Hello, everyone, welcome back to another AI video. In this one, we're in chat GPT, and we're going to be trying out and using the new CapCut plugin. And the reason why we're doing this is because we can now create one prompt, one click videos that we can upload to social media. We can upload them to YouTube. I'm going to be honest with you, they're not the greatest. But the plugin just came out, and I'm going to show you how to do it step by step. So let's get going. So I'm in chat GPT. I'm just going to go over to the left side, and when you do that, I've got the CapCut plugin installed. There's a good chance you don't have it installed, so let me show you how to do that. All you need to do is just make sure you're on GPT4, click on No Plugins Enabled, go to the Plugin Store, and then just type it in. So I'm looking for a CapCut. So I just typed in CapCut, and presto, there it is. I would just click on Install it right now. There you go. I'm just going to click on Install, reinstall it. There we go. Come back out. You'll see CapCut is installed. The next step, and it's really this simple, I'm just going to type in Create a video about Pearl. I don't know. Let's go with a war. Let's go with a battle harbor in square format, because maybe I want to publish this to Facebook or something like that. Literally, that's it. I'm going to go ahead. I'm going to hit Enter, or I'm going to press the little purple arrow, and now watch this. It's going to go ahead. It's going to say it's using CapCut. If you click on this little dropdown arrow, you'll see exactly what it's doing. It'll even tells you what the content is, so you can sort of see it as it gets created. Crazy stuff.  And then once this is done, we get a link. And then literally, that's the end of it. I've created a video about the attack on Pearl Harbor in a square format. You can watch it by clicking the following link. Let's go ahead and watch the video. Click on that, and here we go. Now, you'll see here that it's loading. It's going to take just a few seconds. I'll come back when it's done. I'll show you what we got, and then I'm going to show you how to make some basic edits while you're there. All right, here we go. So it has loaded up CapCut for us already. You do need to have an account with them. If you don't, it'll ask you to sign up for free. But this is all free. Look at this. I'll hit space bar.  I'll show you how to reduce that volume, and then we'll show you how to make some of these little edits here as well. So I'll hit space bar, show you what we got. The attack on Pearl Harbor was a surprise military. Okay, so not bad. I'll scrub through. You'll see here, it's got another image here, and it's got some pretty relevant images. This is not going to, again, this isn't going to win you an Oscar. Let me tell you that right now. And I'm not sure what that is. That looks like a mistake of some sort. But let me show you. So when I hit space bar, prize military, the audio is too loud. So this is what we got out of the box. Let me show you how to make some edits. So I'm going to go ahead and just sort of pull this up a little bit. Now at the bottom layer here, this is the audio track, and you'll see it's 56 seconds long. It's the exact duration of the video. I'm going to double click on it, and when I do that, I get the option to adjust the volume under basic, and I'm going to pull this down to about minus 14. Now this is totally up to you, but what we're doing is we're reducing the background music audio, which was too loud. So now I'll hit play, listen to this. The attack on Pearl Harbor was a surprise military strike. Okay, and it sort of sounds like we're not at a World War II battle, we're at a ballet class. So the audio isn't quite correct in my opinion. So what are we going to do? Simple, I'm just going to go over here to the left side and you'll see audio on the toolbar here. And then I'm just going to go ahead and maybe pick something that's a little bit more to my liking. So let's listen to this. Okay, maybe not that one. I'm looking for something that's at least 56 seconds long. Okay, maybe not this one here. All right, cool, whatever. It doesn't matter what it is. I'm going to go ahead and delete this track by just clicking the delete button and you'll see it's gone. I'm going to grab this one here, I'm going to left click on it and then I'm going to try that again. I'm going to drag that in and I'm just going to start it right at the beginning. Now listen, we've swapped out the audio track. The attack on Pearl Harbor. Again, it's too loud, so I'm going to go ahead. I'm going to click on the audio track.  So that the voice generation is more, you can hear it a little better. So now hit play. The attack on Pearl Harbor was a surprise military strike by the Imperial Japanese Navy against the... Okay, so it's a little bit better but let's say we don't like maybe the visual. So for example, if I move my play head forward to here, this visual does not look correct. It just says Adobe Illustrator with a bunch of script that I don't understand. So I'm going to go ahead, I'm going to click on that and then I'm going to delete it. And you'll see here that it has gone ahead and closed it for me. So it's gone ahead and removed it and it's gone ahead and ripple deleted. Now I'm going to commands it or controls it. I can also replace it. So if I go to the left side, instead of saying this image, I'm going to go to stock videos and then I can search for something like, I don't know, battle and let's see if we get anything. Okay, not so good battleships is not what I wanted but let's say I like this panda and this is the panda that I want to use.  So I'm going to drop it right there and then I'm going to go ahead and I'm going to make sure that it fits. I'm going to delete any volume, click on done. And now watch this. 103 American lives. The event. Okay, so we've got some weird stuff going on here. This has actually gone ahead and been replaced as well but let's say I want to replace that with this, I don't know, school of fish. I can just drag and drop it right on top, Presto. Can't replace media needs to be shorter than the original material. Oh, okay, sorry about that. It needs to be, so I'm going to take this one, drop it right on there. There we go. See that? I've got a top down shot and basically I'm just going in and replacing footage that I don't like. I've replaced the audio. I've reduced the audio decibel level so that it's not as intrusive and it's not so nasty and then I was at play and listen to this. From interfering with Japan's plan military actions in self. That is literally it. And let's say I want to maybe make sure that it stops everything at the edge here. I'm going to pull this song to the edge. I'm going to delete this piece. I'm going to delete this piece. And then I'm just going to pull this to the edge and literally we've got the entire thing done inside. Now, this is one prompt, one click but these are the techniques you need to use to create videos without even having to really know how to edit video whatsoever. That's all there is to it. Thanks for watching.


two and a half thousand euros how good is that yeah two and a half grand giving away between now and nine o'clock and then after nine we have greg uh giving away that big ten grand mountain whoo lovely all right can i say hi to our grants only that's glenn park in balamacool letter kenny ten today and that comes in from granny let's have a look at your facts of the day today human thigh bones are stronger than concrete there you go fact of the day monopoly board is the most popular board game in the world hey i wonder what it'll be next what would be the next board game i don't know this loads of them isn't it um every acre of american crops harvested contains a hundred pounds of insects yes eat up your breakfast okay in the 19 in the 1980s well the mid 1880s i should say aluminium was more valuable than gold that is fact of the day hello greg hughes here inviting you to join us for a special two-hour show this sunday as we broadcast live from the london irish vintage clubs 10th annual vintage day on the grounds of chacomal gales we've loads of special guests live music and plenty of crack that's this sunday live and highland radio from 12 noon or watch the show on our website youtube or facebook and if you're in london come along and say hello nutritious lamb feed offer directly supported by the arivo fodder support fund is now available at your local homeland store by 10 bags of nutritious intensive lamb or nutritious lamb creep crunch get one free contains intake booster for higher intakes better thrive and faster finish season sees apply contact your local homeland or farm commercial specialist today visit new trace dot ie for full product detail do you know how green skills can benefit you as an employer or as an employee skills to advance has a range of highly subsidized green skills courses for the workplace that will help future proof your job and make irish business greener for more information contact your local education and training board or visit skills to advance dot ie boost skills boost business with skills to advance brought to you by solace and the etbs an initiative of the government of ireland what's your dream job lawyer engineer criminologist or software developer do you want to develop renewable energies create the next great drama or improve people's health anything is possible at ulster university negi campus where you can study in the heart of dairy london dairy clearing places are now available to help you realize your ambitions it's time to say hello future apply now for september 2022 at ulster dot ac dot uk let's have a look at this weather forecast 13 minutes away from that i can tell you about the studio it's heating up we're gonna be giving away money very shortly rather cloudy day some hazy sunshine a few showers in places temperatures well it's 21 degrees anything from 17 to 21 degrees but it's more like 42 degrees in here at the minute nine minutes away from nine o'clock friday oh i love fridays and the burger sings the blues well we're gonna be singing we're gonna be singing from the rooftops very shortly we're joined in studio with shon quince shon good morning to you hi lee how are you this is the first time you've been offices office me uh in the studio and we're giving away money way too early for me lee way too early you were in writing early this morning you're in about half seven can be here a big day today big day and it's it's been mad this morning already um last minute phone calls at buying tickets but wow two and a half thousand and then a chance of winning ten thousand what a friday it's going to be a great friday and it's a good time of the year and a good climate to be winning some money and all of that and we're joining studio with greg greg you're gonna be giving away ten grand very soon very soon as well not what what make you on you which one this one here oh no that's not much better great to be quite honest you'd see that oh the the ah no genie mackerel i just that yeah just go no no you stay there because you've got a very important job to do shon so two and a half grand out this morning right we're gonna do the drawing big thanks to everybody bought tickets absolutely yeah absolutely and everybody's got a chance yeah there's quite a number of tickets in theirs but uh there's going to be very very happy people very shortly we're going to be ringing one person for two and a half thousand their name will go back in the to the drum that's right so they win again yeah and the lines will open again shortly about nine o'clock so if you haven't bought a ticket or you want to buy another ticket you can't shon he has trouble shaking that roller drum thing me digger there tombola give it a good burl there now look there's one trying to sneak out already as we speak well this is all excitement this isn't how many is facebook beast live if you want to check in as well and see all the action live as we do so shon yeah so is reggie i think that's a good burl there now greg oh we are well we get a bit of a drum roll here i think so oh come on take your time take your it's a long drum roll hey not just yet not just yet just keep it going there you can't open it where's that where's arson kelly when you need him where is he let's see let's see can you flick the camera over so we can see are you under pressure oh he got it open he got it open good on you fair play to you right good luck to everybody that bought the ticket that's a big day for everybody it's nice to be in the draw what number take it one two one five do you want to pass that to shon shon and give the good shout out because he's close to the microphone yes so it's one two one five show it up to the camera there can you flick over to shon show him his handsome looks there let's see oh flick the camera over there now so there we go so it's all legit can you flick it there oh it's on is it all good on you oh yeah so it's one two one five and shon tell me this right does the ticket holder know their number do they some of them do you some of them forget some of them don't have a pen oh my goodness entered a few weeks ago so will have forgotten so but good thing is we have all the details we'll be on to them now we'll be making a quick phone call if they answer the phone they may come on and speak to us yeah bro please go on and greg has got 10 grand to give away very very soon we'll be back in a few minutes time foilside at the center of more for your euro with great value gifts so it's worth making the journey wherever