Hello, everyone, and welcome to Hawaii, the state of clean energy, and I'm your host, Vichuan. Our underwriter is the Hawaii Energy Policy Forum, and that's a program of the Hawaii Natural Energy Institute. I'm very pleased to welcome our guest once again, Toby Kinjade, who's a regular now on my show because he has so many good ideas and lots of good, lots of good flies that he actually customizes them himself. So Toby is a pioneer in the solar industry. I knew he was, but I didn't realize he's been in the solar industry for over 40 years, so about as long as I've been in the hydrogen industry. So today we're going to talk to Thorin about solar energy powering Oahu. Can it be done? Can we make that case? Toby, welcome back to the show, and I got to tell you I'm one of those people that doesn't, I'm not, I don't believe we can do that, but I want you to tell us how we can do that. Fair enough. Aloha, Commander, and thanks for having me. Well, okay, this is a really big question. I mean, and it's such a big question that the whole world turns on it. Can we actually power a modern grid, an energy intensive grid like Oahu with solar energy? Is that a pipe dream? Am I pulling your leg? Am I am I fantasizing or being aspirational? Well, I'm a realist. I've been in the solar industry for a long time and we deal with reality. We deal with water pumping systems, home power systems. So the question for me is all about reality. And that's the only thing I'm interested in. So what we have is, well, I'd love the slide that you have there. You're showing how you can double use land. And with agrivoltaics, like you're showing, this is something very useful. But what's interesting is when you solar power an island or an island grid, we want to use no new land. We want to do it with a footprint so small that you don't even notice it. So is that real? The other thing is we don't want to use our good agricultural land. It's not like we have gobs of gobs of spare land. So we don't want to cover all our ice fields up with arrays, like you saw on the opening slide here. Absolutely. So the big question is, can you actually solar power a grid as intensive as on the island of Oahu? Now, the funny thing about that is that's a big order. I mean, just at first glance, that's a big deal. So hey, let's jump to the first slide and let's just kind of set the scene. So our world, everyone I think would probably agree in reality is a fossil fuel economy. We're fossil fuel run. And so on this slide, we want to talk about very quickly or just briefly, what's the reason we need to change? Well, what are the characteristics of fossil fuels in our civilization? Well, it's toxic. You know, dose makes the poison. So if you have too much toxicity, there's going to be a point where it's going to knock you down. So toxicity, that's a problem. You have the expense. Nobody ever fills their tank up and says, oh, that was cheap. Not lately.  I'll just growl at you. It's big. So that's a problem. And then what's the cost of fuel? How do you know? What's the cost of diesel next week? I don't know. How would anyone know? What's it next year? What's the price of fuel five years from now? So all of the industry all around the world is suffering with these kinds of conditions. So what else do you have inequity? You have a volatile price. You don't know the inequity is everyone needs it and only a few people have it. Well, that's a lot of tension. That's not easy. And then it's scarce because they're not making it anymore.  So that's a lot of tension too. And so it's on a carbon cycle and burning this carbon.  It begs the question, hey, can we really keep doing this? So what's amazing is in the last 30 years, we've burned half the carbon that we've ever burned. So, you know, from the 18th, 19th to 20th century, we've all been burning carbon for 300 years. And yet after all of the public money and all of the outcry and all of the commitment to clean energy, have we actually even moved the dial? Not one bit. The No, it's actually the dial is going backwards. Using more fossil fuel than ever, despite all the investments we've made in wind turbines and solar arrays and all the other, we're just barely keeping pace. Yes. And the key to what you just said is we're not, you know, dropping the emissions by 50%, like the IPCC says, you better do it six years or you're not going to stop what's going to happen. No, we're not going the right way at all. So I think any rational individual would have to look at that and say all of the policy, all of the analysts, all of our work has amounted to nothing. It's in effect. So what's the problem? Well, we just went through a list on the fossil fuel side. So what we're going to propose is that, and for example, in the big question of how can we power the grid of Oahu, we're going to propose a clean hydrogen economy for the world. What does that mean? If we could pull it off, what would it mean? Well, it's the opposite of what we just said. It's non-toxic. It's available to everyone. It's inexpensive because there's no fuel costs. It's just hardware. So you could stretch the cost of the hardware out so the price goes down. It's available to everyone. It's equitable because everyone could do it. It's stable. The price is known. It never goes up because it's just hardware. And it's about a water cycle for the 21st century. This is our future. Now, I'm going to make this case that this is the only path actually that we could take that solves all of our problems everywhere, all at once, basically using the same equipment for everyone. Okay. Now, that's quite a claim. Is it true? Am I kidding you? Is this a delusion? Am I just saying this so I can sleep better at night? Am I aspirational? No, no, no. I don't care about any of that. I'm a realist. I want to know, can we actually save this world? Can we actually get off of this industrial drug that we're addicted to that is going to actually tear down all of our species, acidify our oceans, and we're going to end up as a civilization like every other civilization in history relegated to the dustbin? I don't know. This is the turning point. So today, we're going to ask a really big question. Can we actually power the grid on a walk? Okay. But before I answer that, I'd like to see the next slide. Let me throw that up in you because I want to make a point. The terrible of the spilt milk. The spilt milk. Okay. Imagine, I'm going to ask everyone a question. Imagine you have a table and you put a big jug of milk on and then you're setting the table or doing something and your dog jumps on the table. If you've ever had a dog, they always want to jump on the table and they knock over the jug of milk. Glug, glug, glug, glug. And it's spilling across the table, falls off the table and pools in this little puddle. Okay. Here's my question. What do you do first? Now, if you're an oil company, you'd say, oh, forget about the glug, glug, glug. No problem. It's falling across the table. What are you worried about? You got milk on the table, use a straw. And if it's falling on the floor, hey, you're worried about it. I'll get some tissues and sponges and I'll start mopping it up. Glug, glug, glug. Hey, no problem. In fact, here's the best idea. How about I pull up the shag carpet and let the milk drip into the floorboards? Hey, solve some problem. Out of sight, out of mind. That's called carbon capture and storage.  As long as you have the glug, glug, glug, glug, we're talking about 40 billion tons of air pollution. Air pollution doesn't weigh very much. So 40 billion tons. It's acidifying the ocean since our disaster. So I would submit to you some common senses left the building because all the policymakers, all of the consultants, all of the fossil fuel companies, watch you to look at the puddle on the ground and disregard the glug, glug, glug. So paying no attention to the man behind the curtain. Just don't worry about it. What I would say to you that any third grader has the critical thinking skills to be able to understand when the jug tips over, you don't start going to get towels, you write the jug. Wouldn't anyone agree with that? That's got to be job one. Unfortunately, we're in the situation here because the fossil fuel companies, hey, they're not going to give up. They're not going to roll over. In fact, it's worse. They went to the public money and said, look, look, look, look. We don't, you want you interfering with our business. So get all the public money to go for a clean economy. And I tell you what, give it to us. Yeah, yeah, yeah. We're going to pull up the carpet so you have the sour milk and go on the floor. It'll be someone else's problem. And in fact, this explains why we're in the situation we're in. We're not thinking properly. And so I just want to throw that cartoon up to kind of illustrate, hey, come on, let's bring some common sense and reality to this situation. So thank you for indulging me on that cartoon. Well, okay, so how big is the problem? How much load do we have to take care of? Yeah, that's now that we're going to get to the real question, which you just brought up. Hey, it's a mountain of energy, and it's such an enormous amount of energy, it's extremely intimidating. And anyone who just kind of, you know, at a glance would say, oh, you're talking about Mount Everest, what are you talking about? Everyone knows solar panels are diffuse, not very effective, not a very high energy density, what you want to run an industrial grid, get out of town. Well, really? Well, I'm in the real world. That's where I live. And so let's take a look at that. So let's, let's throw up the first slide. Now, let's answer your question. What are we talking about? What's the point here? Now, I tried to look up, what is the total energy of the grid on a walk-through? And it's a big number. I got a number from 2018, and it said about five and a half era watt hours, that's really in watt hours per year. That's really big. But some people have said from other sources I've read that it's as high as six or maybe even seven, they didn't have references, but I'm going to say A, it could be. But what I'm proposing is, not only are you going to solar power, this is the question we're going to ask, not only solar power the grid during the day, but you have to have enough solar to produce clean hydrogen to provide power at night. And when it's stormy, that becomes when you, when you factor that all in, it's a really big number. And I'm going to say you want to really do this for real. You don't need six or seven terawatt hours. You need 10, really in watt hours per year. So that is a real number. And it's a big number. So I can understand at first glance, you might be like, Oh, what are you talking about? Yeah, it's big. Oh, I don't want all my nice agricultural land covered up with TV either. Yeah, I mean, if we come to the end of our analysis and it takes 50% of the island, no one's going to do that. And all of the critics who have said, you will never solar power a Wahoo. And there are many people that say this, I've heard public utility commissioners say it, I've seen public policy analysts say it, I've heard engineers even say it. I've heard solar advocates say it because they want to be reasonable, I think. But I don't know what their motivation is, but we're going to actually answer this question to everyone's satisfaction. Can you really do this? Okay, so on that slide, if I say 10 terawatt hours, you know, what does that mean? It's hard to put your arm around what that means. Okay, 10 terawatt hours means 10,000 gigawatt hours, which means 10 million megawatt hours, which means 10 billion kilowatt hours. So now we can understand a kilowatt hour because that's on your bill. What's the cost of a retail electricity delivered on a Wahoo? Need 45 cents a kilowatt hour? Not around there. Oh yeah, but every slide where you watch things. Yeah, it's going up and down, but it's a number and everyone understands what a kilowatt hour is. Okay, so what I'm saying is in reality, if you want to have a solar-powered grid 24 seven, 365 days a year, you need 10 billion kilowatt hours per year. Okay, that is big. So so far the critics probably have it on me, but let's get to the end. Okay.  