error correction and consistent results, we have a much bigger problem. The bigger problem we have is that the entire world's classified communications, confidential communications, financial systems, all depend on classical cryptography today. We would need to upgrade all of that in order to make it quantum resistant. If anything, many of those systems, these legacy systems, are a lot more difficult to upgrade than an open public, open-source blockchain like Bitcoin is in terms of its signatures. In fact, we're already seeing a proposal for a new signature system called Schnorr that is being introduced as part of Segwit v1, hopefully in the next year or so. But while Schnorr isn't quantum resistant, it is demonstrating the capability of developers and the ecosystem as a whole to upgrade and switch the digital signature algorithm. Therefore, we can imagine a time when quantum computing starts approaching the practicality of breaking classical cryptography, at which point we do upgrade and we will need to upgrade the digital signature algorithm. We can see now that that's possible. It's something that practically can be done within the ecosystem and it will be done probably in the next year for Schnorr signatures. Now, the other thing, of course, is to realize there's a difference between quantum computing being applied to digital signatures and quantum computing being applied to hashing. Hashing algorithms are fundamentally different from digital signature algorithms. While the street logarithm problem that is at the core of digital signature algorithms is, in fact, very approachable by a quantum algorithm, a quantum solution, and if you have a quantum computer with enough bits, you will be able to break it thoroughly, reverse it in very, very short time. That does not apply equally to hashes. So, hashes do not use or are not based on the discrete logarithm problem. They're not based on factorization. They're not based on these one-way functions we see with digital signatures. Instead, they use a series of permutations, binary shifts, and things like that, which are not actually that easy to develop a quantum algorithm to break. So, hashes are a lot less sensitive to quantum computers than digital signature algorithms. And just there, you might notice this little trick that Satoshi introduced, which was to never present the public key naked if you like to the system until you're spending from that address, and instead present a double-hashed version of that called the Bitcoin address. This is a genius little trick, which not only provides level of abstraction, but it actually provides a level of protection against weaknesses in the elliptic curve algorithm. Weaknesses that can be introduced through a malformed signature, such as reusing a random number in a signature, or weaknesses that are introduced because of quantum computing, or vulnerabilities in the algorithm for digital signatures that are discovered later. No matter what the weakness might be, if you follow the best practice, which is to use a Bitcoin address once and only once, and to immediately spend all the funds in it the first time you sign and never sign for that address again, what you get is a much higher level of protection because the moment the public key is recorded on the blockchain, it's recorded because the funds just got moved and are no longer in that address. The address effectively is empty, zero balance. So the only time the public key is shown is when, even if you exploit it, you get nothing because there's no money in it anymore. And that's a really smart trick because even if quantum computing gets the point where digital signatures can be compromised, now what that means is that someone would have to be able to compromise the digital signature in a time between you transmitting a transaction to the network and it getting confirmed and the money spent and then break it, sign a different transaction and double spend it in less than 10 minutes. So it actually puts another layer of difficulty, if you like, on top of the public key and digital signature algorithm itself. So this question comes up a lot, so I'm just going to make it very clear. A lot of people keep getting worried about quantum computing. We are nowhere near the level of quantum computing that would be needed. There is a small possibility that some intelligence agencies are a decade or more ahead in their ability to do quantum computing. If that is the case, which we can't really tell, then they might be able to do things in the next decade to break Bitcoin's digital signature algorithm. But if they have a secret that important, they're unlikely to use it to attack something that's not that important. When that secret is also important for attacking confidential and secured and classified communications of military adversaries, or even the keys that control the nuclear weapons systems in order to disarm them and turn them off. So that's the kind of secret you keep very, very close and you don't reveal it by attacking some Bitcoiners just to cause some damage and a cryptocurrency. More likely, however, we are still at least a decade, if not several decades from quantum computing being able to break digital signature algorithms. And more importantly, if we get closer to that point, the digital signature algorithm and even the hashing algorithm can be modified to make it more quantum resistant and more secure against quantum attacks. Eventually, you get to the point where quantum computing is a commercial reality for everyone, where everyone has the ability to use commercially available quantum computers on the desktop, on the laptop, maybe on the smartphone and or whatever other device. At that point, you can use a quantum computer to do quantum encryption that cannot be broken by a quantum computer trying to do quantum decryption. And therefore, if everybody has quantum computing, the threat goes away. The threat only exists if very few people have very powerful quantum computers and nobody has any ability to change the algorithms, which is a rather extreme and unlikely scenario in any way. So I'm not worried about quantum computing. It's a fascinating topic to talk about. It's fun to speculate. But in practical terms, these are not the threats that the Bitcoin faces. There are much more practical ways to attack Bitcoin and other cryptocurrencies than these exotic types of threats like quantum computing.


Everybody forced here ripping a box of 2022 mosaic football hobby for Steve are here. We go Steve I believe this is our last person before the new year. Let's go Steve Close in us out buddy Let's go Alrighty This is also the first mosaic box. I've ripped so you got to pull some heat now And honestly, I don't think the check was too much to auto for box on average New ultra-air razzle dazzle insert and tessellation parallel Let's do it Let's do it That's scary thinking about it right now on Robinson mosaic Red mosaic Jair Alexander Reds will be hobby exclusive not numbered parallels though a Straight fire purple mosaic Tom Brady 14 at 49. Okay, that's really nice look with that purple Number Brady first pack. All right, Steve. All right, that's what I like to see Purple Brady 14 of 49 straight fire the straight fire is like so good this year. There you go, man. All right Let's keep that momentum going Here Wilson base Shot white Desmond Ritter mosaic Ezekiel Elliott Silver Howie long Nick Chubb montage breeze hall blue chips rookie the insert oh Nice number Brady for a person at PCs. That's that's what it's all about See some more. Yeah Wondell Robinson, David O'Jabba mosaic rookie Desmond Ritter a Red mosaic rookie Devonte Wyatt blue chips Malik Willis mosaic in it to win it Simeon Rice James who Williams Kair Elam a Justin Jefferson mosaic red white. What is that all about? as a flag as a Is that what is that a parallel? I don't see that on anything else. That's a parallel, isn't it? Maybe I don't want to go crazy if it's not but let me see here national pride Okay, so that's like a base variation then Right, so yeah, okay. That's what I was thinking. Okay, so not everybody has a national pride, but that's pretty cool. Okay I'm sure wasn't crazy rare silver deer and waller Montage Josh Allen mosaic blue chips rookie Traylon Burks That's it. Yeah, so I was thinking Steve. Yeah, pretty pretty nice. I Want to make sure wasn't like something huge. I Want to give it a dude do justice whether it's a super rare card and on Malcolm Rodriguez Jelani Woods mosaic Kevin Bayard a Purple Hassan Haskins rookie mosaic number 17 to 49 All right, purple is the way of this box Henry Stairmasters who overdrive Shannon sharp. That's cool overdrive Shannon sharp The insert took a lot better this year and they have the past couple Malik Willis base What is that coming up? What is that? What is that? No way Gold rookie Otto Brock Purdy seven of ten Steve Oh man Gold Purdy No way That is a hot card right now That is a hot hot card right now Was a Robert Quinn. That's awesome dude silver Donovan people's Jones. Oh My goodness, I couldn't I had to stop pause for a second You might want to list that when you get it Take a snap try to be holding it. Yeah Steve Corral got to Cade Otten rookie mosaic Yes, seriously Red mosaic Fred Warner silver montage Trey Lance Yes, just as he doesn't he doesn't have autos and everything so what is out there is going for like even more Immediately brushed it dude. Heck. Yeah, there's a lava base Skull Philips pro mosaic Denzel Ward silver Eric is a conma rookie Introductions Drake London a mosaic overdrive Jerome bettis Yeah, for sure Dario exactly. That's another reason is like a gold mosaic. That's I mean that is a true like to 10 Like there's no other You know rookie autos and mosaic. You know what I mean? There's not like variations of that There's a is that gates? Yeah, it's right there. Antonio gates on the mosaic auto Hunter hen fro hunter hen fro ren fro Damian Harris red mosaic James cook Derek Henry mosaic mass or meta mastery Honestly, that'll go that'll do really well mine too, especially right now A blue mosaic Darren Waller a lot of color in this box. This is awesome. Man 62 and 99 silver rookie Garrett Wilson I'd sit at the door toward the ride Silver center stage 20 rumbo. That was a really good box. He was a lot of good color Number Brady Gold party dude sell sell sell that's awesome. Congrats. Steve. There you go, Steve last personal of the year delivered I heard that Appreciate it, buddy


