oice. When I was around 11 years old I went with my older brother who was 16 to his girlfriend's house. She lived in the country and they had a small pond out in the back of the property. I wanted to go because they were going to be swimming and even though I did not know how to swim my brother promised to teach me. We all swam around for a while and my brother tried to teach me to swim. He told me it was like I was trying to swim while towing a load of bricks behind me, in other words, he was giving up on me. Just then a car pulled up in front. We were still close enough to the road to see the car so my brother and the others went over to see who was there. He told me to stay where I was which was the shallow end of the pond. However, there was a small dock across the pond. It was probably only 60-70 feet away so I decided that I'd show him. I'd prove I could swim by swimming over to that dock. So I started off and soon discovered my brother was right. It was like I was towing a load of bricks behind me. The water was only a few inches over my head but it may as well have been feet as I was floundering in the water and I knew I was in trouble. I touch bottom and jump up just barely getting my head enough out of the water to get some air in my lungs and scream for help but no one could hear me from the road as they were all talking. I could see them each time I jumped. I was very scared and knew I was going to die. Then I heard a voice. The voice told me to go underwater and swim. I got as much breath as I could in my lungs from one of my jumps and dove underwater. I was no longer afraid. I could swim easily underwater, no hauling bricks. I would just have to stick my head above the water every few strokes to get air, then back under and continue to swim. I felt confident I could make it to the dock and no longer thought I was going to die. I did make it to the dock and climbed up and looked across to the area I had come from and felt proud and safe. Of course I got all over my brother and blamed him for my near death, haha. But I knew I was in the wrong and if it had not been for the calm, still voice, I would have drowned and my poor brother would have carried the guilt for the rest of his life. So, I guess when I heard that same voice in May of 1972, I knew I was in good hands and would be, once again, safe in his hands. What an amazing story, two amazing stories, Janine, that is incredible. And it is so biblical too. That phrase, still small voice, is found in 1 Kings 19-11-13. It is also translated as the sound of a low whisper, a gentle whisper, a soft whisper, the sound of a gentle blowing. But if you want to know, here is where it comes in the King James. He said, Go forth and stand upon the mount before the Lord, and behold, the Lord passed by in a great and strong wind, rent the mountains, and break in pieces the rocks before the Lord, but the Lord was not in the wind, and after the wind an earthquake, but the Lord was not in the earthquake, and after the earthquake a fire, but the Lord was not in the fire, and after the fire a still small voice. And it was so when Elijah heard it, he wrapped his face in his mantle and went out and stood in the entering inn of the cave. And behold, there came a voice unto him and said, What dost thou hear, Elijah? So perfect this phrase, this passage in 1 Kings with your story. Here it is, you are talking about the wreck, the tumbling over, around, around, around, the possibly drowning, and yet here it is saying that God wasn't in the diesel, God's voice wasn't in the crumbling and the crunching, God wasn't in the water, but you heard the small voice. Here it is in the passage in the Bible, the Lord was not in the wind, and after the wind an earthquake, but the Lord was not in the earthquake, and after the earthquake a fire, but the Lord was not in the fire, and after the fire a still small voice. Amazing, Janine, what a great story. So that's Church of the Undead material right there. I'm not going to do that again. I did that last week, last Sunday, with using a message from Chamber of Comments. I won't do that again for the upcoming Church of the Undead, but man, that is so on the nose. Okay, this next one comes from Amanda saying, My whole life I have been fascinated with the paranormal, but thought I never encountered anything of the sort. Maybe I wasn't sensitive, or maybe I just had never visited any place that was haunted at the right time, me being a bit of a chicken. The fascination made me perfect for the Weird Darkness community, but while listening to your podcast about shadow people, memories started coming to me. The most vivid memory was about 10 years old, growing up in rural Southern Alabama. My mother and I were staying with my grandmother at her home because my parents were in the middle of a divorce. I was outside my grandmother's home on her cordless phone talking to one of my friends. I looked towards her garage, behind the house, as we chatted. I saw movement behind her garage, and noticed this black figure moving to go behind the garage. I immediately got my mother, as no people should be there. She convinced me that there was nothing there to be seen, it was just a black, angus cow, and indeed there were cows in the pasture behind the garage. However, I refused to accept this explanation for several reasons. This figure appeared to me to walk on two legs. It had a hat on, similar to what I thought of as a cowboy hat. And while the cows had access to this area, they were not actually near this location at the time. Also noteworthy, this pasture had an old cemetery behind it. As a younger child, I lived a mile away from my grandmother's. I frequently have night terrors of someone standing over my bed watching me sleep. After I cried about it to him enough and he realized it was not just going away as I got older, my father got me a huge touch lamp. I now could reach out and easily touch it and supposedly reassure myself that nothing was there. I never saw anything when I did this, but I suppose if something was standing over me, I gave it a warning. I left the area I grew up in after high school and went to college. Looking back, I can almost certainly say that I did not continue to have night terrors in college. I moved back afterwards though and moved into a family home that my great-grandfather built with the help of his father and the law. Three times, once when I had dental work, and two more times when dreadful things happened in my life, I saw this figure, but this time near the doorway of my bedroom. It was all black, but I had a close-up view to see that it was dressed in black. And as for a face, my mind did not want to comprehend it. It had one, but to put it together, it seemed to know that it was terrifying. Finally, in this same house, my husband and I almost lost ourselves. We almost divorced because of the trouble between us and I don't know what would have happened to us without each other because it would have been a mistake. We pulled ourselves together and decided to move, not because of anything we thought the house was doing, but because we had defined work and that area had none. The last night there, we were getting ready for bed after finishing packing our U-Haul. All was dark outside, a single light quicker than someone walking, but not flying, mind you, went floating from one window to the next and out of sight. It was not anything we had ever seen in the years in the house, not headlights from cars on the road, we did not hear anything that could cause it, nothing. I could never get my mother to talk about any of this with me. She always shuts me down and tells me that Christians don't talk about this stuff. But keep in mind, she still lives in the area and still owns all three properties that I experienced this on, so she might be afraid to talk about this and get the attention of whatever is there. I always have a strong aversion to this area. It's a lot about the living people there, but what if it's also about this entity or entities that visited me? I don't know of any deaths that have occurred on the properties, certainly not in the houses. I know this was long, but I really enjoy your podcast. You aren't the first paranormal media I've listened to, slash watched, slash read, but you are the first to open my eyes and make me finally drop the veil of fear and connect the dots to see that I too have experienced something. I'm still terrified of this entity, but I have since moved to Tennessee, so I think it's time to start exploring my experiences. Thank you, signed Amanda. P.S. I'm just getting started on the podcast and can't wait to listen to some more. I'm so glad there are so many to listen to. I won't run out for a long time. Thank you for doing what you do so well. Well, thank you, Amanda. Those are very, very kind words. I appreciate that. And yeah, that shadow person almost sounds like the Hat Man, which is not necessarily a shadow person.  