you live with convenient parking late night and all your favorite brands you're better off shopping at foilside see foilside.co.uk great means fairs across the transport for Ireland network have been reduced which means traveling with TFI local link will cost up to 20% less visit transportforirelands.ie Fleming doors not flamin doors you know garage doors agri doors insulated doors milking parlor doors Fleming providing engineering solutions since 1860 Fleming 91 48 234 don't just do okay in your career do great at all state don't settle soar don't procrastinate innovate don't do mediocre do magnificent at all state great work great life great people where great opportunities await search all state and i careers today new farm schemes get the green light for more in your farmer's journal his paul mooney EU commission approves Ireland's cap plan we reveal how grain growers get just one cent from a pint of stout fears over fertilizer supply for next spring farmers will foot bill for extra tb testing have we reached pig milk price demand for firewood soars and ram prices increased by 46 euro ahead plus don't miss our focus on autumn animal health inside this week's irish farmers journal on sale now breeding's furniture coulee maville with over 40 years of experience in furniture and floor coverings they've got irish main furniture suites beds mattresses dining flooring slide robes and occasional furniture great prices and large selection now available at breeding's home furnitures coulee maville i'm all over the place all these microphones this morning break my soul beyond see three minutes always kind of coming up to two minutes away from nine o'clock so well we have a winner shan we do have a winner this morning for two and a half grand it's in carry go can you remember the lady's name i'm putting you on the pressure now it's megella gavigan and carry guard you just won yourself two and a half thousand euro congratulations of highland radio and there's going to be 10 grant the thanks greg these are all good at clap and they're very good you know we won't we're so happy you're paying them too much you're paying shan they take it back in the name goes back into drama gavigan carry guard is just one two and a half grand and within the hour possibly or a little bit over it greg he's going to be giving out how much again ten ten grand we'll be in touch with uh with magella to let her know the good news all right and there's greg he's doing a bit of work before the show real work all right shan have a nice day thanks for popping in cheers bye


Goten Education Department promises no glitches for 2019 online applications. Johannesburg, the Goten Education Department will go live with its 2019 online applications system for grade 1 and 8 pupils on Monday. The online portal saw problems in 2016, but the process was a lot smoother last year. The department's Steve Mavinis says applications will close at the end of May. We have enhanced the system. We'll be able to have 30 hits per second, which means parents won't have technical glitches.


Welcome to JSA TV and JSA podcast. I'm Jamie Scott, CEO and founder of JSA. And joining me today is Mr. Paul O'Hara. He's the senior strategic account manager for data centers of Cummins. Paul, welcome to JSA TV. Thank you. It's good to be here. It's great to have you. And for our viewers who may not already know, could you tell us a little bit about Cummins data center business? Yes. Cummins provides backup power to data centers. And so we sell our larger engines coupled to generators. These are diesel engines, mostly, to provide power during power outages or any type of curtailment when the data center needs it. And so it's always on standby. It's always ready. And it's ready to run for up to the limit of the fuel that they have. So usually about 30 hours before refueling. So we usually sell to, in our group, we sell to the larger data center accounts. That is the household name accounts. So there'll be data centers global. And there's teams of people that sell and provide project management for these data centers. And looking over your website, your latest news, you guys have had a banner year. You ended 2021 with high ratings from the Wall Street Journal. You're in the Wall Street Journal's 2021 management top 250, as well as Newsweek's 2022 most responsible companies list, both amazing lists to be on, as well as other sustainability accolades. So can you tell us how to come in sustainability commitments, really roll over to your data center practice? It's kind of seamless. All of our products are made in factories that have sustainable policies and procedures. All of our field crews that go out and service these things are under the same type of sustainable guidelines. And so, and we work with our customers too, as some of these larger customers are pretty far along on sustainable development. And so the call I just got off of was R&D for field cells, putting them running the data centers on hydrogen. So we're working together for long range projects where we can be green and sustainable, environmentally friendly for the years to come. Yeah, and that actually brings me to my next question. Smooth transition, I appreciate that. But of course, we all know Cummins is one of the top global providers of backup power solutions, particularly for the data center industry. But, you know, what are you hearing? What are the new trends that you expect to see in this space, particularly for 2022? More hydrogen? Well, more talk of hydrogen. Hydrogen is still a ways off because we need some infrastructure. We need the products to be developed. We need the costs to come down, and that'll come down with volume. And so right now, it's more the current products, making them less expensive, smaller, cleaner, and then running on some sustainable fuels like the HVO, which is a synthetic diesel that's made from plants. And so we see a lot of data centers as they look at being able to get rid of fossil fuels in the data center and decrease the signature that they have of carbon, going to those types of fuels in their existing equipment as a stop gap until we get to a hydrogen type backup system. This is exciting. I feel like every conversation I'm on these days, I'm sure you do as well. We're talking sustainability. We're talking about what can we do to reduce those carbon emissions. And I know, of course, Cummins being very active in this role. But what can we expect to see from Cummins, particularly your data center practice in 2022 and beyond? What are you particularly focused in on? I guess all of those things that I talked about for our current products becoming less expensive, that is a power density, getting more power out of the existing product. Becoming faster to market, that is, our production cycles need to squeeze down so that we give our customers more time before they have to order equipment. Our costs need to come down. And so it's tough time for reducing costs because everybody has cost pressures nudging the prices upward. And so in spite of that, we're trying to reduce our costs. So these are the times that we need to do that, those things. That's the next 12, 36 months. Yeah, it is. It's a crazy time. We're trying to decrease cost, streamline processes, make it more efficient, effective, and sustainable. That's not an easy order for sure, but certainly Cummins is moving forward and embracing basically this new way of doing business. So where should our viewers go to learn more about Cummins data center arm? Well, Cummins.com and we go to the power systems area of Cummins.com and there's some data center marketing materials there. All of our products can be found there. And yeah, you'll find out all sorts of things about Cummins that you never heard of. Well, thank you so much, Paul. It was a pleasure having you on JSA TV and JSA podcast. And thank you viewers for tuning in to JSA TV. As always, stay safe and happy networking.


What I had to do was work on the basics and the fundamentals. Well, they relied on athleticism and their natural ability. And because I stick to the fundamentals, it just caught up to me. I'd always heard this urban legend of how insanely intense Kobe's individual workouts were. Well, when I found myself on his camp staff, I figured this is my chance. This is my shot. So my earliest opportunity, I walked up to Kobe and asked if I could watch one of his private workouts. He was incredibly gracious. He smiled and he said, sure, man, no problem. I'm going to Mark IV. What a professional courtesy.  I just sat down to watch. And for the first 45 minutes, I was shocked. For the first 45 minutes, I watched the best player in the world do the most basic footwork in offensive moves. Kobe was doing stuff that I had routinely taught to middle school-aged players. And don't get it twisted. This was Kobe Bryant. So he was doing everything with an unparalleled level of intensity and was doing everything with surgical precision. But the stuff he was doing was incredibly basic. His workout went on for a couple more hours. And when it was over once again, I did not say anything to him. I did not say anything to his trainer. I just quietly left. But my curiosity kept nipping away and it eventually overwhelmed me to the point that I had to know. So later that day at camp, I went up to Kobe again and said, I don't understand. You're the best player in the world. Why are you doing such basic drills? And he flashed that million-dollar smile. He gave me a very friendly wink. But he said in a very serious tone, why do you think I'm the best player in the world? Because I never get bored with the basics. I never get bored with the basics. I had a summer where I played basketball when I was like 10 or 11 years old. And a very prominent summer league in Philadelphia called the Son He'll Be. My father played my uncle play and they were like all-time greats and stuff. And here I come playing and I don't score one point the entire summer. Not a free throw, not a nothing, not a lucky shot, not a breakaway layup, zero points. And I remember crying about it and being upset about it. And my father just gave me a hug and said, listen, whether you score zero or score 60, I'm gonna love you no matter what. Wow. That's the most important thing that you can say to a child. Because from there I was like, okay, it gives me all the confidence in the world to fail. Right, and from there I just went to work. All right, what I want to work on first. All right, shooting. All right, let's knock this out. Let's focus on this half a year, six months, nothing to shoot. All right, after that, all right, creating your own shot when you focus. So I started creating a menu of things. When I came back the next summer, I was a little bit better. And then the next summer I came back I was a little better. And the summer came back, the next summer I was a little better. I scored. It wasn't much, but I scored. And this is 12, 13. 12, 13. And then 14 came around, back half at 13, 14 years old. And then I was just killing everyone. And it happened in two years. And I wasn't expecting it to happen in two years, but it did because what I had to do was work on the basics and the fundamentals. Well, they relied on their athleticism and their natural ability. And because I stick to the fundamentals, it just caught up to them. And then my body, my knees stopped hurting, I grew into my frame. And then your athleticism, once you have the fundamentals, the hard work, the mindset, and you tack on the athleticism, it's game over. Then it was game over. Kobe Bryant, the best player on the planet, and someone that has truly mastered his craft, said that his entire secret is that he never got bored with the basics. And as obvious as that may be to you all right now, that was a life-changing moment for me. See, in that moment, I finally realized that just because something is basic, it doesn't mean that it's easy. Those are not synonyms, and yet people mistakenly use those words as if they're interchangeable. Just because it's basic, it doesn't mean it's easy. If it was easy, everyone else would be doing it. And you all know that we live in a world that tells us unconsciously that it's okay to skip steps. Tells us we should always be looking for a shortcut or a hack. Tells us we should constantly be chasing what's new and what's shiny and what's flashy and what's sexy. But I'm here to tell you when you do those things, you are making a huge mistake. And that's because of the basics work. They always have and they always will. Extraordinary accomplishments come from doing ordinary things for extraordinary periods of time. But when I was younger, my action showed that I was more impatient. I thought there was some shortcut I didn't know about. I would see the guys on stages being like, this is what we do, we do the basics, we do a lot of it. I was like, they're just trying to hold the secrets back. And I'm here to tell you that that is not true. Working out and doing reps is not extraordinary. If you saw one set of a workout from me at the gym, you wouldn't think, oh, this guy is amazing. It's the fact that you do it for 20 straight years. Nothing about that is extraordinary. It's the dedication to doing it over and over and over and over again, despite feeling like there's better opportunities, despite people approaching you with new and exciting things, but saying that you're committed to the original vision you had and that, not the action itself, but your commitment to the action is what makes someone extraordinary, not the thing itself. And to me, that made being extraordinary possible.