What's up everybody? Jason here for jazzy's case breaks dot com. 21-22 Pinini Donner's Optic Hobby basketball four-box break Pitching number 14 just sold out number all-card ship and 14 and 15 of the last sets of four boxes from the case So again, if you have a little character set that means you want that from a team random little or a pack filler I should say Give away some teams there So again, we have two sets of four One two three for love four five six for the right Let's uh choose which one we're gonna use here and it is gonna be Six One two three four five six. We're gonna go to the right side. This left side will be for Pitching number 15 Let's go. So again the next break is like I said still on the website down to 10 left.  Oh, actually hold on. Let me take up a strategy over phone.  I don't think it's actually a hurricane. It's still a tropical storm But yeah, I don't know when that's happened, but I guess they're saying that because of the heat wave it actually pushed it up But no, it's not really Sharif Cooper, I think it's tropical storm, but they're obviously calling it hurricane.  Look at that Kate all silver for Jeff My name. Yeah James Wiseman eight Cuita name is Cuita And a reigning threes Luca Donchich.  Yeah Galen Green Mariners lost Well, they're playing well at the Braves, right? I think we knew that already come into it They just better not lose to the teams that they should be winning Dominator's Lakers and a Brandon Boston, Jr. 53 About a little Scotty Barnes Opti grabs that is going to Toronto and Brian There you go. So Scotty Barnes is live in the stickers. It's not the on-card All right guys two more to go It is what it is All right, next one Ben Simmons 76ers Gonna run we got Terry Rose here Joel and B Anna Jason Preston off the collar. We had she more than 100 dominators DC going to Jeff Moses Mooney Giddy Zach Levine Was that a green? Oh, man, is that a Raider rookie green? Snikes Think it is. Oh my god, I think that's Evan Mobley No, it's not. Wow. I thought I seen the hair Kyle Lorry who that actually got me a little nervous there. I see the top of the head. I was like, oh man, it's Evan Mobley Whoo, false alarm guys. Sorry That is Kyle Lorry though for the heat Sean out of five green two or five I could have sworn that was a Raider rookie. There's Mitchell, Scotty Barnes Last box. Yolkic Rusovic Gary Trent, Jalen Green Alright, I'll check it out, man Corey Casper this Jason Tatum light it up. It's a blue looks awesome 21 of 49 LeBron start gazing there light it up curry Derek favors Excuse me and Raymond Felton. It's our first little vet right there, right? I feel like That's Nick's autograph Wayne with that one. We got a kawaii Leonard Pink Velocity and a day wrong sharp. Thank you, Rex. They run sharp hollow. There you go guys That was the break. All right, and there you go guys. So that was the break here. I'm gonna light it up there Felton, Ruto Jamura. Actually, that's our second bit. Scotty Barnes, Kate Cunningham Hollow, Davey on Mitchell, Zyra Willems, another Kate Cunningham Hollow out of five Kyle Lorry. So here were your four autos Appreciate you guys. Number 14 in the books. Now, let's just switch things really quick I want to do the left or right for those all-star games or all-star duels We'll go left or right Let's roll Two and a five seven times one two three four five six seven Stays with the left side. So left side will get all those there and Appreciate it guys. Like I said, next one's in the store. Jasby'scasebreak.com


August is hot and so are the savings at Appalachian Wireless, where all smartphones are 50% off. Even the 64, 128 and 256-gauge models, two-year agreement required, better service, bigger savings. That's today's Appalachian Wireless, an East Kentucky network company. Millions were looking to the sky today in anticipation of the most talked-about event of the year, the 2017 Solar Eclipse. A few thousand of those spectators descended upon the East Kentucky Science Center and Planetarium in Prestonsburg, as the center gave out thousands of free solar eclipse glasses. Officials with the center say the turnout is an example of the growing interest in astrology. We're not even looking at this from a scientific standpoint. This is just having people come out, enjoy the science center, enjoy the eclipse, so people are interested in this stuff. So this is a great crowd. It shows what they support we're doing here, and the crowd's out here, but the Planetarium dome is full of people watching it, the classroom is full, the exhibit hall is full, so it's great. We appreciate everybody coming out. Although the day was about the eclipse, those in attendance were able to learn something and just be a part of this spectacular event. It's basically an event where the moon moves in front of the sun, which makes it basically dark in the daytime. And it could basically cook your retinas, so you need safety protective gear. That way you won't go blonde because it's very, very strong. It's a very pretty thing with the sun blocking the moon. A couple minutes ago it was like I did Orange of All, now it's just like a little moon. It's been the first full solar eclipse that's going all the way across America in 99 years, so I wanted to see it. For more information on the East Kentucky Science Center and Planetarium, visit the web address listed on your screen. Reporting from Prestonsburg, I'm Shawn Allen for EKB News.


Throughout the month of May, the Department of Defense celebrates Asian American and Pacific Islander Heritage Month. Over 65,000 Asian and Pacific Islanders serve in the U.S. military, displaying the type of diversity the armed forces have. I think it's great. I think it's really inspiring. It basically shows that no matter who you are, as long as you have the drive and the skills, you can succeed and there's that opportunity for you and the Navy. Members from Allied Joint Force Command Naples displayed many pieces of Asian culture, including Jiu-Jitsu, as well as various traditional food items. For one sailor, showing off these things is important. A lot of cultures, you know, you kind of do forget your original culture. I mean, I'm looking at a poster right there with origami and I'm just having flashbacks of me being like six years old and my grandma teaching me how to do origami. So to me, it's kind of bringing back the old family traditions that you see, you know, when you were kids. Petty Officer Matt Wright, Naples, Italy.


Gymraeg gyddonğachent yn rydyn ni. Bydd yn ddig yn rhoi iawn o fath oeddool bydd yn y wneud. Gwyn am ychydig o'r fath o fath cael angen o gyhoedd conservellolgau. Gwyn am ychydig o'r fod y程yd o'r hunain, bobl o'r fath o'r curwmiadau. Mae hynny yn dechrau sydd yn gwneud o'r periyfau cymdeithasol o'r falch yn soliolol wneud. Bydd yn gymryd ei nór o'r dim yr ymddiolol. They desire to be liked, admired and supported by many people. As I have said in a previous video, the narcissist is dependent on our reactions. They are dependent on what we do, feel or think in response to what they are doing. It feeds them. It makes them feel as though their false self exists. when you respond to what they are doing, you are acknowledging their false self, you are supporting their delusional fantasy, this is why they have to be the centre of attention, they have to be the person who excites everybody's interests or concern even if they are the centre of an argument and everyone hates them, it still makes them feel as though they exist, Mae gennym gweithio ti'n ni ddweud o arferfodaeth yn ei gweithio. Ond yn ddweud yn gwneud hynny, i ddyn nhw'n gwneud yn ddweud. Yn ni'n byw yn ddweud o athodd yn ei wneud. Felly mae hynny ddweud heb gennu gweithio amthodd yn ôl a dyna'r ffyrddau. Mae'r ffonn ac roedd yn ddweud o'i ddweud o fewn o'r ffaith. Rwy'n ddweud gweithio ar y gyfer yma, It's not because it's something they are passionate about. They're just doing it because it gives them the attention and admiration that they need. They're just doing it because it makes people notice them. They're not passionate about what they're doing. They're addicted to people's reactions to what they're doing. They don't really enjoy their work. They enjoy the results of their work. They rank the quality, proficiency or value of the hobbies, activities or people in their life. Not by the satisfaction or enjoyment that they provide, but by how useful, profitable or beneficial they might be. They only care about things that will make them publicly acknowledged. In the narcissist world, everything can be divided into two groups. Things that leave them deprived of attention and admiration. And things that make people react to them. They have no care or passion for these hobbies, activities or people. All they really care about is the reactions that they might generate. The attention and admiration that they might provide for them. They want to be famous. They want to be known by many people. It makes them feel powerful. It gives them the ability to direct and influence people's behaviour and the course of events. It provides them with a constant source of narcissistic supply. It boosts their false sense of self-worth, self-esteem and self-importance. It makes them feel special, as though they are better, greater or different from what is usual. They don't really care about bad publicity. They don't care about things that might cause harm to their reputation. As long as they remain the focus of public attention, that's all that really matters to them. What bothers them is when they are not getting attention, when they are not getting publicity. They hate it when people do not take notice of them. It makes them feel as though they don't exist. And when they begin to feel that way, they will do whatever it takes to get it back. They will do whatever it takes to get people to focus on them again. They will become disruptive and excessively extrovert. Some narcissists will even physically attack you, just to get you to pay attention to them. Narcissism is everywhere in the media these days. It's all over Facebook, Twitter, Instagram and Snapchat. They all want attention and admiration. They don't care about you. They only care about how you react to them. They act as though they are happy and they love themselves. But secretly they are miserable and they regard themselves with disgust and hatred. Just as they secretly feel worthless and insignificant. Yet they portray an image of superiority. However these narcissists portray themselves, it's always the opposite of how they really feel. They are trying to trick you into believing that they are something or not. When what they are trying to be isn't something that they are even interested in. All they are really interested in is people's reactions to what they are doing. It is the reactions that make them feel alive. It makes them feel as though their false self exists. They don't really care about being famous. They are more focused on people's reactions to their fame. That's what they are addicted to. That's what makes them feel special and important. But it comes from external sources. It's not generated from within. They are living a lie. But if they can get people to agree with it. It makes it more believable for them. And that is why they obsessively promote it. That is why they buy likes or subscribers. They already know that it isn't real. They know that's not who they really are. They know that's not what they are really about. But if they can get people to believe it. It makes it more believable for them. If people react to it. It's then how they will react to it. Which is why all they are really focused on is people's reactions. That is the only thing that has any value or significance to them. Because it is the only thing that makes them feel alive. Thank you for watching. I hope this video resonates with you. Please like, comment, share and subscribe. Click the bell icon to receive notifications for my future videos. If you would like to donate. My paypal link is in the video description. Coach the inquiries you can email me at www.coachingatjima.com. Thank you for watching and I'll talk to you soon.