Some people think that it's more of like a leader of the shadow people, or just a completely different entity entirely. Once in a while, shadow people can be, or at least feel, benign. They're not malevolent at all. Sometimes they're just there. You see them. The Hat Man that is is almost always malevolent. In fact, I don't remember reading any any experiences of people dealing with the with the Hat Man, seeing him or whatever, and not feeling evil in his presence.  That's what makes me think. But you said that he was dressed in black, not necessarily all black like shadows. I don't know. I have no idea what it was that you experienced. But definitely creepy.  Christians don't typically talk about this stuff. But I think that's because they just don't want to talk about it. Because a lot of them are afraid to talk about anything evil.  But I mean, we're not messing around with evil here. We're not we're not messing with Wisha boards and spirit boards. You know, we're not opening up tarot cards or to look into the future or crystal balls and stuff like that. For those of you who do that, fine. But if you are a born again Christian, that's something that you probably shouldn't be doing because the Bible teaches specifically against it.  So she probably is afraid to even talk about it. So yeah, I think you're probably right. She doesn't want whatever's there to get her attention that that I think that's probably a very good guess on your part.  This next one, let's see here. There's really no name attached to it. But the subject line is maybe clairvoyant. So I'll just call this person clairvoyant. This happened to me when I was around seven years old. We lived in a small country town in New South Wales, Australia. Every year on New Year's Day, the town holds a rodeo that kicks off the rodeo circuit. This year that I was seven years old, my mother took myself and my two younger sisters to the rodeo. We picked seats on the bottom row of one of the grandstands so that my little sister could stay in her stroller and mom could easily tend to her. It was starting to get quite hot in the sun and all the bulls and horses were really getting worked up and bucking really well. Mom says that we should move over to the other grandstand that was in the shade while the officials were getting the next round of bulls ready. We moved over to the other grandstand with fresh drinks and fairy floss and got settled in to watch the bucking fairy floss. I'm guessing that's probably something like cotton candy. Anyway, the first bull to come out was named Chainsaw, and he was the meanest bull in all the rodeos. As he was bucking and snorting and throwing himself around, he had moved closer to the fence. All of a sudden, Chainsaw gave a huge jump trying to throw off the cowboy, and he actually three-quarters jumped the fence, right where we had been sitting when mom said we should move. I don't know if this was clairvoyance of some kind or some kind of precognition, but I like to think that there was some benevolent spirit prompted my mother to move us before getting hurt by the bucking bull named Chainsaw. I'd love to hear this read on the podcast, and I absolutely love your podcast as it goes well with my gothic personality. Thank you so much. Well, thank you, clairvoyance, whoever whatever your name actually is. And yeah, you know what? That could very well have been a benevolent spirit, although I'll admit I hate the sun. I'd much rather find a place in the shade. So that would just be selfish reasons that I would be looking for a place like that. But you never know. I mean, that could have been, you know, some spirit that's kind of kind of putting that thought into your mom's head. Why not? We talked, talked just a couple of minutes ago about that still small voice. Sometimes it's not an audible voice.  So that very well could be the case. So thank you very much, clairvoyant. This next one comes from Jason. He says, Hello, my name is Jason. I've had quite a few unusual happenings throughout my life, but for now a quick creature feature will do. This happened in Huntington, West Virginia in the summer of 2014. While letting my phone charge, I was bored one night and must have looked outside my bedroom window at the right time. I saw some sort of glimmer glide by. I was trying to process what I saw, but couldn't compare it with anything I knew. But I knew someone who may know more about it, someone who is a published author. I got in touch with him and explained what I saw. His answer surprised me. According to him, I saw a pterodactyl fly by my window, saying their wings are nothing like they appear in film. Not only was he surprised I saw a potential dinosaur in modern day, this was the first time I saw a creature that defied normal explanation. But that's another story in itself. Jason, this actually comes to me at the perfect time. I am in the middle of listening to an audiobook by Linda. Hold on, I've got my phone right here, I'll find out the name. I don't know why I can never remember her name. Because I have several books by her too. Let me check here real quick by Linda S. Godfrey. There we go. The book I'm listening to is called American Monsters. She actually has a couple of chapters in there regarding flying monsters. And she talks about, I mean this is the chapter I'm on right now. As you email this to me, I'm listening to the chapter about the flying reptiles and the dragons that people are supposedly seeing. So it's interesting timing on that. And I believe that that's possible. I don't see why, you know, we've got Bigfoot, who a lot of people think is just a gigantipithecus, like an old believed prehistoric ape that may still just be around. There are reports of McKellie Mbembe in Africa, which is a like a Diplodocus kind of dinosaur, Brachiosaurus, Vegisaurus from Jurassic Park. That's being reported at places that they had no idea what dinosaurs were because no dinosaur digs had ever happened there. But the locals were saying that this creature not only did live there, still does. There are still sightings of it. So why not flying dinosaurs, the pterodactyls and Pteranodons? Why not? I think that could be part of why some Native American cultures have stories of thunderbirds or those might have been actual real giant birds, I don't know. But I believe you. I believe you and I'm kind of jealous because that's something I've always kind of wanted to see. And let's see here, let's do one more. Okay, and we've got one last story, and this one comes from Nancy, and she calls it My Dad. Hi, Darren. I'm 25 and the middle child of five. Ever since I was old enough to understand the world, I knew I was different. My parents always sheltered me even though I have two younger sisters. My two older brothers were the same. They always worried about me more than their two-way younger sisters. I always found that odd. My dad had some sort of disagreement with his parents that kept me from meeting them until I turned 12, even though we lived in the same town about two miles away. My father is a longshoreman in the port of Louisiana, I'm guessing that's what that really means. My mother is a doctor. It wasn't until my dad had a near-death experience that he decided to mend his relationship with them. I finally went to their house and met them. It was such a wonderful moment. My grandma brought out a photo album and I saw a picture that would change