The conflict and uncertainty surrounding U.S. elections has pushed Bitcoin up by over 33% in the past two weeks. Bitcoin has even broken the 18K resistance level following the news of Pfizer and Moderna's vaccine development. The question everyone's now asking is, how long will the bull run continue? Hello everyone, this is Michael van der Proppen. From Cointelegraph Markets, welcome to a new Cointelegraph Markets weekly update. It's going to be one of my friends, Alex Saunders from Nuggets News from Australia. Welcome, Alex. So let me start off with the DXY that you've been talking about a lot. We're just right on this level that I'm sure you've touched on plenty of times. And if that breaks down, then we're expecting all our favorites to perform pretty strongly. Alex Saunders is the founder and CEO of Nuggets News. He has been researching trading and investing in cryptocurrencies since 2012. He has also served as a board member of Blockchain Australia. Australia's leading community association dedicated to blockchain education and innovation. He joined our markets analyst, Michael van der Proppen, to discuss the bullish factors that are pushing Bitcoin towards its next parabolic move. As we've just had the elections, I think it's one of the topics we should be addressing. And the first question I have is, how did you experience past week with the all discussion between the two and the tweets from Trump and everything? How did you experience it? I think it was pretty much to be expected in the end, wasn't it? Like you knew Trump was going to make a lot of noise. And probably if he got beaten, probably say it's not fair. But just lately, he's kind of taken it on the chin. And it seems like even his party around him is saying, look, let's just move on and get on with it. So I'm kind of glad it's behind us. I think the markets are really going to like it. I think there's a lot of money that's just sort of waiting to know where to go. And it's not that attractive to go into bonds. So I think there's going to be a lot of movements and a lot of volatility, but in the good ways, there's going to be a lot of bullish conditions and sectors to play. I think, are they going to get the three trillion stimulus across the line? That's probably the big question now. And if they can't, are the Fed going to step up? So the US in particular, they need some money. People that are losing their homes, commercial mortgage back security failures, all those things that have just been put on the back burner and they didn't get it for the election. I think the question just becomes, what does it look like in terms of the stimulus and that money that gets handed out now? But is there any difference between any president? I mean, Trump was also shouting that they would be doing tremendous stimulus packages if he got elected. Well, he's not, but Biden is going to do almost the same as they literally don't have any other option. Right? Well, like I was saying, the markets love a narrative and it was a two trillion versus a three trillion stimulus. But that was like up or down as far as the traders were concerned.  And when Trump got elected, it was, oh, it's going to be chaos. But then he cut taxes. So that ended up with higher profits on paper. Biden's talking about raising taxes, but then he's talking about doing a bigger spend and changing other things. So I think it also comes out in the wash these days with passive investors and the Fed increasing liquidity. You just look at the relationship of the M2 money supply. It's those other things that ultimately dictate the market. It's not what one politician does, says. Yeah, I mean, that was one of the questions that I wrote down. It's like, we all know in what type of financial system we are living. And now we're taking it the broader perspective. We are living in this inflated financial system since the 1980s, 1970s, when the Golden Standard got released, right? RDS is the new president and Muppet in our financial system. And what I mean is we all know that they have a tremendous amount of debt in the markets right now that has to be paid off at some point. And that's why they are trying to get that inflated economy just continuing. Is he a Muppet or can he change anything to the financial system? What I mean is if there is going to be a crisis, can he change anything? I don't believe so. No, and I probably would have said Biden more so because he's so old as well. And a lot of people are going to be able to tell him what to do more. I think he's going to be way more malleable and just not out of control in left field. And that's going to be good for the VIX again in the markets. They're not going to be worried about a Biden tweet, for example. But I think what exemplifies the global market conditions is what happened in Australia in the past couple of weeks. And you guys might not have been following it. But basically the head of our central bank has always been about, we will never go to zero. We will never go negative and we won't start QE here. And he's literally done a bigger backflip than Jerome Power did when he got into office. Remember, he was saying that he's going to raise rates and they're not going to be at the mercy of the market. And then he backfliped. Well, our RBA head just did the exact same thing. So he's cut to zero. They're doing 100 billion of QE in six months. That is massive for Australia. It's on par with what the Fed were doing at their most aggressive. So our markets have started to take off. I think property is going to go up. But you stop and think about, well, what's actually happening when people are losing their jobs? And we're about to go through what happened in America for the past 10 years. It's going to drive wealth inequality. It's going to push out property prices. Stocks are going to go up. And the thing is that our central bank chief said all this. He's been writing articles about how it creates wealth inequality, low interest rates, money printing, QE pushes up asset prices and causes speculative behavior. And he even said this in his speech the other day. The board recognizes that low interest rates can encourage some additional risk taking as investors search for yield. It also recognizes that low deposit rates can create difficulties for some people, particularly those relying on interest income. And these issues will need to be closely watched over the months ahead. So to me, that just shows you the world we're in. That the guys at the top, they actually know that it's bad, but they're just going to do it anyway because we're all doing it. And having a bubble is better than having a crash. I think that's all they've got to fall back on now. Yeah, but you know, every bubble ends up in a huge crash. We have experienced it with 2017. Bitcoin was inside a large bubble. It crashed in 1929. It was a big bubble actually surrounded by tech back in the day as well as radio was coming up. It crashed dot com bubble. It crashed. So what we see right now with every, well, the banks saying negative interest rates are going to zero is that they're just extending as long as they can. They know that once they are going to raise the interest rates, it's going to be bad for the markets, right? It's just extending and they extend it already since 2008. That's how far I can see right now. So, and we've also seen that Jerome Powell is actually going to loosen their inflation target. They have to fix target of 2% per year. And what he stated a month ago is that they're going to loosen it in which it's going to be a variable 2%. So one year they can just have one and a half. One year they can have two and a half. For me, it's that he's saying, we don't know anymore. We're just doing whatever we can right now and go ahead with it. Are we going to continue with this inflated market? Or is there a point of return in which we're going into deflated markets as they can't keep up anymore? Yeah. So I've got this analogy that I joke about it. I say the picture of the economy as a sausage machine. Okay. And we're just putting in all the loose ends and the sausages come out. And the Fed and the government are just pushing in all this money and these changes and they're pushing on a string. And they keep looking at the other end and saying, why isn't it like good? Why are economic conditions good? There's no wage growth. There's all this inequality. We can't figure out why this is happening because our textbooks tell us that it shouldn't be the case. And that is exactly the point where we're at. So you can actually stop every bubble from crashing if they want to print 10 trillion in two years' time when it crashes. And then two years after that, they might have to print 100 trillion just to stop the huge deflation, which is the natural forces, which is the way that things want to go because of demographics, deflation. I think decentralization. Technology. Technology, decentralization, moving out of cities because of COVID. Everything you look at is basically deflationary. But all the money printing, it's causing inflation of the assets and it's not getting into the real economy. So that's where they're looking to bypass the banking system and do these retail central bank digital currencies. And that is what can cause inflation if they start putting heaps of money or universal basic income into people's pockets. So that's kind of where the system is at the moment.  And the straining of the system comes out as civil unrest or inequality or food prices go up, but wages won't go up because it's going to the owner of the big corporation. So they're all the things that it kind of shows you that the system is broken. But my sausage machine analogy, they're not changing the machine at all. They keep doing the same thing and wondering why nothing different comes out the other side. Yeah, it's also if you have the numbers here in the Netherlands, the income levels have almost not increased in the past 25 years. I guess it's only around 50% the incomes have increased. The overall housing prices here in Amsterdam raised by 600%. So that's the whole point you're describing here and what they are trying to do. Exactly. In Australia, house prices have doubled in the past decade and wages haven't moved up at all, not 1%. Oh, and that's the old variable that can actually push the economy down because these people will lose their job and have no ability to pay their mortgage anymore. So that's the variable that they have no power of unless they're going to do UBI or anything to. That's exactly right. But just the other day, we had more rate cuts and the big banks didn't pass on any of the cut to the mortgage holders. And then the banks came out and said, we're not allowed to kick anyone out. So there's this moratorium where they're not allowed to kick people out of their homes for a year. And so that way there's no homes hitting the property market for sale. And if no one's allowed to sell their house, then prices can't go down. So we're just sort of pushing on all these holes as the boat's filling up with water, trying to check these things as they happen. The overall question I also have is, I know you have been discussing this with Roel Paul and Roel Paul has also been debating that if we get deflation with these debt levels, that's going to be very bad for the markets. But my question is, hyperinflation and the US dollar, I just don't see it occurring. Do you see that as a potential option that occurs? The world's reserve currency going into hyperinflation. And what effect would that have for Bitcoin? It'd be great for Bitcoin, but it can't happen in America because America and again, essential bankers don't even realize this, I don't think are a lot of the other bankers around the world. There's actually two economies. There's two US dollar economies. There's the United States.   