Welcome back to Toffy TV. It's the road to glory. We are back after a slightly mixed last episode the last game We blew we blew it against Sam Dory We really did we had a big chance to jump up that league and get in the European places and We blew it one nil up and we ended up getting B3 one So we're now playing Real Madrid. Oh Well 2004 yeah, it's the Galacticos area. Nice one. No jar Galacticos here Well, oh there is two two expo players with this so Talk us through the team but Even Campo Fernando Iero Who's just actually being the Spanish man? Roberto Carlos Blackburn Ex-Lester City Cambiasso Got Salari who was a good player David Beckham Zinedine Zidane But But we have to bounce back Yeah, we've brought Fandanao in. We have to show Beckham out. We have Readies back after a tight groin reckless back rats is back in there because We've kept faith with age when if although we need to maybe chip him with a couple more goals because We need that physical presence if the not scoring so listen. We're at the Bernabao It's gonna be tough But we've got a good game faces on I'm off for the game face on right here. We are in the Bernabao It looks magnificent Game for you bless game face. It's game don't the face I have for the 80s. No, why cuz I think we've been going 87 kids Come on Patience make the right decisions at the right times Okay, keep it That she's taken her off on it as you would. Yeah, I'm there in the middle. I'm there Give you it. I'm up to it. Go on get in the middle. I'm gonna get on. I've gone. I've gone shady Benz here in for you Oh, she's always missed it go on I thought I did just because I'm catching cold. Oh, he's just made what? Why is that see that look at that come on? This is absolute hundred miles an hour stuff from the start It's all right. I'm in Beckham territory. That's you that you come to it. Come on. Yeah, mate No, and it's Beckham. It's Beckham. It's Carlos. It's a Dan. It's anyone any of them the goalie probably could score for me I Shine one down. No, I'm not saying that I'm not saying that and that's not a file as he clearly wins the ball all day I'm a all day Keep it keep it keep it What's the file? That's not a file Refer is your home area.  Oh God. Oh It was never a corner for the start it's never never a free kick absolute Rob. They robbed Really poor defense. It is poor defending I think that some of the lads out in shock of the red that it wasn't a foul Come on, let's turn this round to turn those frowns upside down space that can't because there's no There you go. There's a file referee Say 2018 it's not the dad, you know the dinosaur ages Dead man one it we did well a just to get men into around them Could that go on little one little one. There you go. Go on. Go on. Keep going. Keep going. I'm calling me. Don't worry Come inside now. Come inside. Come inside attack the space In the middle oh Never pass when you stretch in No I'm gonna pass it there. I'm having a no I mean, that's that's the rule of living most in your life. Bye. Yeah Look navel Oh Come on, right when you like get the crosses in it's not a game for I Like to like to like get in the middle Right, alright, alright, not a great arm. No, it's an understatement. We haven't made any chances We have we haven't made them we've had one we had the one half only by linear Karelia. Come on. Just raise this just raise it It's got full boys on Little one go on pop it off it's a Yeah That was most better. That was most better. I expected that to hit the back of the net.  That was never a free kick and Carlos is just Mr. Boom Bastik is a shop come on I found out what to go 40 minutes ago. Really would stop. It's a All right, go and banches just turned up better. Come on It's your box service. Come found yet. Oh, why did it go there?  I still can't believe to keep a push me at the answer about how that was in the game changer Can't give our man a space No Nothing nothing nothing nothing is going for us Haven't got near these let's be honest. No, but they're just doing exactly what we should be doing. It's just fast fly I think the difference is the place they Will it make people way? Little triangles big all in past part and then trying to keep it going you across the pitch You need to come back the other way We might not win the league. No, no, that's never come into my head Go on Get get something and the songs the game. Oh, that's amazing. Yeah, that was amazing I'll tell you what when we implode we implode It's weird cause I just feel as though this in the Chelsea defeat of being the only two Yeah, we haven't really been in as a game and yet we lost the last game 3-1 Just kill me. Oh wait. Oh Honestly really this is this is how we're playing it, are we? Come on 11th Starting to slip down that table again and those losses have become more than the wins halfway in the season number nine this game 19 Playing against Rome with the likes of Montella Candela Candela Emerson I'll die and obviously taught you for us. So we've we're just going with what we've got Straight in there. We want to win this game right here. We go. Come on back a good of some park Let's do these. Come on. Come on. I'm ready.  Come on straight in. Come on. No messing just get do everything right Okay, back to doing things the right way See pop I'm gone Superb We in there. We're in there. There's loads of us. Don't get out. Get in. Yes more like it That's what I'm saying. That's what we're about. Did you start getting it? It is Getting all the way through the middle.  Just let us get on with winning Cut that out if we get beat Together we are strong. Yeah, see that see that Are you with us or again us again is yeah, it's a great It's a great Stevens look at I just dug your awkward then she's get yourself a goal and she said yeah, I like to will come on build on this now build on it Bob Bob do Build on it. Of course you would. That's what we have to do Be aggressive be aggressive without running out of position. That's what it's all about So I've just luckily for us. I was there Not These are a little bit worried about us.  We've absolutely mulled them everywhere with the passing super you've got one-on-one and you've just done nothing That's what you think over the key now. I'll one-on-two Come on only appellate. Okay. Come on win as a team Bad missus. That was great. Yeah, that was great football. That's it. It was aggressive. Yes Oh, I'm off for this. Come on. Let me get it again.  I'm on the hype train. Go on.  Come on Yeah, yeah, yeah, yeah, yeah, yeah Great Sight give it sight Just suck it off my take candy from a baby. Can you play to yourself? Yeah, go on there All right, you know what's good for the goose is good for a gander. No idea what all that reference fits into what you've just done, but That's how it fits inside the side Oh That's fine. That's fine. That's fine. That's fine. It's fine. It's like power of up. There's the ball All right What a run by power there. I love his little shooing to himself.  Give a pen. Yes, give a pen. No. No, it's a free kick. It's a free kick, isn't it? It's got to be a... It's not in the box. It's not in the box. Shit, that's a free kick. Yes, big wall. Second ball. Jump, hit him. Yes! No! Win it! Yes! No, come on! Come on on! Yes! Don't, you can't see. Don't. Go on! No! That was awful! That's because of this yes thing in my head. Together we are stronger, remember? Powerballing! Yes! New. I'm talking nonsense. Keep your head. The head is kept on the shoulders.  Yes! Go on. Go on. Go on! Thanks to be honest with you. What a save. I kept it alive. Shit, it will light it well. And again. We've pummeled. Yes! Good off! Come on, it's chasing down. It's nail, nail. Nail, nail. Nail, nail. It's chasing down that negative goal difference. Come on. It's all right. No, no, no. I've got no criticisms. With the manner of the football. Get it in! Get it in! And the great is absolutely screaming. Screaming! Screaming! Oh, the lads are hungry. They are hungry for like a wolf. There's a song in that somewhere. Oh! He's not tonguey as a wolf. I feel like he's terrified. The noise is your making. He's absolutely keeping it up. Good save! Good save! It's hearty, innit? It's OK. OK. It's not OK. This is really not OK. Time to just control ourselves. Save your energy now. Less grunting. Less wolf-like behaviour. That's a fair point. It's a fair point. Less wolf-like behaviour. I don't know how many times that's been said on the football field, but do you know what? For this occasion, it's right. And again. The body's in there. Still loads of bodies in there. I'm just throwing it.   She's just gone. She's just gone. He's opened the space for you. He's opened the space for you, Baz. The space has been opened. Ten, finish! I didn't think it was on first time. Go on! Get in. Best save I've ever seen, I mean! Let him worry about the next save, Baz. That's what I say. That's what I say. Go on. Throw it in. Go on. I don't believe that. I honestly don't believe he saved that. Get in the middle. Get in the middle. At least I'm in the middle. You might have been in there. I was in there. I put him off. I put him off with my wolf breathing. That's what I did. That's what I did. That's what I did. A goal. You're in. Of course. Come, go pass me. Don't lose. Did you get far? Because that's what it was. It was far. I'm trying to get with him. I'm shaking. Be yourself. Oh! How's that? He doesn't understand. The wolf just fine. Yes, yes. Nice little sweeping crossfield ball. Keep running. I'm running. Play me. Don't lose it. I haven't done it. I haven't done that. And I'm back on. I'm back on. Baz, don't lose it. Go on. I pinched it. Pop it. That's how for now. Days. This is you. This is you. This is you. All right. You're in the 90th minute. If it drops, smash it. Wolf-like football. You fell over. Your legs are wide open. And I'm not going to finish the reference. Oh, my. Weird. Clear. We were tuning up after 12 minutes. We're hanging on in the 90th minute. Clear. That'll do. Confidence. I was quite confident about that. Confidence. Because of the wolf. Just get that wolf. And launch. You launch that ball as far as you can kick it. Jump, jump, jump, jump. Wolf-y. Yes, yes, yes. To be fair. Right. To be fair. The keeper has stopped that from being able to 8-1. Some of the saves he's made there. With the bat. Unbelievable. In the 9th. 19 games in. 27 points. We're 11 off the top spot. We are, what, 6 from the European place. 8 from the European place. 8 from the European place. It's alright. We're in the semi-finals with the Classic Cup. We're 9th in the league. We've won 8, lost 8. We're minus 5 on the gold difference. We know that. But you know what? If we lane from our mistakes. We'll be alright. I think we can win the next 19 games. I'm just going to say. We've gone above Roma with that win, so come on. I think we can win. A lot of games, yeah. Europe is the minimum I want. I'm not going to lie, but 8 points offer at the minimum. It's the minimum, so that's what we've got to hunt down. Let's do it. Let's do it. Starting in the next episode of Road to Glory. Let's do it. Let's go win some games of football. We'll see you then.