my life forever. It was an old picture of dad when he was maybe 12, standing tallest to shortest, with my uncle Wally standing the shortest, my dad in the middle and a taller boy standing the tallest. I never met the tallest, but I had seen him before in my dreams. He would randomly be placed in my dreams, like if I would dream about being in McDonald's, he'd be the cashier, or if I dreamed about getting pulled over by the police, he'd be the cop. There was always a lesson in those dreams. The cop might tell me, don't speed, you might kill yourself or someone else, or the McDonald's cashier might say, take it easy on that fast food, you might gain some weight. Anyway, my grandma kind of took notice and she closed up the album and she had me help her make cookies. I overheard them speaking outside with my parents in a low voice talking about, she has the right to know. I didn't butt in, but that picture stayed in my mind for days. I finally asked my mom, who was that tall boy with dad and uncle Wally? She called my dad to the room and she brought out the same picture. My dad says, that's my older brother Eddie. He said more than just that, but at the moment it was a vivid memory that came to me, but I was about ten and I got my appendix removed. I woke up after surgery to somebody who I thought was my dad sitting next to me watching a baseball game on television. When I asked where my parents were, he said, they went to get something to eat, and I told him, I'll watch you. How are you feeling? I said, good. He said, you want some water? They brought me a cup. He had a smile on his face the whole time. I was loopy from anesthesia, so I couldn't stay awake long. When I woke up, my parents were there. I asked, who was the man watching TV? They said, maybe a doctor or nurse. I didn't ask further questions, but seeing that pic, I realized it was him. My mom gently put her hand on my shoulder and said, maybe I'm sorry. I said, for what? She said, we should have told you sooner, but you weren't old enough to understand yet. I said, what are we talking about again? My dad says, that tall dude, that's your dad. I didn't understand, so he explained that he was my uncle. My real dad was his older brother, Eddie. My real mom was pregnant with me when my dad was shot and killed, so I never met him. My mom was a drug addict that lost custody of my two older brothers and myself. They adopted us when I was about six months old. By the time he was finished, we were all crying. My dad said, I understand if you hate me.  You're my dad. I'm sure my other dad is happy you took care of us. Then I explained to him the dreams. He said, yeah, he's still around protecting you. I was happy to know that my dad, Eddie, was still in my life, even if it took so long to realize that. When I got a little older, my dad took me to his old neighborhood for a birthday party. I found out that he and his brothers were heavily involved in a gang, which led to my dad, Eddie's demise. That's the day I met a good friend of theirs that I'll just call Unk. He's much younger than all of them, but he was a dear friend to them, not the least. He greeted my dad with a hug and asked, is this Nancy? He said, yeah, that's her. Unk had tears in his eyes. I asked, are you okay? My dad told him, she looks just like Eddie, right? He nodded, tears rolling down his eyes. I asked, you knew my dad? He said, yes, he was awesome. I was amazed to see a guy with so many tattoos that looked so hardcore a few seconds ago, now crying. Sorry Unk. I could tell he had a lump in his throat.  That made me so happy. He and my dad had swapped stories about my dad, Eddie. He told us about how he inspired him to start working with cars and how he saved him from a tragic end. He also told us about the incident that led to my other mom finally losing custody of me. I was actually relieved that I was adopted.  Unk started coming over to the house every so often. I began calling him Tio, which is Uncle in Spanish. I'd always ask about my dad, Eddie. He was happy to talk about him. One day I finally asked about my real mom. With an expressionless face, he said, she's around. A few years went by. My dad, Eddie, still showed up in my dreams when I was 16 and started driving. I took my mom's car to the old neighborhood my dad grew up in to a really good Mexican restaurant for some tacos. Unk happened to be there. We spoke while he waited for our food, and then his demeanor changed completely. When a woman walked in looking terrible, she ordered something but was being extremely rude. She looked at Unk and said, Why aren't you dead? You should be dead, A-hole. I could tell he wanted to say something back, but he bit his tongue. His food was ready, but he waited for mine to be ready too. I asked who that woman was. He says to me, Baby girl, I'm sorry, but that's your mom.  I was in shock, but I fired back at you bee. Unk pulled me outside to calm me down. I started yelling, I don't care if that's my mom, no one's going to talk to us like that, Unk. Right then that woman walked out and said, You're my baby girl, and tried hugging me, but I said, No, I'm not, but thank you for giving birth to me. I was proud that I handled the situation. I was wondered what it'd be like to meet her. When I turned 18 and graduated from high school, I made a list of things I wanted to accomplish before the age of 21. I accomplished all except one. I wanted to have some sort of relationship with my real mom. I told my parents that and they supported it. I tracked her down a few times, but she only got deeper into her addiction. It's a shame for her, but that was her choice. Still, the dreams continued, always the same, but finally, I had one where I had a real conversation with my dad, Eddie. We talked about my business, I was getting ready to start. He said, Go for it. I'll help any way I can. It was such a vivid dream. It felt so real like he was never gone. He even asked about Unk. He said, That's my little guy. Tell him I'm proud of him. The reason I said that I always knew I was different was because I never found much attraction to boys or men. I was 17 when I realized I liked girls, women. When I told my then-boyfriend that I didn't want to be with him anymore, he slapped me and attempted to rape me. I was able to fight him off. He called me crying later, apologizing, but I ignored him and I reported him to the police. He was arrested. He went down to the police station for further questioning. That jerk admitted to what he did, but said her dad and uncle should be arrested too. They attacked me and chased me down with a baseball bat. They didn't believe him. I kind of did. I asked my dad, but he denied it. He was actually at work when it happened. I found out a few days later it was Unk. I can't say I was happy he did that, but I also can't say I was mad either. But who was the other guy with the baseball bat? I asked Unk. He admitted to it. I asked who the other person with the bat was. He said he acted alone. Then he stopped and smiled. He said, did I ever tell you that your dad, Eddie, was a huge baseball fan? Well, he told me he was buried with a bat. Took me a few days to realize it was probably my dad, Eddie. Please don't get me wrong. I really don't condone violence. I'm a minor in social science, and I've come to learn and understand that in certain environments, that's the way things are handled. Recently, my girlfriend and I rented an apartment in the South Bay of Los Angeles. Unk lives nearby. She asked me to marry her. I said, yes. We decided on something small, we're saving up for a home, with just our immediate family, a few close friends, and of course, Unk. A few weeks before my wedding day, I dreamed of my dad, Eddie, again. I was a Starbucks, and he was the barista, of course. We spoke briefly because he had a long line. He said, you should have your first dance to Will You Still Love Me Tomorrow by the Charelles. I love that song. I said, yeah, for sure. I asked, will you be there? He smiled and said, yes, I'll be there. I told my parents and grandparents that I wanted that song for our first dance. They stared at me surprised. Then my grandma said, that was our