So they can keep doing this money and it's just this battle about is it going to go out of America to these emerging markets that are just being crushed because they so desperately need dollars to pay back the debts, but they can't get any as well as their own economy, which is experiencing deflation. So they would have to print so much money and they'd have to point it at the rest of the world and point it at their own economy before they'd risk getting inflation, let alone hyperinflation. So I just don't see that happening. And then that brings me to one company that's recently announced that they'll be accepting Bitcoin, which is PayPal. Are they jumping in on the hype because they know they are losing or how do we should be seeing it because if you buy Bitcoin through PayPal, it's not your Bitcoin as you can't hold your keys, right? I think they watched the documentary about Blockbuster when they got pitched by Netflix and laughed him out of the room and they don't want to be those guys. But at the other end of the spectrum, there's all these motor companies that are making the switch to electric vehicles. So I think it's that sort of thing where they basically know they can't compete and it's past ignoring them and laughing them off and now you've got to join them as the same goes. So they're trying to stay relevant. That's why Visa, Mastercard, all the big banks, FinTech was already coming for payments and banking. But now you've got the crypto world that's removing more intermediaries and overheads again. So that's better. So it's going to be an unbelievable experience for the end user, just like you and I can have a video chat like this. Money will basically become free and valued to send anywhere over the internet. So I'm really looking forward to that. And yeah, they can't compete.  Well, they went bust because technology made a better product and PayPal and Natagame basically go bust. So it's not a bad thing. There's just other jobs and technology comes along and disrupts industries. Yeah. I mean, it's also if you send money through PayPal, you've got the insurance, which is like a 4% fee on every transaction you make. But there's also projects, building insurances on Ethereum and stuff. So all in all, PayPal is just an ending story in that regard. But it's funny that they are jumping into the markets the moment that they know that they are losing. And probably they knew it already, but now the moment is kicking in where they accept that they are losing and try to make a buck out of it. All right. Let's round off this show with expectations of the markets. And I would like to know, where do you see Bitcoin and crypto going the next months? And what, when do we see a new all-time high? Is it going to be this year, beginning next year? Let's share some bull bear thoughts on the markets. I think I honestly think that Bitcoin will hit $100,000 in the next five years. And then it is going to become about, well, do they actually try and shut it down or how do they regulate it and treat it? And if it's allowed to just continue on its mission and absorb all the money from around the world and become a global reserve currency and gold, you know, I think that it can get to $1 million per coin in the next, whatever that is, 15 years. Ethereum, I think that that is going to pass its all-time high by the end of next year as well. I think Bitcoin can pass 20,000. And it's even going to keep going like right now, the next few weeks, and get to 20,000 very quickly. Or it's going to have its bit of a range now and then hopefully positive into next year. But you just look at the forces around you and it's not stopping, it's just accelerating more and more countries, more and more money. There's no way out. So yeah, it's a perfect backdrop and it very much feels like 2016 heading into 2017. Yeah, I agree on that point. I mean, I don't believe we are in such a parabolic move yet as in 2017, as the euphoria under retail investors is just not there yet. I kind of believe that we're currently still in that rangebound construction period as we have seen in 2016, in which we just flip levels, go sideways for months, build further and break out after a few months. So in that perspective, I do agree that it's, I guess it's more likely that we're just having a little bit of correction, but that's still very healthy for the markets. I mean, Bitcoin is doing really well. I think it's one of the best assets to have for the next years, Ethereum next, especially as what Ethereum can achieve, what Bitcoin can't. So I agree on that point. And it's just a matter of time before more bullish news comes out of listed companies buying Bitcoin for their cash allocations or it's just coming. It's the wave has started and the wave is only accelerating from here. That's what I expect to occur, right? Yeah. People are going to be really surprised by how much money's out there and how small Bitcoin's market cap is once wealthy investors start to try and get on board. Yes, I agree on that. That's a nice conclusion of this show. Well, everyone, thanks for watching. Thanks, Alex, for being here. Thanks for sharing your thoughts. Don't forget to subscribe beneath here to this channel and we'll see you back again next week for the new show. Coin Telegraph, like, subscribe, and hodl.


How thick can I pour Art Resin? Art Resin was designed as a surface application as opposed to a casting resin. This means that it self-levels at about an eighth of an inch, so long as you pour in 1 eighth inch layers, you can go as thick as you like. Keep in mind that the cure time is based on that 1 eighth inch layer, so if you do pour thicker, it may take longer to cure. The reason we recommend this thickness is two-fold. First, it allows the bubbles to escape properly, and secondly, it avoids any excessive overheating of the resin. So for best results, always pour in that 1 eighth inch layer, and you're good to go.


Let's take a look at how we can use AI and particle chat GPT to help us understand research articles. My example article is this article by Sapienza on early internationalization. I use the same article in other videos where I talk about how you read theory articles. And how would we go about understanding this article? Traditional way that I teach students is that you need to first identify the key concepts and the key terms, and find the definitions for those terms. They're often defined in the article. Sometimes you can find them in Google. Sometimes terms like AIDS and growth are something that you have intuitive understanding of. Using AI and chat GPT allows for taking a shortcut with some big caveats. So what you can do is that you can copy face the article text into the GPT and ask the AI to summarize it for you. So how we would do it is that we take the article, we copy the text. This is a bit inconvenient to copy because it contains two columns, but you can find a single column version of this article. The third skate, for example, which is the version before it was formatted for the journal, and copy-paste the text from there. So I copy-paste the text to a text editor. And why I use text editor instead of copy-pasting directly to GPT is that there are page numbers, there are footnotes and other things that I want to take away so that they don't throw off the AI. I don't know if they do, but I just thought that it is good to make sure that there's nothing extra. So I just have the text of the article here. Then we put it in a chat GPT and we ask it to summarize it to us in simpler terms. And the answer comes, it's pretty much correct. Only one small detail is that this is not about establishing your company in a foreign country, but it's about expanding an existing company into a foreign market. So there's 95% correct, only the first sentence contains a small error. What's more, we can also ask for follow-up questions and clarifications. So we can ask, what does imprinting mean? That was a term I saw used in the article introduction, and it tells that correctly that imprinting refers to things that happened early on in the company, company life cycle that have long lasting consequences. Then I wanted to know a bit more about what role the imprinting process plays in the theory, and I asked that question, and then I get an answer about mice. So this is not the biology article. How would the AI suddenly pivot to talking about mice? Well, it's because how the AI is trained, it's trained to predict what word comes next, and then that allows you to write answers. And it's trained with large amounts of data, and imprinting is often discussed in the context of biology. So it was answering based on the context that we gave, but based on also the training material, and now it gets it horribly incorrect. So this is a good AI for generating text, but it is not very smart, so you need to be very critical of what it gives you. It might give you a correct answer, or it might give you something that is completely incorrect. Let's take a look at another example. I wanted to understand what is research fungibility. It means that the resource can be used for different purposes. That's correct. And then I wanted to ask what is moderator, and what are the arguments that the article makes for supporting moderators. So moderator is a variable that affects their relationship with two other variables. The first sentence has something correct, but then there is a renewable energy comes up. We don't know why. And then it says that there are three moderators, which is correct. One is fungibility, that's right. Another one, imprinting, not true. Social norms, not true. So resource fungibility is correct, and then there are other contain things about renewable energy, which is not related to this article at all. So is this a useful way of reading articles? I think with some caveats it is useful. So the caveats are that this is very good at summarizing text. So you can tell the AI that this is a big piece of text. Make it shorter so that you just tell me the main idea. And it does that job really well as we saw in the first response. Then you can explain new terms that when you ask follow-up questions, but it might get those incorrect. So it can explain things, but it can go also horribly wrong. So reading, using this AI and asking questions about an article is definitely not a substitute for reading the article, at least at the time of recording, but it can be a useful thing as a first step when you want to have just like a big picture understanding on what the article really is about. And there the summarization function can be really, really useful.


You need a financial advisor with the freedom to focus on your specific needs. Whatever your goals, Reid Potter can create a game plan tailored to you. Call Reid at 432-0777 at Pikeville, Kentucky to learn more. The Johnson County Public Library will begin their Color Your World Spring Story Hour for children newborn to five years old on March 7th. A lot of children, especially since the COVID-19 pandemic, have not had a lot of socialization with other children. So that is one of the great benefits about coming to the library and getting that interaction with peers. It's also really important for pre-reading skills and kids that have exposure to books and stories and literacy. They're more likely to do well in school once they start. Children will be given the opportunity to hear a story, do crafts, and interact with other children in a fun and educational environment. We try to do different programs here at the library that bring all kinds of different families together so that way the kids can get the social interaction with peers their own age.  The Color Your World Spring Story Hour will begin on March 7th and run until April 20th. There will be two sessions held every week, one on Tuesdays at 6 p.m. and one on Wednesdays at 11 a.m. Pre-registration is required and can be done by calling the Johnson County Public Library at<PHONE_NUMBER>. For Mountain Top News, I'm Kelsey Dean.


