Everybody forced and jk ripping 2023 prism football hobby. This is six box half case number 34 Just a second here There they are picker team style. I've got Adam L Steelers Andre with the Lions Blair has the Jets Bob M Falcons and Bucks Brayden Vikings Brian H Chargers Charlie P Ravens Chuck F Broncos Daniel D. Bills David I dolphins Eric M Packers Fairhawn Pats Gerard has the Raiders James C Rams Jason B Browns Jason W Cowboys Jeffrey H Texans Carrie has the Chiefs Matthew a bangles Matthew M Colts Peter a Titans Randall C with the Niners RJ has the Giants Rob and Washington Rory C Cardinals Saints Ryan B Seahawks Sean M Panthers Thomas W Jags Todd C Bears and Wallace S with them with the Eagles birds as they say Let's do it everybody Gotcha JJ H. Yeah, I always just assume no sarcasm when I when I see text here.  Yeah, a lot of people hate the Cowboys. So make sense Hyper numbered to 175 rookie Titans Tyjay Spears Bijan Robinson All right, Silver is gonna be Hunter Henry Patriots red wave the 149 Rogers Jets The weather spoon and Anton Harrison to 99 James Winston Saints blue ice Kendra Miller, Sam LaPorta Playoff touchdown tonight so far Hey, man, Puka, then Wicks earlier Rishi rice last night Rookie score. We like to see it. We like to see it shrouded really well Silver rookie is Camus Smith Dolphins blue ice Justin Jefferson at 225 for the Vikings Colesia Kansy chase Brown Bro prism break blue ice to 99. I say tank Dell rookie Texans 95 of 99 of the tank. That's a cool one. That's good to get him back healthy next year Canada Van Ness Probably the most explosive playmaker that Stroud has silver fireworks is grease hall for the Jets blue wave the 199 Damian Pierce Texans Eric Gray KJ Henry Hyper to 175 is Mooney for the Bears trade Tucker Miles Murphy Silver Jerry Judy Broncos followed by a Bryant young autograph to 1 to 49 San Francisco We're back Bryant young Julius Branson Paris Johnson be sure to like to stream everybody Got a good amount of people watching You nice to get a like from each one of you Subscribe if you haven't and definitely subscribe if you if you haven't already follow on twitch Genetic Mahomes red wave to 149 Bolton for the chiefs both free Gibbs rookie and Eric all pretty soon. Hope we get some more content going on the YouTube side of things. So that'd be fun We'll have a brand new loyalty program rolling out as well in the near future All good things for y'all silver Damian Pierce Texans orange out of 249 Deandre Swift Eagles Charlie Jones Jake Hainer nice a prism flashback rookie Anthony Richardson Colts Purple ice that win that of 225 the Cowboys and Wilson and Zach happens to the new Tron roller coaster at Disney is kind of sick Yeah, I wanted to go on it before it has expired We got over it that way But got to do guardians in another time. That was fun The queue is insane You get inside The Guardians queue is kind of cool, too Well, I got it. I imagine the tron was crazy.  Welcome to the grid or whatever. We've got a Press proof. There's a Zay flowers prism flashback there Press proof of Eric Gray for the Giants on the SP Congrats Giants It's a pretty rare dude purple ice to 225 cam acres to the Vikings. I Don't know if they ever announced like odds or rumored odds, but those are rarer than snakeskins for sure We pulled like two or three of those However many cases that's very nice with the Giants all-purpose fields purple ice to 225 a conco titans and Yeah, why Brian branch it goes through like welcome to the world the grin or ever and then it's like And then the screen opens up and it's clear and you see like the launch That's kind of cool Silver rookie tannery key Eagles blue wave of 199 Joe mixon mangles The one Johnson and Roshan Johnson the launch is I would like to write that right red wave rookie to 149 Paris Johnson jr. For the Cardinals. You'll you'll understand what I'm saying Like it was intense Silver prison break rookie Gibbs for the Lions Jordan Addison rookie autograph Minnesota Awesome hit Vikings. It's going to Brayden base Addison and a mazy Smith digs Bailey zappy purple power number to 49 Patriots BJ Ojolari Justin shorter. You've been to Universal recently No, I think I haven't out of there's velocity coaster Silver is Derek Carr for the Saints orange it at 249s burns for the Panthers Anthony Richardson rookie for the Colts do it at night I was up with it at night Jake. What happens at night get the scary lost Raptors at you at night Get scared now That's a Ram is Benz Quranic autograph. I got fired Mims and Davis Hyper sake one out of 175 for the Giants John all right case I'm beauty Never upset when coaches get fired that should have never been hired in the first place Silver Holland for the Dolphins They dang in the bottom left. Just want to point that out. It's a Silver Holland No purple Jordan love out of 125 Packers Yeah, it is on it Interesting. We'll let us I don't guys hadn't seen it in years. I hadn't seen it since he was a rookie in 2021 looks like Cave law That's what I was gonna say. I haven't seen it forever blue ice Rookie Jack Campbell for the Lions out of 99 the key in Wharton. You've been on the resistance, right? Yeah I Ride sick but on the other one. I'll be pretty surprised at Belgium goes to Dallas prison break silver CJ Stroud for the Texans Michael Vic legends out of the 149 Falcons smear Gibbs Derek Hall That kind of get traded to Spicy off season purple ice out of 225 is Rashad penny Eagles will be traded Jones and Hainer Back to the menu. Let's do it. I'm down. I Can see it down Silver for Washington John Riggins Hyper to 175 Mike Williams Chargers Wilson and Evans. Who do they go with? Tray Lance You wouldn't get a coach like Belichick and then trade Dak though like that's just taking a step back You're gonna have to find a quarterback then purple to 125 is Barrett for the Bucks Tillman and Hyatt, he's gonna trade for Bailey Zaffi making the starter for the Cowboys He's gonna do straight up Dak for Zaffi silver rookie commanders KJ Henry fair to me Orange wave the 60. Yep. Me too as a Patriots fan. Yep the 60 Joey Bosa Chargers. It seems fair John Robinson Tyree Wilson with the Jeep up purple ice nice will Leviss rookie number to 225 Titans 35 out of 225 Yeah, exactly brunette they trade Dak their cap is murder and like that's just not something you do if you want to Belichick's probably gonna want to win now. He's probably not interested in going somewhere where they're gonna not have a quarterback He's leaving somewhere where it's It's a lot of his fault that they don't have a quarterback, but Blue ice to 99 prismatic Debo Samuel for that for the Niners Orange to 249 is Calvin Redley for the Jags Paul pro-debo, right? KJ Miller, Sam Laporta It was him almond raw and Hill right the three receivers Purple ice to 225 Donald for the Rams Foskey and Campbell Silver is Richie Grant for the Falcons. So 149 Roger Jones rookie red wave auto for the Steelers That's in Bennett. Tyree Stevenson Rogers there red wave the 149 Vaughn Miller for the Bills Eric Gray KJ Henry Ram silver Matthew Stafford purple to 125 chiefs sky more not less On Robinson got a pass Roquan locked down purple power Wiggins. They wanted to number to 49 for the Washington Redskins as they were Brentson Pierce Johnson Yeah, I feel like if it's if there's multiple teams interested bill. It's actually gonna be like let me go to the Chargers because they have a Arguably top five Yeah, post Brady wants to win the not ready, but a bell check once a wins record. I Don't think he's definitely he's like a hundred percent not gonna retire He would have stayed in England if If craft is like, yeah, dude, we want you back red wave the 199 Moseley for the Jets Musgrave and Yahya Diabe Honestly at this point Paul I think I'd rather my favorite NFL team have Harbaugh as the coach then Bella check at this point Silver mink if it's Patrick Steelers red wave the 149 Matthew for the Saints Over shown in McDonald's all-purpose Wilson orange wave the 60 Lamar Jackson for the Ravens 27 of 60s a flowers rookie and a Devon a chain silver rookie Giants Eric Gray purplish the 225 is back to you for the Packers Macintosh and I'm a decade Osama Time out first time in the second half. Oh boy. Did Ann Saffron's noses busted up now, too? Geez Blue wave rookie a chain dolphins out of 199 got Bryce young and Jane Reed bridge club down a bokeh That's funny.  