Hi, I'm Rob Cussner, the candidate for Amherst Town Council, seeking one of your three at-large votes. We're about to embark on a new experiment in governing the town, so it's important to elect counselors who have experience serving the people of Amherst, who work constructively and persistently to accomplish long-term goals, but who also can think critically and independently about the future of our community, both within the borders of Amherst and beyond. If you've walked or cycled an erotic rail trail recently, you've seen a good example of my efforts and accomplishments in this direction. Back in 2006, then-Congressman John Olver called upon me to revitalize the advisory committee that would oversee the redesign and reconstruction of this 12-mile-long conservation, recreation and transportation corridor, serving Amherst, Hadley, and Northampton. I was unanimously elected chair by representatives of these three communities, and then over the next decade led the committee through a complex regulatory and design process, negotiating consensus among multiple conflicting interests, listening carefully to, and working with public and professional individuals and officials from throughout the Commonwealth. After several years of monthly meetings, we arrived at a design which we and the public were very happy, and then we worked to ensure its implementation. I hope you enjoy the results next time you're out on the trail. I've also served on the Amherst Select Board between 2005 and 2008 on the Conservation Commission between 2000 and 2003, and as chair of Amherst's Public Transportation BISEL Committee. On the Select Board, I take some pride in helping to negotiate the agreement between the town of Amherst and the UMass police department to cooperate on coverage of the town and the university, which saved the community a great deal of resources. I helped with former town manager and then finance director John Musanti to turn around some of the problems with the Health Claims Trust Fund that were present at the time I took office, and I led efforts while on the board to acquire farmland and conservation land for the town, among many other things. During my time on the Conservation Commission that was during the era of Pete Westover, I worked on both the implementation of various trail projects, including the Swift Way, what's now called the Swift Way, and that was done with Arthur Swift and Pete over many years of discussion and planning and design, and also the routine work of enforcing the wetlands laws of the town and the Commonwealth. On the Transportation and BISEL Committee, I served with Arthur Swift for a number of years and then chaired the Transportation Committee during the last couple of years, and during that time we essentially designed the present bus system and the bicycle infrastructure of the town. I've been a town meeting member since the early 1990s. In fact, my maiden speeches to town meeting concerned both the public schools where I was advocating for universal preschool and also for world language in the elementary schools and also petitioning for bus service. One of those worked out pretty well, the other one didn't. Professional I'm a mathematician who's conducted research worldwide and observed how people live and organize their societies. I've been on the faculty at UMass since 1988, raised a family, my son attended the preschools, elementary schools, middle schools, and high schools here at Amherst. If elected, I predged to bring my long experience, my creative skills, my intimate knowledge of the town, and my deep passion for Amherst to serve you as a counselor. Thanks for your attention. Thanks to Amherst Media for this forum, and thanks to all of you for coming out to vote on September 4th and November 6th, and please stay active in governing our community. Thank you.


Hi and welcome to week three of Off the Hill at the ANU. The second week of the election campaign and I think for me personally it's the week where I realised just how long this election campaign is going to be. Today we're going to talk about immigration, we're going to talk about polling and what we need to know and then we're going to talk about the campaign narratives and how this sort of seems to be playing out for the two major parties. So let's get started. Ryan. Jill, you're the expert on polling. We've been seeing on the ABC for example this week a lot of stories, news stories about their vote compass feature which is this device that lets you figure out which parties policies you're closest to, which just seems fair enough, but there's then been these news stories about what that polling, as they call it, means about what Australia thinks and what Australia values. Should we be paying attention to that sort of polling and really I guess in general what should we be wary of in polling, what should we be looking out for? Okay, it's a big question, right? My short answer with regard to vote compass is no, it's not a news story, right? Things like vote compass are designed as voting advice applications so that people who don't really know where they sit can log in all their information about how they stand on various issues and it will spit you out an answer, saying where you are closest to a party. The problem with things like vote compass is obviously I think we've all done it probably, our colleagues have done it, but regular voters don't do this. They just don't log on to the ABC website and feel like answering a self-administered poll. So while they wait to demographics and we try to make those samples look as much like the population as possible, what they can't account for is the lack of political engagement, right, that most people have and you know, this is your job is to reach out to people to make them more engaged. When we see opinion polling generally, that doesn't have that problem because you're actually reaching out to people and asking them whether they want to be asked or not basically. The thing to look out for with opinion polling is that we're seeing this sort of convergence on 50-50, right, that they've been up and down and up and down and we're seeing a sort of reversion to the main, you might call it. And it's another way of describing it is a homing tendency that, you know, we kind of flip with the idea of changing parties between elections, but really what we do is we come back to our home party. And that's what we're seeing now, is we're getting closer to 50-50 and so that's what we'll see at the election. The bookies have it not so close, though, I mean, the sports betting agencies seem to think that the coalition are favourites. Are they wrong? Are they better at football than they are at politics? The bookies tend to do pretty well on this, right? I mean, they've got money at stake, so they've got sort of skin in the game. What the bookies tend to take into account is, well, first of all, money, money that gets put in, and also that there's a seat by seat game, you know, and that 50-50 opinion polls don't mean that that's going to be distributed evenly across the population. You know, that moves up and down in electorates, which is all kind of stuff that we know. But, you know, in terms of Vote Compass, what are your thoughts as a marketer? Yeah, no, not very good at all, because as a marketer, I'm looking for engagement. I don't know that you're awake and listening to what I'm talking about, and that I'm actually engaging with you directly and I get some sort of feedback from the market on that. So Vote Compass doesn't do that. I mean, the gauge is where your interest might be, but at the same time, that's just where you are at that point in time. How do you think of that point in time? It can change really quickly and, of course, it will change quickly because of a policy announcement. If you make a certain policy announcement, you go, oh, yeah, now, hang on, that really gets to me. Even though these other things are important to me, I know I have to still live life. Yep, absolutely. So, you know, a good example, that's the child care policy. You stay, anyone with kids who has to put their kids in child care, that's probably usually up there as rating number one on the issues. Yeah, absolutely. Yeah, and the other thing I'll be looking at too, which doesn't get picked up a lot by the surveys, which as a marketer, I find distressing, but I know the parties do it, is recall of information. So, if I'm doing all this big messaging campaign, I'm doing the advertising campaign, I'm getting the leaders out there every day of the week. What are people talking about really? Yep. What are they really recalling about what I've just talked about that week? So, on that sort of, you know, more substantive issues, right? We've seen Peter Dutton this week say, I think what we would all think of were fairly uninformed, maybe ill-advised comments on humanitarian intakes and what refugees contribute to society. What are your thoughts on that, Ryan? Well, the comments were offensive, and I think offensive to a much broader group of Australians than Peter Dutton might have expected. And I just can't help but wonder, as some people have been speculating this week, whether this was what's known as a dead cat strategy, which Linton Crosby and the UK Tories perfected, whereby when the campaign is in a way talking about things you would prefer it wasn't talking about, where it wasn't going as you'd like, you throw a metaphorical dead cat on the table. And all of a sudden, the dead cat may be unpleasant, it may not be pretty, but everyone's talking about the dead cat. And we saw this morning, Malcolm Turnbull had a Fairfax op-ed all about border security. This was not what Malcolm Turnbull was talking about a week ago. And so the hope must be that the debate has changed. Absolutely. And I think here we need to remember that it's, we don't always represent the population at large, right? I mean, there is a genuine concern among the community that immigrants, you know, may take jobs from Australian-born, you know, workers, to the extent that this is a genuine threat is, you know, I think we might say questionable, but this is a legitimate concern that voters have. And I think we really need to at least acknowledge that, right?  