Modern society has brought many advancements from personal computers and factory lines to artificial intelligence and smartphones. One unexpected side effect of modernization is burnout, which was coined in the 1970s. It was originally described by psychologist Herbert Freudenberger as becoming exhausted by making excessive demands on energy, strength, or resources. But in recent years, the definition has expanded. Burnout is physical, emotional, or mental exhaustion that causes decreased motivation, low performance, and negative attitudes. It can result from prolonged stress or enduring overwhelming physical or mental exertion that is beyond one's workload. The concept of burnout has recently made a comeback thanks to the pandemic. With the transition to remote work, many people are experiencing burnout at home. Remote work makes life easier, but without healthy boundaries in place, your personal life could suffer. You may experience typical emotional or mental cues of burnout, such as reduced focus or lack of motivation, but there are physical signs of burnout as well. We'll explore seven signs of burnout and possible solutions. Number one, overall fatigue. The most obvious sign of burnout is fatigue. Feeling drained and exhausted from working is understandable, but fatigue is a bit different.  The fatigue you feel when you're burnt out isn't just linked to the amount of work you have to do, but also to the amount of stress you're under. Stress consumes a lot of energy, and chronic stress is like contracting a muscle for a long time. It gets fatigued and isn't very useful. The same happens to us. Chronic stress affects your sleep and eating habits, but also overproduces cortisol, which takes up a lot of energy and doesn't replenish it. In a dangerous situation, this is a normal response. Your body is priming you to fight or run away. However, we don't confront tigers on our way to work or have to run miles from our enemies. We sit at our desks. Without using up the glucose, our stressed out bodies have produced. PhD researcher, Robert Jester, explains that burnout causes extreme exhaustion that isn't solved by getting sleep. A great way to prevent fatigue and an increase in cortisol levels is by taking 10 minutes out of your day to relax. Just close your eyes and focus on your breathing. Try to detach yourself from all the things you think you have to do or know you have to do. Two, sweet cravings. Craving sweets is common when you're chronically stressed and burnt out. It's your body's way of replenishing the glucose it spends. However, constantly adding glucose into your body and not actually using it can cause future problems down the road. A common result is type 2 diabetes which occurs when your body develops a resistance to insulin according to the National Institute of Diabetes and Digestive and Kidney Diseases. Also, it doesn't help that when you're stressed you naturally crave sugar because your body uses that as fuel to confront the dangers that have got you stressed. Number three, tension headaches. Another common sign of burnout is tension headaches says psychologist Dr. Adam Borland. Tension headaches are also a side effect of too much stress, but can also be caused by a change in eating habits. When you're stressed, you might forget to eat or drink water throughout the day. Or worse, you might also eat unhealthy junk food because they're either faster to cook or less fussy. No judgment, but those types of foods can be culprits for your headaches. If cooking isn't your thing, try to prepare food ahead of time. Take 30 minutes over the weekend to plan your meals for the next couple of days. Make sure you include plenty of vegetables in your prepped meals. If you don't like the idea of preparing food ahead, cook simple meals. Simpler meals usually involve few ingredients. There are a lot of ideas online, so go ahead and do some research for inspiration. Four, high blood pressure. High blood pressure is both a consequence but also a sign of burnout. It can be a consequence of burnout if it is caused by the foods you're consuming. But as a sign, high blood pressure is caused by chronic stress. Stress can cause temporaries by some blood pressure that can damage your blood vessels, heart, and kidneys, according to the Mayo Clinic. And your hormones can damage your arteries, which can lead to heart disease. But don't fret. You can do things to help manage your stress and improve your health. Exercise is a great way to keep your blood pressure more stable and it also can help you de-stress. If you're burned out and fatigued, opt for a lighter exercise. Do some yoga or tai chi. You're still moving but not moving in a way that stresses out your body more. Number five, heart problems. If your high blood pressure is left unattended, it can cause heart problems. Having high blood pressure can put you at risk for stroke and other cardiovascular diseases. So please find a way to decrease your stress-induced high blood pressure. According to Dr. Mary Nualangan, who specializes in cardiology? It's common to have heart issues like palpitations or a racing pulse once in a while but if they happen a lot or for longer than just a few seconds, you should get checked out. When we're dehydrated, our body may also increase our heart rate to keep our blood volume the same, which could lead to more serious issues. If high blood becomes a concern, reach out to a medical health provider for help. Six, stomach problems. One surprising sign of burnout is stomach problems. Burnout heavily taxes your gut and can produce intestinal inflammation. If left untreated, it can lead to stomach heavily taxes your gut and other complications. In recent years, there has been a plethora of research supporting the link between your mind and your gut. When stressed, your body halts digestion in order to produce more energy. If you're constantly stressed, you're not digesting your food properly and putting your gut health at risk. But there is a silver lining. A 2013 clinical review published by the University of Chapel Hill found that psychological treatments like CBT were effective in treating gastrointestinal disorders caused by stress. So if your burnout is too much to handle, reach out to a licensed professional for help. And number seven, insomnia. The final sign of burnout is insomnia, which is a common symptom when you're stressed. Insomnia is what happens when you take your worries to bed with you. They loom over you and prevent you from getting a good night's sleep. To combat insomnia, it's best to try to put your mind at ease. A great practice is to exercise. In 2013, Chicago's Northwestern University discovered that older women with insomnia were able to improve the issue by doing a half hour of exercise three times a week for four months. They ended up getting an extra 45 minutes or more of much needed sleep. Burnout is serious. It's not something you should overlook. If you need a break, ask. Talk to your boss about taking a few days off. Find ways to make your life just a bit easier. For example, if you feel like your to-do list is too much, plan your time for each task and also plan your breaks. If you feel overwhelmed by the amount of tasks on your to-do list, prioritize them, but don't exceed more than five. Once you've made your list, calmly work through them and get them done. If you don't finish, that's fine. Just add it to the next days to-do list. But try not to exceed the five items to keep it manageable. Can you relate to feeling burnt out? Share with us in the comments. Keep an eye out for Sai for more content and see you next time.


Women's marches where the China's on their hats and they're just so angry and so mean and so Miserable, you know, like the trolls that come after me. They're just so miserable and it's just so far from God's plan for them and That's all they know and I think the breakdown of the family is The culprit the the bottom line is the culprit the breakdown of the family these yeah women not feeling because they do want Tension even though the feminist have tried to convince them not to get it. Yep. They do want male Attention, they want to be love and they think this is love. It's not yeah. Well, it's all little hearts, right? It's a heart here and a heart there. It's this little loving heart, but yeah, it's not love I think about this to these girls, you know, especially Yeah, but especially for girls beautiful physically legitimately very good-looking They have thousands and thousands and thousands of I call them beta males like every photo they post on Instagram and everything They put on Twitter and everything they put here and it's like you're you obviously love male attention and validation But it's not number one is digital. It's not real number two You could have a hundred million of these beta males like your stuff and you're still gonna feel empty inside because none of them actually love you So it's all fake.  What is that gonna matter? Even for us how many people watch our videos or how many likes we get on Facebook at the end of the day What does it matter? It's that? Having a husband that loves me and children and grandkids that's love me want to be with me That's what matters, you know love and family exactly. Yeah. Yeah, they're giving up the best the chances of them ever of course if they repent and And you know believe in Jesus Christ he can make all things new but they they'll have so many scars in their lives to have to over


There's some concern from people who think that we're not going to have a work stoppage, that we're not gonna have a strike, but I can assure you that if they try to play something like this with us, we'll make that West Virginia stoppage look like a game of patty-cakes.  The current bill is a cut to retirees and they're asking folks to work longer basically for less pay to pay off the state's debts with our contract that we already signed, they can't do that. And so if it does get to the point that the house does pass it, I would expect very quickly and I'm not talking, you know, just a few counties. I mean, I think Eastern Kentucky will lead the way because it affects us more than the rest of the state. I guarantee you 120 counties will come together and it will be unbelievable. You know, it'll be the biggest one probably in the nation.


You would not meet an early sleep, just so we'd lament and weep. Flowers blossom from your wounds and from your wounds lessens leap. You would not meet an early sleep, just so we'd lament and weep. Flowers blossom from your wounds and from your wounds lessens leap. I came to the mosque yearning, not just with my eyes heavy. My soul wants its charity, lessens it yearns to carry her. Yes, my tears they flow like before, but they aren't in a hurry, because I know that Zahra lives on and she sits in heaven, it's lady. I've cried for so many years, time to make use of these tears. I've cried for so many years, time to make use of these tears before her father's one faith, before her father's one faith burns away and disappears. I heard women were oppressed and were equal half of men. And yet here was a young lady that tore down Caliph's reigning. I heard that a veiled woman was submissive and hidden, and yet she rebelled against a tyrant, demanding what was taken. Men and women are not the same, look how great she became, and me a man worth nothing, unless I'm serving her name. There are lessons to be learnt, hearing of your defiance.  You were not veiled by cloth, rather you were veiled by patience. On earth the stars guide our roots, when stars look to you for their guidance. Come and see a Muslim lady, her anger makes God angry. Heaven lies beneath their feet, our whole purpose basically. That man did not just break your rib, that man he did not break your rib, instead you had broken his throne. They came begging for forgiveness, knowing that they cannot atone. Every woman that is slapped, that is abused, and alone, has you wiped away their tears, whether or not they have known. Every woman that is slapped, that is abused, and alone, has you wiped away their tears, whether or not they have known. And if strikes his wife, a husband, Fatima, shall break their hand. And on judgment day, if he's shia, with that caliph, he shall stand. Because I don't believe that she died.  You'd see her by Rosa Parks refuse her seat on that bus. In fact, maybe she was there and Rosa succeeded thus. I don't believe that she died.  You may see her by Rosa Parks refuse her seat on that bus. In fact, maybe she was there and Rosa succeeded thus. By God, if only they knew of her silent would be few. The lead of all women, they'd stand by her through and through. Truly, we've oppressed Zahra by keeping her within our wars. Truly, we've oppressed Zahra by keeping her within our wars. And yet she stands thanking her Lord whenever a tyrant falls. Women are looking for her. Women are looking for her all over this earth. There are calls, but they have not found her yet. And truly this thought, it appalls. Take Fatima to women. And deep in love they shall fall in. She's the leader of all women, both now to come and have been.  I say peace of the heavens and earth be on the Lady of Light, on the flowers that blossom, from cries that turn day into night. Your flag of Zahra is held by your daughter's steady righteous and upright. Why are veils being banned? Because they are a source of might. No veil threatens countries' heads, unless worn by Muslim heads. Because when freedom is worn all over earth, freedom spreads. She sits somewhere here today.  No doubt when this mosque has closed, to feed someone poor she'd vow. And once she has fed someone poor, she would find tyrants anyhow. This is simply who she is. Oppression, she can't allow. This is the house of Ahmed. They don't sleep until, they don't sleep if poor are unfed. And where there is injustice, you'd find them right at its head. Now, the time has come. Now, that I have found lessons, I find my eyes twitching. Now, that I have found lessons, I find my eyes twitching. Tears are begging me to flow. Her tail they are remembering. They remember that her rib broke whilst Allah's house was burning.  The Qur'an's chapter kotha left burnt and wounded, with a scar how many women lament her unborn taken from her. The sons of Zahra are known as the men who would strike their chest. Both for her broken rib and for Hussain's chest by the horses pressed. Our revolt is eternal, both internal and professed. It screams out when to our Lord, Mehdi's return we request. All of those who in goodness relied shall stand beside Mehdi's side. All those left hurt by this world, no more will they have to hide. O Zahra, you'd not meet an early sleep. Just so, you would lament and weep. Flowers blossom from your wounds and from your wounds lessen sleep. Flowers blossom from your wounds and from your wounds lessen sleep. And from your wounds lessen sleep.