I would want Bill Belichick. I think Silver emergent Kenneth Walker Seahawks hyper to 175 is what for the Steelers Skaronsky and Johnston which team and Madden is gonna have Peter. What's his name as your coach now? What is his name what's the coach's name? They don't have his rights or something something like that. Yeah, I feel like that forever Stafford flashback for the Rams. Yo, I mean They didn't flag that No Are we watching the same game right now? Red wave the 149 is tonsil for the Texans crazy. What are we missing here? Well, she can't see and chase Brown Holy cow. Did they think that he ran into the other defender first? I mean if that happened against the Lions that for sure we called silver Chad Johnson Bengals and Ivan page junior rookie out of for the Vikings Vanaconda Van Ness Oh boy, I was a make-up all maybe for the for the off-sides or something They called the false start blue ice out of 99 Jared got for the Alliance The first the defender that didn't jump. Yeah, but like Dude silver and Jonathan Taylor for the Colts orange to 249 is Lance for the Cowboys And there's no consistency. We'll never know purple ice out of 225 Justin Tucker for the Ravens Wicks and Smith and Jigla. That's a thick patch auto coming up Hype silver Micah Parsons for the Cowboys and it is and Aiden O'Connell 11 of 49 purple power RPA Going to the Raiders awesome hit Raiders Dotson Deontay Banks The defender that jumped and landed on on Robinson's back Man, I get a y'all room for the Lions, but holy cow. That was a make-up call to 175 Marvin Jones for the Lions I was a make-up for the That false start they called Whoa, whoa, whoa, whoa another one another one Silver Michael Thomas for the Saints and a Sam LaPorta 10 to 49 purple power RPA for Detroit Got his playoff touchdown tonight. That's going to Andre big card Andre. Nice, dude Sick that's going in a mag. I said the jigbook and tank big speed Silver rookie is McDonald for the Jets Purple Pat Pete to 125 Steelers Pat Pete picks six tomorrow. It's like old times. I know Josh Allen's gonna turn over once or twice Orange wave to 60 Montez sweat for the commanders Yabi Dotson Josh Reynolds help out the Rams once again with that drop Purple eyes rookie to 225 Ken J Miller for the Saints Donald Musgrave All right, orange out of 249 is Harrison Smith for the Vikings a chain and over shown Too easy all pro Prismatic B John rookie purple eyes to 225 Higbee for the Rams He's on rookie Tyree Wilson for the magic play tomorrow. He's a Papa John's Blue ice in 99 burrow for the Bengals Red wave to 149 Tyler Smith for the Cowboys Whether spoon and Anton Harrison. Oh Boy RJ. Oh, boy. Just that Flashback is rookie weather spoon for the Seahawks purple of 125 Minshew for the Colts For the base do it another patch. Oh, what it's happening here Is that in the same box Jalen Hyatt's a 49 purple power RPA That's three in the same box, right? Two in the same box, right? What was in the previous? Oh, is it okay then? All right, then. Geez, which one are we getting that shorter and chase brown? I Silver Lamar Jackson for the Ravens and Forest camo Derek Hall Seahawks rookie Six of 15 Van Ness and Kansy. I thought the Packers had a real chance. I didn't think the Dallas defense would be that bad They looked totally lost Silver for the Jags mark Bernal purple eyes to 225 Fant for the Seahawks All right last box. Yeah, I think a lot of people expect the bucks to win. I am Jake will be there right We're the bike jersey tomorrow Purple eyes to 225 Venn Ness for the Packers. We're gonna wear a bigger problem. Nice You know why Brian ran she I was gonna say walls all the All the Eagles fans. There's a couple of those fans of the shop. They would not be surprised whatsoever Hey RJ, so that's how we're doing. Look at that, dude. So that's how we're doing Jake RJ. How are you doing? That's more important than how we're doing silver McKinney for the Giants Tray Tucker to 75 rookie auto for the Raiders The one Johnson Roshan Johnson, you're gonna cover this dirt down. Okay, I was expecting it again Silver rookie a chain for the Dolphins blue wave waddle for the Dolphins to 199 Tank Del rookie and Chris Rodriguez I think what the Dolphins need to do what they need to do to adjust in the offseason is Tell the defense to just stay healthy. There you go Hyper to 175 is Shakir for Buffalo Steelers could knock out the I miss the playoffs. They could win O'Connell Washington Definitely tougher for a team like them to win but I don't even know anymore Yeah, but the cold might even it up a decent amount Purple ice out of 225 is key Klee for the Panthers Hull and tune It could happen it could happen a while But the bills are the team that like I feel like they're either gonna lose or win by like 30. I don't know man They're just they're that kind of team You got Josh Allen as your QB. It's just chaos reign supreme at all times to 175 Packers rookie done TV on wicks nice wicks And hooker Joey Porter surprised at how bad the Browns looked I thought the Texans won this over how bad the Browns looked Forest camo is overshown rookie for Dallas at a 15 There we go 11 or 15 you'd say in mims Here's not about your cold. You and a lot of people up north man To me it's gonna be brutal here tomorrow. You're gonna be brutal here tomorrow You don't even want to know Don't even want to know Silver fireworks Tyree kill for the Dolphins orange to 249 dat win for the Cowboys Young darn alright, we are you gonna throw three picks against the Steelers and still win Lockdown in Hutchinson blue wave the 199 is Herbert for the Bears. I Know Sean a room against the bills Horton and lettuce I'll be rude for the Steelers let's go Steelers a Byron young on the rookie auto for the Raiders Maisie Smith tanner McKee So can't believe we had three purple power RPAs He's wild That's very possible to Jordan They might not need to pass a lot at all silver mile Sanders for the Panthers Red wave rookie to 149 KJ Henry for the commanders Addison cook and Allen could just have a bunch of rushing or it's together Blue ice sky more number two ninety nine for the Chiefs Jones Anthony Richardson Yeah, it's It's not that cool here Especially in like the on the coast It's not gonna be that cool. It's freezing here. I'm talking about actually I was freezing. I don't think I I don't think I've put on sleeves I put I wore jeans last night, but I thought it was nice dude. I was freezing last night the windows open It is a good windows open weather around here. I guess if you're at Disney and you're outside all night It's kind of chilly Makes sense. I Can see that So we're talking we're not talking like It's a three-weather. It's not it's not cold here. It's very nice here recently Okay, we had the press proof rookie for the Giants Eric Gray It's 99 unless we had burrow Debo James Winston Jared Goff sky more tank Dell Jack Campbell Lamar Jackson Montez sweats Joey Bosa Bailey Zappi John Riggins Derek Hall DeMarvian overshown autos of scurroneck young Darius Davis Ivan Page jr. Jordan Addison Broderick Jones Michael Vic. That's Bryant's young Trey Tucker a no Connell RPA for the Raiders on the Purple Power same with Jalen Hyatt for the Giants and a Sam LaPorta for the Lions. That's so freaking weird that there was three Purple Power RPAs and a sixth boxer I don't think I pulled more than one Purple Power RPA outside of this break overall That's crazy, man. Congrats to those who hit and now thanks again everybody moving on out to more football


Backslash in the paper theoretically substantiates the methodology of development of information and communication competence, ICC, in teachers of the military education system applying the distance form of learning. Scientific approaches to the concepts of methodology have been analyzed and the author's vision of the methodology of development of ICC in the military education teachers has been suggested. In particular, they determine the methodological approaches to the methodology of its development, as well as its main stages, purpose, tasks, content, methods, types of training sessions, tools and organizational forms of learning. Backslash in.