No, I'm already not sugarcoating anything. I'm saying, you don't need six terawatt hours, you need 10 billion kilowatt hours a year. Okay, now that's per year. So let's divide that number by 365 days. And that gives us a daily amount of what you'd have to produce with solar panels. Now I'm not talking wind, I'm not talking about anaerobic digesters or anything else. I am talking about the question we're asking. Can you solar power the island of Oahu as far as my brain is concerned? Okay. So when we do the division, it comes out to you need just north of 27 million kilowatt hours per day. Okay, again, first glance, everyone might just walk out of the room and say, you are never going to get there. Oh, really? Let's find out. Okay, so now we've defined how big the mountain is we have to climb. Right. And it's a big one. Everyone understands that. Okay, so let's go to the next slide. If we're going to propose that you're going to power this island, this whole grid, this 10 billion kilowatt hours per year, this 27 million kilowatt hours per day with solar, let's talk about what is the solar resource on Oahu. Okay, so let's bear with me now. Let's just talk about solar energy for a moment. Solar energy, of course, comes from the sun. Little is reflected off the top of the atmosphere. And what we're concerned with what happens when it finally hits the ground. So what the scientists did is they took a square meter of surface area and they put pyrometers all over the world. This has been done for years and years and they measure the two kinds of solar energy that happen. You know, diffuse light, like from clouds or through the atmosphere, and you have direct beam, and they integrate that and they add it all up and they come out with some an equivalent. So what they've done is they made a whole map of the world and they said, well, you have dawn and dusk and then you have peak sunlight. So if we integrate all of that, any location on earth has a solar resource defined by the peak hours per day. So for example, for example, let's say you're in Tucson, Arizona. I've been in Tucson. I know Tucson. It's hot. They have months soon in the summer. This is all accounted for. Now in Tucson, you get about six and a half peak hours per day of sunlight. Pretty good. In Portland, where I am, you get about three and a half hours of peak sunlight. We're in the Willamette Valley. It's a microclimate. Yeah, yeah, yeah. Okay. Now the question is on Oahu, what is the solar resource? Well, now we understand that you have the windward side, kneeward side, and microclimates, but they've all been analyzing this for years and years and the most reliable number, and I got to pick one. If you look on the map, in the Pacific Rim, and especially on Oahu, it's about five and a half hours per day. You have to give clouds. It's not as good as Tucson, but you've got a lot of good solar energy. Now that's half the equation. That's the energy part. Now we have to talk about how much power and therefore how long will it actually be able to produce power? Okay. So what they've measured is that one square meter of sunlight in 39 and a third inches on a square receives about 1,000 watts of energy per square meter. That's at peak. That means when the sun's up high and you measure what's coming on that square meter of surface, you got 1,000 watts. But that's optical. We're talking about electricity. So what we're going to do is take that same square meter and we're going to put a solar electric photovoltaic panel in that square meter. Now what would you expect the power to be? So again, I have to call out and make a number. What is the efficiency of a solar panel? Well, for our purposes, I'm conservative. I'm not interested in any hope. I want to know what the real number is. So I'm going to choose 15%. Now, is that reasonable? If can we get solar panels that are 15% efficient? Well, yeah. Actually, nowadays you can buy panels. I've seen that 22% advertised, 24% possibly. Anyway, I'm not going to sugarcoat it. I don't want a good answer. I want a real answer. That's the world I'm in. So I'm going to say 15%. Okay, we're almost there. Bear with it.  You get 15% of 1,000, which is 150 watts. That's the power in a moment in any instant at peak you would expect on a Wahoo from a photovoltaic panel that's one square meter. Okay, now energy is power times time. So to calculate how much energy we could produce on a Wahoo from one square meter solar panel, we simply multiply power times time. 150 watts is the power times five and a half hours, which is what the government says you've got that they've worked out. This is true. It's not a myth. Now, when you multiply the two, what do you get? You get 825 watt hours per square meter per day. Now, what does that mean watt hours? Nobody uses watt hours. So we have to convert this one about apples to apples into kilowatt hours. So 825 watt hours is eight tenths of a kilowatt, 0.825 kilowatt hours per square meter per day. Okay, thank you. You're all buried with me. Now we have two numbers that we know. We know how big the mountain is. We've got to climb in terms of energy, 27 million kilowatt hours per day. And the question is, can you do that on a Wahoo and what would it take? So we're looking at that. And now we know what the solar resources, we know that it's about eight tenths of a kilowatt hour per square meter per day. All right. Now we know the size of our mountain we're going to climb and we know how good our equipment is in climbing it. So now let's answer the question. So let's go to the next slide. Now, on this slide, you'll see we need 27 million kilowatt hours. And we know that the reference, the resources, eight tenths of a kilowatt hour per square meter per day. So you divide and that tells you if my proposal is correct with that 10 billion kilowatt hours per year, you're going to need 33 million square meters of solar panels. And that's a fact. Now the problem is for me is I don't know how big that is. I don't have a feel for that. What does 33 million square meters mean? Is it helpful? Yeah, it's helpful. Yeah. Well, I thought to myself, okay, what do I know? Well, I do know that one square mile has two and a half million square meters, 2.59. So we just simply divide the 33 million kilowatt hours by 2.59 million square meters per square mile. And what you get is a result of 12.82 square mile. Now it doesn't need all in one place. It's all over the island. We'll get to that. But I still don't have an answer that I haven't satisfied myself yet. Is that big? Is it small? Is it impractical? What do I know? So I thought, okay, here's how you find out. You take your 13 square miles and I'm going to call it 13 because I'm conservative. I think you need more. You need 13 square miles. So everyone, this is real now. You divide that simply by the whole area of Oahu, which is just a hair under 600 square miles. And the answer comes out much to my astonishment just over 2%. Now, hey, yeah, that's not entirely impractical. Now, if the math doesn't care what I want, the math is to the math. And it's good to have said, hey, you need half the island. If it was half the island, I would concede that all of the people who said you can't do it are correct. If it's a third of the island, I'd still say you're right. If it's a 10% of the island, I'd say, yeah, you're still right. That's not practical. But 2%. Hey, the problem is, anyone who has ever asked or tried to talk about solar power in Oahu, immediately gets thrown out as a crazy guy. Because of course, we know when you can't possibly power a modern electric grid of 10 terawatt-hour to do all that, it is never going to happen. Well, let's look at that. 2%. Now, what's amazing, I'm trying to calculate how much of Oahu is already paved. And I think you could argue that's more than 2% because you've got parking lots, you have the tops of big box stores, you have the port authorities, you have government buildings, you have schools. The largest consumer that's not military of electricity is the University of Hawaii, Manoa. You have all kinds of roofs, not with a little panel. I'm talking about you didn't put all these panels. The wonderful thing is, it's already paved. It's already zoned. I can get trucks out there, I can do the whole thing. And the magic that goes to your opening slide is that you could do all of this with not one lick of new land. You already have the land, we're just going to dual use it.  