Your Coca-Cola bottler presents Claudia based on the famous play and novels by Rose Franken Brought to you transcribed Monday through Friday by your friendly neighbor who bottles Coca-Cola Relax and while you're listening refresh yourself. Have a coke and now Claudia anything the matter Hartley anything the matter. I should say not Everything is superb wonderful Static No, I'm glad you have country relative. I always was David why is it? I sleep better in the country. I eat better in the country. That's because you don't live in the country David How about another cup of coffee? I don't mind about do that's true. Claudia Hardly comes up to the country spends a day with his farmer brother, and he has absolutely no worries. What worries of you Must I count them so early in the morning? Yes, David. Go on be explicit. Well, hardly I worry that the pipes aren't going to burst summer. They don't burst in the summer if pipes could mine would Then I worry that the cow isn't fed properly. We haven't got a cow dope. That's really why I worry Some place somewhere in the state of Connecticut. There's a cow for me. Where I don't know exactly where so I Can't do anything to see that she's gotten the proper care. So all I can do is worry. That's a nice realistic approach to things What other fantasies are on your mind David? Well my fields aren't tilled my barn isn't completed my driveway has to be Regravelled and marred may rain. No wonder you look so well. Those kind of worries will keep you young man It's the old story about the grass on the other side of the fence. What about it? It's greener Don't you believe a word David saying he hasn't got a thing on his mind except problems And they don't have a thing to do with living in the country even problems can't seem so serious up here when you're a Million miles from no place exactly 53 miles from Times Square and 42nd Street It's a million miles when you've gone the 53 While the sky looks different up here everything feels different look at us We're having breakfast and there are no morning newspapers on the table. That is one thing that is always rather Irritated David likes to read the papers at breakfast So he doesn't have to talk if I could have breakfast every day without having to read about the stock market or the threat of war Or conditions in Europe. I think I could really enjoy the meal If I could come home every night to Eastbrook from Wall Street the prices stealer the fluctuation on the cotton market They wouldn't worry me at all. He's making me feel as if I were a lighthouse keeper Living on a small bundle of rock completely remote from the outside world. That's the luxury of it David You're not exiled when necessity calls you hop in your car and there you are Boiling in the cauldron with the rest of us But afternoon comes and you're away freed independent living your life alone back here The outside world beyond your fences and it's up to you whether you choose to cross them or not I do not choose but there is a choice and that's the beauty of it and today I'm exerting my right to choose. I shall not go into New York. You won't. Oh, how wonderful After all it isn't every day my one and only brother gets back from Europe comes up to see us Oh, I don't want you to put yourself out for me. I will not don't worry. I'm putting myself out for myself I'm going to hide myself away in the spare room and work on some plans for the freight terminal How is it coming along? So far so good.  Just say the word. All right I say the word now Will you excuse me? Of course delighted. Claudia and I shall have a fine talk While you work and my wife sleeps. I'll show you the phone. That's what I hope you'd suggest Don't let her run around too much. Don't you worry about it. Where do you want to begin, Hartley? Right out the front door. I've been itching to get my nose into some of this morning country air There you are. The door opens, the curtain rises, our farm stretching as far as your eye can see And I was traveling through Europe just days ago walking through the ruins of capital cities Watching the hunger lines Hearing the cries and complaints of a destroyed civilization. I try to imagine what and where the perfect world would be I Think perhaps you found it. I think perhaps we have too hotly Here for all those who want it. Oh there, Mrs. Norton.   But now you're sorry this far made another hundred miles from New York Oh, that's a nice thing to say right in front of the visiting relative. I'm just starting to show Mr. Norton around the farm. Why don't you join us, Mr. Tucker? You know a lot more about it than I do Well, I don't want to be intruding. Wasn't here on your land to snoop in, mind you. Come June mornings with the mis-line on the road I kind of like to climb up on the crest of that hill behind your house It gives me a feeling in my bones to see the world gathering up on the doorstep. Keeps fellow young. Stand up on a hill Top of a hill is the center of things, you know. The center of things, eh? I thought I'd be running away from the world up here. The world, Mr. Norton, is really run from places like Eastbrook See that ribbon road stretching out down there beyond the farmhouse. Yes, I see it. That there's River Road That's the road we live on. To my way of thinking and to them what knows is right The world is really run from River Road. You don't say it. I do say it, do. There are two ways of life, Mr. Norton. The city way and the country way. Broadway versus Main Street. Wall Street versus River Road Broadway and Wall Street don't run the world. They ain't even in the middle of it. Why, they don't even know what the world's all about They're so busy busy just just stayin' alive. Just just just fighting for air. I must say I can't quite go along with you on that Well, I haven't even seen the morning newspaper up here. We don't have to read the morning newspaper, Mr. Norton We know what's in it. We're the ones that's in it. So what's use of reading them? Look here, Mr. Norton. Do you know your congressman? My congressman? Well, of course I know him. He's uh, what's his name? I voted for him three times. I see. What's his name? How would you like it if I told you I know my congressman personally? I was drunk up with him. If I know him Well, I know what kind of a man he is know the girl he married when he's sitting down there in Washington in the House of Congress I can depend on him to talk a certain way Because I I know the kind of me teach made up and I hear Mr. Tucker talk hardly it makes me wish I've been born in Eastbrook that have having just migrated up your fitting in Mrs. Norton You can't tell me that living up here isn't getting away from the world Why even in the middle of a war your doorstep could be peaceful talk about war say We're the folks who know what war is about Every boy that put on cocky up here in Eastbrook. I knew him Yep, knew him ever since he was a little thought knows father knows mother I seen them go down to the office there and sign up for the draft then I seen him get dressed up one fine morning leave town The whole time he was gone. I knew I knew I knew where he was and what he was doing and what was happening to him I stayed home sure and I growd McCorn and fed McAddle did my share to keep things going But I was with every one of the 300 boys from Eastbrook who went to war because I knew everyone up But he got say that yes, there is a difference watch their names in New York. That's more boys signed up and went off You didn't miss sitting next to them in the grains meeting. You didn't talk to his father every day See his hands shaking holding on to a letter it just received and you didn't know every boy who didn't come back He didn't go there and sit in on church service or wait up through the night with a wife who's having a child and Who had no husband who never would have him?  Tucker You say you're on Wall Street Well, who raises a corn that makes the market go up in the wheat and the cattle in the pigs They're the backbone of the country. You got a raised corn first Then you can buy or sell to your heart's content Maybe you just better stay in New York hot and take life easy. Maybe you're right. Of course. I'm right. I heard you Them ears of mine have been hearing for 78 years got pretty good at two. Don't miss a thing But I ain't finished look here at mrs. Norton. She's having a baby I'm afraid even somebody's eyes who haven't been seeing for 78 years could see that fine thing happened a baby Up here. We're all waiting to see him been watching and worrying along with mr. Norton for the past months Glad to see everything coming along fine What what what's what's a baby in the city nothing but a birth certificate? Seven million other people bustling around and nobody cares one birth certificate more or less, but up here Her baby is important. I've been thinking it was just an event between David and me Well, not up here. It ain't here. We know the baby's dad So we expect a lot of his boy cause his dad's a good man We're all waiting to watch this baby grow up and Get on a few years and then go down to the school house his dad's got built Even if he goes away and stays away for 20 years, maybe and then comes back. We'll know him We won't have forgotten him That's there that baby yours better grow up to be a fine boy or I'll show like the hide-off in him Now you know what you have to live up to That wonderful to know just what it is isn't it? Yes, it is I guess I woke up in a kind of dream this morning I didn't realize that the sun shining in my room was lighting up the pulse of the world I was wrong and that post is beaten fine and strong Mr. Notting never you fear as long as the world is made up of e-books a post will keep on a beat Well guess I'll be toddling along sure was nice to a bear to have listened to you talk, Mr. Notting. Yes was um, I'll see you again when you decide to get away from the outsides of things and come up here Man, it's funny the idea city folks get but can't blame them. They're soapy puddles and bamboos at all time Darling, what are you doing way up here?  I want to tell you What is it darling David you have any idea how nice it is That our baby's going to grow up right here in Eastbrook on the crossroads of the world Radio listening is a sociable pastime and it can be made even more sociable if you get out several bottles of ice-cold coca-cola That holds good whether you have guests or the families alone of an evening in fact You'll find coke is pleasant even when you pause during housework or your favorite daytime program Take a tip from me and listen refreshed Well, mr. King it looks as if Jared Tucker sold me a bill of goods really convinced you did he absolutely I've started feeling as if I'd never really lived before They tell you the truth. Maybe I haven't there are different ways to live. I suppose so Well, it's a glorious morning up here I think I'll walk across some fields and pick Julia bouquet wildflowers. Why she'd love it any wife would by the way Do you know why Claudia and David haven't raised a flower garden yet? Well, I I suppose they've had so much else to do first I suppose so but for me home isn't home without flowers living in the country Yes, and Claudia agrees with you and I have a feeling she's going to try and talk David into a few roses and Petunias, but that's for Monday. Let me know if she succeeds with I will I will mr. Noton on Monday. Bye. See you then As I was about to say Every day Monday through Friday. Claudia comes to you transcribed with the best wishes of your friendly neighbor who bottles coca-cola So listen again Monday at the same time And now this is Joe King saying or a foyer and remember Whoever you are whatever you do wherever you may be When you think of refreshment think of coca-cola Or coca-cola makes any pause the pause that refreshes And ice cold coca-cola is everywhere These programs star Catherine Bard as Claudia and Paul Crabtree as David and the entire production is supervised and directed by William Brown Maloney and now here's a word from your friendly neighbor who bottles coca-cola