But if they're not in work, they don't stay. Now we're assessing the degree of energy in all of Sweden. The United States is very good. And the three nations are positive. We have seen almost as many of them as non-northodox democracies in three institutions, but the best of them is Wall Street, and the best of them is North Korea. We would, once again, elude us. And then we would have to begin again to stand together again in facing the menace of war and totalitarian tyranny. Even then, we did not lose heart. At Stand Together, we have. When the first few were here, we were referred to a decade before. At this point, for our age and for all time, we're continuing this work. It's profoundly reassuring to me that we're all in parallel at three. That you will continue to share with the American people our thanks, our affection, and our determination to stand with you until freedom has triumphed. Sir Winston put it very well when he said, the day may dawn when we will never flinch, never weary, never despair. After eight historic years of your presidency, one of the blitzest in America's history, Peterson, look, all that you have accomplished over these eight years on behalf of this great nation and of three people everywhere. And an opportunity to look again to the bright promise of the future. Two things. That the decade would be less dangerous if the West maintained the strength required for peace. But above all, Mr. President, you have restored faith in the American dream. A dream of boundless opportunity, built on enterprise, individual effort. Give it a go, Mr. President, and the world's best office. You start.


Course in Miracles Live. We are walking with Jesus, holding his hand and listening and following and opening our hearts to Jesus and really to the precious love of God that is our birthright, our inheritance. So today we're going to go into Lesson 45. God is the mind with which I think. And this is so beautiful because really the Holy Spirit was given as the Comforter and the Instructor. And basically this lesson is really telling us God is the mind with which I think. The Holy Spirit is the mind with which I think. It is that identification with that thought system of love that is real and true. That is truth. Today's idea holds the key to what your real thoughts are. They are nothing that you think you think just as nothing that you think you see is related to vision in any way. There is no relationship between what is real and what you think is real. Nothing that you think are your real thoughts. Resemble your real thoughts in any respect. Nothing that you think you see bears any resemblance to what vision will show you. You think with the mind of God. I remember I've emphasized a lot of times too that Christ is an idea in the mind of God. He's starting off the second sentence. You think with the mind of God. Therefore, you share your thoughts with Him as He shares His with you. They are the same thoughts because they are thought by the same mind. To share is to make a like or to make one. Nor do the thoughts you think with the mind of God leave your mind because thoughts do not leave their source. Therefore, your thoughts are in the mind of God as you are. They are in your mind as well where He is. As you are part of His mind, so are your thoughts part of His mind. Where then are your real thoughts? Today, we will attempt to reach them. We will have to look for them in your mind because that is where they are. They must still be there because they cannot have left their source. What is thought by the mind of God is eternal, being part of creation. Our three five minute practice periods for today will take the same general form that we use in applying yesterday's idea. We will attempt to leave the unreal and seek for the real. We will deny the world in favor of truth. We will not let the thoughts of this world hold us back. We will not let the beliefs of the world tell us that what God would have us do is impossible. Instead, we will try to recognize that only what God would have us do is possible. We will also try to understand that only what God would have us do is what we want to do. And we will also try to remember that we cannot fail in doing what He would have us do. There is every reason to feel confident that we will succeed today. It is the will of God. Begin the exercises for today by repeating the idea to yourself, closing your eyes as you do, then spend a fairly short period in thinking a few relevant thoughts of your own, keeping in mind the idea of the day. After you have added some four or five thoughts of your own to the idea, repeat it again and tell yourself gently, my real thoughts are in my mind. I would like to find them. Then try to go past all the unreal thoughts that cover the truth in your mind and reach to the eternal. Under all the senseless thoughts and mad ideas with which you have cluttered up your mind are the thoughts that you thought with God in the beginning. They are there in your mind, now completely unchanged. They will always be in your mind exactly as they always were. Everything you have thought since then will change, but the foundation on which it rests is wholly changeless. It is this foundation toward which the exercises for today are directed. Here is your mind joined with the mind of God. Here are your thoughts, one with his. For this kind of practice, only one thing is necessary. Approach it as you would and alter dedicated in heaven to God the Father and to God the Son. For such is the place you are trying to reach. You will probably be unable as yet to realize how high you are trying to go, yet even with the little understanding you have already gained. You should be able to remind yourself that this is no idle game, but an exercise in holiness and an attempt to reach the kingdom of heaven. In the shorter exercise periods for today, try to remember how important it is to you to understand the holiness of the mind that thinks with God. Take a minute or two as you repeat the idea throughout the day to appreciate your mind's holiness. Stand aside however briefly from all thoughts that are unwordly of him whose host you are and thank him for the thoughts he is thinking with you. God is the mind with which I think beautiful. And to support us with today's lesson, the section that came to me from the text is from chapter five, section two, the voice for God. Healing is not creating, it is reparation. The Holy Spirit promotes healing by looking beyond it to what the children of God were before healing was needed and will be when they have been healed. This alteration of the time sequence should be quite familiar because it is very similar to the shift in perception of time that the miracle introduces. The Holy Spirit is the motivation for miracle-mindedness, the decision to heal the separation by letting it go.  And although you can keep it asleep, you cannot obliterate it. God himself keeps your well alive by transmitting it from his mind to yours as long as there is time. The miracle itself is a reflection of this union of will between father and son. The Holy Spirit is the spirit of joy. He is the call to return with which God blessed the minds of his separated sons. This is the vocation of the mind. The mind had no calling until the separation because before that it had only been and would not have understood the call to right thinking. The Holy Spirit is God's answer to the separation. The means by which the atonement heals until the whole mind returns to creating. The principle of atonement and the separation began at the same time. When the ego was made, God placed the mind in the mind, the call to joy. This call is so strong that the ego always dissolves at its sound. That is why you must choose to hear one of two voices within you. One you made yourself and that one is not of God, but the other is given you by God who asks you only to listen to it. The Holy Spirit is in you in a very literal sense. His is the voice that calls you back to where you were before and will be again. It is possible even in this world to hear only that voice and no other. It takes effort and great willingness to learn. It is the final lesson that I learned and God's sons are as equal as learners as they are as sons. You are the kingdom of heaven, but you have let the belief in darkness enter your mind and so you need a new light. The Holy Spirit is the radiance that you must let banish the idea of darkness. His is the glory before which the association falls away and the kingdom of heaven breaks through into its own. For the separation, you did not need guidance. You knew as you will know again, but as you do not know now. God does not guide because he can share only perfect knowledge. Guidance is evaluative because it implies there is a right way and also a wrong way. One to be chosen and the other to be avoided. By choosing one, you give up the other. The choice for the Holy Spirit is the choice for God. God is not in you in a literal sense.  When you chose to leave him, he gave you a voice to speak for him because he could no longer share his knowledge with you without hindrance. Direct communication was broken because you had made another voice. The Holy Spirit calls you both to remember and to forget. You have chosen to be in a state of opposition in which opposites are possible. As a result, there are choices you must make. In the Holy State, the will is free so that its creative power is unlimited and choice is meaningless. Freedom to choose is the same power as freedom to create, but its application is different. Choosing depends on a split mind. The Holy Spirit is one way of choosing. God did not leave his children comfortless even though they chose to leave him. The voice they put in their minds was not the voice for his will for which the Holy Spirit speaks. The voice of the Holy Spirit does not command because it is incapable of arrogance. It does not demand because it does not seek control. It does not overcome because it does not attack. It merely reminds. It is compelling only because of what it reminds you of. It brings to your mind the other way, remaining quiet even in the midst of the turmoil you may make. The voice for God is always quiet because it speaks of peace. Peace is stronger than war because it heals. War is division, not increase. No one gains from strife. What profiteth a man if he gained the whole world and lose his own soul? If you listen to the wrong voice you have lost sight of your soul. You cannot lose it, but you can not know it. It is therefore lost to you until you choose right. The Holy Spirit is your guide in choosing. He is the part of your mind that always speaks for the right mind, the right choice because he speaks for God. He is your remaining communication with God which you can interrupt but cannot destroy. The Holy Spirit is the way in which God's will is done on earth as it is in heaven. Both heaven and earth are in you because the call of both is in your mind. The voice for God comes from your own altars to him. These altars are not things, they are devotions. Yet you have other devotions now. Your divided devotion has given you the two voices and you must choose at which altar you want to serve. The call you answer now is an evaluation because it is a decision. The decision is very simple. It is made on the basis of which call is worth more to you. My mind will always be like yours because we were created as equals. It was only my decision that gave me all power in heaven and earth. My only gift to you is to help you make the same decision. This decision is the choice to share it because the decision itself is the decision to share. It is made by giving and is therefore the one choice that resembles true creation. I am your model for decision. By deciding for God, I showed you that this decision can be made and that you can make it. I have assured you that the mind that decided for me is also in you and that you can let it change you just as it changed me. This mind is unequivocal because it hears only one voice and answers in only one way. You are the light of the world with me. Rest does not come from sleeping but from waking. The Holy Spirit is the call to awaken and be glad. The world is very tired because it is the idea of weariness. Our task is the joyous one of waking it to the call for God. Everyone will answer the call of the Spirit, the Holy Spirit or the Sonship cannot be as one. What better vocation could there be for any part of the kingdom than to restore it to the perfect integration that can make it whole? Hear only this through the Holy Spirit within you and teach your brothers to listen as I am teaching you. When you are tempted by the wrong voice, call on me to remind you how to heal by sharing my decision and making it stronger. As we share this goal, we increase its power to attract the whole Sonship and to bring it back into the oneness in which it was created. Remember that yoke means join together and burden means message. Let us restate, my yoke is easy and my burden light in this way. Let us join together for my message is light. I have enjoined you to behave as I behave but we must respond to the same mind to do this.  