There's no room. So what an astonishing result. Hey, if unless you think 2% is unreasonable, then I would concede. But if you think 2% is not all that crazy, since you already have land, I'd say the answer now finally to the question, can we solar power the grid of Oahu is? Yes, you can. If yours 2%. So that's an interesting result because most experts that I've listened to just dismissed the question out of hand. They take it off the table. It's not even discussed. They're going to say, well, you've got to do other things that are going to know I'm only talking about solar panels, solar electric panels, and electrolyzers that make clean hydrogen, which is what you demonstrate in your project on the big island. That's what you're doing. You're using sunlight and water to run an electrolyzer to make clean hydrogen fuel to run the fuel cell buses. Wonderful. It's just an extension of that. Maybe add some fuel cells to put back that hydrogen energy on the grid, as we're talking about the grid. But you have proven the case that this technology is not arm-waving. It's demonstrable. It's real. We know everything about it. And what I tried to illustrate in this calculation is I'm not sugarcoating it. I'm not saying you need 6 trillion watt hours a year. I'm saying you need 10. That's the number. And we've proven now mathematically that if you can live with 2%, you are completely energy ingrid. Right. Well, okay. Now let's, okay, sorry. No, no, no. I just wanted to segue into, well, let's talk about the money. Because now that we've established you physically could do it, what about the money? I mean, if it's a trillion dollars, how are you going to pay for it? Right? So let's go to the next slide. Because That's your milk, sir. Yeah. You see, the fossil fuel industry is basically there in command. They're selling you something. They're just like a drug cartel. They want to keep selling it to you. So they're going to tell you a story. And what do they tell you? They say, well, you know, if you want to convert, you want to transition to clean energy, it'll cost you a brilliant dollar. Ooh, that's scary. Well, a trillion. Do you have a trillion? I don't have a trillion. Ooh, I guess we can't do it. And then they're going to say, well, and the cost is they're only giving you half the inflation. They don't talk about what you make. Just the cost will be so detrimental to you.  Ooh, now I'm really scared. That sounds terrible. All right.    I'm not going to get a paycheck. I'm not going to get my, oh, this is terrible. No, no, no, no. What do you want, Mr. Oil Company? Well, we want you to give us the billions of dollars. This is going on to pull up the carpet and let the milk sink into the floorboard, this carpet and store it. So this is really kind of barbaric and kind of criminal and basically tough guys around the world say, you're going to buy our oil or forget about it. We'll bury it. So, so this has been great. Toby, let's wind it up. We have one last slide. We have about 30 seconds. Okay.  What they do is they lease the computers.  They make the lease payments and then after three or five years or whatever the term is, then they say, okay, we're going to give them back and we want to lease new computers just like with a car. So the point is you're not going to pay for this huge system we're talking about. No, no, no, no, no, no. An underwriter is going to pay for it. And if it's established and put in and working, then they get tied, they get the, the lease payments and you can stretch it out so that the lease payment is 15% less than what you pay now. So here's the bottom line. How much does it cost to transition the island of Oahu to a 100% clean solar powered grid? Nothing. Doesn't cost you a penny. You don't buy it, you just use it. And if the lease payment is set to be 15% less than what you pay, then I don't know what everyone's electric bill added up on Oahu is, but let's say it's a billion dollars. Well, that means if you did it under this lease program, you're not coming up with the money and you're going to save $150 million each year, day one every year that you're leasing it. So the big lie of the fossil fuel company is, well, you can't do this because it'll ruin you financially is an absolute deception. You absolutely can do it. And this is how. So let's bring up the last slide and tell us all the good things about solar. Right. So what we're talking about is a real solution. You're not going to be devastated financially. You don't have any of the pollution. You don't have any of the vulnerability or the variability that caused you so much trouble. You don't need new land. You have no fuel costs. Hey, tell me where the downside is.  All you have to do is use it.  So. Which isn't bad. That's not bad for my car. Actually, that's my thing. Yes, it's a win-win-win-win. The underwriter makes money. The user makes money. At the state of Hawaii saves 100, if it's a billion a year, saves you 150 million a year. That's not bad. The money stays in your pocket instead of going off island. So to all the energy experts out there who completely dismiss this idea, please, please have a look at the math, at the math we're suggesting. And if you can find where I'm wrong, I'd like to hear from you because I've been in this army for a long time and this is how you get to a real result. So the answer to your question is yes. No, no, please. It's a yes. Okay. Yes, you can do it. Okay. I love Lila, Olivia on a positive note. Though I believe that they've been watching Hawaii, the state of clean energy on think tech, Hawaii and Yi. And today we've been talking story with Paul Vickincade, though powering Oahu on sunshine. Many people think that's impossible. Toby told us how we can do it. Thanks for participating, Toby. I really appreciate you, especially since you're recovering from COVID. God thank you. It's clear out of your sick bed to put on the show to us today. Hey, we all have a future. Let's do it. Yeah. Both the mariners try. Thank you. And thanks for all our viewers for tuning in. I met you on moving back in two weeks with another edition of Hawaii, the state of clean energy. So Aloha, everyone.


With an estimated net worth of $700 million, Paul David Houston, better known as Bono, has cemented his status as one of the wealthiest musicians in the world. While most recognize him as the legendary frontman of the rock band U2, Bono has also established himself a businessman through his private equity firm Elevation Partners. This profitable venture shows Bono's instincts and has contributed handsomely to his fortune over the years. Amidst all his fame and success, Bono remains an unbashed family man at heart. The rock icon considers Dublin, Ireland his true home and continues to maintain deep roots there.


In 2018, we worked in partnership with the Decho First Nations to establish Edejei, which became Canada's first indigenous protected and conserved area. Edejei, in the Northwest Territories, represents over 4,000 square kilometers of land, protects headwaters, is home to boreal caribou and is of cultural and spiritual significance for the Decho and Clichodene. This was and remains a major step forward on how we not just protect nature, but do it right. By working in partnership with indigenous communities, and by listening when indigenous people tell us that walking forward on reconciliation and making progress on conservation must go together. And if Edejei was a major step on a shared journey, then what we're announcing today is a leap forward. This morning, I announced that Canada will invest up to $800 million in four major conservation projects led by Indigenous peoples from one end to another in the country, which represents a total of almost 1 million square kilometers. There will be a conservation project in the marine zone of the Grand Ours in the west, another in the north of Ontario, the conservation project of the Omoshikego, a project in the region of Kikitani in Nunavut, and a last project in the Northwest Territories. Each project is unique in its genre, and in a few moments their representatives will talk about it more. Canada will provide up to $800 million to support four major indigenous-led conservation projects across the country, covering almost a million square kilometers. Each of these projects is different because each of these projects is designed by communities, for communities. On the Pacific Coast, in the Great Bear Sea, we're supporting the group representing 17 First Nations to deliver an integrated bioregional marine conservation and sustainability initiative. Heading north, you'll reach protected boreal forests, rivers and lands across the northwest territories in a project that's a partnership between 30 indigenous governments across the territory supported by the Government of Canada. From there, east and even further north, we'll take you to the Kikitani region, and a vast network of protected waters and land safeguarded through Inuit stewardship and governance founded on Inuit Kaoyama-yokutokangit. And then south to western James Bay and the world's third largest wetland, and the vision of stewardship led by the Omochikigokri. These projects aim as much to ensure the well-being of communities as possible to protect nature. We know that in order to create jobs and to favor a strong economy, we must absolutely fight climate change. Well, it's the same thing for the protection of nature. Communities have been clear. Building lands and waters will help build a strong future for generations to come. As a government, our role is to listen and support that vision. That has been our approach with other initiatives, like the Indigenous Guardians Program, which supports First Nations, Inuit and Métis people on stewardship over traditional lands, waters and ice. The role of partner underpins our approach with these new projects, too, alongside others who have committed to join this work. I want to recognize also the leaders from foundations and philanthropic organizations who are part of stepping up to help make these four projects a reality. After all, conservation and protection take money for land planning work, for funding for communities on stewardship and monitoring, for supporting local jobs and growth. It takes resources to create a conservation economy where caring for the land is a career opportunity and where people can be stewards of their own homes. By working together, we're able to deliver that major necessary support no individual partners could provide alone. That brings me to why it matters that we be announcing this today, as Canada welcomes the world to the largest nature cop we've ever seen. Today, we have made a big step in the protection of nature, and we have done it in partnership. Here, all together in Montreal, we call on everyone to take action. The whole world has to do more to protect nature, but we also have to do it well. I hope that other countries will mobilize. We hope that the next two weeks will confirm our collective commitment to carry out a global action. We have a lot to do to protect nature, and we have a lot to do to advance on the path of reconciliation. But today, we show that we do it together.


kwa kenyakin kwa wuna betawi to do this than to do it with one of our own this is your trophy this is your national treasure two guys do you know a deadly zone and we need to keep alive kwa jani in everything to my father it is you guys that got us there when I was getting I'm not on the phone ya pakitamonya let's go he's pointing me to the end of the


I'm Kaya, this is Lou, and we're going to do two songs for you today. The first one is a widely redemptive song. Hope you enjoy it.