Going on everyone Sam here with late sports cars from in two boxes for Brian P We've got one 23 tops chrome update series jumbo and one 23 24 prism collegiate draft picks hobby basketball So good luck here Kyle. Let's do it. They're not Kyle Brian. He's talking to Kyle. Sorry about that Ricky Council Palo nice one there too at 149. That's dope to 149 mo bomba on the red auto for Texas Seth Wendy Silver Timmy Allen prism Isaiah Thomas nice Zeke to 199 and Shrey young fireworks still one being this Trisha Vukovic Jordan Walsh actually pretty solid on the prism auto now a Celtic that's on that one there Brian Silver on men Thompson prism Jeremy Sohan red our blue wave Trey young to at 249 the Hawks and a prism break of Nick Smith Jr. Sydney or city Susoka mark price to 99 that's cool man nice one there one of those underrated players in NBA history longtime Cleveland Cavalier gave Cal shore Omaha bill you case on Wallace red 299 and Keontay George now the Utah Jazz pack here good luck Brian that is a national pride Hakeem Elijah one on the short print nice one there congrats buddy that's cool really cool actually Nigeria Omaha bill you case on Wallace not a nice one now a Oklahoma City Thunder member of the Oklahoma City Thunder grads on that one again Brian Berry solid J. Dub Silver also of the Thunder Hakeem Elijah one Benedict Matherin blue ice to 99 and Nick Smith Jr. all right TC update jumbo now good luck here mr. Brian come on geez true gold of Joey Wendell the new met the 50 correct Dylan Dodd Jordan Walker and Brent Hedrick Ray legend for sure big time Ray legend yeah that's he's all star there we got radiating rookies coming up I'm gonna say it's gonna be Taj Omar Narvaez Christopher Morrell nice on the short print take that for sure that is not bad not bad at all Cody Bradford Rafi devours and Dylan Tate Freddie Freeman blue Bob and Anthony Volpe refractor debut blue auto Corbin Carroll there we go congrats Brian let's go man god man to 150 hell yeah let's go true blue good stuff Brian let's go gonna Henderson and Christian Vasquez so sweet I'll be going into a mag for you my friend Jake Sable Christopher Morrell prism Oscar Colas and Tyler Anderson Corbin Carroll and hops out says it's Colas Prism Sadie Alcantara and Peyton Battenfield Sam says he's losing it we are losing it it's been lost dude oh what happened to this car like a antenna like an old car there's Danny our Donnie Sands to 499 on-card refractor rookie on it fills his hair channel to is a full of steamers really staying that is Kodai Senga Ronald Cunha and Vaughn Grissor I still lost the game so it even matter dude what are we doing I don't know Omar when it comes out we probably will I just don't think an F1 product isn't come out in a minute and same with same with UFC Francisco Alvarez thanks for the follow there Anna Joey Ortiz Max Scherzer gray rod stop Anna that last auto as well J rod prism of Trevor Williams Adley Ruchman and Nick Ahmed locks of the day tomorrow God that I come into work yeah true my lock of the day I don't know what my lock of the day tomorrow would be Vulpie Bobby Miller not gonna lose yeah the Ravens will not win or lose tomorrow but I haven't looked at the lines for tomorrow to be honest lock the box losing that's a lot might happen you and Kyle are getting wrecked for that Dustin May you're a fan of a team who just lost the ones and that's the case Dustin May Joey Ortiz nice one there for the Orioles to 299 or 199 in the teal wave auto Austin Riley Bryce Harper and Nick Martinez break there mr. Brian thanks again for the sport buddy get him right out to my man


The mystery surrounding Madeleine McCann has gripped the world for 15 years. It was during one of my checks that I discovered she had gone. I can't really go into any details about that. Very honest, very truthful what we're saying here. I agree 100%.  I think that's a very difficult thing to see. There are pieces of their body language that you would have a hard time faking. That lack of energy in this face of an interviewer who's asking you these questions, which of course the police would say, no matter what, don't say this on camera if you want your daughter back. We love our children very much. Did you see anything suspicious? We didn't. If we did, we wouldn't tell you. Most people will think that's dooper's delight or dooping delight. I don't see that at all. There's no dooper's delight in here. I would stake my reputation on. Mark, what do you got? Yeah, so, you know, what's amazing is compared to many of the videos that you've seen us, you know, comment on is we've got no resume statements here. We've got no obfuscation. We've got no distractions. We've got no, hey, look at the squirrel over there. A lot of people use the word congruent. This is a congruent message. German Christian Brookner formally declared the number one suspect. The official suspect status ensures he can still be brought to justice. Criminologists watching on believe it's only a matter of time. It is very significant he's been named as a formal person of interest, a formal suspect in this case, because that is in likelihood a prelude to charges coming in the next few days. This is Brookner at the wheel of his camper van on the Portuguese coast five weeks before the three-year-old vanished from a nearby holiday resort. The vehicle has since been seized by police and pulled apart by forensic experts. They're really looking to build up that evidence so that they can convince their equivalent of the Department of Public Prosecutions that charges are in fact solid enough to be laid. Maddie's parents, who themselves were once considered suspects, may get the justice they deserve. The authorities confident they have their man. They just have to prove it. It was during one of my checks that I discovered she'd gone. I can't really go into any details about that. Very honest, very truthful what we're seeing here. I agree 100%.


So this is my very first atelier. So I'm very excited to be participating in this new kind of event, even though I missed the first half. So what I'm going to be talking about, it's really about a pair of natural moduli spaces associated to a closed-oriented surface. That's the S and a league group G. In fact, it's really one space that has two different presentations. So with any topological space, maybe the first question you might be interested in is the number of components. And that's really what I'm going to be talking about. And we'll see that for these moduli spaces, this question about components has two aspects. What I've called here a mundane aspect, where there are obvious topological labels that are obviously constant on connected components. And so there's no surprises there. But there's also what I call an exotic contribution to this number of connected components, where you see components that are not detected by the obvious topological invariance. And that's really the subject matter. These moduli spaces, I'll say more about what they are in a minute, but one of the perspectives is from Higgs bundles. And that's what I'll be talking about here. So these will be moduli spaces of Higgs bundles that we'll be talking about. So really, I hope the high point of the talk will be discussion of a new result that's joined with Brian, Brian Collier, Oscar Garcia Prada, Peter Gothen, and Andre Oliveira, where we have detected some new exotic components in the moduli spaces associated to this group, SOPQ. And we'll see that part of the reason that this is interesting is because there are two well-known sources of exotic components for various classes of groups. And this group doesn't lie in either of those up till now, well-known sources. So there's something new going on here. But so before we can get to that, there's a certain amount of scene setting and background that we'll go through. So the plan is basically to tell you a bit about these moduli spaces. Say something about the ways that exotic components are known to arise, and then tell you something a little bit more about these new results. So that's the plan. So let me try and tell you a little bit just what we need to know about these moduli spaces. So the two of them, the one is representation variety. So homomorphisms from the fundamental group of the surface into the group G, up to conjugation. And the other one is a moduli space of Higgs bundles. So Higgs bundles I'll say more about in a minute, but these are holomorphic bundles with some extra structure. So this seems like a very different sort of beast. The relation between these two, I mean, this bridge is the result of big theorems. This is the non-Abelian Hodge correspondence. So this is the non-Abelian version of what Alex was talking about earlier today, the linear Hodge theory correspondence. So these spaces are not linear spaces. So that's sort of related to the non-Abelian. G, right. So S is a closed oriented surface of genus bigger than 1. And G is in principle any reductive Lie group. But for this talk, really, these are the ones we're going to see. So it's going to be SLNC, or special orthogonal group, or some of the real forms of those groups. But this does fit into a wider setting, but that won't appear much in this talk. You'll notice that in both of these modular spaces there's some adornment here. Here there's a plus, and here there's this word polystable. And I'm not going to say too much about either of those, except that so these are the conditions that you need in order to get nice modular spaces. I mean, both of these are parametrizing equivalence classes of objects. Here are the equivalence classes by action of conjugation of G on space of homomorphisms. And in general, if you just take the full set of equivalence classes, you don't get a nice geometric space. You have to impose some restriction. And this means reductive representation, and here the restriction is inspired by geometric invariant theory. So those are the two spaces. And we want to understand the connected components of both because they're homeomorphic. Sorry? Any n. Doesn't matter. We'll see that when n is 2, it's a little bit special, but any n. There was one other thing I was going to say. Oh, yeah. So notice here that all we have is the surface and the group. So this is purely topological data. Here we have to fix a complex structure on the surface, so turn it into a Riemann surface. And that's what allows us to talk about holomorphic objects over here. OK, so as I said, the focus for today is going to be on the connected components of pi 0. And we're going to be focusing mostly on this side of this correspondence. OK, so let me tell you a bit about what Higgs bundles are. So there's a notion of them for any g. It's somewhat easier to state for slnc. And then I'll say something about how you modify that for subgroups of slnc. So we've fixed the complex structure. So now we have a Riemann surface. And then a Higgs bundle on the Riemann surface is a pair consisting of a holomorphic bundle, so rank n holomorphic bundle. The determinant is fixed to be trivial. That's related to the s there. If that was a g, then you wouldn't fix the determinant. And the Higgs field, the extra piece of data, he has one description of it. It's a holomorphic map from the bundle to the bundle twisted by the canonical bundle. So this is the holomorphic cotangent bundle of the surface with a restriction on the trace. So this locally is just a matrix of one forms. The trace restriction here is also related to the s there. It's telling you that this lies in the Lie algebra of slnc. The important things we need to know about these Higgs bundles are, so here we have a holomorphic bundle, a Higgs field. If we add an extra piece of geometric data, a metric on the bundle, then that combined with the other defining data allows you to define connections. Here the d is the churn connection. So that's determined by the