first dance. The day of my wedding, we left an empty seat for my dad, Eddie, next to Unk. He was there. I felt him the whole night. He's still around. Probably always will be. Nancy sent a second email to follow that up, but honestly, I think that was just the perfect ending just right there, so that's where I'm going to leave it. Thank you so much, Nancy. What an amazing person your dad sounded like, both dads. And Unk. I think we'd all be blessed to have an Unk like that. Well, thanks for listening, weirdos. If you like what you just heard, please share Weird Darkness with somebody that you know who loves the paranormal or strange stories, true crime, monsters, or unsolved mysteries like you do. And please leave a rating and review of the show in the podcast app you listen from. You can email me anytime with your questions or comments at Darren at WeirdDarkness.com. If you do, you might end up in a future fireside frights if you're telling a paranormal story, or if you just have a question or something, you'll end up in my chamber of comments. WeirdDarkness.com is also where you can find all of my social media, listen to audio books I've narrated, shop the Weird Darkness store, sign up for monthly contests, find other podcasts that I host, and find the Hope in the Darkness page that I spoke of earlier. If you or somebody you know is struggling with depression or dark thoughts or trying to get over a paranormal experience that you just can't seem to get through, all stories in Weird Darkness are purported to be true unless stated otherwise. Weird Darkness is a production and trademark of Marlar House Productions. Copyright Weird Darkness 2022. And now that we're coming out of the dark, I'll leave you with a little light. 1 Peter 3, verse 14. But even if you should suffer for what is right, you are blessed. Do not fear their threats. Do not be frightened. And a final thought from Don Hutchison. What you do every day should contribute to giving your life meaning. If it doesn't, why are you doing it? I'm Darren Marlar. Thanks for joining me in the Weird Darkness.


Hi, your own. This is Andrew. Hi, Andrew. How's it going doing? Well, I have some psychological observations. I wanted to run by you. So since you brought up the attitude towards businessmen and your frustration about objectivists kind of not being able to respect businessmen who don't have the same political beliefs, something that occurs to me is you and I have talked before about being derepression in emotions and the importance of being derepressed. And it occurs to me that, you know, there might be an issue going on with a lack of independence in self-esteem. So self-esteem is at its very base love for oneself. And I think the objectivist conception of that would be you earn that through efficacious action in reality. But it would seem to me that if somebody is repressed, they're potentially missing, you know, both in positive and negative emotions. They're missing the ability to feel that love for themselves. Psychologically, self-esteem is the most important value, I would say. So there may be this seeking, kind of constant seeking for somebody else to complete that feeling. And something I've observed in people's attitudes, some people's attitude towards Ayn Rand. And I've seen this routine go on where they adopt Ayn Rand as, you know, somebody that they respect. They subscribe to objectivism. And then after a certain period of time, I think that they kind of adopt her in a way that's almost more of an idol rather than this is somebody I respect immensely. And I think that that introduces, again, somebody who has a dependent self-esteem, a kind of inferiority complex. Second hand, they then they then see something or observe something she said or did that they don't agree with. Now they can agree with 99.9% of what she said or did, but that 0.1% they get obsessed with it. And it almost takes over their attitude towards objectivism and towards her. And eventually they turn against her. They want almost nothing to do with her. They want almost nothing to do with objectivism. And it occurs to me is the same process of what we might be observing as somebody who can see a business leader who they respect, but who has this thing that they disagree with. And so they don't only feel like towards him, maybe they don't love him because he's not perfect to them, but they hate him. They actually hate him. It's this dichotomy between love and hate that may be introduced from this dependence in self-esteem. So I wanted to get your comment on it. I think that that's valid. And if you've observed some of the same things. Well, I definitely observed the same things.  And I mean, I think it's connected. It's all connected to oppression in the sense that I think a lot of objectivists, I mean, think about how difficult in a sense of objectivism in this sense, right? We redine Rand when we're 18. We have no experience of the world, right? Very little, very little as thinking thinkers as adults. We hand it on a silver platter, the truth. I mean, the answers. And it's mind boggling. How do you deal with that, right? Because we don't have enough knowledge of reality to integrate that truth and make it ours in a first-handed kind of way. So most objectivists go through a long phase of rationalism where they understand the philosophy abstractly. They understand all the connections between the things, maybe, right? Certain aspects of it they own because they are living it and they relate to it. But vast parts of it, they have no idea what's going on there because they can't relate it to concrete in reality because they haven't experienced those concretes in reality. And if, as you've observed in the in the past when we've talked, they're also psychologically repressed, then they can't connect to certain parts of it because they don't emotionally they can't go there and they don't allow themselves to go there and they don't experience the emotions. So the philosophy stays for them as a floating abstraction or at least chunks of the philosophy do. And then usually their point of disagreement comes in one of those chunks that is floating a floating abstraction now and they can integrate it and they can't make any sense of it and they obsess about it and then they blow up. And you know what I've seen particularly with Trump of all and it's the same people who obsess about Trump. I mean I'm not talking about people who voted for Trump and think he's okay. I'm talking about people who everything he does is magnificent, right? Is excellent. You know and everything, every criticism is left-wing propaganda, fake news, stuff like that. I mean I have them on my Facebook page every single day. I mean thousands of them. It's unbelievable. And many of them called themselves objectivist ones and maybe still do, which is spooky. But again it's there. It's at some point what they really integrated is politics. They kind of got politics. They got engaged in the politics. Maybe they came from a conservative background and they always kind of had a lean towards the objectivist politics, but they got politics. They never fully integrated reason, purpose, self-esteem.  They never fully integrated values, hierarchy of values. The importance of living. I see this in people and I don't think about people who will professionally do this, but people who are as a hobby obsess about the threat of Islam or obsess about the threat that the latest conspiracy theories about the left or obsess about something and it's always political and it trumps their entire life and their happiness doesn't matter anymore because the Muslims are coming and you know we need to build our bunker right or whatever. And so you know maybe they've got a job, some of them I'm not sure, but maybe they've got a job and maybe they do okay with their job, but they're not, they don't have a career in the false sense of it. They don't have a love life in the false sense of it or a romantic life in the false sense of it. They don't live fully. They obsess about these things and their objectivism when it comes to morality, epistemology is the complete floating abstraction. All they ever integrated was politics and the same thing here with the businessman. Instead of seeing the productive genius, instead of seeing the beauty and the benefit you get from it, it's as an egoist. The benefit I get from my iPhone or from Amazon. Oh, Jeff Bezos. I forgot Jeff Bezos. I'm a huge Jeff Bezos fan. Look at what Amazon has done to their own life or look at what Zoom, we're using Zoom right now, what it has done to their lives and I admire and say thank you, want to say thank you to the people. They go look and check their voting record. As if that's relevant to the experience of, wow, I can use Zoom to do all these amazing things.  