Thank you so thank you all for coming and first I would like to thank the organizers for inviting me to speak at this workshop it's a real honor and I'm very happy to be here. Okay so today I'll be telling you about some recent work that I've been doing with Giuseppe Matone who is a graduate student at USC and it's about these things we call positively ratio representations okay so this is not a real word I'm taking a past tense of a noun but this is the best you could do okay we are mathematicians so what are these these positively ratio representations are a particular kind of surface group representations okay and maybe to to to give you a more some motivation of why we consider this type of representations let me start by explaining to you a class of examples sorry and these are hidden representations okay so for the rest of today I'll let s be a closed orientable surface connected as well I should say that okay of genus at least two okay and I'm going to denote the fundamental group of the surface by gamma all right now the technical space of s right it's an object that we've already seen in this workshop and there many ways you can think about this space okay but I'll be thinking of this in two different ways today the first way is as the deformation space of hyperbolic structures on s okay or if you want the set of isotope classes of hyperbolic matrix you can put in your surface okay but you can also think of this from a representation theoretic point of view and this is the set of discrete and faithful representations row from gamma to the orientation preserving isometries of the upper half plane which is PSL2R okay consider up to conjugation by PGL2R right which are the isometries of each two okay now you can now next we consider this home morphism which I call yota n so this goes from PSL2R PSLNR and this is a home morphism induced the home morphism induced by the linear SL2R action on the n-1 tensor the n-1 symmetric tensor of R2 okay so SL2R acts naturally on R2 linearly and so it acts linearly on the n-1 symmetric tensor as well okay and this as a vector space is isomorphic to Rn okay so this gives you a way to put SL2R into the linear group of Rn right then you can check that this actually lands in SLNR and when you projectivize you get this homomorphism okay it turns out that this is also the unique up to conjugation irreducible representation from PSL2R into PSLNR okay meaning that if I want PSL2R to act on Rn without preserving any proper subspace then this is the only thing I can do okay but for us right what this gives us is a way to put tecno space into the space of home morphisms from gamma to PSLNR okay consider up to conjugation by PGLNR okay and what you do is you take every representation here and you just post compose it with this irreducible guy okay to get it into PSLNR right of course you need to check that this descends to a map on the conjugacy classes okay so now we come to our first definition the n-thitching component which I will denote by hit NS okay is the connected component of this space which I'll denote by X NS here that contains the image of the map iron okay so here it's well known that tecno space is connected thing right it's a cell in fact okay and this map iron here is continuous okay so the image of iron is some connected subset in here and you can just take the connected component of this that contains it okay in other words the hidden representations which are the representations in here are those that you can continuously deform until you get something in here okay so this space of representations was first studied by Nigel Higgin and he used the Higgs bundle techniques to understand the global topology of this space so for example he also shows that this guy is a cell okay just like protecting the space okay but the geometric properties of the representations in here right we're very mysterious until Fonso Laburi came along and prove this what I think is a remarkable theorem okay if you take any hidden representation row okay then row is going to be what we today call a north soft with respect to the power-bolic subgroup the minimal power-bolic subgroup okay so like everybody else I will not explain what a north soft means to you formally okay but you should think of this as saying that your representation has some very nice geometric properties with respect to some particular power-bolic subgroup okay and in particular right we know that this implies that row from gamma to G is a quasi-isometric embedding okay so here you put the word metric on gamma this is a gram of hyperbolic group and here you choose any left invariant metric you want remanent metric on your league group okay also if you take any non-identity element in your group okay then row of gamma is diagonalizable over R right with eigenvalues are having pairwise distinct absolute values okay and I would denote this this absolute values of the eigenvalues by lambda 1 of row of gamma to lambda n of row of gamma in descending order okay so lambda 1 of row of gamma is the eigenvalue of row of gamma with largest absolute value of the eigenvalue of row of gamma with largest absolute value and so on okay now choose a collection of non-negative numbers a1 to an-1 so that ai equals to an-i for i oh sorry okay from now on alright so here I'm just picking any collection of non-negative numbers with a particular symmetry okay so I'm really just choosing half of them and using this we can define this thing called a length function okay and this is a map from the space of closed geodesics on my surface okay so what is this this is defined to be the set of non-identity conjugacy classes in gamma okay consider up to the equivalence relation where the conjugacy class of gamma is equivalent to the conjugacy class of gamma inverse okay so this set here is naturally in bijection with the set of free homotopy classes of unoriented closed curves on your surface okay so if you choose a hydraulic structure on your surface then this becomes is naturally the the set of closed geodesics on your surface right because every free homotopy class contains a unique geodesic representative okay so I call this the set of closed geodesics but really this is a topological thing okay so the length functions are met from here to r plus and what you do is you just take any equivalence class containing the group element gamma so this is the closed geodesic corresponding to gamma and you send it to the sum from i equals to one to n minus one of ai times log of the ith eigenvalue of rule of gamma divided by the i plus one eigenvalue of rule of gamma okay so you can check that this map is well defined okay you need this condition to make it well defined okay so an example of yeah so maybe I should remark that this length function depends on the choice of my ai's okay but I will suppress that in the notation all right so an example of this right is in the case when n equals to 2 okay when n equals to 2 your second hitching component is exactly type in the space okay this is a consequence of the well-known fact that type in the space is a connected component of x2s okay so this is type in the space and in this case I only have one choice of the ai so I just choose a1 and say let's choose it to be one okay then the length then for any closed geodesic in cgs the row length of c is exactly the hyperbolic length of the closed geodesic c okay in the hyperbolic structure corresponding to my function representation row okay so these are the kind of things that I'm interested in studying right I would like to understand the length functions these guys that arise from a hitching representations or in general from these things we call positively ratio representations okay and if this is the kind of thing you're interested in one a natural quantity that you would like to understand is this thing called the entropy of the representation okay so definition for any hitching representation row and for any choice of again you need to choose some ai's right with this symmetry but I won't write that here okay the entropy of the representation is defined the following way okay first you count the number of closed geodesics on your surface okay whose row length is less than some number t okay and for any t this quantity turns out to be finite okay for hitching representations and in fact it grows exponentially with t okay so you take the exponential growth rate okay so this quantity is called the entropy now it's another well-known thing about a space that this quantity here is constant on technical space okay so this is kind of a boring quantity to look at if you're interested in technical space however once you change your lead group right to become a higher rank right for PSL nr where n is 3 and bigger then this quantity is no longer constant okay in fact it's a consequence of my thesis result that when n is at least 3 okay and and let's say you choose okay now you need to choose all the ai's to be 1 okay so in other words the length function I'm considering is the one that assigns to every close curve by the log of the top eigenvalue of row of gamma divided by the bottom eigenvalue of row of gamma okay so if you choose this particular function okay then there exists a sequence in the hitching component right along which your topological entropy your entropy would converge to zero okay and also a senior also produced them other examples of these things so produced other examples right where your entropy is going to zero in the case when n equals to 3 they're using other constructions so for me a natural question then is for these length functions right can I give a criterion on a sequence of representations to indicate when is the entropy along that sequence converging to zero okay so the question when is there a criterion for when this happens okay okay and so we we could not answer this question completely okay but we got pretty close okay if you restrict yourself to just sequences that stay in the epsilon thick part of the hitching component okay then it turns out that we have a criterion okay so first I need to tell you what the epsilon thick part is okay so this is a corollary of some theorems that I'm going to state later okay if so if for any epsilon larger than zero okay you define the epsilon thick part of the hitching component okay to be the set of hitching representations right where the row length of any closed geodesic is at least epsilon okay and of course this again depends on the choice of the ai's right but every time you choose an ai you can define this epsilon thick part okay okay then now you let a row i be a sequence in this epsilon thick part there will I be a sequence in here then the entropy along this sequence converges to zero if and only if for any subsequence of this sequence okay there exists the following things okay first a further subsequence which I will abuse notation and also denote by row i okay a possibly empty collection in the space of closed geodesics there are pairwise non intersecting and simple okay and finally a sequence of mapping classes okay such that alright so before I write the condition let me just say that the mapping class group acts on the hitching component just like how it acts on packing the space okay by remarking right and also for any length function you choose right the the the mapping class group preserves the epsilon thick part of the hitching component as well okay so you need these three things satisfying the following okay right bigger okay okay so first if you look at the supremum over all sequence all elements in your subsequence right this further subsequence okay and you look at the maximum overall curves in this possibly empty collection okay of the fi times row i length of your closed geodesics in here okay this has to be finite and second if you take the limit as I approaches infinity so move along your sequence and you look at the minimum okay over all closed geodesics okay they