holomorphic structure and the metric, but then it's modified by the Higgs field. So this defines special metrics built on the Higgs bundle. The stability notion, which I mentioned as necessary for building the modular spaces, ensures that you can find a preferred metric, actually satisfying a harmonic condition, which has the consequence that this connection is flat. And so that establishes the connection between stable Higgs bundles and local systems and therefore representations of the fundamental group. So that's a key part of this non-Abelian Hodge bridge. You had a higher difference. Right, so then you can, I mean, this makes sense, but then you have to add an extra condition that the square of phi vanishes, phi which phi is 0. There's an extra condition. But I'm not thinking about that. Right, right. Right, right. But we're not going to go up in dimension. OK, so that's for SLNC. If you want to take a subgroup of SLNC, the way this description changes is you impose extra restrictions, which you can interpret as designed so that the holonomy of these flat connections lies in the group that you're interested in. And what that translates into, in some cases, is on this slide. So for SLNC, you have just a bundle with fixed determinant and a Higgs field with a trace condition. If you want holonomy in SONC, the bundle has to have an orthogonal structure. And the Higgs field has to be skew symmetric with respect to that orthogonal structure. And then the connections that you get from that will have the right holonomy. If you want to restrict to real forms of these complex groups, you have to impose extra restrictions on the Higgs bundles. In the case of SLNR, the rank n bundle again has to have an orthogonal structure. But the Higgs field now has to be symmetric, not skew symmetric. And in the case that is really the star of the talk, the SOPQs, the rank n bundle, I mean the rank p plus q bundle, has to be decomposed as a sum of a rank p and a rank q, both of which have orthogonal structures. So the orthogonal structure decomposes in this way. And the Higgs field has to be compatible with this decomposition in this, so it has to look like this. The t here means the transpose with respect to these orthogonal structures. Which objects? So phi is the Higgs field, so that's a map from v to v twisted by the canonical bundle. And q is the orthogonal structure that appears when the group that we're talking about is the orthogonal group. So q is a symmetric bilinear form that defines an orthogonal structure on the bundle. Orthogonal. No, there's no q here because this is the Higgs bundle for SLNC. So here we don't need any more data other than what I've described here in order for these connections to. Phi is a bundle map? Phi is a bundle map from v to v twisted by the canonical bundle. So v is over sigma, right? Phi is an endomorphism. If the k wasn't there, phi would just be an endomorphism of the bundle. Bundle endomorphism. Tough as the identity, right? Yeah, you can formulate those conditions in that way here. Right, right. So locally, this is a matrix of one forms. And those descriptions are related by SLNC transition functions for this case. Because you want this to take values in the Lie algebra of this group. That's related to where you want the holonomy of this thing to lie. Y is what? That's also related to the s here.  Well, no, yeah, g in this line is SLNC in this line. It's sO. Q is the new ingredient that you need to distinguish these Higgs bundles from those. And I'm advocating you interpret this as the condition you need so that the connections you build have holonomy not in SLNC, but in this subgroup of SLNC. OK. So these things have moduli spaces, whose components we want you to understand. The story is different for the case of g complex and g real, as we'll see. Let me start with the moduli spaces for complex g. Here's the classical semisimple Lie groups that are covered by this. So these spaces have many interesting properties. I've listed just some of them here, not all of which are going to be relevant in this talk. They, I mean, they're complex analytic spaces, but they actually have a hypercalous structure. They also have these two. Mg is the moduli space of polystable g-higgs bundles. The Riemann surface is fixed, and the topological type of the bundle is fixed. We'll see. We'll get to that. Right. From one point of view, we're varying the pair, the holomorphic structure, and the Higgs field. And we're looking at isomorphism classes of those objects. If you take just all isomorphism classes that doesn't have a nice geometric structure, but you impose a notion of stability throughout the unstable ones, and then what's left does form a nice moduli space. It's not always smooth. So when I say hypercalous structure, I mean on the smooth locus. In all cases, though, it has a very convenient C star action, which you get by acting just on the Higgs field, scaling that by non-zero complex numbers. It also has a very convenient function. You get by evaluating the L2 norm of the Higgs field. I mean, I put more function in quotes here, because when this isn't a smooth manifold, then you can't really be doing more theory. But you can extract topological information anyway from some of the properties of this function, and that will come up later in the talk. So that will play a role later in the talk. The other thing that we'll be using is this other feature of these moduli spaces, which is a vibration over a vector space, which turns out to be of half the dimension of these spaces, and such that the fibers of this vibration are generically abelian varieties. So generically means there are some singular fibers where that's not true, but away from the singular locus, there are abelian varieties. This map is defined by taking invariant polynomials and evaluating them on the Higgs field. And we'll see that this vibration has many wonderful properties, including being a convenient setting for understanding mirror symmetry. That's not something that we're going to say anything more about in this talk, but this structure will play a role in understanding some of the connected components. So these are all the interesting aspects of the space. The one aspect that is not so interesting for moduli spaces for complex groups is the set of connected components. In the case of the complex groups, there's really only obvious topological invariants that appear, and they do label connected components. So there's not much in the story for those moduli spaces. Oops, sorry. Right, so the critical points for this function are actually the fixed points for that C star action. I mean, it's a more spot function, I should say. I mean, the critical sets, the critical points are not isolated, but there are critical submanifolds. The main thing that we use about this is that it's a proper map, so it attains its minimum on every connected component. And that's the way that it's going to be used. And that statement is true whether the space is smooth or not. OK, so good. So yeah, please keep the questions coming. Slow me down if necessary. OK, so when the group is a real form of those complex groups that we just saw, the story about connected components gets considerably more interesting. A certain amount remains the same. I mean, this is still a moduli space of bundles with extra structure. But since there are principle G bundles underlying these Higgs bundles, those have topological invariance characteristic classes, and those must be constant on connected components. So those are the obvious invariance that partially label connected components of the moduli space. They decompose this space. So I hesitate to use the word component, because the whole point of this is that these labeled by characteristic classes are not necessarily connected. But it does give you this decomposition, where C is just here generically standing for whatever characteristic classes are appropriate for the G you're talking about. But what's new when the group is real is there are several phenomena that appear. On the one hand, in some cases, turns out that there are too many of these invariance. There are bounds that they have to satisfy in order for these components to be non-empty. So not all invariance can appear. The opposite phenomenon also occurs where there are too few of these. In other words, they fail to distinguish all the connected components. There's also a phenomenon that occurs for these real cases, where some of the components that appear have special geometric significance. We'll see in the example that I look at in a second, sort of the first and sort of prime example of this is that one of the components is a copy of Teichmuller space. But there are other geometric structures that are also detected by some of the special components that appear for special real forms. So that's sort of the overall general setting of the type of question that we want to look at lives in. So let me start by describing some of the phenomena in the case of SL2R. So let's look at this example. This is historically maybe the first one, but also we see essentially all the phenomena appearing in this example. So now we're looking at, from the one point of view, it's the modular space of SL2R Higgs bundles, which by this non-Abelian Hodge theory correspondence is the representation variety for the fundamental group. That should have been an S, not a sigma, because we don't need the holomorphic structure here into SL2R. There I've made it an S there. So this goes back to Worker Bill Goldman, who gave the count of the number of components for this representation variety, and here it is. So this is for a surface of genus G. So the number of components is here, this number, two to the two G plus one plus two G minus three. And so we'll understand that a little bit as we go down this slide. So the obvious invariant here is, I mean these representations and these Higgs bundles have associated to them an Euler number, so essentially an Euler number of an SL2R, of an SO2 bundle. So that's an integer. Turns out that there is a bound on this integer. So here we see the topological invariant the too many aspect. The bound is G minus one for the components where the invariant is not maximal, that those actually are labeled by that invariant, so those are connected, but for the maximal ones, there are two to the two G components all corresponding to that value of the invariant, each one of which is a copy of Teichmuller space. So that's what you see from the representation variety point of view, how does this look from the perspective of Higgs bundles, which is the perspective that we're going to extend when we look at other groups. So first of all, what does an SL2R Higgs bundle look like? So you probably don't remember from the table, but this is a rank two bundle, that's the two, but because it's an SL2R, it's a rank two bundle that has to have an orthogonal structure, and that can always be put in this standard form, a line bundle plus its dual. So the Higgs bundle is a line bundle plus its dual, the Higgs field has to be symmetric with respect to the orthogonal structure defined in this standard form, which turns out to mean that it has to be off-diagonal like this. Each one of these pieces in the Higgs field is a section of a line bundle, the line bundle determined by L in the canonical bundle. So that's the data that defines the Higgs bundles. There's topological invariant staring at you here in this line bundle, it's the degree of that line bundle. The bound on that is very accessible from the Higgs bundle point of view because the stability notion which I didn't define for you implies that one or other of these components cannot vanish. Which one depends on whether the degree is positive or negative, but the fact that it can't vanish puts a bound on the degree of the bundle of which it's a section because line bundles can only have sections if their degree is non-negative. So that's where the bound comes