Now another proudly we hail one of radio's outstanding dramatic half hours starring Lee Tracy and presented transcribed by your Army and your Air Force. From Radio City, New York, here is your star and host on proudly we hail the distinguished Broadway stage, screen and radio star Lee Tracy. Thank you, Kenneth Banghart and hello everyone, welcome again to proudly we hail. What's our play about Lee? Well, we have a highly dramatic story, Ken entitled Straight and Level, a story about a man's conflict within himself, a story of his struggle to make a decision that affects his whole life. I believe it will make for excellent listening. We'll begin right after your important message. The United States Air Force needs veterans of any service with training in such fields as radio, radar, aircraft maintenance, weather, photography and many others. If you can qualify, you can now enlist in your old grade or better. For full details, visit your nearest United States Army and United States Air Force Recruiting Station. And now with your star Lee Tracy in the role of Ken, your Army and your Air Force presents the proudly we hail production of Straight and Level. To follow the sun and chase the stars, to source skyward and dwell in the lofty cloud worlds of upper space, to join the eagle and touch the hand of greatness with wings. Let's have it. There's no point in my trying to soften the blow. You'll have to give it up. I see. All of it? Most of it. The testing certainly. You can fly for your own pleasure if you like, but nothing fancy. Straight and level. That's right. Straight and level. What is it I got it or haven't got? In its simplest terms, your body is worn out, particularly your heart. It's taken too much punishment for too long. You've been in this game for a long time and from your record and from what I've read about you, a large part of your flying has been on the rugged side, testing. It's caught up to you. Just like that. No, not just like that. But my last physical, I was fine. What's happened has happened suddenly, but it was a gradual wearing down. Suppose I took it easy for a while. Suppose that I... No, the damage is done. But don't take my word for it alone. Go see another doctor. I'll give you the name of several. I guess I'll take your word for it. You see, you're the fifth I've been to in the past week. What's the matter, Sally's night out? No, I haven't been home yet. Anything wrong? With this baby? I should say not. I was just checking her over. There's going to be a big day. You ought to hit the sack. Yeah. The weather looks like it might get rough, front coming in from the east. You should have been a meteorologist to hear all this. She's a bird. She's what you dream about all your life. And you're the lucky stiff that gets to fly her.  Maybe I should have been the mechanic. I can't ever take what you've got away from you. I don't get you. Ah, ah, keep it.  Hey, what have you been doing? Guzzling beer? No, no, I'm just getting salt, I guess. I keep thinking about all the flights and all the planes behind us and this is what they've led to. Yeah, they shine that light down here. Yeah, we've seen a lot change, Ken, from jennies to this, but let's not cry in our soup about it. We'll see a lot more before we're put out to pasture. Okay, Mac, just got me all of a sudden. Doesn't it ever get you? Well, sure, but there isn't any point in making it sound like a sad story, uh, fasten that cowling there. Yeah. Huh? Guess I'll go home. Well, if I had a gal like Sally to go home to, don't think I'd be spending my nights with this hunk of bailing wire. Ah, the heck you wouldn't.  Don't leave the woman I haven't got out of this and go home, will you? Tomorrow's the final flight test. If she comes through that like she has the others, we've sold ourselves a plane. You mean con aircraft has. They'll get a lovely contract out of this. And we'll get a lovely bonus, especially you. Yeah, yeah, especially me. Is that you, Ken? Hi, kid. Sorry I'm late. I had to see a guy. I've got to supper in the oven. I'm not very hungry. What's the matter? Don't you feel well? No, no, no, no. I feel fine. I just got to get a lot on my mind. Oh.  That sounds good. How'd the test go today? Fine. Perfect. She's our honey. Is tomorrow the last one? What? Oh, I'm sorry. Yeah, yeah. Tomorrow's the last one. Ken, what's wrong? Nothing's wrong. Nothing's wrong, I tell you. You can't fool me, even if you won't tell me. Look, Sally, can a guy be tired? Can a guy have a lot on his mind without having something wrong? Are you going to get me that sandwich? Yes, I'll get it. But there's no need to snap at me. Oh. I'm sorry. I wish I could tell you how much. Can't you sleep? Mmm. Want me to get you an aspirin? No, no thanks. I think I'll get up and smoke a cigarette when nerves are kind of wound up, I guess. You forget you're not a kid any longer. I don't forget that. Well, Chuck Henderson had better give you a good long vacation when this job is done. Yeah, we'll go lie in the sun. Ken, Ken, don't you think it's time we started thinking about the future? You're still a young man as far as average standards go. But I'm in a business where there aren't average standards. Is that what you mean? Well, yes, you know it is. I'm an old man in the air. In the air is for the young. Don't be silly. You're not old.  You've got to give it up sometime. That's right. Give it up while I'm still healthy. Give it up before I go into a dive I can't pull out of. A man with your experience could get a good job on the ground. Chuck would find a place for you. A man with my experience can't work on the ground. You can grow away from the earth. I live up there. I live here with you. And I'm second best. No, no, that's not so. You're part of it. I'm not very good at explaining.  If I told it, after tomorrow I'll have to tell her. I'll have to tell her all. I'll tell Henderson and tell Mack. But not before it's done. She's my baby and I'll see her true to the finish. What a way to finish it. What a way to level. Now one of the first things to get into your head, Murdoch, is that if you want to learn to fly, you've got to learn to keep her straight and level. Notice your wing there in relation to the horizon. Notice the angle. All right, correct it. Ah, that's too much. Okay, now check you know. Get the idea, Murdoch. That's it. Keep her straight and level. You're the guy who's going to test this bird? That's right. Done much testing? Some. He's plenty hot, tricky. The other boys don't like her. I like her. She looks sweet. Looks like she wants to get off the ground. My name is Jack McDonald. Everybody calls me Mack. I'm Ken Murdoch. Everybody calls me Ken. Been flying long? No, not as long as I'd like. Not as long as I'm going to. You're kind of young, aren't you, Ken? I don't see any gray beard on you. Well, I don't have to take them up there. I just have to see if they get there. Well, I'll trust you if you trust me. Okay. Yes, he should fly straight and level, anyway.  That's right. I take Sam up and see what they'll do. Sounds romantic. It is. Have you been doing it for long? No, no, no. About five years now. Flew the mail for a while, did a little instructing and stunt flying with the circus. Was that big man you were with the pilot to? No, no. That was Mack. He's a mechanic. Isn't another one like him. I seem to be asking a lot of questions. Ah, so you're interested. Curious. I've never met a pilot before. I've never met a pretty girl like you before. What do you say we get acquainted? Oh, you're awfully fresh. Hey, how would you like to go up there? See those clouds? That never, never lands. You can go and look. You can't stay. Not like this world. Nobody's there but the wind. You lose the earth. Would you really take me up? No, you bet I would. I think I'd be frightened. No, you wouldn't be frightened.  Would you want to do any stunting, would you? Ah, don't worry. I'll keep her straight and level. You wanted to see me, Chuck? Yes, Ken. Sit down, will you? I had a talk with Marvin. She thinks she's about ready. That's what I wanted to talk to you about. Looked her over? But all yesterday and most of this morning with Mac should be able to start a vaccine test for the end of this week. She's the hardest thing we've ever developed. She's all aeroplane. Oh, Jett. Can you've been chief test pilot for me with time out for the war for 12 years? That's right. Any complaints? No, no, no. One. The best in the business for my money. Ken, let's not kid ourselves.  So, what about letting one of the younger boys do the tests? Are you kidding? I'm just as fit as any one of those kids and I've forgotten more about this game and they'll ever know. I've followed this baby right from the drawing, boys. She's mine, Chuck, and I'm going to test it.  In the first place, she's not just yours. She's everybody's in this shop. A lot of sweat and money and brains has gone into making her. You know, there are plenty of interest in her and if she makes the grade, we'll all profit by it. I don't want to take any chances with her that I don't have to. I've seen the report on your last physical. What about it? Nothing wrong with it. No, but if you compare it with the one you had a year ago, there's a difference. Your reactions aren't as sharp as they used to be. And for this kind of job, you know what that can mean. I know you're acting like an old woman. My reactions are good enough to handle anything that flies. You've never seen me do a thing to prove otherwise. Well, that's true. I haven't. But I've never seen you fly the new ship. Well, just stick around. You'll see me fly it. If you don't like the job, I do tell me then and I'll listen to you. She's my baby, Chuck. I test her. You get yourself a new chief pilot. Now, look, will you take it easy, a hot-headed Scotsman? Just pull out of that dive and fly straight and level for a minute. You'll have to give it up. You forget you're not a kid any longer. It's caught up to you. I'm an old man in the air and the air is for the young. I'm over the field at 30,000. Here we come.  