are in the complement in S of your collection of closed curves okay so s-c is a possibly disconnected union of surfaces with boundary okay and I've defined closed geodesics only for closed surfaces but obviously you can define it for surfaces with boundary as well and I require these guys to be non-parifero okay and then you look at the fi times row i lengths of all these guys okay this has to be infinite okay so this is the criterion so more informally what is this saying right this is saying that if I take a sequence in the epsilon thick part of the hitching component right then the entropy along this sequence is going to zero even only if up to taking subsequences and up to the mapping class group okay you can find some way to cut your surface into pieces maybe something like that okay so that the lengths of the curves they used to cut your surface have to remain finite along your sequence but on the complementary pieces right the shortest I mean the the systole length the shortest non-parifereal closed curve its length has to go to infinity okay so this is kind of a pretty easy criterion I think so yeah okay so maybe I should move on from now okay so this is a statement of a corollary of our main theorem in the setting of hitching representations okay after so this was initially the our target right this was something we wanted to prove and after we prove this realized that our techniques actually generalized to a much more general setting okay and that's how we came up with these things we call positively ratio representations okay so before I move on I would like to say a little bit about the the strategy of proving a corollary like this or theorem like this okay I mean if you try to work on this very quickly you realize that your life would be a lot easier if you knew that your length functions came from a negatively curved metric on the surface okay and sometimes that happens right in the case when n equals to 2 right your length functions I said you know they will arose as length functions of an of a hydraulic metric on your surface okay but in general right there is no canonical metric on your surface that gives rise to these length functions okay but we have something close it turns out that these length functions arise from geodesic currents right and that is good enough right for you to play a lot of this entropy counting this entropy counting game okay and so let me explain what geodesic currents are we kind of saw kind of saw this yesterday during Vivek's talk but in case you missed it I'm gonna there's gonna be a lot of overlap of what I do here with what she did okay so first what are these guys so a geodesic current right is a gamma invariant locally finite or L measure on the space of geodesics in the universal cover of s okay so this guy is defined to be the set of unordered pairs in the boundary of my group okay that are pair wise distinct okay so if you choose a hydraulic metric on your surface then this again is naturally identified with the set of geodesics in the universal cover okay but again this is a topological thing okay this has a I mean as a there's a natural gamma action on this space because this is pair of points in the boundary of the group there's a topology here as well okay because the boundary of the group has a natural topology and so this makes sense all right so and what's an example of this the easiest examples those that come from close geodesics on your those to the geodesic currents that you can associate to close geodesics on your surface okay so for any close geodesic my surface okay let eta be a group element that is primitive so that if you look at the equivalence class the equivalence the close geodesic corresponding to eta to the K this is C right for for some positive integer K okay so in the case when C is primitive then K is one okay but this kind of keeping track of how many times C winds around itself as well okay then in this case the med the geodesic current associated to the close geodesic C okay is sometimes I also abuse notation denote this just by C this is defined to be K times the sum over all elements in your group right of the DRock measure on gamma times eta minus gamma times eta plus right where eta minus and eta plus are the attracting and refilling fixed points of this group element in the boundary okay so let me kind of draw a picture of what I'm saying here alright so you have your surface downstairs okay and you choose this curve C this close curve this close geodesic I want to assign a geodesic current to this guy right so you just look in the universal cover okay you look at all the possible lifts of this geodesic upstairs okay something like this you know it goes the infinitely many okay so something like this is eta minus this is eta plus maybe this is gamma times eta minus gamma times eta plus and so on okay and on this infinite set of geodesics you just put the DRock measure on all of them you take the sum and you scale everything by K okay this turns out to be a geodesic current okay so what you've done is you've taken the space of close geodesics and put them inside of the space of geodesic currents okay now definition Bonheon define a very nice intersection pairing on the space of geodesic currents okay so there exists a continuous a symmetric by linear pairing I okay from CS times CS no negative that generalizes the geometric intersection number meaning that if I take this intersection pairing and I just restrict it to the geodesic currents that I that I get from the close geodesics then this is exactly the geometric intersection number okay so of course you know he gives a definition of how to you know actual proper definition of this but I won't do that in the interest of time okay so for us what this gives us is a way to define a length function given any geodesic current okay so this implies that for any geodesic current that you can define the length function our new and just like the length functions we define for hedging representations this is a map from the set of close geodesics on my surface r greater equals to 0 and you just take every close geodesic and you look at its intersection pairing right with this geodesic car okay now I'm gonna impose another condition on the geodesic currents I'm considering okay this one's important geodesic current new yes okay it's called period minimizing okay if for any positive number t the number of close geodesics whose lengths are less than t is finite okay so examples of period minimizing geodesic currents are the new view currents right those that come from negatively curved metrics on my surface where we saw that yesterday non-examples are measured lamination right for measure lamination you cannot there's no minimum for this I mean for and if you make t up really small this is still always infinite okay so for my purposes you know you should think of a little view currents as a as good measure lamination as bad okay so well what is the advantage of looking at the length functions of geodesic currents versus the length functions coming from representations right and the thing is turns out that when you are when you have a geodesic current that's period minimizing you can pretty much pretend right that you can that the that your length function comes from a length function on the offer met of a negatively curved metric on the surface okay so let me give an example of a proposition of something like this okay so let C be a closed geodesic on my surface okay and I'll add that is non-simple okay and then let C1 C2 C3 be close to the six right that I'll obtain right by performing surgery to my non-simple close geodesic C right at a self-intersection point okay so I'll draw a picture to explain what I'm doing okay so suppose that this is your non-simple close geodesic C okay and you have some self-intersection point right then there is a few ways you can cut C into pieces at this half intersection point right two ways you can cut it vertically like this and this will give you a two close geodesics like this okay I'm gonna call this C1 and C2 okay but you can also cut it horizontally like this right and this gives you a third close geodesic like this and this is C3 okay and the statement is that for any geodesic current without any assumptions at all okay the new length of C is always at least the new length of C1 plus the new length of C2 and it's also at least give me and also at least the new length of C3 if furthermore if the geodesic current new has the property that new of U is positive for every open set U in the space of geodesics in S-theta okay then the inequality is strict okay so if you have a negatively curved metric on the surface then something like this is obviously true right because once you cut your curves do not change length and then you pull it tight so they become shorter okay turns out that in the space if you work in the world of geodesic currents you can also prove something like this it's not so hard okay but proving a proposition like this in the world of representations right it's surprisingly tricky okay so that's why that's one advantage you have right when you think of length functions of geodesic currents versus length functions coming from representations okay and this is a point I would like to emphasize right for most of the other examples that say Vivica showed us yesterday about geodesic currents a lot of them right they arise you know when you really have a metric on your space or not on your surface right for which the length functions come from okay but for here you don't have a metric on your surface right and that's why you know proving something like this right it's easier to do in the world of geodesic currents yeah okay so that's that okay so this proposition gives us the following consequence right if you take a period minimizing geodesic current okay and say a subsurface okay a geodesic subsurface so let s prime in s be a geodesic subsurface okay meaning if you choose a negatively curved metric on s okay then s prime can be homotoped to a totally geodesic subsurface in the usual Riemannian geometry sense okay if you have something like this okay then there exists a pens decomposition okay let me not call it P and commit c1 to ck so it's a bunch of close curve simple close curves in s prime okay so that the new length of cj for any j is at least the new length of c for any close geodesic c in the complement of the previous curves in s prime okay so so what am I doing here right how do I build such a pens decomposition well you start off with your subsurface okay so let me draw a picture subsurface look something maybe this okay so your subsurface can be the whole thing and then you're close right but typically you have boundary okay and then now you look at all the non-peripheral close curves in here and you find the shortest one okay that exists because your period minimizing okay now this proposition right where did it go yeah this proposition ensures that okay so you ensures that either this is already a pair of pens or you can choose your shortest simple guy as your shortest close geodesic that's non-peripheral to be simple okay because if you if your shortest one is non-simple you just cut right to make it smaller okay unless you're a pair of pens then you cannot do it anymore right now you cut your surface here okay and then you get another geodesic subsurface you iterate okay until you get a pens decomposition I don't know this okay and this is called a minimal pens decomposition questions okay with this I can make several definitions some I'm going to define certain quantities associated to a period minimizing geodesic current okay so for this definition I need my geodesic current to be pure minimizing and s prime a geodesic subsurface okay then first I can define the cisto length right so this is L nu s prime this is defined to be the set of closed geodesics sorry is defined to be the minimum right of the new length of C such