Unless it's important. I mean, from that, it never goes to politics.  It just I mean, that's down the road, maybe. Right now, I'm enjoying this. This is a benefit to me and I'm an egoist. So if I'm enjoying it, I want to relish in the enjoyment of the thing, not in some thing irrelevant. So it's, I just saw a comment Jonathan wrote on YouTube. I wish you all had better critics. Yeah, me too. You'll think of a few. Who I wish I didn't, who I wish were better. So it's, it really is this lack of integration. And I think you're absolutely right about self esteem. So all of this comes down to if you're an objectivist, and the philosophy is a floating abstraction, and you obsess about certain things, you never actually gain the benefits of the philosophy and the benefit of the philosophy is self esteem and happiness. That's the, that's the, that's the payoff. That's what it's about. Because you never actually work to integrate it. You never actually have it integrated available to you on a moment to moment basis. You're not living it in a moment to moment basis. It's something that occurs to you when you do politics, but in life, most of the time, you're afraid. And this is what motivates a lot of this interesting politics. It's fear. It's motivated by fear. I've noticed this and people obsess about politics. They're afraid. They're constantly afraid that the world's going to come to an end, that the left is going to come, the Muslims are coming, somebody's going to fire melons are going to kill somebody. Now there's reason to be worried. But this obsession and fear, you got to live. And, and, and they forget to live. So they never get that opportunity to integrate the self esteem. Now, it's a spiral. Everything in objective, everything in life is kind of the spiral. Lenny Peacock talks about the spiral of knowledge, right? And it reinforces so you don't have self esteem.  The fact that you have a weak emotional life makes you obsess about certain things and be more afraid, which reduces your ability to have self esteem, which it just reinforces each other constantly. And a lot of it is connected to the fact that they can't go to morality in a personal way, because they're oppressed, because to go to deal with morality on a personal level, which is the only way really to deal with morality, means to be exposed to exposure. So what am I true motivations? Because look, we all have altruism embedded in there. And we all have to root it out. And it's not easy to root it out.  And that requires the opposite of repression. That requires actually to be connected to your emotions and knowing why you feel what you feel and then finding the source of it and rooting it out. And if you're oppressed, you can't do that. And therefore, you're stuck. And I think that's where people, again, that was a long answer. But that's, that's where people are. So I think, I think these psychological issues are really, really, really crucial. That makes sense. Yeah, the Trump, absolutely. And the Trump, absolutely. And the Trump point, I mean, that is so, that is such an expression of this in that like they're looking for Trump to comply. They love Trump. I mean, I would literally put that way, they, they, they love Trump. And it's like, if you say something negative, you're attacking someone they love. This is totally a lack of self-esteem. They're looking for a Superman to solve their problems. And in this case, if they're political problems, and Trump is their Superman, and he, he, he is the guy who's gonna solve all the problems for them. And because, because you know, what the great virtue Trump has for them is that he seems to stand up to the bad guys, whoever the bad guys are, right? You stay and when you show them that he doesn't really, they whitewash it because they need a hero, they can't live without a hero. They need to believe somebody is fighting their fight for them. Because, because their life depends on it because their self-esteem depends on it. Yeah, absolutely. Jonathan, you want to say something? Just quickly, Ron, I mean, you're getting your back to your point about, well, for me, at least, objectivism is not about winning a political argument, but simply having a happier life, having a living a better and fuller life, not ever, of course, with unhappy moments or challenges, but living just feeling more alive and not, you know, winning a political skirmish on Facebook or anywhere else. Yeah. And feeling alive, which means, you know, and using objectivism to solve problems and deal with reality and deal with people and deal with yourself and, and deal with stuff.  That's what it really is all about. Yeah. What we need today, what I called a new intellectual would be any man or woman who is willing to think, meaning any man or woman who knows that man's life must be guided by reason, by the intellect, not by feelings, wishes, whims, or mystic revelations. Any man or woman who values his life, and who does not want to give in to today's cult of the stare, cynicism and impotence, and does not intend to give up the world to the dark ages, and to the role of the collectivist roads. Using the super chat. And I noticed yesterday, when I appealed for support for the show, many of you step forward and actually supported the show for the first time. So I'll do it again. Maybe we'll get some more today. If you like what you're hearing, if you appreciate what I'm doing, then I appreciate your support. Those of you who don't yet support the show, please take this opportunity, go to Iranbrookshow.com slash support or go to subscribe star.com Iranbrook show and, and, and make a kind of a monthly contribution to keep this to keep this going. I'm not sure when the next


Thank you for the presentation. Thank you. And thank you for the invitation. I have sort of extra sentimental reasons to be happy speaking here. First, because that's an occasion to see old friends like Stefano Lozato, Franca e Cardi, who live in Trieste. And also, I spent here quite a while, more than 20 years ago. And at this time, the story I am speaking about was sort of between conjectures, hopes. And basically, nobody believed that it might be true. And I was very happy to be accepted here at CISSE. And 10 years after, I was in 2003, I was giving a mini course here. And at this time, some conjectures, some first conjectures were already proved. And there was sort of a more general hope that many things might be true. And by now, 10 years after, some major conjectures are already recently proved and the field is still developing. So I would like to illustrate, basically to illustrate the field on one concrete example, not that this example is the most important, but it is sort of the most visible and the easiest one to present in 45 million lectures. So I will speak about diffusion in a periodic billiard as a model problem. So the problem is the iron fast wind tree model. You place rectangular particles in the plane in a periodic way. It is periodic with respect to z plus z. And the rectangles are aligned with respect to the lattice. This is important.  