from. So you get a decomposition of the modular space of Higgs bundles into pieces labeled by the degree of this bundle L. And because like beta is a map from L inverse to L tensor K. Yeah, well no, it reinforces it. It's a map from L inverse to L tensor K. So it's a section of L squared K and the other one is of L minus two K. So when the parameter is maximal then the degree of one of these is zero and so to have a section it has to actually be the trivial bundle. So the structure of the bundle actually specializes. It's not determined just by its degree. It also has to be a square root of the canonical bundle. And that means that the description of the maximal components change is different from the other components. There's now one component for each choice of the square root of the canonical bundle and each one of them is determined just by the piece of information that hasn't been used yet, namely a section of this square of the canonical bundle. So by a quadratic differential. And so this identifies these components with a copy of the space of quadratic differentials. These are the Hitchin components. This is actually the identification of Teichmuller space with the space of quadratic differentials that Mike Wolf was also responsible for originally. But this is seen from the Higgs bundle point of view. And you see two to the two G copies of it corresponding to the choices of the canonical bundle. So there are many different phenomena at work here. And this is really a reflection of the fact that SL2R lies in, you know, it wears many different hats. There are many low dimensional isomorphisms at work here. SL2R is the same as SP2R and it's a double cover of SO12. It's also, you can also identify it with SU11, but that wasn't essential for this story, so I didn't put it here. These groups all lie in bigger families. This one is in the SP2NRs, this one SLNR. This one actually can be viewed, should be viewed as lying in two different families. In the one case, it's the SO2Qs, so keeping two fixed. And the other is SOPP plus one. Both of those are special cases of the group that we eventually want to talk about. So the groups that you see here live in, I mean, there's some special features that are the important things here. Some of them, the ones in blue here are like SLNR are the split real forms of their complex groups. While others, so you should look at this one and that one, this is in purple because it lies in both these families. But these groups are distinguished by, I mean, I've called them Hermitian. What it refers to is that the symmetric space you get from these quotient by their maximal compact is a Hermitian symmetric space. So those, these two features are the ones that are known to be responsible for special components. I'll say more about that in a minute. You'll notice that this one didn't get, wasn't either blue or red. So this is where the novelty is appearing here. Okay, so let me just say something about both of these mechanisms that are at work in the component story. Let's see, okay. So what is special about the split real forms? So here's a list of the real forms that fall into this category for the various classical groups. Here I've reminded you what the Higgs bundles for SL2R look like. In the, what the Higgs bundles that live in the special exotic components look like, I should say. There is some of the bundle is a sum of line bundles, but the line bundle is a square root of the canonical bundle. So some power of K and the Higgs field is constrained to not, I mean, the term that lies over there has to be nowhere zero and this one can be anything and that's where the parametrization comes from. The way this generalizes is, so there's a description in terms of the representation variety and in terms of the Higgs bundles. In this side, you get a generalization of the Teichmuller space. So these are the higher Teichmuller components. On this side, you get a generalization of the Hitchin components that were parametrized by quadratic differentials. And the way these appear just very briefly, the special thing about the split real forms is that there's an irreducible representation of SL2R that lies in each one of these. So if you take a representation corresponding to a point in Teichmuller space and compose it with the irreducible representation into the split real form, then that defines representations in here and the higher Teichmuller components are the components of representations which deform to one of these. On the Higgs bundle side, the Hitchin components that you get generalize the components which were parametrized by quadratic differentials. The quadratic differentials were actually the base of the vibration for the case of SL2C and the Hitchin component was parametrized by these by picking a section of this vibration and that construction, the existence of a section of this vibration which picks out a component in here is the way that these Hitchin components appear for the split real forms. So the one, okay, so don't panic. This is not as scary as it looks, but this is the example that is really the relevant one for our new SOPQ components. So this is what happens for the Hitchin components for the split real forms SOPP plus one. So these are according to this picture parametrized by the base of the Hitchin vibration for SO2P plus one and that contains not only quadratic differentials but differentials of all even powers from one to two P. The bundles in these Hitchin components just like these bundles are fixed, they're constrained to have a particular form, they have to be sums of powers of the canonical bundle and the Higgs field, I mean this is an SOPQ so it has to be compatible with the orthogonal structures on these bundles so it has to be of this diagonal form but the information in this sector that I've called eta, that's what goes where the star is, I've illustrated in this diagram. So what we have in this diagram are we have the summands in the two bundles distribute alternating from the summand of V to the summand of W and then the arrows are the non-zero components of the Higgs field. So the Higgs field is a map from W to V twisted by K. It has many components corresponding to this decomposition. These arrows indicate where they lie. Notice that these ones are all non-zero. These are essentially the identity. You might wonder how do you get an identity from the Pth power to the P minus first power and the reason is that these are Higgs fields so there's a twisting by the canonical bundle which isn't in this picture but if you put that in then that would be a map from K to P to K to the P. So this is the way the Higgs bundles in the Hitchin component for SOP P plus one look and quickly memorize that because this pattern will appear again in a couple of slides. Okay, so that's what happens for the spit real forms. For the real forms of Hermitian type so that was the other type of family that appeared in the case of SL2R and for which I said you get exotic components so there the relevant features are a little bit different so the example we saw SL2R everything is a sum of two bundles and the Higgs field is diagonal with respect to that that's a general feature for these real forms of Hermitian type. There's essentially integer valued invariant here it was just the degree of L in some of the other cases it's not exactly an integer it's a integer multiples of some constant but there's a bound which from the Higgs bundle point of view you can see as a consequence of the stability criterion when the obvious invariant hits its maximal value the structure of the bundle specializes in the way that in SL2R L had to be a square root of the canonical bundle there's something sort of analogous that always happens and so you get new invariants appearing because the structure has specialized so the maximal component decomposes into components labeled by a new invariant. Each piece labeled by this new invariant acquires a new interpretation which we'll see coming up as well it's like a modular space of Higgs bundles for a different group except now the Higgs field is not twisted by the canonical bundle but by the square of the canonical bundle so this is a K squared twisted G prime Higgs bundle for a companion group G prime. So I want to try and work through the way that looks for this particular example because again this is the one that is relevant for the SOPQ so in this case so this is an example of a group of Hermitian type the Higgs bundle this is an SOPQ so the bundle has to be a sum of a rank two and a rank Q the rank two bundle is now an SO2 so again it has to decompose or it can be put in this form of a blind bundle plus it's dual here I've shown you what the orthogonal structure looks like with respect to that decomposition W is a rank Q orthogonal bundle so no special form for that and the part of the Higgs field that I've called eta is now a map from W to L plus L inverse so it has two pieces that are labeled there in a diagram similar to the one that we saw for the Hitchin components where we label the summands in the bundles and put in arrows for the components in the Higgs fields this is what the Higgs field looks like for an SO2 Q Higgs bundle so you get a quiver diagram like this in this case they're actually two invariants one of them is an integer it's the degree of L they're also invariants for the bundle W this is an orthogonal bundle so its degree is zero but it has Stiefel Whitney classes in principle it has two but in this example this little zero down there means that we've taken the connected component of the identity of this group and for that restriction that implies that the first Stiefel Whitney class of W is necessarily zero so we're left with just the second Stiefel Whitney class which lives in H2 with Z2 coefficients so they're actually two labels but this is the one that participates in the Hermitian symmetric story there's a bound on this which you can see in this case by looking at the composition of the map this part of the Higgs field and it's transposed with respect to the orthogonal structure that gives you a map from L to L inverse and at each stage remember we've twisted by the canonical bundle so this is a map from L to L inverse K squared and stability says that that can't be zero so you get a degree on the bound of L from that condition there it is so that's the bound in this case so now we get a primary decomposition of the modular space into sectors labeled by the degree of L and also this second Stiefel Whitney class that's also a legitimate label but we get something special happening when the invariant here is maximal has its maximal value and again it comes from the fact that when that happens this is really a section of a degree zero bundle so it says the bundle must be trivial and the section must be nowhere vanishing and that specializes the structure of the SO2Q Higgs bundle it fixes L to be essentially so it's got to be the canonical bundle twisted by a square root of the trivial bundle so the only thing that you allow to vary is I which has two to the two G possibilities point of order two in the Jacobian if you like also the rank Q bundle has to decompose into I plus a rank Q minus one orthogonal bundle so this picture has become this picture where W has split into I plus W naught and here are all the pieces of the Higgs field that you're allowed so now if you look at this diagram we can reinterpret the information that we have in there in a new way if you look at the information down in this part this piece beta naught is a map from the rank Q minus one orthogonal bundle to the square the point of order two twisted by the canonical bundle that's down there the information here is really I W naught and beta naught and that information apart from the fact that there's a square there defines a Higgs bundle for the group SO1 Q minus one so this part of the diagram defines a point in this space a K squared twisted Higgs bundle for the group SO1 Q minus one if