as C is a close geodesic s prime okay the shortest closed curve that the length of the shortest closed geodesic okay and you can define the crit the entropy right just like how we defined it for hitching representations okay so each nu s prime is defined to be the exponential growth rate right of the number of close geodesics right whose new length less than t okay so this is the entropy and finally we can define a third quantity using our minimal pens decomposition which we call the pented cisto length so third for any minimal pens decomposition P of s prime okay I define k nu of s prime right to be the length the new length of the closed geodesics s prime okay so that the C that is not let's just say this that is not Perry there is not a multiple of a simple closed curve P and non-peripheral okay this is called the pented cisto length so basically I look at my surface I have my minimal pens decomposition I look at all the close geodesic in here and I remove the guys that are just multiples of curves in my pens decomposition or multiples of curves in my boundary okay ignore those guys among the rest I find the shortest one and that's the pented cisto length okay and the theorem that we can prove is the following okay so that nu be a period of minimizing geodesic current I see now pure minimizing geodesic current okay s prime and s a geodesic subsurface right and P a minimize minimal pens decomposition for s prime a minimal with respect to this geodesic current this period minimizing geodesic current new okay this is true in this setting then you have the following inequality okay log 2 over 4 is at most the pented cisto length be scaled by the entropy of my subsurface okay and this is sorry I forgot I forgot the most important thing okay so there exists a constant C depending only on the topology of your surface okay such that for any period minimizing geodesic current and for any geodesic subsurface and for any minimal pens decomposition okay you have this inequality that the the pented cisto length renormalized by the entropy is bounded above by this constant times 1 times 5 sorry plus plus the log of 1 plus squared 5 plus 1 over 2 multiplied by the ratio of the pented cisto length to the cisto length okay so let me point out a few features of this inequality first this inequality right is a is invariant on the scaling okay so if I take my geodesic current I scale it right then the entropy let's say I scale my geodesic current by k the entropy will be scaled down by 1 over k and this guy will be scaled up by k right so nothing happens and of course these are ratios of certain lengths right so that when you scale nothing happens here as well okay so this is really an inequality on the space of projectivized geodesic currents okay so this might look kind of strange right but let me give you some consequences first if I don't care about the if I don't care about the cisto length at the pented cisto length I only care about the cisto length then you can actually make this a much simpler okay there exists a constant c depending only on the topology of my geodesic subsurface okay such that for any period minimizing geodesic current okay the cisto length I renormalize by the entropy bounded above by c okay so in particular I mean so yeah I should say this in the case when s prime is a closed surface and new is the new view current right then I've been told that this inequality can be abstracted from the current literature okay so something like this might not be entirely new but I've not seen anything close to this inequality before so if you have you should let me know as well okay what this implies right if is that if I have a subset of geodesic currents and I know I have a uniform lower bound away from zero of the entropy but then that would give me a uniform upper bound of the cisto length okay and there will have consequences for us later as well okay now a second corollary right is that is that the analog of the corollary I wrote down for hidden representations just now holds in this setting okay so first for any epsilon large and zero I need to define for you what is the epsilon thick part of the geodesic currents right so define a sub epsilon of s prime to be the geodesic currents the period minimizing geodesic currents okay so that the new length of c is larger than epsilon for any closed geodesic in my subsurface s prime okay and then okay I did not name that that corollary just now but I guess you guys know what I'm talking about so let's say the the Hitching corollary the H corollary so the corollary in that I first wrote down right in the setting of hedging representations right holds okay when I'm with let's say with the epsilon thick part of the hedging component it replaced with this guy here a epsilon s prime right and you just change all your sequences of representations with sequences of geodesic currents okay so with this corollary okay so maybe I should say a little bit of why these things are true okay so basically if you are in the epsilon thick part right of the of the geodesic currents right this ensures that l new s prime cannot be too small okay then you divide your inequality throughout by k new s prime right then on both sides right you're gonna have some expression in terms of k new s prime right that goes to zero if and only if k new s prime goes to infinity okay and then using that you can you can play around a little bit and you can show this corollary holds as well any questions so far okay so with this corollary right the only thing we need to do now right is to relate is to bring my representations into the picture okay everything is in geodesic current world so far right so how do you associate how do you relate the length functions you have from your representations to the geodesic currents right and this is where we realized that what we did actually works in a much more general setting okay and so these are positively ratioed representations okay so I'm going to give you a definition of this but to do that I need some basics from Lee theory okay so I'm gonna so if you're not so familiar with Lee theory right don't worry because I'll do a running example alongside what I'm gonna do okay so I'll let G be a semi-simple g group of non compact type okay and K is going to be a maximal compact subgroup okay and then a is going to be the maximal abelian subspace of the perp of the le algebra of K which lives inside of G okay so this guy is the le algebra of K this guy's the le algebra of G here I'm taking the perp with respect to the killing form okay so this guy is lies in here is a maximal abelian subspace okay and then I have a choice of simple roots which I call Delta right and on the set of simple roots I have the opposition in evolution right which I denote by Yoda okay so example you can take G to be PSL4R okay K to be PSO4 running out of chalk again okay and then a in this case is going to be the set of four by four diagonal matrices they are traceless okay so this is a three dimensional vector space alright and what else do I need so Delta right in this case you're gonna have three simple roots alpha 1 alpha 2 alpha 3 okay and each simple root is a map from it's a linear map from this vector space to R can what you do is you take your diagonal matrix and you send it to AI minus AI plus one okay so each route takes the difference in the adjacent entries along my diagonal okay and these are the simple roots particular linear functions on these guys okay and then the opposition in evolution in this case right takes each simple root and sends it to alpha 4 minus I okay so you take alpha 1 to alpha 3 alpha 3 to alpha 1 and alpha 2 is fixed okay so this is the running example now for any data subset of Delta okay you can get two things from data one is this thing called a parabolic subgroup okay in fact all parabolic subgroups are going to be conjugate to one of these guys right and you also get this projection pi data right from a to a data which is defined to be the subset of a right so that alpha of X is 0 for any alpha simple root that is not in data okay so let me do our example again so suppose that data is all of Delta okay then in this case a p data p Delta is my upper the upper triangular matrices yes l4r okay this is my parabolic subgroup and my projection is just the identity map in this case from a to a okay but suppose that in the case when Delta is alpha 2 then p of alpha 2 right is the the set of matrices right where the bottom two by two corner is 0 so this is in ps l4r okay and your projection right now it takes a a to a alpha 2 and what you do is you take your diagonal matrix and you send it to the following okay you you basically take the first two entries and you take the average and you put it there so a1 plus a2 over 2 a1 plus a2 over 2 okay and the last two entries you take the average of these two and put them there I'm sorry I think I wrote too small here okay but let me just in case you can read I'll just say that you take a1 a2 you take the average and you put the same thing in the first two entries here take a3 a4 you take the average and put the same thing here okay so this lands in a data a of alpha 2 right because if you evaluate alpha 1 and alpha 3 here you always get 0 right okay so one last thing before I well two last things before I I tell you the definition okay I need this thing called the Jordan projection and this is a map I denote by lambda from g to the positive valve chamber in my l e algebra in my in my maximal abelian subspace okay and in the example okay so this is given to you by the Jordan decomposition theorem okay and what this is in the example so let me make it simple okay suppose that a g in PSL 4r is diagonalizable over r okay with eigenvalues with absolute value lambda 1 to lambda 4 okay then lambda g right is going to be the the diagonal matrix which is the logarithm of the absolute value of lambda 1 the logarithm of the absolute value of lambda 4 okay so I mean PSL 4 the product of these guys is one so when you take the log the sum is zero okay so this is a traceless matrix as well last thing okay last one last one yeah I'm going to define for you a notion of cross ratio okay okay there are many definitions of cross ratios out there but the one I'm using right is the one by lat rapier okay so a cross ratio right is a continuous map okay from quadruples of points in the boundary of the group okay so that the first two points and the last two points do not share any common points okay so a can be b but a cannot be c and a cannot be d okay so it's a map from this set to r okay satisfying a few conditions the first one is a is a symmetry condition so b of a b c d is to be equal to b of c d a b okay for whenever this is defined and second you have a co-cycle condition so b of a b c d plus b of a b d e equals to b of a b c e so this is all the setup I need next I'm going to state a proposition which is probably the deepest thing I'm stating it right now and this is a I mean okay so it basically follows okay from the work of can I say that right okay some burino okay I'm and start and let up yeah from so let up yeah okay so the hardest part of this proposition is really to understand what these four people did and combine everything together okay and what it says is the following okay so let theta in Delta be a subset of the roots right set is such that is invariant under the opposition evolution okay and then also for any alpha in data I let the symmetrized alpha alpha sim right to be defined to be the sum of alpha and its image under the opposition evolution okay all right so we have this now then the statement is that if row from gamma to g is p data and also okay so you know having nice geometric properties with respect to this probably subgroup okay then there exists a cross ratio that you know by b data alpha okay with the property that be data alpha of such data let's say let's say for any gamma non-identity element the