He teaches you how to keep me as the model for your thought and to behave like me as a result. The power of our joint motivation is beyond belief but not beyond accomplishment. What we can accomplish together has no limits because the call for God is the call to the unlimited.   The Holy Spirit is the mind with which I think. In heaven, I have no thoughts apart from God and when I think only with the Holy Spirit then I remember God is always for me, for the real me, for the Christ me, for the waking up to the Christ me. It's the capital self, the Christ divine self that is where the truth is. That's what's real and true, it's innocent, it's eternal, it's changeless. It's an amazing, glorious experience. It's the direct experience of I and the Father are one. Yeah, it leaves the world behind with a smile, gentle nod.  So this is the inspired life. I remember that's how I got into this whole teaching. I was raised Christian, a loving Christian family, a beautiful church. In the summertime going to Bible camp and then when I got into my teen years working towards what seemed to be called confirmation in the church when I was 18 and then a period of real deep soul searching, you could say, searching for my soul, searching for my spirit through my university days in Cincinnati and basically it was a lot of clearing, a lot of emptying out, a lot of, you might say trial and error experientially, opening up my mind, opening up my heart with an eagerness to be shown the way and then the culmination of those 10 years of full-time university study, then, yeah, it was like an answer to a deep prayer in my heart for a real direct fast shot, a real direct path to God. And that's when the Course in Miracles came into my life, into my hands for the first time. Yeah, that was really beautiful in La Jolla, California, Southern California, not too far from San Diego. It was just like a tsunami wave of love washing over me and a very strong feeling that my life would not be the same, that my way was set in towards God, towards truth, towards freedom, towards happiness in a deep way that I couldn't fully understand at this point, but I actually could feel it very strongly. I was like being pulled. I felt the call. I was inspired. I was swept away. I was swept off my feet and thus began this, yeah, wonderful, amazing journey without distance to a goal that has never changed. It's always the present moment. It's always now, no matter what our body seems to go through or what situation, circumstances seem to come before us, it's really just answering the call of our heart to remember God, remember this divine love. So, and yeah, then you go along and it seems like with each phase it comes from studying the Course to practicing these deep teachings of Jesus to letting go and then letting go some more and then letting go again and yet again and yet again. It's just a journey of letting go of everything you think you think and think you know. It's a journey of emptying your mind of all thoughts, of all beliefs, of all, you might say, ego scraps of ego and scraps of ego emotions, all doubt thoughts. You're just letting them up and letting them out, letting them go. So, we're here to cheer each other on in this journey without distance, in this recognition. It's just a present recognition is all it is. As Jesus says, enlightenment is but a recognition, not a change at all. It's just a remembering, it's a recognition and we give thanks for this pathway that has been given us. We pray for help, we pray to escape the distortions and deceptions of the world. We pray to escape the pursuit of illusions and say, no, I would only accept the truth. Show me the truth. Take me to divine love. Let me feel a love that never ends. Let me have a glimpse of eternity that I may remember the fullness and the expansiveness of eternity. So, thank you for being on this journey together. We are walking hand in hand, arm in arm, side by side to the light of truth. Every day it's a let go. Yeah, we've come to see that. Every day we have to let go. Let go of believing that we're in charge of anything. Let go of trying to control anything or anyone. Let go of our egoic ambitions for the future and let go of our egoic regrets from the past. Ha, time goes by slowly, softly, gently, without its touch upon us in this present moment. We sink beneath the timeline. We come within to the acceptance of the love of God. God bless you. Go forth today. Remember our lesson for today. God is the mind with which I think. Love you. I'm gonna be stepping away from my gadgets again, my computer, my phone and iPad and just having some beautiful prayer time gently going within and going with the flow of that. So, thank you. I'll be back though. I love you so much.


It's fun to be here. Um, so I think a geometric group theory we enjoy, we like to talk about groups that act nicely on nice spaces, where the nice spaces are usually kind of combinatorial objects, you know, cube complexes, curved complexes, graphs, uh, things that kind of have a combinatorial flavor. I'm going to talk about groups that act on the circle, and this is not a kind of discrete object at all, except secretly it is. Um, and although I won't kind of highlight why secretly it is in my talk, um, because that involves co-cycles and things like this, um, I do want to highlight some parallels between groups of homeomorphisms of the circle and sort of some of the other groups that we've seen so far and approaches to- to- to talking about both. Um, so my plan is, here's the plan. Uh, in the first half, so that's this hour, I'm going to talk about some basics, let's call them fundamentals. These to prove a theorem, um, that would be the end of this half of it. Um, I'm going to prove in quotation marks, a Titz alternative for groups that act on the circle, or for the whole group of homeomorphisms of the circle. Okay. This- this says orientation preserving. I like oriented things. So, uh, this is the group of orientation preserving homeomorphisms. No- no flips. Um, and, uh, I'll say in a second what that is, and then in the next hour, I'm going to prove a theorem about a particular action of a particular group. Um, um, about the group of automorphisms of the fundamental group of a surface. So I want G to be at least two here.  You can look at the group of all automorphisms of that group. Great, this is an object we're all good with. Oh, look, and- and this is even a hyperbolic group, so there's some geometry going on there. Um, so this group being a hyperbolic group has a boundary. Um, and the group of automorphisms acts on this boundary. Hopefully that's something you can dig out of your mind from last week. Uh, and what is this boundary? It's a circle. It's a circle. Topologically, it doesn't have any further structure than just being a topological circle. So this action is by homeomorphisms. Okay. And so I'm going to prove a rigidity theorem about this action. It turns out, uh, for this particular group, any action of that group on the circle is- looks like this one exactly. Okay. Uh, so that's going to be an exciting theorem and a way to teach you sort of some- some- some tools for talking about these objects. Uh, and hopefully relate it to things that we've thought about, uh, last week. Um, the Titz alternative. So you- many of you or some of you might know, uh, what this theorem is for linear groups, right? It says that- Titz's theorem is that if you take a finitely generated subgroup of GLN, some linear group, then there's a dichotomy. Either that group contains a non-Abelian free group, or, um, virtually up to finite index it's solvable. Okay. So it sort of looks free or it looks belian. Uh, this is- and- and so in generally we say that, you know, a group G satisfies a Titz alternative if something like this is true. Finally generated subgroups are either looking like solvable or contain free groups. This isn't quite true for groups acting on the circle generally, but something very close is in- due to Margulis. Uh, so he showed that any, not even finitely generated, but any subgroup of homeomorphisms of the circle, um, either contains a non-Abelian free group, either contains an F2, um, or the amlog being solvable or looking kind of like your Abelian or in this case amenable. Uh, the best analog you can have here, um, is that there is a measure on the circle invariant under your group. Okay. So, or there is a gamma invariant probability measure on the circle. This isn't a dichotomy. It's- it's possible that you could have both happening at the same time. But I'll explain sort of what all these words mean. Uh, when we get there. Uh, well, let's start right at the beginning. How about some examples, uh, of groups of homeomorphisms of the circle. I'm going to give you some examples to convince you that, uh, this is a- this- this big group is a rich and interesting object. Okay. Uh, so we saw one already, uh, odd pi 1 sigma g. Uh, that's- that's a pretty complicated group. Uh, let's maybe try and go, um, for- for- for other things. Uh, how about like just the group of rotations? So I can think of the circle as like, I don't know, you know, uh, unit vectors in the plane I can rotate. Um, another thing we've already seen so far, let's see what have we seen. Uh, last week we saw PSL2R acts on RP1 by Mobius transformations. And I mean RP1 is just a circle. So that gives you, uh, a nice action of this group on the circle and the SO2 subgroup inside here is acting by rotations. What else have we already seen? Uh, oh, this morning we saw Thompson's group, uh, group F. This acts on the interval by piecewise linear homeomorphisms. And if I take my interval and I just glue the two sides together, I get a circle. So this acts on the circle with a fixed point. Uh, but there's even a- there's a- there's a slightly bigger Thompson's group, uh, that acts on the circle. This is contained in a group, um, T, uh, which is in between F and the group V advertised before. Okay. Uh, T is what happens. It's a group- subgroup generated by F acting on, uh, S1 which you think of as this- this interval with the two endpoints identified. Okay. By piecewise linear, that gives you a nice linear structure. Um, by piecewise linear homeomorphisms plus, I- now I'm gonna allow you to move the endpoint that I glued. So you can take this group to be generated by this and there's an order two rigid rotation. Okay. So, uh, silly way to write this if you like more- more- this kind of notation is the transformation x goes to x plus, uh, a half mod 1. Okay. Where I think of the circle is, uh, this interval mod mod 1 or r mod z if you like. Um, and this group is, uh, interesting and special because, we thought that this group was not simple, although it's