Give it up. Give it up. The action's on its job. Give it up. Give it up. I can't. We won't come out. Lee Tracy starring in the role of Ken Murdoch in the cloudy, we hail production straight and level will return in just a moment for the second act. Registered nurses. The United States Air Force Medical Service offers you a great opportunity to serve your country and further your own career. Yes, you can become a commissioned officer with good pay and allowances while you receive postgraduate training and anesthesia, operating room management and techniques, nursing administration, and other fields. Nurses with special qualifications may train as flight nurses at the Air Force School of Aviation Medicine. For complete information, right to the Surgeon General, United States Air Force, Washington 25 D.C. You are listening to Proudly We Hail. And now with your star Lee Tracy in the role of Ken Murdoch, we present the second act of straight and level. Right. You've been dreaming. Where? Oh, yeah. Wow. That was one for the birds. I don't wonder. You fell asleep in the chair here. You were all crammed. Yeah, you can say that again. What time is it? It's about six o'clock.  Well, I'll try to be more quiet this time. I got about time for 40 winks. You'll be in great shape to test that plane. I'll test it no matter what shape I'm in. And nothing in this world could stop you. That's right. Nothing. Well, honey, I'm off. Well, you ate a good breakfast anyway. Do you feel all right, Ken? Never better. Come on. Give me a smile now. Just think tonight we can go out and have ourselves a good time and tomorrow we'll make some plans for the future. Future? Ken, what do you mean?  We'll go with bells on. Oh, well, be careful, darling. Remember, I have a world, too. And if anything ever happened to you, I'd lose it. I don't live up there, Ken. I only live here.  Don't worry now.  You're the best wife a guy ever had. I don't deserve you. If I've been, you know, jumpy with you lately. Well, it's been because of... I understand, darling. Now, you better get going or you'll be late. Well, honey, I thought it might rain today. Well, give me a kiss. Wish me luck. My love goes with you. No man ever had a better good luck charm than that. Keep her straight and level, Ken. You bet. I'll just put the horizon in a different place. Morning, Ken. Morning, Chuck. Looks like a good day for it. C-A-V-U. How are you feeling? Peachy. Why? You don't look so hot. Listen, we had this out before I started the test. Up till now, my reactions have been all right, haven't they? Yeah, I know. She's still in one piece and so are you. Not just the way we're going to stay. I told you if you found anything wrong without my flying this baby, you could talk to me and I'd listen. Otherwise, leave me alone. Let me do my job.  Now, just simmer down a minute and listen to what I was going to say. Okay. I'm sorry. What were you going to say? And if you felt like putting it off for a day, it would be all right. Put it off? Isn't that all your brass over there? They didn't come out here to have you delay things. Well, it's up to you. Well, there's no time like the present. You know, if I didn't have sympathy for your wife, I think I'd fire you.  Oh, well, do you feel like a cup of coffee? No, thanks, Chuck. I've had my breakfast. I want to talk to Max. He said he wanted to see you, too. I'll walk down to the hangar with you. You know, Ken, I've sunk everything into this. If she makes it today, things are going to be humming around here. And if, for some reason, she doesn't? That hasn't happened since when was it, 37? Yeah. That one was a bird, all right. All she could do was flutter. I suppose we'd just start again, you and Max and Marvin and I. Those were pretty good days, weren't they? Yep. We didn't have a beam. Just a lot of hope, dreams. We had the stuff to make them work. We've been making them work ever since. War isn't much good for anything, but we learned a lot from it. No, just grow, too.  Well, I guess so, little. Back in 37, there were only the four of us. If she doesn't live up to... She'll live up to anything. There isn't anything as sweet as that child anywhere. Well, you should know, I guess. I must be getting old or losing my grip or something. But it's all or nothing. And if we don't make the grade, a lot of people are going to be out of jobs next week. You really shot the works, didn't you?   All I've been able to think about is the bonus I planned again. That or we'll be without that beam again. I'll be willing to bet you that bonus if she comes through with flying colors. Well, if she does, I think you and I should have a talk. What do you mean? Well, I need help, Ken.  You mean give up flying for an office job? Now, don't take off. It would be a good job, a well-paying one. You wouldn't have to give up your flying, but you could step down and let some of the younger boys handle the rugged tests. You could be put out to pasture to graze with the sheep. But why have you got to look at it like that? Because that's what it amounts. Now, look here, Ken. I'm not going to argue with you now. When you finish today's test, we're going to have a talk whether you like it or not. You go blinded by your own stubbornness and pride that you can't see the runway for the trees. I'll see you later. I'm going to act when you check the shoot. Anderson, tell you I was looking for you. Yeah. I guess I forgot about it.  You and ten other guys. What's on your mind? You all set to go? She's all set to go, but you're not. What are you talking about? I happened to run into an old friend of mine last night, Doc Royal. We got to talking about this and that. I told him about the new plane, the test we've been running. I told him about what she's got to do today. He asked me who the pilot was. I told him. Then he told me. I see. He wanted to call Henderson right away to stop you. He said a terminal velocity dive would finish you. I asked him not to call shots. I said I knew you'd figure it out for yourself before you went up and you wouldn't go up. I said anybody trying to stop you would run into trouble and you'd never forgive them if they tried because you're thick-headed and stubborn and you've got too much pride. That's what I told Doc. He said he'd leave it up to you or me. If you didn't come out of that cloud bank you'd been flying around in. You got it all figured out, haven't you? I have. I still hope you have. I don't want to go out there and tell Henderson, but I will if you don't. Max, listen. I've got to fly this test. I just can't quit now. I know I've reached the end of the line, but I want this last flight. I've got to finish it right. Royal said there was only one way you could finish this flight. How does he know? Because he stuck some gizmo on my chest. He said, breathe deep. I always had my luck. It won't run out on me. It won't because you're not going to give it a chance. Mac, you and I have been friends for a long time. We've seen a lot of rough weather go by. We've always understood each other. Can't you understand this? Of course I can, you dope.  But not like this. Not like this? Well, how then? A headline in a newspaper, a lost contract, and Sally all alone. Up there you don't have to think about anyone but yourself. Well, you've done it so long now you can't think any other way. Oh, can look. I know you must be in a flat spin, but do your head, man. It's too late, Mac. It's too late. It's not too late till you climb into that cockpit and you're not going to do it.  You wouldn't forget about this for friendship's sake. That's one of the reasons I won't forget about it. You win. I'll go tell Chuck. You go out and really do the town. Yeah, yeah, sure. Hold this thing, will ya? It sure takes... Sorry, buddy. Blame it on the heat of the moment. Go sit, Ken. Right. You know what to do? Yeah. Where's Mac? Oh, he got a telephone call. Oh, all right. We can't wait any longer. Good luck. Thanks, Chuck. Now I don't take any wooden props. Up we go, baby. Up to the sun. Climb, you sweetheart. Climb! The only thing up here is freedom. And they want to take it away from me. This blue space is my world. These clouds my kingdom and this my clean and light. There's no time. Only to silence the universe. 1248 Murdoch, do you read me? Over. I read you, brother. Like a book. Murdoch in XK1248. XK1248 Murdoch, come in, come in. Cast altitude reached at 1048. Might as well make it as neat and hardly as we can. And now, the time has come, the walrus said. Here we go, baby. Hold on to your wings. Down and down we go. We don't make the grade. A lot of people are going to be at a job this week. He's all technical, but you're not. But if anything ever happens, he's all or nothing. XK1248 over. And what the devil are you doing? Why didn't you continue the dive?  Get Richard's. Tell him he'll have to finish the test today. I'm coming in for a landing straight and level. To follow the sun and chase the stars. To soar skyward and dwell in the lofty cloud worlds of upper space. To join the ego and touch the hand of greatness with wings. Tracy will return in a moment with a word about next week's show. You veterans who want to get ahead listen to this. The United States Air Force needs thousands of trained men today. Veterans of any service, as well as non-veterans trained in such fields as radio, radar, engine maintenance, weather, armament, camera work and many others. And here's where your training and experience counts. Veterans, if qualified, can now re-enlist with their old grade or better, skip basic training and be assigned initially to a nearby Air Force base. You non-veterans, if you have technical training the Air Force needs, can get a grade that matches the skills you have when you enlist for basic training. Get all the facts at your nearest United States Army and United States Air Force Recruiting Station. Volunteered today with the United States Air Force. This has been another program on Proudly We Hail, represented in cooperation with this station by the United States Army and the United States Air Force Recruiting Service. Proudly We Hail stars Lee Tracy. Straight and level was written by DeWitt Cobb. The music was composed and conducted by John Guarnieri. This program was produced under the supervision of Charles and Rogers Productions and directed by Charles Wilk. This is Kenneth Banghart speaking, and here again is your host and star, Lee Tracy. Join us next week over the same station for Proudly We Hail, won't you? Melee Peninsula for a fast moving adventure story in the wilds of the Melee jungle on a besieged rubber plantation. Our play is titled Story at Ten Direct. Hope you'll be with us. Until then, goodbye.