group a b data alpha of gamma minus gamma plus so this is the repelling attracting fixed point of my group element in the boundary of the group and then gamma x and x okay so x here is just any element in the boundary that is not gamma minus and gamma plus okay you can check that this quantity here doesn't depend on the choice of x okay this here is for example this is called the period of the cross ratio okay this quantity is equals to the following okay you take your group element you evaluate it using the representation okay then you hit it by the Jordan projection to let it land inside the positive valve chamber and then you hit it by the projection pi data okay and then you hit it by the root okay so this thing looks really scary okay but what is this really this is a length function okay if you work out the examples that I wrote down the example of PSL4 just now that I wrote down right you can see that in all those cases right this quantity this quantity here is just the logarithm of certain ratios and products of eigenvalues of row of gamma okay so this here is really a length function so what this is saying is that whenever you have you make these choices right choose a theta and alpha like this okay then the length function that you get right always comes from a cross ratio okay no it does not like if PSL nr for n even do you wait wait say that sorry aha no no so so theta is not a fixed point a fixed subset so you can take theta to be all of delta then that will be fixed right okay okay so with this proposition now I can write down my definition okay p data and not soft representation row from gamma to g okay it's a p theta positively ratio yes don't use the oh I'm sorry this is alpha sin yeah that's important here I'm I hit it with a symmetrized room okay thank you any other questions okay so if you have a not soft representation like this then this is called p theta positively ratio if for any alpha in data okay the cross ratio b theta alpha which is guaranteed by that proposition okay evaluated on any four points in the boundary of the group in the order so this is this has to be greater equals to zero for any a b c d in the boundary of the group in this order so the boundary of the group is topologically a circle right so I'm just saying that if you take a b c d four points like this and I evaluated using the cross ratio this always has to be non-negative okay and the upshot from this right is okay I mean this is a very fundamental principle it's you know something like if you have finite additivity right and you have non-negativity then you have countable additivity okay so let me just state state this as an observation by Ursula Hamunstad okay that if right is a cross ratio okay such that b of a b c d is non-negative for any quadruple a b c d in the boundary of the group okay then there exists a geodesic current okay so that when you do the intersection of this current right with any closed curve corresponding to the group element gamma okay this is the period of your cross ratio of your cross ratio is b of gamma minus gamma plus gamma x x for any non-identity element in your group okay so whenever you have a positively ratio representation right then the cross ratios you get you can always associate geodesic currents to them okay and this is how everything goes through everything you do in the world geodesic currents right will work here okay so I'm out of almost out of time but maybe I should say what the examples of this positively ratio representations are okay so you have hitching representations which are first introduced and these are positively ratio refers to any parabolic subgroup but you also have the maximum representations in there my these are the ones that Beatrice Posetti was telling about us telling us about on Tuesday okay those are all positively ratioed respect to the stabilizer of a Lagrangian subspace okay then you can build more by taking certain direct sums and tensors of these representations can the basic reason you can do that is because if I have two cross ratios with this property and then you add them you get another cross ratio of this property again okay so you can build many many more representations that way all right so but an important non-example quasi-fusion representations okay if you take a quasi-fusion representation that is a north soft respect to the line and hyperplane stabilizer and you can define the cross ratio as well okay but as once you're non-fusion right then this cross ratio cannot be positive right basically the the how bad your limit curve is right ruins the positivity okay and so this kind of trick doesn't apply for quasi-fusion representations okay so I should stop here thank you


It probably happened when I was a graduate student and was studying things and I realized that I really enjoyed learning and working with other people who also enjoyed learning and studying things. And then from one or two experiments we did, they actually worked out. They had failed previously and then I realized that failing in things is really important than to guide you in what you do subsequently. And I think probably then is when the light went off, this is really fun and I think I would enjoy doing this and I'd like to do more of it and see if that's true.


Yeah Stupid reactions Hey welcome back to our Super wor AXS units Corbin Rick any false on Instagram Twitter for more juice content It's so cheese. It's so cheese. The most important thing in the game. Squad, break out! And follow us on our personal YouTube channels, links are always in the description below. I do some cooking and other things. And Rick, you just did some reviews and you're also doing after thoughts, correct? Yes, I'm doing after thoughts of things we do here, like interviews. Like when we don't get the time to talk about the behind the scenes kinds of stuff. So I did one on our interview with Anurag Kushev, the one on Nawazid and Zadiki. And I'm just catching up and I'll continue to do those on things we do on the channel when I wanna expound because I can pontificate and be verbose. Gross. And so today we are reacting to a trailer and it's an Assam trailer. Yay! After we watched that fantastic film, Amiz, I believe it was called, right? Yeah, Amiz, I'm S-A-A-M-I-S, I remember that. Phenomenal film, once again, if you haven't watched that, go watch it. I believe it's on like, I think I put the link in the review of where you can go buy it online because it's in- And did you see comments and messages from Stupid Babies? I am so proud of you guys who stopped the reaction, went and watched the movie, and then- There was a lot. A lot.  Great job. Go support not only the big Hindi cinema, but support smaller, good cinema. Doesn't matter the language. Anyways, but this one was recommended a lot after we watched that in the comments. For Amiz, I believe it's the language, it's called Village Rockstars. It's a 10-year-old girl living in poverty in a village in Assam, India, dreams of owning a guitar and forming a band with the local boys. This one, a National Film Award for best feature film, best editing, best child artist, and best sound recording. India's official submission to the Oscars in 2019. Wow. Really? In Assamese? It said best sound. I'm wondering, we don't need to check it right now, but I was so impressed with the sound on Amiz that I wonder if it's the same person because the sound was Oscar level sound on Amiz. So that sounds great. I can't wait. Let's watch this. Three, two, one, go. That's beautiful. Yeah. Holy cow, that's beautiful. The sound, hear the water? Beautiful. She's adorable. Beautiful little girl. I don't want to see it. I want to see it. Beautiful trailer. It reminded me immediately of, and you might agree, a Florida project, right? Yeah, the kids running around and just being themselves and being, yeah, absolutely. At 100%. Boy, that looks beautiful. Reminded me of that. And the shots was just beautiful. Like, incredible. And I see why it would go, it would get nominated. I mean, it would get so many awards. I think somebody in the comments says it won 44 awards all over the world. And obviously official selection to the Oscar obviously didn't get there. We know, because we didn't, it wasn't there. But some films are so good that you can tell from the trailer because you're immediately immersed into the world and you don't want to leave. I didn't want the trailer to end. I wanted to stay there and watch what happens in this story. And they said so much by just showing us and not telling us, great trailer. Yeah, really great trailer. Reminded me, like you said, a lot of Florida projects but also like the beginning of Combatology Nights. Very fly on the wall ask, you're just following these people around. You might not even know what the story of this film is.  That's enough for a film. Yep. And it looks beautiful. And so I can see why Stupid Baby just would recommend us. This looks like our cup of tea. Yeah. Also, those little kids didn't look like they were actors. I'm wondering if they just found them? Maybe so. What were you gonna say? No, I was just gonna say, I love the recommendations. We've got the Stupid Family is not just the most loving and supporting but they're the most cinematically intelligent. The things that they recommend, the way that they don't do spoilers in the comments, the way that they'll stop over reaction, go watch the film and come back. Absolutely the most cinematic, loving, wanting the world to see great cinema, irrespective of region. And one of my favorite things that's happened with our Stupid Reactions is we not only have this passion for seeing the bridge gapped between Hollywood and Indian cinema, but when we get messages from Indians saying, thank you for showing me the cinema in my own country that I didn't even know was there because it's not talked about. And we're slowly getting more Stupid Babies watching these films that no one knew about because they don't have the legs of the films that are Hindi films. And I love that this is happening. That's not thanks to us. That's thanks to you guys for recommending it to us.  Yeah, obviously, because we just react to what we are, as what's requested. And that's because obviously we know nothing. We only know what you recommended, but I do feel like our subscribers are, one, they know us very well. And so like other Reactive Channels, pretty sure get like recommendations for films like Dahum 3 or films like Dry, or like big action ones. But like, except if they're new, I feel like they know that that's not our cup of tea and we like good, and we like cheesy stuff too.  Like we enjoyed a lot of war, like the big action parts of war, but our cup of tea, and I feel like we've kind of molded our subscribers and the ones that have stuck around are those same ones that appreciate cinema like we do.  Also, the cast has five Dosses in it. So I'm wondering if this is like just a family affair. I don't know if Doss is a very common name either. I'm just assuming. So this is clearly a passion project for her. That is deeply impressive. Yeah, I bet it was done on a shoestring budget too. Oh, I'm sure. One, if you have to do all that, that usually indicates that it is, but it didn't look like it. The cinematography looked freaking gorgeous. Beautiful. And so you guys can let us know what all that's about, but that looks phenomenal. Should this be our next film from Assam that we watch or recommend others down below? Cause you know, we explore everything. We explore every single region because one, we don't just stick to one because we have to read subs for them all. So they're all the same to us. We just like good cinema. If it's good cinema from Hindi, if it's good cinema from Assam, Lallium, Telugu, Tamil, wherever. Bunjab, we wanna see it. And you need to recommend it to us, please. That is your job. Let us know. The. The. The. The. The. The. The. The. The. The. The. The. The. The. The.