Yeah, but let's have a debate that's not involved in vilifying people and denigrating people. We can have a debate about migration. We can have a debate about asylum seeking numbers. And maybe we should have those debates. But the way that Peter Dutton is doing it is just a terrible way for the country. So we've found, we've found when we've asked people in the ANU poll, which I author on behalf of the ANU, that Australians are really good at distinguishing between refugees and the sort of regular migrant intake, right? What Australian voters really don't like is the idea of regular arrivals. So say, last year we asked an 83% of Australians believe immigrants make positive contributions to the economic and cultural life of the country, but then 65% support tougher measures to exclude illegal immigrants. So there is a disconnect there. But people are pretty good at passing out the differences. Yeah, and I think this is like, this is a really important part of messaging to during a campaign. You're speaking, you might be speaking to everyone, but everyone is made up of distinct groups and identities and segments and everything else in between, right? So when you speak, you can actually end up upsetting people you didn't intend to upset. But also, sorry, messaging off of TAC all together. Like for example, Turnbull wants to be known as progressive. How can you be progressive when one of your ministers speaking like he's from the 1950s? Well, because it's jobs and growth. So this is the narrative, right? Yeah, exactly. But at the same time, like, getting back to what Ryan was saying, if you throw the dead cat in front of people, for sure they'll look. But also, that could be horrified run the other direction back to where they think, oh, hang on, they're not talking like that. They're talking like, you know, a lot better talk than what I'd like to hear. I'll go back to them instead. So I think that's been picked up a bit in the polls, too, that Labor really at two party preferred, they've just edged ahead by one or two points and some of the polling out this week. I think that's picked up on that sort of thing, too. I know it might drift back again during the campaign. It is a long campaign. It's a long campaign. Long campaign. Labor want to talk about Medicare, the coalition want to talk about jobs and growth. That's it. What's what's going to, who's going to win out? Yeah, look, and this is where it comes back to maybe where people go to vote compass for is identify, you know, where people are at right now between the social investment and the financial investment. Who do you go for? Do you go for somebody who's talking about economic leadership? So all the polling picks up on the economy being really important. And therefore that's a coalition strategy now based on that importance. But does it really reveal that importance? The polling here can be playing tricks with the parties. Right? Is it really that important preference? Why? So as a marketer, I know there's certain things where we call them decision rules. So it could be one up the very top here, which dominates everything. So for example, buying a car has to be fuel efficient. You look at the fuel efficient cars, right? So whatever it happens to be. But then next one down could be something else the stereo system, not so important as fuel efficiency. So if you find a car, great stereo system, bad in the economy, you'll go for the one always going in the economy. And it's all about the economy. So where does that leave Medicare for the ALP? Well, look, I think on Thursday and probably into Friday this week, we've seen a debate where Bill Shorten is saying, I'll save Medicare. I'll save Medicare. And Malcolm Turnbull is saying, I'll cut the taxes for certain companies in Australia. And I think Labor will be happy if that's where the debate's happening. I think the Labor is very proud of its record on Medicare. Thinks that that will matter to voters. And I think they'll be happy if that's where the debate is. Final thoughts for the weeks? Well, look, I want to say this to the people out there. Number one, get out and make sure you enroll to vote. The polls, they're all sorry, close very soon. Make sure you enroll to vote. I think they close actually next week. So make sure you're on the list. It's your voice, your vote, your voice, your vote counts. So yeah, get out there do it. The political scientist says that, you know, it's never that rational to vote.  You know, you get to be part of the festival of the election. There are lots of close seats. Everyone should enroll to vote. Don't listen to the political scientist. No, definitely enroll to vote. My final thought is this. It's been about 18 months now since the Liberal National Party room said that there would be a plebiscite on marriage equality. Labor's promising that it'll be legislated in the first 100 days. There are still almost zero details from the coalition on what that plebiscite will look like. And that's going to matter to a lot of young people in some crucial seats of this election. My final thought is that I've been in the US for a week and we should be very, very thankful for our democratic system, the way our elections are set up, the kind of candidates we get. You know, sometimes this can all seem like a bit of a drudge. You know, it can seem like we don't get the best choices. Good grief will be lucky. So thanks, you all. Thank you all for listening and for watching.  Yes, I am. We're very lucky in Australia. And we'll see you all next week for week four of Off the Hill and week three of the campaign.


Okay, so let's start. Last time I wrote an exercise and I wrote, so this exercise, you have a covering, and you have a mapping from X, continuous mapping. And if you prescribe one point, if you prescribe one point for this mapping, okay? You will see with F tilde of X0 equal E0, okay? And then we want to show that this is unique. And it seems that I wrote B connected, okay? That's, but we need X connected, yeah? So, F tilde is unique. So it's not B, it's X. If X is connected, okay? And probably I wrote B, that's X, we need X is connected, okay? So the idea is the usual one for connected spaces. You assume you have two liftings, okay? Two different, well, no, two liftings. And then you take all points in X where they agree, okay? These two liftings. And there's one point where they agree because we describe in some point here, we say what we want. So this set is non-empty. And then what one proves is that this set is where they agree, the two mappings, is connected, is open. And also the complement, or it's closed. Also the complement is open, and it's non-empty. So if X is connected, it's a whole X, okay? So they agree everywhere. That's the structure of the proof. And so we need X connected for this, okay? I missed, I wrote B connected, but we need X is connected, okay? But the idea is this, okay? Take all points where they agree. This is non-empty, open. The complement is also open. X connected implies it's the whole space. So they agree everywhere. So they are equal, the lift-to-liftings. So this was just the correction to this exercise. Now I prove a fundamental group of some surfaces were too many folds. But a little bit more general. This is surface. We start more general, so here's a theorem. We want the two sphere. We have S1, okay? The circle, fundamental group are the integers. And we gave two applications. One is the Brouwer fixed point theorem. And the other is the fundamental theorem of algebra, okay? Now you want the other spheres, S2, SN. So this is a special case of Fonkampum's theorem. I will not prove the general case, but Fonkampum's. It's very known under the name of Fonkampum's theorem. So if X is the union, A union B, AB open, the intersection pass-connected. That's important. So we have X is the union of two open sets. And the intersection is pass-connected. And we choose the base point in the intersection, X0. So then this special case says if pi one of A, X0, this is our base point. And also pi one of B, this is base point, are trivial, trivial groups. Then also pi one of X, X0 is trivial. This is a special case of Fonkampum's theorem. So the general Fonkampum's theorem describes the whole fundamental group, also in non-proven cases, okay? So proof, so what we have to prove is that every path, so let W be an element of the fundamental group of X with base point X0. So W is a path, no? W from zero one to X is a path with a closed path with initial point and final point. It's a base point X0. We have to prove that W is homotopic to the constant path, okay? We have to prove W is homotopic, path homotopic to the constant path. So this is constant path in X0. That means the fundamental group has only one element, which is the trivial element, class of the constant path, okay? So it's a trivial group. Okay, so it's again the lemma, by the way, if W, the just a remark here, if W from I to S to, no, sorry. That comes later, that remark. I was already in an application of this theorem, okay? We will apply them to SN, to the M sphere. It's homotopic to constant path. So the Lebesgue number lemma. Applied to the open covering. So we have an open covering of X, which is AB, so we take pre-image, W minus one of A, W minus one of B. This is the open covering of the interval I, zero one. This is a compact metric space. So there's a Lebesgue number. Associated to this open covering, that means every subset of diameter smaller than the Lebesgue number is contained either here or here. That means the image is contained either in A or in B, okay? That we applied already for the homotopy lifting lemma, for the path lifting lemma, then the homotopy lifting lemma. Okay, this implies, so we subdivide the interval. Into small intervals, and such as the diameter of small interval is smaller than the Lebesgue number lemma. So this is subdivision. So this is T zero equal to zero, smaller than T one. Smaller than, smaller than T L minus one. Strange. Smaller than T L, the last one, which is one. There's a subdivision of the interval I, such that W, of each of these small subintervals, let's call them T I, T I plus one, sorry, interval, closed interval, T I, T I plus one. It's contained in A or in B. It's contained in A or B. Either it's here or it's here, okay? For I, for all I. I equal from, what is the first interval? Where are all the, from zero to L minus one, no? Zero, alternately in A and B. Alternatingly, if one is in A, the next one is in B. Otherwise we can forget the point in the middle, okay? Alternatingly and B, so alternating E, alternating E means if one is in A, then the next one is in B, then the next one is in A. If two are in A, you just forget the middle point, okay? Then we have a larger interval and we are in A. So this is okay. So we can say alternating in A and B. That means that the end points are all, and the initial points of this are all in the intersection, okay? If one is in A, the next one is in B. So this implies that VI of zero and VI of WI, oh, sorry, it's not zero one, that's a problem here, okay? Because that means that W of TI and W of TI plus one are in A intersection B. The initial and final points are all in the intersection. X zero is in the intersection, it starts in the intersection and then it goes on to the intersection and then it starts, okay? Maybe I'll make this picture look like this. So our picture looks like this. This is A, this is B. Here's the base point in the intersection which is path connected, by the way, that's important. So we start, maybe in A. Then the next is in B, okay? So the end point of the first one is in the intersection. Then comes the next one. The end point is again in the intersection. Then comes the next one in the next one, okay? And finally, we come back to X zero. Here are three pieces, okay? Divide it in three. So we have a problem with the product a little bit, but so I have to, the product of these is not defined in our notation, okay? Because our products are defined on the, it's zero one, okay, of passes. This is not zero one, okay? So we reparameterize linearly this W on this to zero one, okay, each of these small. So let W, I, W restricted to TI, TI plus one, okay? So this is our original, this is this small little piece here, but we want zero one, reparameterized to zero one, okay? Reparameterized to zero one in a linear