Okay, anybody interested can come in and sit down, we'll start the second lecture. So the first lecture was somehow a motivation overview of the field. And as I said, what we are going to do now is give you the foundations to understand some of the topics in the first lecture. We will not talk about all the applications. We will not talk again about the Oppenheim conjecture, but we'll talk about analog, more simple to comprehend things. And so in this lecture, we will talk about the geometry and dynamics of one of the simplest examples of a league group that is important in the applications, and that's SL2R. So we talked about translations, that was of course a very simple example, in particular because it's a commutative group. This is a non-commutative group, and as you've seen in the lecture, there are end dimensional versions of this. So just to fix the notation GLNR is the group of invertible N by N matrices with real coefficient SL and R are those matrices with determinant one, and of course SL2R is a special case. So today, we'll talk about the case N equal to two, and we'll deal with this group in general in such a way that you will see, not see, but that will allow you to generalize these things to those higher ranked groups, and this viewpoint is that I'm going to present is particularly useful for that. So SL2R is then more explicitly, of course, two by two matrices where the coefficients are real, and let me just write out what the determinant is here in this case. It's just this. So this group will be the main character in this lecture, and it's very intimately related to hyperbolic geometry. So what is hyperbolic geometry? Hyperbolic geometry is a geometry like the Euclidean geometry or the spherical geometry. And it is the hyperbolic plane in two dimensions. So if we look at two dimensional hyperbolic geometry is a model for the hyperbolic plane, the analog of the Euclidean plane. And it can be realized by the upper half plane, sometimes also called Poincare upper half plane, or Lubashevsky plane, depending on where you have your office. And that's the space, and we equip it with a Riemannian metric that makes this a Riemannian manifold. So I'm going to write every point in this complex upper half plane as x plus i, y, x and y are real variables, y of course is positive. And this is the element. You can also write it in terms of the quadratic form in the tangent space that gives you this line element. So the Riemannian metric for two vectors, tangent vectors, in the tangent space at z, you can write this metric as v dot w divided by imaginary part of z squared, where this here is just the standard Euclidean inner product. So you have this funny scaling factor here. And so let me make a picture. This is a picture of the upper half plane. Here's the point i, and we look at all these points. Now this metric gives you a distance, right? That's what it means. So you get the distance between two points by looking at curves that connect these two points. You can calculate the length of this curve. And then you can minimize it. And if you do that, you see that the curves that are minimal, these are called geodesics, are actually semicircles that are perpendicular to the boundary. Or if you have two points that have the same x-coordinate, the curve will just be a straight line. So this is how geodesics look like in this Riemannian surface. It's infinitely extended. And it's a beautiful geometry, of course, and you have to switch off your Euclidean mind here because points that have a small y-component can be very far away, because you see when y is small, the line element will be very large. So small Euclidean distances here might not be small hyperbolic distances. Okay, now how does SL2R come into this? When you have such a beautiful geometry, you can ask, well, what if I apply transformation to this picture, say, coordinate transformation? And you want those transformations that leave the Riemannian metric invariant. These are called isometries. And the group of isometries is precisely given by a particular action of this group SL2R. And that's why SL2R is so important for the hyperbolic geometry. So SL2R acts on H by fractional linear transformations. So this is a map from H to H, where you take a point in H and you map it to something I call JTZ, which is simply defined as AZ plus B over CZ plus D, if T is a matrix of this form in SL2R. Okay, these are called fractional linear transformations or Möbius transformations. And note here that the imaginary part of JTZ is given by Y over CZ plus D squared, exercise. So as I go along and you get bored, just do these little exercises. Think about if you have any questions for the tutorials. We'll give you in the tutorials a few problems that are related to this. Now, I talked about isometries. So why are these fractional linear transformations related to the isometries of the upper half plane? Okay, so we'll define isom plus as the group of orientation preserving isometries H. And by H, I always now refer to as H as a Riemannian manifold, so equipped with this Riemannian metric. And then there is an important classic statement that says that if T in SL2R, then JT is an isometry. And every isometry can be obtained in this way. So if F is an isometry, then there exists a T in SL2R such that F is equal to JT. Okay, now let me show you how this comes about, this theorem. I'm going to sketch this. We've given you a little bit of a reading list and there is a very nice book by Einstein and Ward that covers this. In fact, we can give you a slides of the pages and PDF of the relevant sections if this is according with UN law. The title, it's on the reading list. So have a look at the reading list. If I write the title, I will finish the lecture too late. Okay, so just as a notation, it's sometimes not so convenient to always write JT. I'll often write just T of Z or even TZ and that will always mean, or sometimes of course, because it is a matrix, I'll just write it like that. And this is understood as this action, CZ plus B over CZ plus D. And this, as I said, defines a group action which you can check. So this means that if you apply first JT2 and then you apply JT1, that's the same as applying J of the product of the two matrices. So what this means is that the map that associates with every matrix in SL2R, a fractional linear transformation JT is a group homomorphism that's onto. So it's surjective and I think these are, I always forget, epimorphisms or whatever. So it's an onto and that what you can find out is that the kernel of this, so you take a matrix A, B, C, D and you associate with it this fractional linear transformation. If you change the matrix A, B, C, D by minus A, B, C, D, so you just put a minus sign in front of all the coefficients, you get the same fractional linear transformation. And that's the only way you can get the same. So that means with kernel plus minus one. So this means that the matrix one, you can check it, right? Matrix one gives the identity transformation. Matrix minus one also gives the identity transformation. So that's the kernel and there's no other matrix that gives you the identity transformation. So here I mean identity matrix. This is identity matrix, yeah, it's a good question. Two by two identity matrix. And so this means that if you remember your group theory that now we have an isomorphism between the group of fractional linear transformations, the group isom plus, so of H with the group SL2R modulo the kernel, right? And we call this group, we give it a special name, PSL2R where P stands for projective. So what this means is that the group that we have indeed here, what this theorem really says is that the group of isometries is isomorphic to this group PSL2R. And one of the things we can talk about in the tutorials is how you can prove this. So we'll give you a long list and you can pick and choose, okay? So we don't expect you in the tutorials to do all the problems we write down on the board. We want you to pick a problem that you fancy and then you can try it and we'll go around and we'll help you if you get into difficulties. Or even before that we can help you. The next important observation is that you have a nice way of writing every group element in SL2R and therefore also in PSL2R in a very nice way and this is very useful for what we are going to do. So let me just see if I skipped something here. And let me, sorry, write first what the theorem is called. This is called eva-zava decomposition. And let us define the following subgroups of SL2R. Some of them you've seen already. So let A be the subgroup of diagonal matrices and capital N the subgroup of these upper trigonal ones. So these are called unipotents. These are not called unipotents. These were examples of unipotents in the first lecture. And then the group of rotations to, so these look like this, cos phi minus sin phi, sin phi cos phi where phi, well phi goes between 0 and 2 pi and because it's periodic let me write it like this in the way you've already seen. So I'm taking R but then modulo 2 pi z. And now what the theorem says is that every element in SL2R can be written as a unique product of these three matrices. So for any G in SL2R there exists a unique Nx, or just say N in N, A in A and K in K such that G is equal to N, A, K. Why is this useful? Well what's the dimension of SL2R? Because we have four parameters and one condition, the determinant being equal to one, so this means it's a three-dimensional space. And this decomposition here gives a nice parameterization of this three-dimensional space. Each of those elements comes from one parameter subgroup with a very nice structure. And so this is a very good way of putting a coordinate system on your league group SL2R. So for every T, X and Phi I get a point in SL2R and for every element in SL2R I can uniquely read off the T, the X and the Y. And the Ha measure on SL2R can be written as so if I have an element like this G and X, A, T, K, Phi then the Ha measure has a very nice formula which is e to the minus T dx dt d Phi. Because remember our eventual goal is to do equidistribution theory for orbits on this space and the Ha measure will be the space with a measure with respect to which these orbits can become equidistributed. Now if you remember the Riemannian element that I've just erased of our hyperbolic metric, it was this, and you can read off the Riemannian volume or area element for the hyperbolic plane. So this is d mu of the hyperbolic plane. It's just dx dy over y squared. And if you look at this equation here and you set y equal to e to the T, you see you get exactly dx dy over y squared times d Phi. So that's the nice thing here. You somehow see the Riemannian area element appear in the Ha measure and this is not a coincidence which is what I'm trying to explain now. Why does the Riemann volume element of the upper half plane appear in SL2R? Okay, and in order to do this, let's go back a little bit first to the first part of theorem 1 where I would like to explain to you how the group of isometries acts on the tangent bundle of the upper half plane. So the tangent bundle is simply the collection of all the tangent spaces. So at each point you have a tangent space and the tangent bundle is just the combined space or the bundle where the fiber at each point in the upper half plane is your tangent space. Okay, now what do I erase first? Maybe this thing here. Okay, so this gives a few hints on the proof of theorem 1A. So we will call V a tangent vector at the point Z where Z is a point in the upper half plane. Okay, so there we have a tangent space. That's a two-dimensional space. We can think of these guys living in the complex numbers. It's a two-dimensional space. They are two really, but we just identified with the complex numbers. That will make some calculations easier. So then the Riemannian metric, once I think of these as complex numbers, unless I've made a mistake and Andreas will immediately correct me if I make a mistake or anybody else. Okay, I think you can then write them like this. So this is just the standard Euclidean inner product that we had before. I said these are two vectors, and this is just the Euclidean product, but now I am interpreting V1 and V2 as complex numbers. So this should give just the square of the sum of the coefficients, not the square, the inner product. So if we look at our upper half plane, we have a