Hi, I'm Joe for jaspyscasebreaks.com and all that excitement from the previous video the high roller mystery crate number three I forgot to do the wax party invite to actually To wax party invites in this breaks. Let's gather everybody's names one through ten new dice new list and Name on top two names after three and a four seven times gets into the wax party Good luck one two three four five six and Seventh and final time after seven Hilton and Ryan Congrats to you after seven. You're in the wax party. Check out the details of that on jaspyscasebreaks.com Thanks, everybody. I'll see you next time for the next one. Bye. Bye


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I think that's not true. That's not true. It's ironic. So, well, technically it is falling. Everything in this picture is falling. The pain is currently moving upwards while decelerating downwards. So everything is falling. So I was a, as part of the NASA initiative for education, I was part of the bomb counter program. I helped design this robot and got to fly it. So we got to fly 30 parabolas of about 25 seconds of zero G. And the reason they called the bomb a comet? Well, they don't like that. They prefer to wait this wonder. The reason they called the bomb a comet is not the 30 parabolas of 25 seconds of zero G. It's the 30 pullouts of 1.8 Gs. That's what it really gets to be. But me being a student at the time of the prestigious University of Madison being of Wisconsin-Madison known for some things including partying I wasn't affected by that. Anyway, a few years later this is me. I'm working for Orbitac on the outskirts of Madison, Wisconsin and this picture is actually taken in Bedford, Pennsylvania. I helped design, build and test this mining piece. So as far as I know, it's the only on-flying variable geometry cut ahead in the world. And it's actually designed to mine on the moon. Pretty neat, huh? So a couple of other projects I worked on at Orbitac, by the way if your industry has a lot of acronyms I ask you to refer to the aerospace industry. So this is the loop like low energy planetary excavator which is the mining machine I just talked about the advanced animal habitat which is a rat cage in space the plant research unit the humidity validation eventually I don't think that's actually an acronym I worked on the Bigelow Aerospace Sun Dancer pressure relief system pressure control system which is very different and the temperature humidity and ventilation system I also worked on the prototype of the rocket engine behind the Aquarius launch vehicle So basically when asked what I did I got to say a number of things I could say I'm an engineer, aerospace engineer I'm a lunar excavation engineer I'm a life support engineer for next generation space station I could say I'm a rocket scientist These are actually in a certain order for the majority of my time as an aerospace engineer I was single So they're kind of in the order of likelihood of picking up a weapon on top actually rocket science just seems a little over the top Regarding rocket science a lot of people in our industry like to equate web development with rocket science I have the unique ability to speak professionally of both industries I would like to unequivocally say this is not true This is now your Stokes equation It describes how fluids are fluid So on top we've got unstable accelerations and then a bunch of other stuff I stands for Inviscid Viscous There's the u-vector We've got some Inviscid stuff There's the internal energy at the bottom We've got our viscous stuff with a bunch of shear stresses Those are the tows and some cues that represent energy Here's our axial shears and our transactional shears And then our heat addition and at the end we've got our mass diffusion due to combustion This is not rocket science This is not rocket science This is really cool But it's not rocket science This is tractable Now your Stokes isn't usual Anyway, this is a ruby cap That would be your cap I was meeting with Steven Anderson for lunch for about a third time We're talking about how this software industry was doing it wrong And he asked me if I wanted to start a business with him I said to think about it and so we went on our separate ways and finished off our day job and then I called up my dad and told my dad this guy wants me to quit my dream job and my dad said maybe you should sleep on it We started this phrase before maybe you should sleep on it I thought that was Holcomb I wish it right there It was actually pretty true You should definitely do that when faced with life changing decisions So I started to examine what was going on at my so called dream job I would ask for random things from my boss knowing that they would increase my productivity and my coworkers productivity You know, version control We love version control You know how we did version control in engineering and copying files This is good times So PR stands for copy release Anyway So I had this dream job and my timeline for the day kind of looked like this I'd sleep, work, and I'd either write code drink or play phrase V or some combination thereof So I realized hey this isn't the dream job that I was thinking about but I ended up doing this starting this experiment called venue works with Steve I didn't really know it at the time but I did kind of basically jump in with the goal of much like Matt's increasing program or happiness This is important to me because I'm leaving my dream job We rubious love this phrase It means we can do very powerful things very simply That makes me happy I went on a journey This is kind of in the middle that's architect So after moonlighting for a few months I quit this dream job I quit this amazing job where I was making cool shit for space So I went from the outskirts of Madison, Wisconsin all the way downtown where I began setting up the roomy environment Welcome to my talk My name is Bradley Griziak and Aaron so eloquently avoided pronouncing my last name So when I talk about the ruby environment I'm not talking about the environment that we're used to It's not the cool output we get when we type rvm info I know this is going to freak you guys a little bit out a bit out but the things that go on outside of a computer, they're actually kind of important I don't know if you recognize that but at the end of a week Friday night you could go home and start complaining about how awful your weekend at work was Man, that's terrible, I just got out on the line Or you could spend Friday night at the office hacking away with your coworkers on some side project or have them forbid bringing your kids into work on a Friday evening and teach them how to cope Woo! Wouldn't that be cool? Anyway, so when I was looking at the the ruby environment I saw three components to it So I looked at I looked at three components I looked at the outside which is the easiest to discern which is the change I looked at the inside which is really quite easy to change and then I looked at people on their tablets Now before I get into those, I'm going to have a quick disclaimer slash caveat Perhaps you're a lowly minion the many rungs of the corporate ladder with my friend here here at PT&D I thought you were going to be the one heckling me I don't know what that's about or you're stuck in a three release who knows but what I'm offering here is a grab bag take what you will and try to apply it if you're not entirely happy with where you're at so basically use what you can we'll be back to the outside who knows the three rules of real estate location, location, location alright, who said that alright, that guy can you do that guy at your ticket would it be on the other side of the room location, location, location those are all the same rules it doesn't look very dry either is this so this is this is a picture of the state capital in Madison this is taken from approximately the same distance from the capital except it's located about 180 degrees from where our office is so I can't exactly suggest everyone relocate their office a black away from the state capital what you can do is relocate into a population center and if you're inside of a population center you get all kinds of things for free you get things like coffee shops coffee shops around us you can have restaurants which play a huge role later on in my talk you actually get design firms not as many but quite a few, that letter B up there that's swing they're literally across the hall from us really nice synergies going on and then of course there's the bars we have a lot of bars around our office it's really nice we've got a bar below us one floor down and we have a group of that has probably around 10 or 12 beers that they're brewing at any given time they're across the street so it's important to have interesting businesses around you that are also enterable usable you could be in a business bar you could have a aerospace company a couple doors down really does that bring you joy day to day? probably not what could bring you joy day to day is the ability to go to a restaurant for a bunch by walking there or out to work anyway, so those are some of the things that are outside of you but if you're in a population center you also get a bunch of activities outside so in Madison what we have is semi-weekly farmers martins not bi-weekly we have one on Wednesday and one on Saturday I think they do that because they're making up for the cold months it's really nice to just go outside and get some fresh fruits and vegetables and meats and cheeses good times we also have lunchtime live so every Tuesday around noon a live band plays on the Capitol Steps it's really kind of neat you have all these kinds of events that happen inside a population center but you have to seek it out I didn't even know that this lunchtime live thing was going on until after we had moved in to our office so you gotta look for what's available to you another important thing to us at least is have some nice scenery so we have late people see it? actually yeah can anyone not from Beniworks name that lake? no that's not I'm sorry the answer is lake Manona not Mendota Manona alright I was expecting to keep that beer taking myself but why don't you so if you don't have a lake hey mom these are cool try to locate there but scenery in general have that it's nice to have one of the biggest problems though with being in a population center is transportation it becomes much more expensive as you locate closer and closer to hubs of activities so you can look into alternative transportation you might think I as a co-owner of a rail shop I roll into work every day in my SLS the truth of the matter is the SLS isn't my computer more to the point I don't really have one 90% of the time I'm actually riding one of these two loads of transportation the bus or the bike Beniworks hands out free bus passes for every employee to use however they want they don't even have to use it to go to work they might use it for whatever it's cheaper way cheaper than buying a parking space and frankly it's much more green if you're into that kind of thing that's what that's to me as a side note about bikes if you're going to do bikes if you're going to have employees that like to like to work I highly suggest you find some way to provide secure storage for us we just haul our bikes into the office but that's about all I've got for outside again do what you can you don't necessarily have the ability to move your office tomorrow but if you have the ability to get some input into the next decision of where the office is going to be consider some of these so next is the inside this is the easiest category of the three to affect because frankly all you got to do is bring stuff in so the question becomes what does the ruby environment look like when applied to the interior of an office first thing to do is to embrace everyone's extra curriculars we've got two developers who are pretty decent at Rubik's cubes at Beniworks I have maybe one of them the other guy he's really good at them and he has a bunch of them that are not just cubes they're various other types of puzzles so he brought them in you can do all of those except for the really big one in the center he hasn't had the time to do it apparently that one takes hours and hours to do even if you know exactly what you're doing so we're also fans of LEGO as you can see by the spaceship guess who that is another thing that