way, but it doesn't matter, okay? So zero goes to this, one goes to this, and then in a linear way, you're interval. So that means WI now goes from the interval zero one, okay? And then, so this is W one, no? This is W two, and this is W three. Here are just three of these, okay? Now, then, now the product is well defined, okay? W is homotopic, not equal, of course. It depends on the parametrization. So W, now we have the product, it's well defined of all these because we have to find the product only for this interval, not for our particular interval. We could define it also for others, but it's more or less the same. So W one, no W, where does it start? The first one is, sorry, maybe I missed something. WI goes from TI, so we start from T zero, unfortunately. So this is not W one, but W zero, no? This is W one, and this is W two, okay? This is our notation. I should, because we start with T zero and not with T one, okay? So now the product is defined. W one times, times W, how many do we have? L minus one, no? L minus one, yes, L minus one. W is not equal to this. This depends on the parametrization also, okay? On the product, in which order, which is associative. Because this is just a reparametrization of W, okay? This is a reparametrization of W. It's the same path, the image is the same, but we go with different velocities, okay? This is a reparametrization of W, so they are homotopic, okay? This is, if you reparametize, the paths are all homotopic. So they are homotopic, the paths are homotopic. This was a homotopic. This was an exercise, okay? If you reparametrizize the paths, then the homotopic class does not change. Okay, so what? Now we have all these small paths. This is in A, this is in B, this, or, let's start in A here, okay? Doesn't matter, A, B. A, this is in B, this is in A, and so on, no? B, A, B. The fundamental group of A and B are trivial, okay? However, these are not close paths for the moment, okay? That's the problem. So we make them close, okay? That we can do since A intersection B is path-connected. That's the important hypothesis here. It's path-connected, okay? That's, otherwise it's not true. So what do we do? We go with W, zero, okay? Then we are not in the initial point, unfortunately. So we choose a path which goes from X zero to this end point and we go back. From here we go back to X zero. Then for the next one, we go back this path, then we take W one. Then we go back to the X zero, then we go back to this again, and then we take the last one in this example, okay? So let whatever is called, doesn't matter. Let K I from I to A intersection B be a path from X zero to, now we want the end point, okay? So W I the end point of one. So this is W zero of one, no? This is W one of one, the end point of these. And here we are close anyway, okay? So we have to do this from I from zero to L minus two is sufficient. Because L minus one, the last one, finishes anywhere in the right point, okay? So L minus two is sufficient. So I will not make these passes here because otherwise we cannot see anything. Okay, and then of course I may write W is still on the topic to what? So it's W zero. So what you said is, W zero, we close it, okay? So we close it, how is it called? It's K zero minus one, okay? K zero minus one, we go back. And then we take K zero, okay? And then we take W, the next one is W one. Then we go back with K one to the minus one. Then we go back, and so, so where does it finish? The last pass is W L minus one. This goes back to the right point anyway. And this before, what is it? K L minus two, all right. And why is this homotopic here? Because this here is homotopic to constant, okay? And the constant, so we use our lemma, okay? So this is homotopic to a constant pass. This is homotopic to a constant pass, and so on. So this doesn't matter, I can forget this, okay? This is like, it's not a close pass. But if you take any pass with a constant pass, then it's homotopic to this pass, okay? So we can cancel this, and then we are left with this, okay? This is the lemma, we prove the lemma. Two lemma, lemmas, no name, associativity first, no. We use lemma one, two, and three, okay? When we introduce fundamental group, one, two, three. This is associativity, this is neutral element, okay? And this is inverse. So you look at this lemma, and now, of course, associativity, we may do it in a different way, okay? So I should use colored chalk, which I don't have, but it doesn't matter, so now I make this, of course, this kind of, this, and the last one is, okay? And now all these passes, here, here, here, and so on, these are closed passes, okay? So these are all closed passes. In A or in B, okay? First one in A, next one in B, and so on and so on. They are all closed passes. But the fundamental group of A and B is trivial, okay? With base point x zero. So they are all homotopic to a constant pass. They are all homotopic to C x zero.  So all homotopic to x zero. So I may write C x zero, C x zero, C x zero, C x zero, okay? So w is homotopic to C x zero, which means that w is a trivial element, okay? Of the fundamental group. So w is equal to C, that's the only element of the group. So this means that the fundamental group is trivial, also of x. This is a proof. Again, the Lebesgue number lemma is the main point. Application, that's the end of the proof. Application, so theorem. For n bigger than one, the fundamental group of SN with some base point x zero is trivial. So recall what is SN? SN is the n-sphere, unit n-sphere. So these are all points x in Rn plus one, such that the norm of x is equal to one. This is the n-sphere. So this is the subspace of Rn plus one. That's the topology, of course. And this is the unit n-sphere, okay? So this was, okay, so proof. So we divide SN into, let SN plus be all points in xn. No, let it just be SN minus the south pole. What is the south pole? Zero, zero minus one. This is south pole. And SN minus be equal SN minus the north pole, zero, one. Okay? Then SN plus is homeomorphic. So this is homeomorphism to what? SN minus the south pole is homeomorphic to what? N, yes, Rn. And SN minus is homeomorphic also to. What is the mapping? Famous mapping? Yes, stereographic projection. From the south pole or from the north pole? And this of course implies that a fundamental group is trivial, okay? So this implies that pi one of SN plus with some base point x zero and pi one of SN minus base point x zero are trivial. These are trivial groups, okay? SN plus intersection SN minus is pass-connected. This is homeomorph to what? So this is equal to SN minus zero both. Plus minus one, no? So this is homeomorphic to Rn minus one point, zero. Which corresponds to Rn minus zero. So it's Rn minus zero, which is pass-connected. Why? Since N is bigger than one. Of course R1, the real minus zero is not pass-connected, okay? It's not connected even, okay? But R2 minus zero, that's pass-connected. So since N is bigger than one. Not for an equal one. This is, but otherwise we prove that the fundamental group of S1 is trivial, okay? That's not the case, that's the integer. So here you see that's the hypothesis that intersection is pass-connected. It's important in this theorem, otherwise not true, okay? Otherwise we prove something also for N equal one. That's the point here, okay? It's pass-connected. So the theorem then implies, Fankampian theorem then implies pi one of the union with x zero is trivial. Okay, that's all. So that's the application of the calculus theorem. Now I can, if I want, we have this remark, okay? For remark, S2, pi one of S2. Let's take the two sphere, which we see, okay, in three space. With some base point x zero is trivial. That's a surface, that's a two-manifold. That's why I write, that's a surface, S2, okay? That's a two-manifold. This is trivial. So I wanted to make this remark, take any pass here, okay? So W is again from I to S2, okay? If surface initial point and endpoint or base point, wherever it is, okay? If W is not subjective, then it's easy that W is homotopic to a constant map. If W is not subjective, that seems strange, no? W from the interval to S2. If W is not subjective, then it is easy to see that W is homotopic to a constant map. Why? So here you have, this is a two-sphere, no? S2, here's some base point x zero. So it's not subjective, okay, W. By homeomorphism, there's some point not in the image, okay? And then we can assume that it's the antipodal point, doesn't matter too much, okay? You just take a homeomorphism and you can assume that it's not so important. I mean, some point is not in the image, okay? And we assume that this point is the antipodal point of x zero, that minus x zero is not element in W or phi. It's not in the image, okay? The antipodal point, which is this one. And now we have a path, so they avoid this path, W, okay? So we have a path here, which is very complicated, but it never comes to this, okay, to this point. And now what we do? We don't have straight line homotopy here, okay? But now we project W along great circles back to the constant path, okay? That's like straight line homotopy, but on the two-sphere. These are not straight lines, but these are great circles, okay? Now project W back to x zero to the point. Contract, contract, contract W to x zero along great circles, okay? Along great circles as two. So here's a point, maybe here's a point, W here, some point, so here's a point of W, so I'm projecting along this, okay? I have to parameterize from zero to one. At zero, everything remains at one half, I go one half at one, I go all the way back to x zero. This is like straight line homotopy, but on S2, which are not really, what is this straight line on S2? This is like straight line homotopy, but now along great circles. We don't have segments, okay? Like we have in RN or something, okay? Along straight line, but along great circles. But it's very similar, no? I hope you see that, okay? In RN, you go along the segment back. Here you go along the circle back, okay? So this implies that W is homotopic to the constant path. This is homotopic, you see the homotopy, okay? You see it, you go, this is missing, and so you shift everything to x zero, to the point x zero, the whole two sphere, okay? Between terms zero and one. So this is a nice example, okay? So you say, but now it's not surjective, okay? Then we can do this, okay? Unfortunately, passes are very complicated. So there exists surjective passes. This would be, otherwise, a very easy argument, okay? Unfortunately, there are surjective passes from I to S2. But, so this proof, I mean we proved it, okay? But Fankampian theory, which is not so trivial as this, but there are surjective passes. That's something, seems strange, or no? W from I to S2. Because this seems one-dimensional, no? And this seems two-dimensional, so how do you can get everything? But there are these strange phenomena in mathematics always, okay? So the first famous name as well, they are surjective passes from I to the square in R2, okay? There's a name associated with these kinds of passes. Somebody already saw this name. What? Piano curves, yeah. Type piano curves. He constructed such strange surjective mappings from the interval to the square. And then you can go to S2, that's no problem, okay? So this is a remark, okay? In some cases, it's clear if it's not surjective. But what is bad are these bad paths, okay? So then you have to work in some way, okay? And that's, you make it non-surjective, maybe, but then you have, it's technical, okay? Anyway, we proved this in general, using the back number number, okay? So other examples of surfaces. So we should, we take Rn minus zero now, example, other example. Rn minus zero is homeomorphic to Rn minus zero. It's homeomorphic to Sn minus one, which is subspace of Rn minus zero, times the real numbers which are bigger than zero, okay? The positive real numbers. What is homeomorphism? So you take a point X here, and then, what is this point? You divide just by the norm, and then you take the norm, okay? So these are polar coordinates, kind of stuff. So X norm, okay? That's just what you do. This is the subspace of Rn, okay? Sn minus one. So this is homeomorphism.  