The ball can go far away, come back, go further, come back. So as a measure of the speed with which the trajectory spreads in the plane, I suggest the following sort of the most simple minded estimate. Let's take the diameter of the trajectory or length t. It is some complicated curve in the plane and I consider it as a set in the plane. So I measure its diameter. I take the logarithm I normalize by logarithm of t. So the first theorem due to Delacroix, Huber and Lelièvre is that for all parameters of the rectangle and here it's really for all parameters of the rectangle and for almost all initial directions for any starting point, the billiard trajectory spreads in the plane with the speed which is roughly t power two thirds where this roughly t power two thirds means that this quantity, this limit is equal to two thirds. And this diffusion, well this measure of the speed with which trajectory spreads in the plane is called the diffusion rate. So this diffusion rate two thirds is given by Delacroix's point of certain renormalizing dynamical system associated to the initial one. It is really the pleasure to give this talk at this conference. So please forgive me a part of the talk, probably 35 minutes would be school and 10 minutes would be conference and it's a pleasure because basically everything was already presented. Renalization was presented, homogenous dynamics was presented, modular spaces were presented, even Zygielowicz constants were presented. So all the ingredients of the talk were already mentioned. So before going on I would like to make several comments on the statement of the theorem. First, well what is surprising here? Surprising is two thirds and it is surprising by the following reason. So there are several classes of billiards with convex cutters starting with the works of probably Bunimovich Sina and Chernoff and with the work. So in one context and in the other context, which is closer to this one, by the work of Delacroix and Thomas Vario. So when the obstacle is convex, then the diffusion rate is one half. So the particle roughly moves in the plane as a random walk. So it propagates with the speed square root of t and here it is faster than square root of t. So the person who is moving by the random walk, he is not quite sober, he doesn't go straight, but he is not as drunk as random work when square root of two. It's something in between. So, and this is funny. Another thing which is, however, I should insist that here I measure the diffusion rate really roughly. I pass to logarithms so I cannot control what is in front of this t power two thirds. So I don't know whether it's logarithm, constant, log, log, no idea. So it is really sort of rough estimate. So this is one more remark. And one more funny thing is that it is really cute that these two thirds do not depend at all on the parameters of the obstacle. You can take the rectangle very narrow and very long, leaving just tiny corridors, or you can even take a chocolate plate and leave narrow corridors between these pieces of chocolate plate. So roughly it is still one of the same two thirds and also you can make your rectangle really tiny little bit still two thirds, roughly. I repeat, I sacrifice the factor which is sort of sub-pollinomial. I cannot control it, but roughly it's still t power two thirds. So a natural question is what would be, what would happen if we changed the shape of our wind tree? So now we plant our trees of different species and it was a question, actually it was a question of Jean-Christophe Yorkosso, the thesis defense of Vincent de Lacroix where the one can make the diffusion rate arbitrary small, changing the shape of the obstacle. And the answer is yes, one can take more corners where all the corners are rectangular and keeping the shape of the tree doubly symmetric with respect to vertical, with respect to horizontal axis. We get the diffusion rate which is like this. So for m equal one we get two thirds. Oh, I forgot to tell that double factorial here is product of all even numbers from two to two m in numerator and product of odd numbers from one to two m plus one in denominator. So when m is equal to one we get two thirds, when m is larger we get other rational numbers and when m is large enough we get this diffusion rate arbitrary small. Here the statement of the theorem is already not for all parameters of the obstacle but for almost all in particular for rational parameters of the size, sizes of the obstacle. The number might be different but still for almost all it's one of the same and you can, if you fix the number of angles and you change the shape in this way it would be one of the same number. What is important is the number of angles, that's it. So it's also cute. Now my goal is to tell how can one obtain these values, very explicit rational values of the diffusion rate and also I cannot help mentioning saying that I am especially happy to give this talk in the Institute for Theoretical Physics because once that's mathematicians who realized that the diffusion rate is not one half as a lot of experimental physicists believed but two thirds which is really different. So there were plenty of experimental works trying to prove that the diffusion rate is one half. You have to trap, no it would change the diffusion rate. So here it is important for me that this is a connected polygon. So basically for me it is, so this thing is connected. No, this is part of the theorem, just a second. You can have periodic trajectories. You can have periodic trajectories both in the initial wind tree and any of these kinds. Whether you can have a periodic trap trajectory, I don't think so. I have to think a little bit, I don't think so. Yes, so I have not a slightest idea and I have not a slightest idea and now I have to make a confession if we are speaking about this. So I was so proudly saying that for any parameters of the rectangle and so on we have these two thirds and so on, if you turn all these rectangles by some angle so if you have periodic obstacles but not aligned with respect to the lattice but turned by some irrational angle there is no approach to attack this problem currently and this is extremely humiliating. There are some sort of hoops but this is a sort of very long and complicated program how to even approach this question. Currently everything is purely two-dimensional and it would be clear in a second why. So I would reduce this problem to a problem, in a minute I would reduce this problem to a problem of surface foliage. So once again this is part of this 35-minute school part of the talk. I apologize there are plenty of experts for whom it would be extremely annoying but I have to remind how to pass from a billiard to a surface foliage. So let's forget about the initial billiard, let's take the simplest possible billiard, the billiard and a square or in a rectangle. And it comes actually from mathematical circles for the middle school that it is much easier instead of following trajectory in this billiard it is much easier to unfold the billiard. So let's take a mirror copy of our table and then in the mirror copy the billiard trajectory would continue just a straight line. And if we have a complicated trajectory, every time trajectory hits the border of the rectangular billiard we take a mirror copy of the billiard and we unfold the billiard table along a trajectory. So what we gain is that now trajectory is a straight line. What we lose is that now we have this unfolded billiard. But know that for a rectangular billiard the trajectory at any point goes in one of the four directions. So red, green, yellow and blue here. And I suggest to identify parts of, well, partons of our unfolded billiard tables which correspond to the same color. This is the same as, for example, mark the corners of the billiard and identify those partons which can be identified by parallel translations. So there are four different colors or four different partons if you wish. So we'll get four copies of the billiard. And after identification these four copies are transformed to a torus because we have to also identify the opposite sides. And our billiard trajectory in the initial rectangular billiard is unfolded to a straight line on this flat torus. And if you take these four copies and fold them back you would get the initial billiard trajectory. So basically what I'm saying is that this is two absolutely equivalent problems. The problem of billiard initial rectangle and tracing a geodesic and a flat metric on the resulting torus. This is basically the same problem. And in particular if we want to understand how far this trajectory would go inside the plane