point here, Z, with a tangent vector V, and now we apply a fractional linear transformation and that will map the point Z to a point TZ here, and it will also change the vector, right? And so I'm going to write that new vector as DT, differential of T times V. That's my new tangent vector. So think of a curve going through this and then you map the whole curve over there. So the new tangent vector will be the tangent vector to that curve that you've mapped. And now you can, of course, ask yourself, well, we already know what TZ is, but how do we have a formula for how the tangent vector transforms under this isometry? And yes, we have. So this is just complex analysis. The tangent vector at the transform point V is simply the differential of T at Z, and you can compute this. And the formula is so pretty and easy, and that's not absolute values. And you get this as the answer for T prime Z. So here's the formula. Now, this is the proof of theorem 1A, right? We want to show that JT is an isometry. So what does JT have to satisfy to be an isometry? The length of this guy has to be the same as the length of V with respect to the Riemannian metric that we're using, right? So let's make it a little more general. Let's take two tangent vectors at the point Z, and we look at the Riemannian metric at that point Z. And remember, this is just written here. And so now what we need to show is that this is the same as this. So this is the question. Is this the same as this? And we can just compute it. Evaluate it now, of course, at the transform point TZ. And you can show this, right? So that's another very little exercise that this is equal to that. That means it's an isometry, but even angles are preserved, right? So this is an inner product. This is actually a quadratic form on the tangent space at Z. And the angle between V1 and V2 is the same as the angle of the transform point. So fractional linear transformations are not just isometries, they also preserve angles. So this is a two-line calculation here. And again, try it in the tutorial, and we'll come and answer questions if there's a query. So I'm not going to talk about the proof of the second theorem, because what I wanted to show you here is that we understand how the isometries act on the tangent vectors. So you have a tangent vector at the space Z. You apply your isometry, you map that to the tangent space at TZ. And this is the formula on how the new tangent vector looks like. Remember, V is now thought of a complex number, right? Very good. Now we want to define dynamics on this tangent bundle. And in fact, the kind of dynamics we look at is the dynamics of the geodesic flow, which preserves the length of the tangent vectors. So you can think of the tangent vector as a direction in which you want to travel, you apply a geodesic flow and then you just follow it, right? Just like in Hamiltonian dynamics, you have momentum and position, you're sitting in some position, you define the momentum and then that specifies the initial condition completely and you can calculate the trajectory. So this is just Hamiltonian dynamics when you talk about the geodesic flow. And so the right space to study the geodesic flow is not the tangent bundle, but the unit tangent bundle. And unit just means that you're looking at tangent vectors that have length one. So let me write down. The unit tangent bundle. It's called T1 of H. In two dimensions also sometimes called unit sphere bundle. So it's the pair of points where Z is a point in the upper half plane and V is a point in the tangent space at Z. So that will be the tangent bundle and the unit tangent bundle has the additional restraint that we only look at vectors of length one with respect to the Riemannian metric. So this guy here is just as usual our Riemannian metric evaluated. All right. Now comes one of the really pretty things in this subject what normally in Riemannian geometry is a subject of geometry, the understanding of geodesic flows and so on. I will now show you how this can be translated in this particular setting to a completely algebraic formulation. And from then on we never need to worry about geodesic flows and so on anymore. We only have to worry about actions of 2 by 2 matrices on 2 by 2 matrices. Fractional linear transformations will disappear from this. Everything will become so much more easy. And the key observation is that you can identify the unit tangent bundle with this group PSL2R that I erased. The group of orientation preserving isometries and this works like this. So here is my claim PSL2R. So we map it to an element in the unit tangent bundle in the following way. So we take a matrix. So remember PSL2R was simply 2 by 2 matrices with determinant 1 where we identified plus and minus 1. So we take such a group element and we have understood the action of PSL2R on an element in the unit tangent bundle or in the tangent bundle here. We understood that these guys are isometries so we've just calculated how such an isometry acts on the vector V. So let's just take this guy and let's just act on the point I and on the unit vector that's pointing upwards. So here's the picture. So this is the upper half plane, here's the point I and this is the unit tangent vector I. That's like somehow the origin in our unit tangent bundle. The simplest point you can think of. And now we just define this map from here to there. Why not? We have an action. We know that this acts nicely on this space. So we just map this point to that point. Now what one can show in fact that this is a bijection. So you can find the inverse. Given any point in here you can find a matrix in PSL2R. And let me show you how this inverse map looks like. And what you're using there is theorem 2, the Eva-Zava decomposition. So here's another claim and that's another nice exercise that you can now do is that we know we can write every element in SL2R and therefore in PSL2R in this way. Log y, okay, phi. So this guy here is simply square root y, 0, 0, 1 over square root y. I don't like the e to the t here so much. For now you will see y. So if you choose this, so every element can be written in this way. And now just a little subtlety here. If we end PSL2R it's not unique because if we replace phi by phi plus pi you get the same element, modulo plus minus 1, right? You see this? And if you don't see it it doesn't really matter for what I'm going to do. So let's calculate this. Let's calculate it. What is this? What is this? And what is that? And if you do that calculation something very beautiful happens. So for this G as chosen here and as I said we can write every G in this way. For this choice of G you get the following answer. X plus Iy comma e to the minus 2 pi I phi, okay? That's the answer you get when you do this calculation. And that's exercise.  Apply it to the point I. Hint, I give you the rotation matrix. K phi acts on this point here trivially. Which means it leaves it invariant. So K phi applied to I doesn't do anything. It gives you I. And then all you have to compute is what does this do to the point I? And then what does this do to the result you get from the first calculation? We can talk about it in the tutorial, yeah? Try that. That will give you a real insight into fractional linear transformations and so on. Now on the other hand on the tangent vector the rotation matrix will rotate the tangent vector. And you can just plug it into this formula here, right? C will be cosine of phi and D will be sine of phi. And just try to work it out how it looks like. Yes, yes, yes, yes. Yes, yes, yes. I know. There's probably something wrong in this formula. Divided by whatever. The right power of y. There's something here so there's a question mark. No, no, no, I don't want you to give the, I want to do it in the tutorial. Who will the modulo ask this question, right? Where are you? I recognized your voice, I didn't see you. But it's your challenge for the tutorial. Because I've written it this way, forgetting that of course a unit vector in hyperbolic geometry is not a unit vector in the usual complex plane. So that has to be divided so that that thing here actually becomes a unit vector with respect to the hyperbolic matrix. So that's a good exercise, yeah? Okay. And then on the tangent, sorry, on the tangent vector, you can then see that this guy, the other two matrices act very simply. Let me put it this way. So this is a nice calculation. Okay, so what have we seen here? Is that we now, well in fact what's going to happen is that if we call the angle of our unit tangent vector theta, then theta is related to this rotation parameter by minus 2 phi, right? Because it appears here and this is really the rotation angle. That's for the tutorial to do. It won't help you if I do the calculation for you. That's something you can do. And what have we learned now is that, well, if you give me a point in the upper half plane and a direction, a unit tangent vector, then I can give you a matrix G. Namely, I just read off here the point where we are. That gives me an X and a Y and you give me a theta. I compute the phi and then I know what G is. It's just going to be this thing. Okay, so that's why it's an isomorphism between the group PSL2R and the unit tangent bundle of H. Right. Now we have the setting on which we work. We want to define now the geodesic flow and the unit tangent bundle, but since we've now identified it with this very nice and simple group of 2 by 2 matrices, I can forget about the Riemannian geometry completely, okay? And I'm just going to define the geodesic flow in an algebraic way and then claim that it actually produces the Riemannian geodesic flow. Right. And of course, I can't go through everything because you can do it. So the geodesic flow is a solution to the Euler Lagrange equations and what the isometries help you to do is to start with a very particular solution of the Euler Lagrange equations which is very simple and get all solutions by just applying the isometries. So what do I mean by this? Let me again make a picture of the upper half plane and let's see. So I'm not going to solve these equations. I'm just going to draw pictures and I'm going to use some colors now. I can find some colors. No colors. So let's see. When we have a, let's suppose this is our starting point and we want to look at the geodesic that emerges from this point in direction i. What you can believe me if you solve the Euler Lagrange equation or if you even just solve the variational principle, you'll find that it's this curve. That's the curve that it is. And if you're moving with unit speed after time t here, you'll be at the point e to the i times e to the t. That's where you'll end up with and t is the time. And what you will see is that if you just remember your hyperbolic metric, line element, oh god, b x squared plus d y squared over y. If you calculate the distance between those points, it'll be, well, this doesn't appear and you just integrate, you know, from here to there and what you will get is that the distance, the hyperbolic distance is just t. So everything's consistent, okay? So that's a geodesic arc from going here to there. The infinite geodesic that you get by moving t to plus and minus infinity will just be this line up here to that. So that's something you can believe me properly, probably. Now let's write this and furthermore, the vector v is invariant, okay? So if I draw the resulting unit tangent vector up here, it will be the same. So here is direction, we're going in direction i. And here will be simply i again. Not quite, right? We need the right normalization factor, which I'm not just guessing. Is it e to the t or something? I don't know. Yeah, so that will be a unit vector in the tangent space with respect to the Riemannian metric, hopefully, okay? But you can check that. It has to be a unit vector. So in effect, it remains the same vector. It's just the length is distorted because of the choice of the metric. That's a geodesic flaw. Represented in the unit tangent bundle description, right? And using theta or using the vector v. Now how will this flow look like in this geodesic in the description of 2 by 2 matrices? Well, I'll just start with the identity matrix and then I will apply a t. So this will give me the matrix e to the t over 2, e to the minus t over 2. So I'm declaring that this now is the geodesic flow applied to the point i in PSL2R, the identity, and