And now there's a general proposition. If you have a product, the fundamental group is easy. If you know the fundamental group, so what? For proposition, the fundamental group, pi one of a product X times Y, con base points X zero times Y zero, it's isomorphic to the pi one of X, X zero, times pi one of Y, Y zero. So this is nice to say, the fundamental group of a product is a product of the fundamental groups, okay? The fundamental group of a product. It's almost nice to write informally. The fundamental group of a product is a product. Well, finite product here, okay? And if it's infinite, you have to think a little bit. Then you have a product, but it's not the product of the fundamental groups, but probably the, well, whatever, two. Of a product of two spaces is the product of the fundamental groups, or finitely many. Otherwise, there's a difference between, if you have a building group, for example, with product and sum, okay? The product is large, the sum is not large, okay? So one has to be careful with this kind of thing. But that's not our, you have two spaces, there's no problem. So I will not, the proof is, this proof is boring, because you just define the natural maps, okay? So we define, there's no, nothing, the quantum is interesting, okay? The back number, lemma you use, and so on, this is, so you define a mapping from here to the product, pi one x, x zero times pi one y, y zero, and you do it in the most natural way. So you have w here, okay? And then you have the two projections, okay? Pi one, pi one is not so good, because we have too many pi ones. P one and P two, P one from x cross y to x, and P two from x cross y to y, the projections, okay? First factor, second factor. And then what you do, we just project our paths, we have our paths, we have w, and then we project it to x. And we take the path, the class of this, this is here. We just project it, and we project w also to y, so P two. So we have this element, okay? So, what we have to prove, well, everything is well defined, maybe, okay? It's well defined, and we're not proof. I mean, if w is homotopic to w prime, then w, P one, w is homotopic to P one, w prime. And you take the homotopy, P, homotopy, okay? You compose everything with P, just so. It's well defined, homomorphism of groups. Well, if you have a product here, w one, w prime, then what you may first take the product, and then project, or first project, and take product, then the result is exactly the same, okay? The paths are the same, okay? Even the paths is not the homotopic, the paths are the same, okay? First the product, then projecting, or first projecting, then taking the product. The result is exactly the same path. This means this is a homomorphism of groups. It's injective. So what is the inverse? Then we have the inverse, of course. Inverse homomorphism, that's maybe the easiest. Inverse, isomorphism. Phi, phi, inverse isomorphism of phi is psi from pi one x, x zero, times pi one y, y zero, two, we're finishing the blackboard, x times y, x zero times y zero, okay, so what you do is you take w one here, and you have w two here, and what you take? You take just the product of the passes now, okay? W one times, in this sense a product, okay? Not the product of passes, in this sense a product. So w one times w two of t of a point is of course w one of t times w two of t, okay? So if you go by this and then you project, then you get back w one and w two, okay? And also the other way around. First this and phi is the identity. First phi, then this is the identity, okay? That's the whole proof. So there's no idea, it's just formal, okay? However, this means then some corollaries. Pi one of r two minus zero. So it's isomorphic to pi one of s one times pi one of r positive. I forgot the base points. I have a problem with base points now, because when I take x zero and s one anyway here, okay? It takes the same x zero, and here I don't know. What group is this? What is this? So this is, yeah, and this? The positive is trivial, no? Straight line homotopy, trivial group. Well, I don't write zero. I don't know what zero is, okay? Trivial group, whatever is, okay? This is again straight line homotopy. So this is pi one of s one x zero, okay? This remains, so this is that. And I mean you project onto s one, right? So the isomorphic is again given by rotation number, okay?  Which doesn't change. So isomorphism is given again by the rotation number. You define it in this way, okay? That's it, and then you call it rotation number. Around zero here, okay?   And that you get by lifting to the reals, and the path from zero to some integer, and this integer is the rotation number, okay? That's the isomorph. Okay, so this, this is also surface, of course, but non-compact. The other fundamental groups are trivial, no? I don't write. If you take s n minus zero, okay? R n minus zero, then it's trivial, okay? The fundamental group. The normal corollary of another surface is s one cross s one. This has a name, the surface. Yeah, the torus. Except it's not so clear where is this living, this torus, okay? If you have it in this way, we have s one cross s one is contained in c times c, or r two times r two in this version, no? c times c, which is, if you want r four, okay? It's a subspace of this, okay? This is one version. But of course, you can do it in r three, okay? But not, this is a very canonical way, okay? And then you have this in r three, and you make this picture. This is one s one, the first. And the second one is this. This is another s one. And if you take the product of this s one, with the second s one, you get exactly the torus, right? So this is not equal maybe, but it's homeomorphic to s one cross s one, okay? But this is living in r three, okay? Formally, it's living in r four. Did you do differential geometry? Not yet. Second semester, yeah. Because this is interesting torus. There are many torus, okay? And then you have all types of curvatures, okay? Flying around. All types of curvatures that we will do. Toe is the main example, okay? In differential geometry. S two is the first one, and the second one is the torus, okay? At the surface. And then you have all kinds of curvatures, okay? Positive, negative, where you have this kind of points and zero, okay? How are they called? Yeah, curvatures, okay. Whatever it is. This torus in r four is a very special torus, because this, what is curvature here? Well, whatever it is, this is a s curvature zero, okay? It's a flat torus. This is a flat torus. This is not flat. Here's curvature, okay? So, seen from point of view of differential geometry, this torus in r three, and this torus in r four are very different, okay? From differential geometry. This torus in r four has curvature zero, okay? It's a flat torus. Flat means curvature zero. This torus in r three has all, it's not possible with curvature zero, okay? You always have some curvatures, okay? But this is differential geometry. Anyway, that you will see. The most interesting torus is maybe this one. It's a flat torus, okay? It has no curvature, this one. Whatever two curvatures. It's locally like r two in r four, okay? It looks locally exactly like r two. Where? What? Well, you want to make this identification. Well, anyway, pi one of s one cross s one, whatever, one times one, I can write now it's also morphic to pi one of s one, one times pi one of s one. One, so this is isomorphic to z times z. So this is written as z squared. That's the fundamental group of the torus. And you see the isomorphism. Because you project a path first on, if you have a path here, first you project onto this s one, and see how many times do I run around this.  And this is, you have to get two integer, two rotation numbers now, one around this, and one around this with the two projections, okay? From s one to cross s one to the first s one, and then to the second s one, okay? Two rotation numbers, and this is z two. That's the isomorphism. That's okay? More or less. Well, the projection is defined, no? From s one to the first, and then you have a path, and then you project this path. And then you are running just around this s one, no? And you count how many times do I run, okay? And then you do the same here. So you get two integers. And these are the two rotation numbers, okay? And this is the isomorphism. So corollary, so this is the corollary. It's a fundamental look at this, okay? And so you have another corollary, that they are not homeomorphic, no? And this is completely clear to everybody, but s two and the torus, s one cross s one. So as a surface, of course, we want to be in r three. That's our space where we see what is a torus, so what? So we make the picture here. In r four, you have to imagine how it looks in r four. Anatomyomorphic. Why not? The fundamental group of s two is trivial. And the fundamental group of s one is z two. So they have different fundamental groups, okay? Of course they are path connected, so it doesn't depend even on the base point. So this is a proof that they are not homeomorphic. S two and the torus, okay? If you want to do that in a direct way, it gets somewhat not so easy, okay? The fundamental group is okay.  