you look at what's left the only thing that's okay so we've got the K is fixed and this I whatever I is A2 here is a map from I to I let's see did I write it correctly? Yeah you can write it as a map from I to I K twisted by K so that's a quadratic differential for part of the story it would be convenient to reinterpret it as a K squared twisted Higgs bundle for the group SO1 one but that's not so important for us here so what the bottom line is that the maximal component where the parameter tau has its maximal value decomposes into pieces labeled by the choices of the point of order two and for each choice the space that you get is a product of this one with that one this one is a moduli space of K squared twisted Higgs bundles for this group and this one is a power is a space of sections of quadratic differentials so there's the exotic components in this example so okay so let me let me with all that preparation now tell you about what we what we find for SOPQ so this I mean this includes SO2Q and it includes SOPP plus one or we'll see more relevantly SOP minus one P but it's those are just special cases of this and outside of those special cases this is not in either of those special classes of groups and so here's a here's a crude summary of the results so this is again let me just acknowledge my collaborators Collier, Gothen, Garcia Prada and Oliveira so what the result is the result says is that for any of these SOPQs there are extra components not detected by the topological invariance whose union is isomorphic to this space so this these pieces look like what we saw for the SO2Q and the SOPP minus one except here instead of being twisted by K squared it's twisted by K to the P and here instead of just having quadratic differentials we have a sum of all the even powers up to two P minus two so let me tell you what I want to do in the next 10 minutes or so is give you some idea of why why there was sort of some expectation that these components might be there or what the evidence was that there ought to be some exotic components in this case and then just say a couple words about sort of the precise thing that we actually proved but before we get there right I was going to make this a little bit more precise so let me, sorry I forgot about this so here the result said that the exotic components the union all looks like this space the invariance, the obvious topological invariance in this example SOPQ are Stiefel-Whitney classes for the orthogonal bundle V and the orthogonal bundle W so in principle four invariance two first Stiefel-Whitneys and two second Stiefel-Whitneys but because of the S over there these first Stiefel-Whitney classes are linked so there's really three invariance one first Stiefel-Whitney class and two second Stiefel-Whitney classes and so there's a component for each of those which is detected just by those topological invariance and the exotic ones don't occur for all possible values of this in the cases where the second Stiefel-Whitney class of V is not zero you only get the component that is labeled by the obvious invariance the exotic ones occur in these sectors where the second Stiefel-Whitney class of V is zero so these, this component that I've labeled with a zero are the ones which are labeled just by the topological invariance and another way of characterizing those is to say that those are the Higgs bundles where you can deform the Higgs field to vanish and you're left with just orthogonal bundles in the others, you can't do that in the others, so in the exotic components the structure of the Higgs bundles all is captured in this diagram so like the SOP P plus one example there's a part that is determined just by powers of the canonical bundle that fully describes V W has a part like that and a rank so this is a rank P minus one piece this is a rank P piece so this has to be a rank Q minus P plus one and that's the analog of what we got in the SO2 Q example so this is the structure of the Higgs bundle of the bundle and the arrows indicate the structure of the Higgs field for every point in these exotic components and you'll notice that some of these arrows have ones on them so they can never vanish so you can see that you can't deform this away for the Higgs field to vanish completely okay so let me just look a little bit more closely at this diagram and this is the way the Higgs bundles look in this exotic component if you look at the part at the top of the diagram that's exactly what we saw for SOP P plus one except now this is SOP minus one P but so this part of the diagram parameterizes I mean represents a point in that modular space and the rest of the diagram the part in red here encodes data for an SO1 this is the rank of W naught Higgs bundle but now this is a map from W naught to this power of the canonical bundle with an extra K because everything is twisted by K so that's why there's a K to the P over there so that's where the two pieces of the, in the description of the component come from or how they're encoded in this diagram okay so now I can tell you about some of the evidence that there was for why there ought to be these components so the first is evidence from the function that I mentioned at the beginning of the talk that serves as a Morse function on the modular space of Higgs bundles so even though these spaces aren't necessarily smooth the fact that this is a proper map means that it attains its local min on all connected components so you get information about the connected components by identifying the local minima okay so in the case of SOPQ this was, I've written this for SLN here's what it looks like for SOPQ the function is really the ultronorm of this part of the Higgs field so this function either in this case or that case clearly it has a global minimum at zero and that's attained on the components that are labeled by zero before where the Higgs field can be deformed to vanish but so if there are other local minima then that holds out the possibility that there could be other components and this was detected by Marta Royo in her PhD thesis in 2009 she was a student of Oscar Garcia Prada and she found local minima for this function on SOPQ where the Higgs field did not vanish and they had this form so this is the diagram that we've had up a few times but with only these arrows all the other arrows have been set to zero so I mean this doesn't prove that there are other connected components but it allows for that possibility so this was the first piece of evidence the second piece of evidence was from Brian's results for these split real forms SOP P plus one so this was a generalization of results for the special case SO2-3 which SO2-3 is almost SP4-R those groups are isogenous but SP4-R is a double cover of SO2-3 and in both those cases so in SP4-R it was known that there were extra exotic components first detected by Peter Gothen and Brian's analysis of the case SOP P plus one found analogous exotic components for NEP for these split real forms so these are components I mean we know that since this is a split real form there are hitching components but these are components, I mean others not the hitching components so in fact there are components labeled by an integer D that, let's see did I put it somewhere up here? Yeah, yeah, it can range between zero and P times two G minus two at the maximum value these are the hitching components but Brian actually described not only detected them but described them for the other values of D as well so this is for every value of D if you take all the components all the even D's and all the odd D's then the union of those actually gives you the special case of the KP twisted SO1Q minus P plus one Higgs bundles that we saw in the SOPQ example so this was a special case I mean now we can say it's a special case but it was an indication of extra components in SOPQs there was one other stream of evidence and I know I'm gonna gobble this explanation but so you can talk to my collaborator Brian afterwards if you want a more coherent explanation but so this was evidence from the other side of the non-Abelian hodge correspondence so what I described before the more theory evidence and the SOPQ plus one came out of Higgs bundle analysis but so Livier Gichard and Anna Wienhardt analyzed the I mean had a prediction for the representation variety the corresponding representation variety for SOPQ which came out of their work analyzing what so what the I mean what these things have in common the Hitchin representations and the maximal representations for the symmetric case those are the two cases where exotic components are known in both of those cases the representations in these exotic components all have the special property known as the you know that there are Nossov representations as defined by Laverie's concept of a Nossov what Olivia and Anna showed was that there's actually a perhaps a more subtle property at work here which relies on a positivity notion for a group and actually a group and a parabolic subgroup of it which both in both of these cases you have but they showed that this property can occur more generally than in these two cases and they were first of all they they conjectured that when the group has this positivity property then there ought to be additional exotic components and it turns out that the only case not included by these two classes of groups okay except for some exceptional groups but the only other case is SOPQ so this suggests that there ought to be exotic components of SOPQ which have this property that they identified let's see I should stop in two minutes so all right so I'll just quickly summarize what we actually did so I mean once we had finally sort of understood what the components ought to look like then showing that you have components like that amounts to finding a map from what the space that they ought to be into the modular space where you think it should live the one slight wrinkle here is that these are modular spaces of isomorphism classes of objects what I've been writing down has mostly been just objects not isomorphism classes in fact the description of the map from data in the space that describes the components into the big modular space is really a map on objects not isomorphism classes so what we showed was first of all that this actually does define a map on between the modular spaces and then we showed that the map is has the right properties to actually define a component so it's closed and it's open and so okay so the first one is showing that that you actually have a map you more or less do just by direct observation direct computation and observation to show that the map is closed the Hitchin vibration actually plays an important role there because you wanna you know you want to show that the map is closed one way to do that is to show that if you have a divergent sequence then the image under the map is also divergent and if the sequence is divergent using the Hitchin vibration really the only way you can detect when a sequence is divergent because it diverges in the base of the Hitchin vibration and that's a sort of the convenient tool for showing that the map is closed showing that the map is open we actually could show really a bit more than that I mean we could show that the derivative of the map is a local isomorphism so that locally this is a diffeomorphism so it's open and but for that you need a good model of local neighborhoods of the modular space and that's really where sort of the work goes so okay that's all I'll say about the proof so you know one obvious question that has yet to be answered is whether these components that we've detected are actually the same as the components predicted by the representation variety picture in the Gishard-Vinhard conjecture so in other words are the representations in the components corresponding to our exotic components are they the ones that have the correct positivity property and stay tuned we'll tell you at the next atelier okay thanks