it's just defined in this way, okay? Now what you have to check to make sure that I have done this consistently with this picture is you have to go back and forth with this isomorphism. Have we got a name for this, Andreas? I don't know. I think you call it i or something. Okay. You will use it? You won't use it. Okay, let's just call it a tilde because we won't use it anymore. It's probably a bad letter. So you now just have this commutative diagram between PSL2R. We're acting with phi t. Then you have this a tilde. Well, let me call it capital psi. Oh, sorry, yeah. Okay, let's do it here. Let me first write another thing down. So this is just to illustrate that this is this geodesic under this identification. And then more generally, you can now define the geodesic flow as a flow on PSL2R to PSL2R by starting with a general group element and then simply multiplying from the right with the flow phi t. That's now the definition. And here I'm just choosing a very particular initial matrix G. That will lead to this geodesic. Oh, 80, I'm sorry, 80. And the claim is, so let me call this psi because this is like a conjugacy between dynamical systems. The claim now is that this bijection psi gives a conjugacy between this algebraically defined flow on 2 by 2 matrices and the geodesic flow. And one way to simply illustrate that that makes sense is just to look at this particular orbit here and see where the orbit of this 2 by 2 matrix curve is mapped to here. And my claim is it's going to be exactly mapped to that geodesic that I showed you. And the same is true for any other geodesic. So if you start with a point here and a tangent vector here, then you'll get this geodesic that will hit the boundary at right angle. And you again will move along this curve for time t. So if you take this point Z and vector V, you find the matrix G in the procedure that I explained. You flow, you'll get exactly this curve. And if you think about it, this curve to describe algebra in terms of coordinates here, of course it's a circle, but you still have to write quite a lot, right? Here I'm just saying you take a 2 by 2 matrix G, you just multiply 80 from the right and that's a simple calculation. That's really just linear algebra. Well, here you're doing already some geometry. You have curved curves. So that's the big advantage. So I've explained to you now that the unit tangent bundle and the geodesic flaw in the unit tangent bundle can be realized as just my matrix multiplication in SL2R. When shall I stop? Quarter past? Yeah? So now the next important... That's the calculation, yeah? That's the matrix that gives you exactly distance t. If I would have put t in, then I would have got 2t as a distance. It's a little bit like with a rotation. There was not so clever. With a rotation I put rotation by matrix with parameter phi and I found out the phi is actually half of the actual rotation angle in the half-a-half plane theta, right? Actually minus theta over 2. So it's just how you relate the geometry to the matrix is just making a clever choice and here I've just made that choice initially. Otherwise I had to half, half that too. Okay. The next important object and the dynamics of the geodesic flow are horocycles. And horocycles, for those of you who are in dynamical systems, are basically the... parametrize the stable and unstable directions for the geodesic flow. But as we will see, the horocycle flow is interesting in its own regard, in particular because for it you have the measure rigidity statements and the horocycle flow appears in many important applications and it's higher dimensional generalizations too. So in the last few minutes, let me just introduce the horocycle flow. And now the picture is the following. So, so horocycles are lines that are either parallel to the real axis or, I'm making a big picture here, or tangent to the real axis. I view this line here as a circle tangent to infinity, if you like, okay? And hyperbolic geometry, straight lines and circles are the same thing, in a sense. As you've seen here, geodesics can be straight lines or circles with a certain intersection property of the boundary being perpendicular, horocycles are the same, just that now we are not perpendicular to the boundary but tangent to the boundary. And there are isometries that can map this to this line as well as there are isometries here that can map geodesics to straight lines to circles. And fractional transformations always map circles to circles or straight lines and straight lines to circles or straight lines. And so the meaning of horocycles dynamically is that if you look at the point of tangency here, there is all geodesics that emerge from this point will intersect this horocycle at a right angle. So that's a geometric feeling. Now here, the point of tangency is infinity, so all the geodesics that run to infinity are these straight lines and you see the same thing is true here as well. And now if you think about the geodesic flow and you remember the way the Riemannian metric worked is that if you flow along this line and you compare the distance between these two, they will come close at an exponential speed in terms of the geodesic flow parameter. And this is exactly the same picture except that now this point plays the role of the point of infinity. All these will come exponentially close. And so this is in some sense the meaning that this horocycle here parameterizes the stable direction of this horocycle flow. And similarly here, these horocycles parameterize the stable direction of everything that sort of flows into this point. Now on the other hand, if you flow in the opposite way, if your geodesic flow runs down here, these points will diverge at an exponential rate. So the horocycle with a vector is pointing downwards parameterizes the unstable direction of this family of geodesics. Okay, and now I'm just going to define the horocycle, these two horocycle flows, stable and unstable in terms of actions on PSL2R. And then that will be it. So the horocycle flow.  There you take, that's going to be called HT. You take an element G in your, so we again, we define this flow exactly in this way. So this, yeah, so this was the definition of phi t. Now I'm just going to define HT again by right multiplication. Now by the matrix, oh sorry, HT, NT where NT is 1T01 and the expanding is just given by H minus GNT minus, where NT minus is 1T01. Now if you again do the same exercises for the geodesics, you will see that, this you see very simply here that this guy here just corresponds to a point starting here, let's say, and then it's just shifting in this direction. So this is vectors pointing upwards. This is why this is the contracting horocycle flow. If you look at this matrix and you start at the point I, you get exactly this picture. So pretend I is here, that's the point I. Then you flow along these lines and you see the equation for this circle you have to compute and so on. So in Riemannian terms it's quite complicated. Here again it's just matrix multiplication and you can convince yourself that indeed this action projected onto the upper half plane gives you a circle if you apply the fractional linear transformations. Okay, so I think I've talked enough. What I didn't cover, what I really wanted to do is to also talk about now quotients by a lattice. So if you instead of looking at H, you look at the coset space H mod gamma and this models hyperbolic surfaces that can have finite volume. And there you can define the geodesic and horocycle flows exactly in the same way because you can identify the unit tangent bundle of these surfaces provided gamma is torsion free with PSL2R modulo gamma. And now gamma acts from the left on this group. The geodesic and horocycle flows act in the same way from the right, right and left multiplication commute. So this is then a well defined flow that you can take from your upper half plane and project it onto this quotient space. And this is exactly our example of a first homogeneous space if you remember the first lecture. Okay, so now we have a setting where we can talk about geodesic flows, horocycle flows, but in an arithmetic framework. Everything becomes much easier in that framework. In particular, all the theorems about measure rigidity are now much, much simpler to formulate and don't exist by the way in sort of many variable curvature situations as far as I'm aware. Okay, and the other thing that I missed is to talk about the modular surface but I think we can come back to that and define that. I don't want to do that in a rush, okay? So we'll have tutorials this afternoon with everybody with all the other lectures together and then tomorrow Andreas will continue with talking about what? Ah, about equidistribution. So then Andreas will talk about geodesics becoming uniformly distributed, horocycles becoming uniformly distributed, and so on. Okay, thank you.


Just just didn't love it and I really scratched my head and I wonder where's God when you need Hi everyone, welcome back to day four of the reading rush. I can't believe it's day four already like it's going by So fast. I love it. I mean I'm I'm so passionate about it And it's what I do your passion all the time today I don't have too much plan other than going for a walk a little bit later So the plan this morning is to finish a song below water, which is the audiobook that I'm currently listening to I have about an hour left just over an hour I might even up the speed just to get through it a little bit faster Then we'll come back and we'll start my next physical book, which is my sister the serial killer I also need to get better today editing my vlog throughout the day because I keep leaving it to like the morning off So I've been editing this morning and it's now 11 because I like I've been busy and so I need to get back to that because otherwise I lose a lot of the day make sure you let me know down below what you are currently reading whether it's for the reading or actual otherwise and Yeah, I'm gonna go finish your song below water. Hopefully it will not take me too long I'm enjoying it, but like I feel like I've hit a bit of a point with it where I'm just still not fully Connected to the storyline. I'm still not fully Understanding I guess the nuances of the story and I think some of that is down to me and so I feel Hard I feel really harsh giving it a three, which I think I think it'll probably be about a three point five But a three is to the good rating I just finished a song below water and I'm actually Disappointed I ended up liking the book less and less as it went along because I kept waiting for world building that never came Like I'm gonna give it three stars Because I think there were a lot of good points to it I like I loved the sisterly relationship as I said yesterday I loved how it dealt with themes like I thought it's themes really strong of racism sexism self-discovery But the plot and the world was never convincing to me You know like we had these kind of magical entities that people were able to be like sirens And there were other things that existed like gargoyles and like sprites But then there were other mythical creatures which never could exist duh that just that inconsistency. I didn't like it I I was confused throughout the whole thing. I have I could not tell you what happened at the end I have no clue like I really was just I was doing other things while I was listening to the end of that book I just wanted to get through it I often like it when a book doesn't explain everything to me like I don't have to have the world building Served up on a plate to me. However, I just think it was just there was just something missing I think it was a really ambitious book and I just wanted more of like the sirens. I don't really know much about Sirens I felt like they kind of had like three traits that we knew and that was it Just just didn't love it and I really scratched my head and I wonder where's God when you need him Just didn't love it and not something I would like Super duper recommend and it's a shame because I think it had a lot of potential and that great Familiar relationship and like I said the themes were tackled really well And it was a really interesting allegory for like black women in our society now However, I just feel like the story itself on the plot itself never took off It was a great idea But it was never like fully actualized in the way that I wanted it to be I am now Hang on one sec I'm off To grammatica I'm gonna start my sister the serial killer which like I said is my last physical book for this read-a-thon I'm actually already 13 pages in so I don't know how far I'm gonna aim to get to this today I think it'll be a really really fast read. I don't think I'm gonna aim to finish it Also, it's 10 years of one direction today. And if you know me, you know I loved one direction so dearly when they were together. I was the I was obsessed with them I interviewed them on TV. Let's not speak about it We don't need to go into that again No one's done it better than them like they are still the best in their field There is a video of some sort coming out at 4 p.m.    