That's what they call, no? And s, the torus, what is this? The orientable surface of genus one. And then maybe you see what is the orientable surface of genus two. Wait, wait, this is a picture. One picture is this, now you have two of these, okay? Oriental surface of genus two, and so on. So you see the orientable surface, okay? Oriental surface of genus two, and so on. They're also non-orientable surfaces, okay? And now I will introduce a non-orientable surface. So what is a real projective spaces? I want to define real projective spaces. So you did geometry, okay? Projective geometry. So if you have a vector space, you have a projective space, okay? So each vector space, that's projective geometry. R3, for example, okay? This is a vector space. So real RP2, RPN, that's my notation, RPN. RP2. In geometry, what do you take as a set? The projective space associated to R3, this is, okay? So you take all lines, so you take all lines in R3 through zero, okay? All lines in R3, zero. These are the points, okay? And in general, for any vector space, you take all lines in the vector space, all subspaces of dimension one, that's the same, okay? All subspaces of R3 of dimension one. That's what it is. In geometry, in projective geometry. That's the definition in projective geometry. Now we notice that, I want a topological space now. All subspaces of dimension one. Each line through zero intersects the two three and two points, okay? So this is a remark. Each line through zero intersects S2, intersects S2, which is, okay? In two antipodal points, clear? In two antipodal. Then as a set, I take equivalence relation on S2, okay? Equivalence relation on S2. So I give the equivalence classes. That's an equivalence relation. I just give the equivalence classes, okay? So equivalence classes, the equivalence classes are X minus X, X and S2. So this is a class of X. Equivalence class of X. Contains two points, which are antipodal points. We want to say this is one element only, okay? And then we have, as usual, we have a canonical projection from S2 to this. So we have a canonical projection P from S2 to the equivalence class and that's RP2. RP2, our definition. What is this? This is equivalence class of X, X and S2. And this is just X minus X, two points. Each equivalence class has two points. So the mapping is P of X is the equivalence class of X, obviously, okay? Each point goes to its equivalence class. So this is, if you want our definition, no? What is corresponds to the division of geometry, no? Where you have lines. But each line intersects S2 in these two points. So assets, they are bijective in a natural way, okay? The lines now on S2 correspond to these pairs, right? So it's the same set. And now we have a topology. So what S2, it's a subspace of R3. So we have a topology, subspace, no? We have a topology here and we want a topology here. So what to define? Quotient, the only reasonable way in general is the quotient topology. So let the quotient topology. So V in RP2 is open. What is the definition of quotient topology? If and only if, yes, P minus one of V is open in S2. So this is what, which topology is this? Quotient topology. This is the finest topology on RP2 such that P is continuous, okay? The finest topology. So this exercise on quotient topology. The finest topology on RP2 such that P is continuous. Okay, that's the quotient topology, okay? And then this is, you can do the same for SN, okay? RPN, exactly the same. I did it for S2, okay? From S2 to RP2. It's a covering. So here's S2. That's easy to make a picture, okay? Now I have to identify antipodal points. That becomes one point, okay? So P projection, okay? Well, I can represent everything on the lower hemisphere, right? A representative. But I still have to identify something here, okay? Because still here we have antipodal points, which we have to identify now, right? And then I cannot make a good picture. This, of course, is homeomorphic to the disk. No, that's the projection, okay? And then I have to identify antipodal points on the disk, okay? So I have to identify still to identify antipodal. This is maybe, so here you have more, still more equivalence relation. You still have to identify to the stock of. This is maybe the standard presentation of projective space, okay? A disk where on the boundary you have to identify antipodal points. You cannot make a picture, a good picture now, three. You can try, but it looks. So what means covering? Covering means just given any point here, I take a small neighborhood, U, okay? And how does the preimage of this U look like? Well, you get one copy here, no? Okay, so, and you get one copy on the other side, no? So here, on the other side, right? This, so this is P minus one of U. Of what? So this is evenly covered, okay? And this is a small disk, these are two small disks, okay? That's all, open disk, open sets. So this is a covering, two fold covering. Each fiber has two elements, okay? That's a covering, two fold covering. Each fiber has two elements. And these are antipodal points, okay? Antipodal points, the two elements. You recall, if this is connected, then each fiber is the same number of elements, okay? That's what I proved, okay? This is connected. This is connected, this is subjective, this is past connected, sorry, past connected, okay? This is past connected. Well, it's connected, so I proved for connected, no? If the base is connected, then each fiber has the same number of elements, okay? Here's two. So theorem, that's a surface. This RP2 is a surface, okay? There's a too many fold, because each point has a neighborhood, which is like an homomorphic to an open disk in R2, okay? You may be saying, these points are strange, this point, no? But it doesn't matter, I can make this where I want, okay? I can make it this. I can avoid my points, okay? That's not the problem. Okay, so theorem, pi1 of RP2, X0. So it takes some class here, no already, okay? It's isomorphic to Z2. So this is, by definition, Z mod 2Z. So a group is two elements, okay? This is a group with two elements. The fundamental group of RP2 is a group with two elements. And in the same way, you prove RPN, I didn't define RPN, but it's exactly the same, RPN. You do everything the same, and this is also, ah, this, well, I have to be careful. Well, I will prove this, but the same for RPN, but we have to be careful with RPN a little bit. So I may give this remark. So we define, in a similar way, define RPN. RPN is all classes X in SN, no, no? So these are, again, antipodal points, X minus X. And then we have also here projection from SN to RPN, the same kind of projection, PX equal class of X. And this is also two-fold coverage. So you define RP2, I did RP2, no? RPN, you define in a similar way. There's one interesting special case. This is RP1. Example, RP1 is equal, so the question is what is RP1? That should be one manifold. One manifolds are very easy, no? So this is X or classes, but X now is in S1. So just, again, X and minus X. So what you have to do is, you have to identify, this is S1, and you have to identify antipodal points, right? This becomes one point, okay? So if you do this, this you can see, what do you get? So what I can forget half of these, then I have each equivalent class still, but I still have to identify these two points, okay? These are the remaining points which I have to identify, to one class, this, this, and this becomes one class. And now if you identify these, what do you see? So you can do it, okay, in R2, and what do you see? S1, okay, S2, S2, S2, S2, S2, S2, S2, S2, S2, S1, you see the S1 sphere, the one sphere, okay? And that's true, RP1 is just the one sphere, S1. RP1 is S1. So remark, this implies RP1 is homeomorphic to S1. That's just the one sphere, that's a special case, okay? And now you say maybe RP2 is a two sphere, no, that's not the case, but how to prove that. Well, I wrote the theorem, which are unfortunately okay. The two sphere is a trivial fundamental group, and the RP2, okay? So a corollary, you have another surface, and this means S2, RP2 is not homeomorphic, RPN. Well, this is true for N also. RP2 is not homeomorphic to S2, why not? Because S2 has trivial fundamental group, and RP2 has Z2, this is not the same group, okay? So RP1 is S1, but RP2 and S2 are very different, okay? In fact, RP2 is a non-oriental surface of genus, let's forget, of genus one, this is one. This is the first of the non-oriental surface. What is the second one, the next one? That has still a name, what, I didn't understand. I still don't understand the name. No, no, maybe spent is with boundary, okay? Here we have without boundary, there's no boundary, okay? Maybe spent is closely related to RP2, okay? But this is with boundary. Surface with boundary, without boundary, no? We have without boundary. Other surface, of course, is a disk, two disk, no? But this has a boundary, okay? So I didn't mention the two disks. S2 are closed surfaces, without boundary, okay? The torus has no boundary, okay? Whatever it is. So not RP2, there's one more famous name of a surface. So you have the two sphere, you have the torus, the oriental ones, you have the projective plane, projective plane, this one, no, RP2. And then the next one is the Klein bottle. That's what it's called, okay? Klein bottle, whatever it is. But maybe you heard this name, no? If your surface is so Klein bottle, I don't define, but this is the next one. Is non-orientable surface of genus two, okay? That's the Klein bottle. There's another name, then I have no name. Then the next one is non-oriental surface of genus three, and so on, and so on, and so on. But without special names. The first one is the projective plane. The second one is the Klein bottle, okay? The non-oriental ones, and the oriental ones, the two sphere and the torus, the next one. Without boundary, okay? Closed, without boundary, only. Well, anyway. So proof. What do we have to prove? The theorem, right? Of pi one of, well, I write even RPN here now. It doesn't matter, X0 is isomorphic to Z2. But here I have to be careful, that's one special case, okay? If N is equal to one, this is S1, so it's Z, okay? So for N, bigger than one, then it's always Z2, for the higher ones, okay? The only exception is, so pi one of RP1 with X0 is isomorphic to pi one of S1, they are homeomorphic, with some base point, whatever it is, X0. And this is Z, that's the exception, okay? Otherwise it's, okay. So now let's prove it. What would be interesting is to do the classification of surfaces, okay? All closed surfaces. However, we don't have enough time for that, okay? That's the interesting point, okay? To classify surfaces. That's a classical argument. Some combinatorial argument, of course. You need some combinatorics here, okay? You have to present the surface in a good way, by a triangulation. So we take the covering, P. I'm doing the case RP2, okay? But the case RPN, for N bigger than one is the same. You will have to see where N equal to one is not good, okay? Maybe. So RP2 to S2, no. That's the wrong direction. From S2 to RP2. So P of X is the class of X. And the class of X consists of two points, which are antipodal points. So this is a two-fold covering. What do we have to prove? We have to prove this one. So I define a map from pi one of RP2 with base point X0. So X0 is a point in S2. I want an isomorphism with Z2, okay? And Z2 I write multiplicatively, okay? So I write, what is this? Multiplicative with one and minus one. And with a product. So this is isomorphic to Z2. The same group, no? With two elements also. So I want to define this mapping. So I take a class here, W. It's the same proof for S1, no, okay? You will see. Not exactly the same. The structure is exactly the same, but we need something else. It's a little bit more difficult, maybe, because we have to use the theorem before. So what is W? W is a path. WI from RP2 is a path with initial point and end point. The class of the point X0, and X0 recall for a point in S2. It projects to its class. So, what did we do? We use a path lifting lemma to get a lifting. Let W tilde from I to S2 be a lifting. So this is a path lifting lemma. Of W with initial point, W tilde of zero. So I have to take a point which projects to the class of X0, so I'm taking zero, obviously, okay? X0, which is a point in S2. So I have to take a point which projects to the class of X0, so I'm taking zero, obviously, okay? Which is a point in S2. Fixed. Let W be a lifting of W with this. Then, what is W? Then, W tilde of one, the end point. What is the end point? If you project, it becomes a close path, okay? W is a close path, right? We lift it, this is X0. So what are the two possibilities for the end point of X0? Two possibilities. X0 or minus X0, yes. So this is epsilon X0, where epsilon is equal to one or epsilon is equal to minus one, okay? X0, the same, or minus X0. These are the two points which project to our base point, okay? And now I can define my mapping.  