commutator subgroup is. This group is another example of a simple group. Um, good. So I've got some particular examples. Let's do a nice, uh, uh, very hands-on example or a couple. Um, uh, free groups fit inside of here. Okay. In lots of different ways. In fact, you can prove if you decide you want to challenge that the generic two in the sense of bear, even generic two homeomorphisms of the circle generate a free group, but I want a particular example that we should hold in our minds for later. You can do this even in PSL2R, but I'm gonna forget that I know anything about PSL2R. Um, I'm gonna produce, uh, free group generated by two homeomorphisms say f and g. Okay. So, uh, what do I want these to do? Here's my circle. I'll give some instructions to f. So I want f to be anything that takes, let's take a little interval here and stretches it out. So I want f to take this interval and stretch it out to being something like this. Okay. So f of i is going to be really long like this. And so what's left over, I wanted to take the complement and shrink it. Okay. So this interval here, I can call f of circle minus i. And g, I'm gonna have to do the same thing to two other intervals. I know maybe I'll call this j. And I'll do the same kind of stretch and shrink. Just take your elastic circle and pull it this way and smooth it that way. So that this interval over here is g of the complement of j. Okay. I claim that these two things generate a free group. Okay. Just as I've described though, I don't need to give you any more information. And there's only one way that we really know how to prove, okay, there's only one widely applicable way that we know how to prove that things generate a free group and this is the ping-pong argument, right? So actually what I claim is that if I take a point, I don't know, say this point x, that is not in any of these colored in regions that I drew. I have a, definitely I have a map that takes a point, that takes a word in my free group. Okay. And sends this to, so it's some abstract word in f and g and f of inverse and g inverse. I just take that word and I apply it to my point x, that this is now a homeomorphism, it sends x somewhere else. Okay. I claim that this map is injective. So not only is this group free, this point has trivial stabilizer. So for some of you, there's a one line proof or there's a one word proof and it's the word ping-pong. And for others of you, here's how you do it. The proof is by induction on word length, okay? And I'll do the, okay, the base case word length zero, that's trivial word, okay, I don't know. Let's do the case of word length one. I want to show that f and g and f inverse and g inverse all send x to different points. So let's check. If I apply f, I'm supposed to take this guy and shrink it into there. So I would go here, somewhere in this region. If I apply g, I'm supposed to land in, I'm not in j, so I'm supposed to land in that region. Okay, so these are different. If I apply the inverse of f, the inverse of f, what takes this and stretches it down, so it'll send x into here. So f inverse sends x into this region, and g inverse, same kind of logic, sends it into that region. And so these regions, the four of them distinguish the image of x under words of length one. And, okay, so that's the base case. And to do this longer, you assume that you sort of have this already. And I can actually take further nested intervals, right, inside of this picture of this interval. If I draw the image of these guys under f, I'll see some little sub-intervals in here. I'm supposed to take this and smoosh it. So I'll see a picture like that. And you can use these kind of refinements, these level two, to distinguish the image of x under words of length two, et cetera, et cetera. So this is a great exercise to do if you haven't had to do one of these before. And I'll leave it as proof by audience. Great, so that's a nice example of how to make a free group. And actually, if you are an expert at this, there's one thing here that people don't often notice. If you use this very by hand induction proof to do this, you don't only recover that this is injective. You specify the location of these intervals, i, j, and then f of the complement, g of the complement, is enough to determine the cyclic order of this point x. If I want to know if some first word of x, if I start reading around to the right, do I encounter a word or a second word, first or second in what order? That's recovered just from this combinatorial data. And that will come out of the proof. I mentioned that because that secretly lives in the proof of Ethereum, I'll quote later, but it's not necessary for understanding everything today. Okay, so there's free groups. Maybe just one more example of very pervasive things. All right, so I went from complicated groups down to simpler ones. Let's go for maybe one of the easiest groups. How about just an infinite cyclic group? It's very easy to produce infinite cyclic groups. Here is one way to do it, where, for example, f could be specified as follows. So I take my circle, I'm going to choose some closed set. So it might be finite, or it might contain some intervals, or it might have a part that looks like a canner set, or some accumulation points, I don't care. Take any closed set, there's k. And on every complementary region, so every interval in the complement, like here's an interval in the complement, I can identify this interval with the real line, it's topologically the same. And under whatever identification I looked, I chose, I can have f in these coordinates, I can make it look like x goes to x plus 1, just the translation. So this closed set is going to specify the set of points fixed by f, and the complementary intervals are going to be the non-fixed points. And for each of them, I have a choice of whether I thought infinity was this way, or infinity was that way in the real line, which way I thought was positive or negative. And that'll determine whether f is shifting points to the right or to the left. So I could choose a function positive or negative to choose right or left, maybe positive there on this complementary interval, maybe I wanted it to go the other way, minus. So this data is really an assignment, or a little more specific, than an assignment of complementary intervals to pluses or minuses, telling you the orientation of the line, which way points are moving, clockwise or anticlockwise. So what's a fancy way to write that? I don't know, pi not of the circle minus k, all right, point regions in the complement to the set containing plus or minus. So that gives you lots and lots of choices of ways to produce homeomorphisms. All of these will be infinite orders, so they'll generate a z subgroup. And the fact is that this data, the closed set in the plus or minus assignment is a conjugacy invariant. So the pair k and this assignment up to conjugacy. So obviously I could rotate my set k and rotate all the places I put pluses and minuses, and I'd have the same kind of looking thing. I'd have a conjugate homeomorphism. Okay, is a complete conjugacy invariant. So it distinguishes homeomorphisms up to conjugacy in homeomorphisms of the circle if k is not empty. Okay, so provided you know you have some fixed points, you can always write it down using this recipe and the recipe. The topological data of this is enough to specify up to conjugacy. So why I like homeomorphisms rather than diffeomorphisms, you can kind of draw a picture of them. Can you draw any homeomorphism like that? No, like here's a bad example. What if I take rotations? Okay, so here's a remark, not generally. Okay, for example, how am I supposed to draw a picture distinguishing a rotation by angle pi and one of angle, I don't know, pi over three or something like this? Both of them are just saying move all your points around some amount. So I don't know what to do. For example, rotations, non-trivial rotations I guess. I'm sorry, does it mean that every homeomorphism of r conjugate to the substitution? Yes, with no fixed points. Every fixed point free homeomorphism of r is conjugate to translation. I mean that is the proof of this fact basically. It's conjugate to translation by an orientation preserving or reversing homeomorphism. And that you can actually do by hand. You don't need a sophisticated theorem to build conjugacy. So that's a nice, didn't put it on the sheet, but that's a nice exercise to do. Okay, so actually let's solve this problem here and try and write down something that looks like, I don't know, a conjugacy invariant that will pick out different rotations, okay? My goal being to come up with a number or something easy to write down that will solve my problem. And I don't have to invent this, this is secretly hidden in work that goes all the way back to Poincare. This is Poincare's rotation number. And it's something that will play a giant role in the second half of this in hour two, okay? So let me set up some definitions. So what does the rotation number do? What I want to do is I want to pick a point on the circle. I have a homeomorphism. I want to capture on average, as I iterate, how far does this point go around? Okay, so you want to capture the average amount, you move around the circle under iterates of f, okay? So to do that, rather than, this guy doesn't have a fixed point of imagining I'm really rotating a circle, rather than remembering every time I crossed over and went around a time and then another time and then a third time. An efficient way to do that is to lift to the universal cover and unwind the circle, okay? So I'm going to start by working there. Okay, so let's set some notation. Here's the definition slash notation.  But if I remember the circle is there, right? I have this deck transformation which is translating by integers. So we'll say homeo of r superscript z, okay? Just like in many areas of math, you write this as the invariance. This is the set of homeomorphisms of the real line which commuting, which commute with x maps to x plus one, integer translation. And hence they commute with all translation by integer amount, okay? If I have something that commutes with this, well then it defines a map on the quotient of r by this map, right? So there's a surjection from here to the homeomorphisms of r mod z.  