in terms of the torus I understand that the question is stupid. I'm suggesting it by purely sort of pedagogical reasons because I would need to reformulate it in a second in a more difficult context. So in topological terms we can do the following thing. We take a piece of our irrational winding line on a torus. We wait for time t. We join up the ends. We get a cycle on the torus. We consider this cycle, well the closed loop. We consider this closed loop as a cycle in the first homology and to understand how far the corresponding line would go on the universal cover on the plane to the right or to the top we can compute the intersection number of this closed cycle with the cycle which corresponds to the meridian or to the parallel of the torus and this number would give us the distance of displacement of this initial straight line in the plane. So why am I saying all this? Because now I would apply exactly just literally the same construction to our wind tree billiard. First instead of our initial wind tree billiard I will take four copies of it exactly the same way as it was done for the rectangular billiard. Then I will use the fact and there would be some identifications between the sides of these copies. I would quotient over z plus z because everything is z plus z periodic and what I get is this surface which is four rectangles where the opposite sides of each rectangles are identified so that's basically for Toray but this rectangle corresponds to fundamental domain of the lattice and the initial rectangle corresponds to the obstacle so you can consider this topologically as for Toray with holes where the sides of the holes are identified by parallel translations and now we consider we send a straight line filiation in this flat surface which in this particular case has genus five and what is responsible for their diffusion rate is the following thing. We repeat exactly the same thing which I pronounced several seconds ago. We take a long leaf of filiation, we close up the ends to get a closed loop. We consider the corresponding cycle in the first homology of this surface. There is much more room now because this is surface of genus five so it's ten-dimensional space and we take this cycle in this ten-dimensional space and intersect it with this one or with this one where these cycles are indicated on the picture. The corresponding intersection number would tell us exactly what is the horizontal or vertical displacement. It is not obvious from this picture. I just pronounced in a very fast way what is the prescription but it is sort of an absolutely doable exercise. You just apply exactly the same technique as with the billiard in a rectangle versus irrational winding line on a torus and you generalize it to this situation you would obtain immediately this description of this story. From now on I forget once and forever billiards. I'm interested in the following problem. I have a surface, a Riemann surface. I have an analog of straight line filiation on this Riemann surface which would be discussed in a second. I'm interested in the following question. I take longer and longer pieces of this filiation, I close up the ends and I want to understand what is the behavior of sequence of cycles in the first homology which one obtain following longer and longer pieces of leaf of an irrational filiation. For this I need to continue with my school part of the talk. I have to speak a little bit about the asymptotic cycles and so on but I want to start with a few morphisms of surfaces and pseudonauts of automorphisms. Forgive me for a too pedagogical talk. So first remark is let's start with the tori. I want to convince you that tori have a lot of automorphisms. Here is one of them.  So this is sort of topological description. If you prefer algebraic description, more algebraic descriptions like this. If you consider the torus as R2 quotient Z2, then any integer map of R2 to itself factors to a map of a torus to itself. So if you consider a map corresponding to this integer matrix, you get a dent twist. And one more way to see it sort of more geometrical is as follows. You can consider torus as a square with two opposite pairs of opposite sides identified. This map maps this fundamental domain of the torus to the parallelogram like this for which you have to identify the opposite sides. But you can replace this fundamental domain by the following one. Just take pair of scissors, chop off this triangle and since B is glued to B, just place it here. What you get is a square, is a unit square, meaning that it's the initial torus. So this is one more proof that we get a map of a torus, automorphism of a torus to itself. Now one more remark. It would be very convenient for me to consider this transformation in the following way. So the space of lattices was considered in the mini-course of Jens and Andreas many times. But let's recall that any lattice defines a torus. Just quotient R2 over this lattice you obtain a torus. So the space of lattices is the space of flat tori. I suggest to consider this transformation progressively. Instead of taking this matrix, let's put a parameter t here and let make t vary from 0 to 1. What we get is a closed horizontal in the space of lattices, meaning that we deform our torus progressively declining the sides. And this picture shows that the corresponding path is actually closed that arriving to t equal to 1, we get back exactly to the same torus we have started with. So we get a closed cycle in the space of tori. Now slightly more complicated diffeomorphism of a torus. Let's apply a composition of horizontal then twist and vertical then twist. Then in algebraic terms, it corresponds to a product of two matrices like this. So it's this matrix. And again, if we want to use the language of fundamental domains, we send this matrix sends unit square to a parallelogram like this. And exactly in the same way as before, we can rearrange this fundamental domain to bring it back to a unit square, meaning that this lattice is the same. The lattice defined by this fundamental domain is the same as the lattice defined by this fundamental domain. Or in other words, this torus is exactly the same as this one. So we get a map of a torus to itself. And once again, it would be very convenient for me to consider this map in the following way. Let's deform our torus progressively. And this time will deform it progressively in the following way. So if you compute the eigenvalues of this matrix, one eigenvalue, well, the determinant is equal to one. One eigenvalue is greater than one. Other is smaller than one. They're expanding and contracting directions. And I will mark this expanding and contraction directions once and forever. And then I will progressively shrink in contracting direction, progressively expanding direction. So at the beginning, I will just move from my initial torus to some other ones in the space of tori. But this picture shows, and all this calculation shows that at some point, when I move along this one-dimensional family of deformations, I come back to the initial torus. So I get a closed loop. Actually, this closed loop is a closed geodesic in the space of tori, which was discussed many times. So it's a closed loop. It's a closed geodesic on the modular surface. So this is just the calculation of eigenvalues and a definition which I would need. These kind of automorphisms of a surface exist not only for the torus, but for surface of any genus. So they are automorphisms which preserve a pair of transversal fallations and which, in appropriate coordinates, are uniformly expanding, contracting in one direction and uniformly expanding in the other direction. And they're called pseudonosses because for high genera there are several critical points. But otherwise, everything is exactly the same as for the torus. So, and this is just the remark which I've pronounced that if you have an onosoph, a pseudonosopholiation, it corresponds actually to a closed geodesic in the corresponding modular space of surfaces. For example, modular space of tori. So this is the promise, the statement. Now, just a word about this family, the space of all tori. It's space of lattices, which