We're being treated.  So yeah, I just got to say that I'm just sitting down to watch the one direction Video whatever this is. So it's two minutes away. I'm nervous like what is this gonna be? So it says music video by one direction performing ten years of one direction. I don't know what that means I don't think I don't think you'll be anything. I can do song. I don't think they love us that much Honey, you've got a big storm coming I'm suddenly 13 again. Oh They're so young. I didn't get to go to the first Tour and I'll forever be Salty about it. Take me home arguably best era in my opinion. It was so young What do you think about it was so young great tour outstanding tour best tour my god The vibes when you were in there like never no one's out. No one else has ever achieved that. Oh my god.  That's it We've seen all this before like I've literally seen that whole thing before okay. I'm glad people agree with me.  I just gonna stand in the middle of the road And if a car comes I ain't gonna move I am now about 70 pages into my sister the serial killer I am enjoying it so far. So I'm about a third of the way in I thought I would check in every third and We've only kind of just set up everything so essentially our main character sister keeps killing guys and she's killed three so far and The protagonist is like well at this point.    Oh, by the way, this is for the prompt of reader books set in a different continent This is set in Nigeria. It's very funny. Actually like the humanness I think it's going to be something I enjoy. That kind of like dark humor. I love like dark humor in books, sarcasm. I feel like the writing is very assured, especially after reading a song below awards this morning, which just the further I got into that I was more fed up I was. This is very knows what it's doing. I have complete faith in it and what it's doing. So I'm hoping to read another 70 pages tonight in between editing. Actually, while we're here, the reading rush literally just posted a reading rush this or that. Twitter sprints are reading on your own. Usually I love the Twitter sprints, but I haven't participated in any of them so far yet. It's just been hard to coordinate them with when I've actually been reading. I tend to get more of my reading done in the mornings and they're not usually running then because I'm in the UK and they're kind of based off US times. I wish there were more running in UK mornings like I don't see why there shouldn't be. Hit me up next time. I can do them in the morning over here. Is that one of your biggest wishes? I haven't really participated in any of them yet, which is sad because I did loads of them last year. Going with the flow or trying to do all the challenges, trying to do all the challenges without a shadow of the doubt. Updating your page count daily or you keep forgetting to, I haven't done that yet. Hopefully tonight or maybe tomorrow morning I'll go and update the first four days because at least I've got the track from the videos because I'm doing my page count in the videos. So I suppose I am updating my page count daily. I'm going to circle both. Vlogging it, obviously. Keep changing your TBR or sticking to it. I've stuck to it so far. I'm just going to have to add to it because we've done so well. Watching reading rush content or actually reading. A bit of both. I haven't watched as much as I want to. A lot of it is sitting in my watch later just because it's so hard to keep up to date with it all as well as read and as well as taking the pictures and posting the vlogs and editing like it's hard. So I'll be watching a lot more afterwards but I have been watching it. So that's my this or that. I hope you had fun. Definitely I thought it was fun. I like the beat. I ain't reading anymore. Yeah, I ain't reading more. I'm not going to. So this is just me saying good night. I didn't get as much reading done as I wanted to today but we did hit a thousand pages today which I know I hit that this morning so I should have recognized it this morning but I only realized now. I only remembered that we hit a thousand pages now which I'm happy with. Like in four days I think that's pretty good. Tomorrow we're going to be going blackberry picking to make blackberry crumble and yeah hopefully it'll be cute. So that's the plans for tomorrow. Read a lot. Get up early early early and love ourselves. I hope you're doing good. Let me know down below how it's going so far for you and I'll see you tomorrow. Bye!


Hello, hello, everyone. My name is Laura. This is my channel Laura's Little Library and welcome to today's video, which are my Pride recommendations. Excited for Pride Month to come around this June. I have been reading so many good books and I have so many good books on my TBR, so if you're interested in that video, make sure you hit the subscribe button so that you can see when that video comes out. I don't know if it'll be before or after this video, but you should definitely check it out and just subscribe. There have been so many good queer based romance and literature coming out and it's just wonderful to see as a queer person all this representation and all of these unique and fun ideas coming into literature and everything like that. So I thought I'd share some of that here with you guys. My first recommendation for you is actually a duology. So if you like dark academia and fairy tales, then I would recommend reading the Grim Rose Girls duology by Laura Pohl. This takes place in this mysterious academy in Switzerland and you have a bunch of these girls who are all kind of reincarnation-esque styles or doppelgangers of different fairy tale characters that we know of. And of course, you've got that drama going on as well as regular high school drama of girls falling for their crushes. There are quite a few queer characters and main characters in these books, which makes it perfect if you are a thriller dark academia really fan that also needs some queer reads for June. Speaking of queer dark academia, which I think should be its own thing because that is really cool. I would also recommend My Dearest Darkest by Kayla Cottingham. This author just came out with her second book and they're definitely like horror queer books that are dark academia. So definitely check the trigger warnings before going into this one. For example, if you're not a big fan of body horror, maybe skip this one. So their second book just came out quite recently and I cannot wait to read that this year. But if you're unfamiliar, My Dearest Darkest follows this. Again, academy kind of on its own little island and there is some mysterious thing under the school. And they may do some, this group of friends may do some little thing to awaken something and oh, it's all dark and mysterious. And I thought it was super well written for a debut novel. And I love the relationships in here. It's just so cute. So definitely check this one out too. Now if you are a sci-fi fan, I actually do have a recommendation for you. There is a book called The Kindred that I read last year. And this was a great romance in our male main character is by and pretty much okay, all of the main characters in here have some sort of queer representation from by to ace to gay and just oh, it's all there. But it's the idea that when you are born and this is in a galaxy far, far away, right? There is kind of this overruling alien race over another race. And the idea is that when you are born, you kind of have a kindred, you are paired up with someone that is like your perfect companion, whether it be romantically or friendship wise or whatever, you have someone that will always connect with you and understand you. And so the prince of this kind of regime era, his kindred is actually someone from a very poor planet and it's really not great for a royal kindred pairing. And so the prince or he's not a prince, but he's like fits in line for the throne until his entire family is basically slaughtered. And he is the only surviving royal. So he and his kindred go and run away to earth and try and figure out what's happening and how do they solve it. And they make a great group of friends on earth, of course. And it was just a really great sci-fi standalone.   It's just not what I gravitate towards. Gravitate space. Get it? So if you're a sci-fi fan and you need something for this pride month, check out this one. Now all the rest of these are going to be somewhere in the vein of contemporary romance, YA, adults, etc., and so forth. So first up, if you are a bit of a foodie like me, I would recommend you chef's kiss. So this feature is a bisexual main character and a non-binary main character. Now just be aware of trigger warnings for like non-acceptance of gender identity and things like that. Because both of these people work in a kitchen and in a commercial style kitchen, they form a relationship. They're very much opposite to tract. Our one main character is very organized. Things have always been done a certain way. She likes her kitchen this way. And then this new person, this new manager comes in and they just change everything around and they're the new boss and what the heck is going on. It's just a very fun, very sweet, but also like very hard hitting at times. Like there are some good conversations happening in this book, but it's still fairly light. If you need an ace recommendation for this month, I would recommend to you I'll Be the One. And this was just a really fun book about a Korean American who is overweight, but she really wants to be a K-pop star. And so she enters in this competition and then she meets this guy and they're really sweet and it's just cute.  This next one was one that I actually read with my book group and we all very much enjoyed it, but again be aware for trigger warnings of non-acceptance within family members. I don't want to spoil it, but I do just want that to be known before going in. But that is the Lesbiana's guide to Catholic school. So here we follow our main character who her father was deported back to Mexico. Her younger brother really gets into a lot of trouble. Like he's been having a really hard time and their mom is having a really hard time supporting the two kids without her husband there. And so some amazing thing happens and she is able to send them both to Catholic school. Her brother is super smart and gets like these great scholarships and our main character has to kind of work her way to keep herself in Catholic school in order to keep an eye on her brother and help him.  They just follow her transition there, her year there, and oh it was so heartwarming and touching and also a tear-jerker and just so much. Like was it a perfect book? Probably not, but I definitely recommend this one for sure. And then finally I'm just gonna, I'm gonna put in there that really anything by Adiva J. Geardar, if you haven't read anything by this author, any of their contemporary rom-coms like their YA, their fun, their queer, they have South Asian main characters who live in Ireland. Like I've enjoyed all the ones that I've read by this author so far so just just gonna throw out there anything by Adiva J. Geardar. Those are all the recommendations I have for you. Thank you very much for watching this video. If you liked it give it a thumbs up. Like I said before make sure you subscribe if you like content like this otherwise I do have bookish social media linked down below. If you want more bookish content also comment down below if you have any recommendations for me of what I should be reading. I am always open for recommendations or just chatting with you in the comments below and until I see you all in the next video I wish you happy reading.