I think you do automatically if you commute with this, but this will make it clear, which is just homeomorphisms of the circle. So this surjects, and the kernel of this map is just this translation. So this actually lies in some short exact sequence like this. Where this one is generated by x goes to x plus one. So lifting to the line, let's me think of elements there up here and that counts how far I move around or how many times I've wound around. Okay, I'm gonna define the rotation number upstairs first. So for what's good notation for a lift, maybe with a little squiggle? For one of these homeomorphisms of the line we'll define the lifted rotation number of f to be, what do I want? I want the average amount, it translates a point, okay? So I'll take a point, let's choose zero. That's a good point in the line. I'll take f, I'll iterate it n times and to get an average I'll divide by n and I want the limit. I didn't leave myself as much space as I hoped for. I want the limit as n goes to infinity of this quantity. One of your exercises is to prove that this limit exists. And in fact, zero wasn't a special choice. I could have taken any point x and I'd still get the same number, okay? So this is, it's annoying to do on the back, blackboard, but I've given it to you with some hints. This is a well-defined number, it's just some real number. That is the rotation number of f. And if I looked downstairs, everything makes sense, mod z now. So for a homeomorphism of the circle, I'll say the rotation number of f is, I'll take a same kind of limit. I'm gonna choose any lift I want, apply it to any point I want. I picked zero, apply it n times divide by n. And this will give me a number, but if I took different lifts, that's like composing this with a translation by one, okay? That will change my average translation amount by one or by two or by whatever power I chose. So this is only well-defined mod integers. And that is the rotation number or the lifted rotation number. So let's do a little mental check, I really, if I have a rotation by, I don't know, maybe that one that's like x goes to x plus a half mod one, that order to rotation, hopefully I would get a half out of this. Well, let's see, let's pick the lift that's really x goes to x plus a half. If I do that n times, I'll add n times a half to zero. I'll divide by n, I'll get a half. I don't even have to pass to a limit, this is great. So the rotation number of a half mod z is a half. And indeed, that gets rotation number a half. So in general, the rotation number of rotations is exactly what you think they should be, rigid rotations. This has lots and lots and lots and lots of nice properties. So let's write some of them down to use. And again, these are things that I'm asking you as exercises to check. I'm going to clear on the definition. Okay, so properties. Okay, so I already said that rotations or their lifts, translations, if I take any real number, this is alpha. That's pretty immediate from the definition. It has a homogeneity property. Meaning that the rotation number, the lifted version of a power of f, k would be any integer, is k times the rotation number of f. Including for inverses. Better than this, so this is a particular abelian subgroup, right? It's a subgroup generated by f. And I'm saying it looks like a homomorphism, an additive homomorphism that are on this abelian subgroup, right? This is true in general for things that commute. That's all you need to prove this. So it's a homomorphism to some subgroup of real numbers as an additive group. When restricted to abelian subgroups, but not in general, all right? As you may even know from say multiplying matrices in PSL2R, you can write a product of two things with six points to hyperbolic matrices. You can write a rotation, a non-trivial rotation of a product of two hyperbolic things. Hyperbolic guys will get rotation number zero, they have fixed points. Your rotation is whatever you wanted it to be, okay? So not, not in general. But it is generally a conjugacy invariant. Meaning that the rotation number of conjugate of f is the same as the rotation number of f. And although it's not a homomorphism in general, it's up to bounded error, which is all we care about in geometric group theory, right? Homomorphism, namely it's what's called a quasimorphism. What does quasimorphism mean? It means something that is bounded distance away from a homomorphism in the following sense that satisfies that if I compare the rotation number of a product, here my product, my, my operation is like function composition, right? If I compare this to the rotation number of f and the rotation number of g, well, if it really was a homomorphism, if the sum of these should be the same as that, right? So let's see what happens if I take this and I subtract off these guys, okay? Well, the claim is that this is uniformly bounded. And in fact, in this case, you can show that it's bounded by one in absolute value. Excuse me? Yes? Are there any non-trivial, obedient subgroups of the group of homomorphisms? SO2, z, z to the n, all kinds of things. That don't eat within things with fixed points, right? You could think of just some little interval of the circle. You could identify with the real line and make a group of translation. Okay, but if there are no fixed points, then there are weirder examples than things in SO2, yeah? And I'll actually, how about I show you one right now, making sure that in a second, once I make sure I've told you all the properties I'm gonna use later on, yeah? When does the quality hold? Good question, if f and g don't commute, typically not, okay? But you can also construct examples where f and g don't commute and the quality does hold. So, oh, sorry, when does the quality hold? Sorry, I meant, I was answering some version of this question where I thought, when are these, when is this zero? Do you mean when are these, when is this one or when is this zero? When is this one? Yeah, when is it zero? Okay, so I answered the zero question. Actually, both questions are interesting. And in general, if I give you f and I give you g and I ask and I tell you what these are and I'm like, what are the possible values can this guy take? That is an interesting problem. And in certain cases, for this particular question, there's a complete answer which I can show you a picture of. There is a graph you can draw, what are the possible values? It has this crazy self-similar fractal structure. This was done algorithmically by Danny Caligari and Alden Walker. And they have a computer program that sort of answers this question in many cases. So yes, you asked actually a very subtle and interesting question. What values does this take when, based on what data? Let me take a slight detour for a minute which might help us over there. I wanna talk about an example we saw later last week. Jair Minsky had a picture where he was trying to describe how you make a lamination that's not a foliation on the torus or on the punctured torus. And I'm gonna be just very slightly less sketchy than he was. He started with, if I take, you wanna understand that the torus are in annulus with foliation of irrational slope. So one way to make an irrational slope foliation on a torus is I'm gonna take, here's my circle, so I'm gluing these two sides together, cross interval, all right? And I'm gonna mod out by an equivalence relation where I'm gonna glue top and bottom of the interval by an irrational rotation or translation. If I do that, well, that'll take this vertical stripes under my gluing. What am I supposed to do? I'm supposed to shift them all over before I think of them being attached back down, right? So that's like, I've really identified my, taken like a slope like this and I've used this to identify my stripes, okay? And if I sort of shear this back over to say, hey, just glue them up and down normally. You'll get an irrational slope foliation of your torus, okay? Then he's like, well, here, instead of doing that, I want one where it's not all dense. And in fact, I only want these sort of going on a canter set of points. I'm gonna like take one of these leaves that goes wrong and I'm gonna thicken it up a little bit and then thicken it, but not so much over here and something like this. If instead you glue by not rigid rotation or translation, but by some homeomorphism F that, say, has rotation number alpha, but not conjugate to a real rotation, okay? The result is that you'll produce something that has a canter set somewhere here identified with itself by kind of a shifting over. And this picture will look exactly like the picture we saw before, okay? Where there's stretched out and thinned out lines following some canter set. So if you did the, if you got interested in his lecture and the Dangeois counter example, as was suggested then, the Dangeois counter example is exactly constructing a function that satisfies this property that lets you build one of these strange relations. I wanna explain sort of what's actually going on there, but I wanted it to tie it to his lecture first. Okay, so consider that maybe motivation for what's about to, what's about to come now. And this is a very useful kind of dynamical trichotomy that says that pictures either look like this or look normal. So here is a theorem which I'll need for what I wanna do next. And so as the following, if you take any group acting on the circle, so any group of homeomorphisms of the circle, then either, this is an exclusive either or. Okay, one of three things can happen. Okay, one, it could have like a fixed point or more generally a finite orbit. A finite orbit. That's one possibility. Another possibility is I don't even have like a very rich large group like PSL2R or something like this or like a surface, co-compact service group in there. Another thing that could happen is that all orbits are dense. This also happens if your group is just like a single infinite order rotation. And the claim is if neither of these two things happens, then something weird goes on. There, an invariant, so a gamma invariant, closed subset. How do I wanna state this so that I don't, let's give it a name, k. It's contained in the closure of every orbit and so that the restriction to k has all orbits dense. All orbits on k, dense in k. It doesn't tell you very much what k looks like, but I can say a lot more. k in case three is homeomorphic to a canter set, and it's unique. It's the unique set with the properties listed here. closed set contained in closure every orbit with the action has all orbits dense. This is often called the action being minimal. There is a gamma advanced closed subset k containing the closure of every orbit with all orbits k, dense in k. So, let's say this in English words with the action on k is minimal. Yeah, so I can forget the rest of the circle ever existed. I now have an action on something I claim is a canter set and all of the orbits of this are dense. So I could make it bigger and not have something like this happen. If I take a minimal such example, I'll get like a smallest possible set that satisfies these, I'll get this property, but it's not necessarily... For example, the whole circle is a gamma invariant closed set contained in the... Oh, contained in... Yeah, you're right. You're right. If I put it this way, containing closure every orbit. Yes, that's true. This is a constant. I didn't realize what I had wrote. You're correct. Let me give you a quick proof of this because it's good to prove some things. I mean, one way to approach this is you say, well, suppose neither of these holds and like let's find k. But very generally, we could start right from the beginning by being like, okay, what am I aiming for? Let's look at... I'm trying to understand closures of orbits. So let's look at the set of subsets of the circle consisting of orbit closures. So this is a set of things that look like gamma on a point, the closure of this where x is some point in the circle. This is partially ordered by inclusion. So I can take a minimal element. So you show that it has the finite intersection property or whatever you want. Take a minimal element. Under this ordering. If that's finite, you just showed that some orbit is finite. Great. So if your minimal element is a finite set, you get case one. If your minimal element is the whole circle, what does that mean? That means that the closure of every orbit is the whole circle. So we're in case two. And if it's not, well, what is a minimal element of this? In general, it will be some gamma invariant set because orbit closures are gamma invariant. It's going to be a closed subset. And I claim it will look like a canter set. So every point should be an accumulation point. So it has some accumulation points. And the set of accumulation points will be invariant on your group action. Your group is acting by homeomorphisms. So if I didn't want this, if there were some non-accumulation points, I could throw them out and I'd get something smaller. So every point is an accumulation point. It's perfect. And the same kind of argument says it has an empty interior. Otherwise, I'd throw out the interior and I'd get some smaller closed invariant under my group set. So the closure of an orbit in that point would be contained in it, would be smaller. So it must have empty interior. So these last two just come from the fact that I took a minimal element. So that says, oh, I already proved sort of my agenda that this is homeomorphic to a canter set. And the contained in the closure of every orbit is just because I could take an intersection, right, if I had something that was