So I introduce a name here, Phi from this, okay? Now I define Phi. So let Phi of W be, this is a definition. So what do you take? Epsilon. If it goes to X0 as a lifting, then we take one. If it gets to minus X0, we take minus one. So it's epsilon in one minus one. We just take this epsilon. We look where it doesn't fit, okay? So that's the definition of this mapping Phi. And now we just only have to prove that Phi. Well, Phi is, now it's the same as before. So then what we have to prove? The first is, Phi is well defined. And this is the homotopy lifting lemma, okay? By the homotopy lifting lemma. Or it's corollary. Phi is very fine by the homotopy lifting. That's the next important step, okay? Then Phi is the homomorphism of groups. How much time do I have? Not too much time. So I don't prove it. It's the same as for the reals. In the reals, one path, you have to translate, okay? Now, if the first path ends in minus X0, you have to multiply the second path with minus one. So it starts in minus X0, okay? That's, it's the same otherwise, okay? So the time is more or less finished, so I don't write it now. So Phi is, this is now interesting. Phi is injective. So this I will do. Well, it's also the same as before, except we have to see the difference between n equal one and n bigger than one. I'm doing the case two, but it's, otherwise it works for every. So Phi is injective. So let W be in the kernel of Phi, okay? And we have to prove it's trivial. The kernel is trivial, no? So what does it mean? This implies that W tilde of one, the endpoint of the lifting, which point is it? It goes to one. So X0 also. It's one times X0. So this is X0, okay? Epsilon is equal. Epsilon of W is equal to one. So this is, of course, is also the endpoint, no? By definition, this is by definition. So this means W is closed. This means W tilde is closed. It's a closed path. Where? In S2. So W tilde is an element where in the fundamental group of S2 with base point X0. And now this is more difficult than in the case of S1. This is a trivial group, okay? In the case of S1, we have the reals here, okay? And this is trivial by straight line homotopy. Here, this is a trivial group. This is a trivial group. That means that W is equal to the trivial element. W tilde, sorry. W tilde, there's only one element, which is the constant path in the point X0, no? That's the only element of the trivial group. So this implies that W tilde is homotopic to CX0 by a homotopy H. They're homotopic. Another project. So this implies that W, which is a projection of W tilde, is homotopic to the projection of CX0 projected. What is this? This is the constant path now in the class of X0, RP2. And the homotopy, of course, is H and then projected. So this is trivial, this step, okay? What is not trivial here is this fact here, okay? This is an addition to that. So this means that W is the class, many parentheses here. W is also the trivial class, okay? And this means the kernel is trivial. So this implies the kernel of phi is trivial. And this, of course, means that phi is injective. And then the last thing is subjective, okay? So objective is trivial again. So the last thing is phi is surjective. So it's the same as before. Maybe I'll do that to two lines. We have to get minus one, okay? Phi goes to one minus one. One we get, no? That's the unit element. We get from the unit element in any case, no? So we have to get minus one. So let U be any path, be a path in S2 from X0 to minus X0. Well, there's no problem, no? It's path connected. We go on the great circle, okay? In one or in the other direction. There are two ways not to go on the great circle. Let U be a path in S2 from X2 to minus one. And W be the projection of U. So this means W is a closed path. It's a closed path. So it represents an element in the fundamental group of pi1 of Rp2 with base point X0, the class of X0. It's a closed path. And now what is phi of W tilde? So W tilde of one. What is W tilde? This is just U because we project it from U, okay? And now we lift back, so we get back U, okay? W is a projection of U, no? W is a projection of U. So we lift back, we get back U, okay? So W tilde of one is equal to U of one is equal to minus one. By our choice, okay? And this implies phi of W is minus one. No, minus, this is minus X0, yes, thank you. And phi of W is minus one, yes, exactly, okay? And this, we get minus one in the image, and this means phi is subjective. And that's the end of the proof, okay? The main difference between this and the case of S1 is that the fundamental group of R is trivial, and that's easy. The fundamental group of S2, we have to prove that it's trivial, okay? And that's not completely trivial, okay? That's Fankampen, we use Fankampen theory in a special case, okay? So this, now we have many surfaces. Now we have RP2, that's a projective plane that's called also, okay? Fundamental group on two elements. We have S2, fundamental group trivial. So this is in some sense to, so fundamental groups, our surfaces, fundamental groups. So we have the surfaces, S2 is trivial. We have torus, this is Z times Z. We have RP2, so this is the surface, and this is fundamental group. So we have what else, RP2. This is called the projective plane, no? Projective plane also, okay? That's the projective plane, real projective plane, if you want. Fundamental group is what? Z2, not this Z2, but this Z2, okay? That's it for the moment, I cannot, we don't have others, okay? So this is, so they are not homeomorphic, okay? They are not homeomorphic. These three are not homeomorphic. Okay, that's application, okay? Of fundamental group. This is difficult to prove without fundamental group, okay? It's not so easy to prove that these are different here, okay, otherwise, okay? The fundamental group is nice for this, okay? And also for the other surfaces, by the way. Fundamental group is the most important invariant, okay? Algebraic invariant. That's the fundamental group. And so Thursday, we have last lecture, no? Thursday. Thursday from, not tomorrow, no? Thursday, from 11 to 13. And then I will finish with homotopy equivalence, probably. Maybe I talk also about mobius band, okay? That's also an interesting example, but it's a surface of this boundary, and we will see also the mobius band, okay? What? Is it? This? Well, more or less distributed, so you have, yeah, it's in the book more or less, but it's in different chapters. I'm not going in a linear order through the book here, okay? Because there's not enough time to do everything in the book, okay? It should be in the book, yes. Projective spaces, real projective plane, that is in the book, yes. Fankamp theorem, in some version, it's later, in the later chapters, okay? Yeah, in some way it's everything in the book, but distributed, you don't, okay? I went a little bit diagonal through this, because we had only five hours of this part, and if you follow one chapter, then, okay? But the important things are these, no? The fundamental group of S1, okay? The fundamental group of S2, SN, okay? Fundamental group of the torus as a product, that's easy. Fundamental group of RPN, okay? Projects, these are a nice example. What we should see maybe also next time is, these are all Abellion groups, right? All Abellion groups, okay? So you may ask them, why do we see only Abellion groups, okay? So we should give one example of a space which is non-Abellion fundamental group. So that I will do first also. At least one example, okay? And typically, fundamental groups are not Abellion, okay? Abellion groups are easy. But fundamental group can be everything, okay? So in general, it's not Abellion. Here it happens that for these first easy cases, okay? It's always Abellion, trivial Abellion groups, okay?  So I will give one example of non-Abellion fundamental group also next time. Okay, that's it.


So while we're doing Top Secret, you can post up your questions in more in Discord, but let's show some Top Secrets. This week, I've got three short videos, and then we're going to do some stills. So let's take our past videos and let the past us show it. Okay, Lady Eater, what is this?  So I've got my Nordic Po Power Profiler Kit 2. And I've got it up here on the computer, and see this is it connecting to Wi-Fi, and then it's sending temperature, humidity, and barometric pressure, and then it goes into a deep sleep, where it uses only about 80 microamps. And here it is updating on Adafruit IO. So one of the tricks I'm doing here is it's got this BME280 sensor, which is like really close to the ESP32. But what I do is I wake up, I don't turn on the Wi-Fi, I read the sensor data, and then I connect to Wi-Fi and send that cache data so I don't have to have as much heating from the ESP onto the sensor. And with 80 microamps, it can last many hours between battery charges. Okay, Lady Eater, what is this? Hey, I'm trying out some new chip-on-board epoxy-coated neopixel strips. So these are similar to the one-color COB chip-on-board strips that we've gotten, but they've actually got, you know, neopixel-compatibles inside. So they've got this like bead of silicone epoxy and like 330 neopixels per meter. So extremely high density, but very smooth color. You can almost not even see the difference between each color along the strip. And I'm driving it with a little QT Pi M0. So far so good. I think they are going to be a little delicate, so you just have to be careful with them. But if you want like a smooth neon look that is very thin and very flexible, this is definitely going to do the job for you. Cool stuff. Hey, Lady Eater, what is this? Hey, everybody. It's got this ESP32 S2 Feather put together. It's got the mini-modules, STEM at QT, and it's got a built-in BME 280 humidity temperature and barometric pressure sensor and battery monitor. So I thought a really great demo would be to, in circuit Python, have that sensor data sent to Adafruit IO. So I whipped up together a real quick circuit Python code demo using all of our libraries. And there you go. I'm sending that data to Adafruit IO, you can see the temperature, humidity, barometric pressure and battery. And if I breathe on this, you'll see the humidity pops up. So I'm sending data every five seconds, and it's working great. So next up, I'm going to try this out in Arduino to make sure it works there as well. All right, then here's some of the photos of more. We got the ESP32 S2 Minis in. That's what we used on those feathers. Yeah, and then we were bringing up the sport. This is the TFT version of the ESP32 S2 Feather. So just verifying that the TFT works. Yeah, and then you can see we got a few boards here, including the Trinky. Trinky. And here's a preview of some of the art for Cyberdeck, which is our Raspberry Pi accessory. We're making these now. Yeah, pretty soon. And that is the top secret for the week. Back in the vault. Back in the vault with you. Okay. Thanks. Thanks. Thanks for watching. Bye. Bye. Bye.