smaller. OK. So that was a little quick and sketchy at the end, but I promise there's nothing too complicated going on. So check that this has the properties that you want. Is the existence of a minimal element guaranteed by compactness? Yeah. You just check that descending things. Yeah. Yeah. OK. Can you give us the measure of space? Nope. You can do one of Yahya's examples if you believe that this really works to make fat canter sets on which you do a shift kind of thing. So things are pretty pathological if you don't, if you just have actions by homeomorphisms. OK. Let me in the remaining 16 minutes or so tell you about what I really wanted to kind of get to, this proof of the, of the, of the Tetz alternative. The rotation number I need, maybe next time, but not right now. And I'm going kind of quickly, my aim is to give you a flavor sort of how some of these kind of arguments go so that you can play with them and imitate them and sort of have a toolkit to play with some problems in the problem session. OK. So let's prove the most general kind of structure theorem that we have for this group. This one, I have a subgroup of homeomorphisms I want to find a free group or an invariant measure. So for instance, if I was acting literally by rotations, my invariant measure would be like Lebesgue measure on the circle. That would be nice. More generally, maybe I could, I won't write the Lebesgue measure one down, but if I'm in case one of there, if I have a finite orbit, I can just put point masses on all of those finitely many points. My orbit has cardinality five. I'll put a point mass of one over fifth on each point. Those points will get cyclically permuted around. That will be an invariant measure under my group action. OK. So you know, so I don't need it to act by literal rotations. All I need to know is that this point, these three, these five points are always, they all stay in their order, right? I'll elements my group, permute these. I can put point masses there to produce an invariant measure. OK. In particular, if I have a single fixed point, then this tits alternative is very easily satisfied. It could have a billion fixed points or uncountably many. I don't care. Just put a point mass at the single fixed point. It never moves. That's an invariant measure. So we're really interested in what happens in two or three. All right. So when this kind of thing doesn't happen, I'm expecting it to look like rotations with Lebesgue measure, some version of that, or to what if I want to prove there's a free group, I better imitate my ping-pong picture, which got erased from before. OK. So I'm looking for one or the other. Well, one thing that would put me in the invariant probability measure would be if I had a group that acted by homeomorphisms where those were equicontinuous, where the action is by some family of homeomorphisms. These are all equicontinuous. I'll remind you what that means in a second if you forgot your analysis. Then the Arzela-Skoley theorem means that your group is compact. It's a fact or an exercise, which you can't do, and I put on the sheet. It's a fun one. Any compact group acting on the circle is conjugate to a group of rotations. So gamma is conjugate. So compact implies gamma conjugate to a subgroup of SO2. So then that conjugate of your Lebesgue measure, putting it back under that conjugate, gives you an invariant measure for the action. You're just acting by homeomorphism. Gamma could be countable or uncountable. Oh, maybe I want it to be countable in what I'm thinking here. No, this is fine. I think fine just in the contact of entomology. Do you have a sort of example that's bothering you? It's not just a finite quantity of compact, but it will be countable. You mean that the closure of it should be... I see. Right. Well, wait. So if the closure of gamma is compact, then its closure is conjugate into SO2.  So that's good enough. Right. Then gamma... Yeah, so that's fine. Then the closure of gamma is relatively compact is fine. So its closure is some subgroup conjugate into SO2. That'll do it. Thanks. Okay, so if you fail to be equicontinuous, if not, okay. So by the book definition says what does it mean?  So that elements of your group stretch them to have a big enough... to have a... bounded from below length. Okay, so let me write that down. So there exists some bounds, some epsilon. There exists intervals where the length of IN goes to zero. And there exists some sequence of elements of your group where the length of GN of IN is at least epsilon. All right, so this is the case that we're worried about and we want to hopefully produce some ping pong out of this and it's looking pretty good because I have like tiny little intervals that get stretched. And that's what I was using in this ping pong picture, stretching and shrinking lengths of intervals. All right, so to make my life easier, I'm going to assume for a little bit that I don't have one of these crazy canter sets that seem like a pathological thing. I'm gonna assume that my action is minimal. All orbits are dense. And then we'll deal with what happens if not else was. So let's assume for a moment a little bit of time that gamma acts minimally. And what I want is to run my ping pong argument.  I want to find something like this and some F that takes this interval to like this big one and then some G that has something like this and it takes this thing and smooshes it down or it takes this one and stretches it over like I had that before. And then I'll have a free group and I'll feel happy. But the problem is that this picture, the way I drew it, doesn't always happen. All right, I might have to draw a more complicated picture. All right, so here's a problem. This picture doesn't happen the way I drew it even in like, I don't know, like SL2R acting on like raised the origin or like the two-fold cover of PSL2R if you like, right? If I wanted to try and do this picture, an SL2R, the thing about SL2R is everything looks the same on both sides of the circle, right? Everything commutes with an order two rotation. It's like an action online. It's not the projectivized version. So I can't have something that takes this giant interval and swooshes it down. Whatever I do up here, a squish is also done down here and I'd have to have another squish. So I wanted to do this in SL2R. I'd have to draw the following picture. I wouldn't, you know, I'd maybe need more of these. And where my F is taking these two points like here and taking these two points there. And so my domains that I was stretching or squishing are no longer connected. And I would want it G to do something like this. And then I could run the same argument, but it would look different. But this is just silly. You know what did I draw? I drew the two-fold cover of this picture. Everything commutes with a rotation of order two in SL2R. Here's the projectivized version. It's the picture I had before. And more generally, if you have some kind of group that acts on the circle, listen, the circle happens to be a K-fold cover of itself. So I could look at all the lifts to a K-fold cover and I'd get a more complicated looking picture. So this is just to say that I might have this silly issue where I had a kind of a complicated action. I could have just taken a quotient and then I could draw my original ping-pong game from before. So this is nice picture. And this is more complicated.  If I could have just been like, listen, you were really acting on this circle and then you took all the lifts to this cover. So the outline of the proof is to first produce that covering map if there was one and then say, all right, if I didn't have this covering map going on then I have exactly this picture going on. I can find intervals in the configuration I wanted before from my step zero of this lecture example of free group and show that there's a free group in it. And I don't have much time left in this hour so I will maybe outline how this works and we can either take it up next time by request or you can sort of fill in details as you like. Okay. So let's see, if I act minimally, I can massage this statement into something else. So let me massage it right away. Put it right there beside, remind us. So if I pass to a subsequence, these G-N-I-Ns, it's a sequence of intervals in the circle that will house dwarf converge to some set after passing to a subsequence. So I could pass to a subsequence so that this condition will say that, you know, all of these contain a very slightly smaller interval, maybe call that J, and then the inverse images of J under G-N are getting really small. Okay, so this says that there is some interval J of length more than epsilon over 2, I can certainly guarantee, with the length of after passing to my subsequence G-N of J, the inverse of J, I guess, going to 0. So I have a fixed guy that gets shrunken. So let's call this interval a contractible one. If my action is minimal, I can move any point to any point. So minimal implies that I could put J wherever I wanted. That says that for any point in the circle, there exists some point Y, okay, so that the oriented interval between X and Y is contractible in the same sense that J was. There exists a sequence of elements that shrink it to arbitrarily small length, okay. Such that there exists some sequence H-N, depending on X, with the length of H-N of this guy, XY, going to 0. I will pick out if I was secretly covered by a cover by the following function. So I'm going to define a function on my circle that assigns a point X to the biggest possible Y where I can do this. So the supremum preceding clockwise around the circle, so that XY has this property. For example, if you're in PSL2R acting by Mobius transformations, this function happens to be the identity. If you give me a point X and you give me an arbitrarily large interval, there's some very, very, very strong hyperbolic element that takes this to being as short an interval as you want. And the claim is that in general, this might not be the identity. I'll state it as a fact that you can check. This guy is a finite order rotation. Or a finite order homeomorphism. And I've cooked it up so that it commutes with the action of your whole group, meaning that phi of GX is G times phi of X. So these are easy to check from the definition. It's saying, when are you in this picture instead of that picture? Here, phi is order 2. But it commits with your group, so I might as well look downstairs. And to say that phi is the identity map says that at every point, I can take as big an interval as I want. Maybe I took this really giant one here. And I can contract it to intervals as small as I wish. And similarly for one containing some point over here or bound with a left-hand point like this, I can contract it as small as I wish. That is exactly what I need to make the ping-pong argument work. So I'll just summarize and say, so passing to a quotient, phi is the identity in which case you can play the ping-pong argument. Okay, that's a good place to stop. Perhaps that's too much. I should say, if you weren't acting minimally, then we have this Cantor set picture. And I can just pretend my Cantor set was the original circle. You imagine, you know, like Cantor's staircase function that takes your interval with a Cantor set in it and collapses each complementary region to a point. This is some continuous function that makes your Cantor set disappear, right? My Cantor set is invariant, so I could apply the collapsing function and I'll get a new circle on which my group acts by homeomorphisms. I can run this case, all right? Either I was equicontinuous on this new circle and so I can pull back an invariant measure to my old one or on this new circle I could play ping-pong and that must mean I contain a free group. Okay, so in very short, the fact that I'm assuming that gamma acts minimally is no big deal because my other option is this Cantor set. And I think that's a good place to end this first bit.