we have seen many times. And if, so one of the ways to parametrize lattices up to scaling and rotation is to say that the shortest vector is horizontal and has length one, then the next shortest is, and choose the next shortest in the upper half plane, then it would be located outside of the unit disk and it would be located between two parallel lines. So this is the fundamental domain of the space of lattices or for me the space of lattices is the space of flat tori. And what is important for me for the space of flat tori is, well, what are the geometric properties of the space of flat tori or other flat surfaces which appear in a second. It is never compact because, just because, for example, in the case of tori, we can always have tori which are very narrow and very long. So the torus can be narrow and long and it can be arbitrary narrow and arbitrary long, nothing can be done with this. So this space is non-compact. Also, it is not quite a manifold, it's an orbifold because there are some tori which are more symmetric than others. So this, if you glue opposite sides over regular hexagon, you get the torus and this torus has extra symmetry which is untypical for other flat tori. And if you glue a torus from a square, it also has extra symmetry which does not exist for other tori. So we get two orbifolding points. In a second we would arrive to flat surfaces. The whole picture would be, in a sense, very similar to that except that we'll get a modular space which would be multi-dimensional. Okay. So let me glue something more interesting than a flat torus. Let me glue a flat surface of genus 2. I start from a regular octagon and I identify a pair of vertical sides and pair of horizontal sides. What I get is a torus with a hole. And then I continue my identification and identifying these pair of sides and then the remaining pair of sides. So after the next identification I get torus with two holes and final identification is surface of genus 2. So we get a perfectly flat surface of genus 2. There is no contradiction with the Gauss-Bernet theorem because to be honest, all missing curvature is hidden in the point which comes from the vertices of the octagon. All the vertices of the octagon are identified to one point so we get a subtle, a conical singularity there. So our surface, our flat metric has conical singularity. But otherwise all the geometric properties of the surface are, well, they mimic the properties of flat torus in particular the holonymy of this metric is trivial. One can generalize this considering any polygon of the same kind with pairs of sides distributed into pairs, with sides distributed into pairs of sides of which are parallel and have equal length. So this is the generalization of the previous picture and you can easily imagine further generalization. And what is important is that there is a group action on the space of flat surfaces like this which can be defined already on the level of these fundamental domains on these polygons. So I forgot to say that I consider the choice of the vertical direction as part of my geometric structure. So for me, from now on and for the rest of the talk, this surface and the surface obtained from this polygon by rotation say by 30 degrees are completely different. I do distinguish them. And then my polygonal pattern is defined inside the plane up to a parallel translation. So when I apply the linear transformation, the result important is also defined up to a linear translation so it would define one of the same surface. My action is well defined. And now the first key theorem of the story is the theorem of Mazur and Vich which says that the action of the group SL2R and of diagonal group are ergodic with respect to some natural finite measure on each connected component on every space of flat surfaces where this notation stands for flat surfaces where I fix the cone angles of the singularities. So recall that our flat surfaces have some cone angles but all the cone angles by construction are integer multiples of 2 pi. So for fixed genus, there is a finite set of combinations of conical angles which are integer multiples of 2 pi and here I denote what are these multiplicities of the angle 2 pi. So here I consider flat surfaces with n conical singularities with angles d1 plus 1 times 2 pi, etc. dn plus 1 times 2 pi.  Excellent. Now why I find this theorem completely magic? Because it claims that you start with, for example, basically any octagon like this and applying this shrink horizontally, expand vertically transformation you can approach at some point any other flat surface in this family. For example, the flat surface obtained from a regular octagon and there is no paradox here, clearly the fundamental domain would never approach but I'm speaking not about fundamental domains, I'm not about polygons, I'm speaking about flat surfaces. So for one of the same flat surfaces there exist many fundamental domains and the true statement of the theorem in these terms is that if you apply this shrink and expand and for an appropriate amount of time and also cut and paste in a smart way, then combining these two operations you can approach or approximate arbitrary well, for example, regular octagon or any other surface in the same family. Okay, now I have to come back to my problem I want to study straight line foliations on these flat surfaces and let's start once again from the torus. For example, one can study the sequences of cycles which correspond to irrational foliation and the following way, choose a transversal, the particular transversal doesn't really, is not important, any. And every time the leaf of the foliation crosses the transversal just join the ends by a segment along transversal, get a closed loop and consider a cycle. We obtain a sequence of cycles in the first homology on the surface and let's normalize this cycle just by, so this is a cycle in the first homology let's normalize this length by the number of returns to the transversal The statement, well, the corresponding limit, this is one of the theorems that almost, that usually it exists, that this limit is called asymptotic cycle and the most general theorem which says that the limit exists is the theorem of Kirchhoff, Maser and Smiley saying that for any flat surface and for almost any direction so the directional flow is uniquely ergodic meaning that there exists well defined asymptotic cycle, one of the same for all starting points. Excellent, now suppose that I suggest to find this asymptotic cycle in the situation when the foliation is not arbitrary but pseudonosoph. This is a very special situation and in this very special situation it is really easy to find the asymptotic cycle just like this. So by the following reason, for example, we can start with the torus and with our favorite Anosov map on this torus, take any closed curve and any cycle corresponding, well, the cycle representing it. For example, one like this or one like this can apply our deformorphism to the surface several times. So in terms of homology, we are applying this matrix to some integer vector and you know what happens when you apply hyperbolic matrix, powers of hyperbolic matrix to a vector. So it is integer vector so it cannot be collinear to either of two eigenvectors but actually your images of your vector get stretched in the direction or in the expanding direction. So to find the asymptotic cycle, basically you take any and you apply several iterations of your map and what you see is up to a very small arrow, you see the asymptotic cycle. In particular you can apply this to first return cycles because in, so suppose that your cycle is the first return cycle to a transversal. Then if we choose the transversal in direction of contracting filiation and, well, this is by definition the leaf of the expanding filiation, when we apply our Anosov map, what happens is that our transversal follows itself but gets contracted and the piece of filiation gets much longer. So we'll get first return cycle under the map or iterations of this Pseudonosov deferomorphism. What we get is, we get sequence of first return cycles to shorter and shorter transversals. That is, we get exactly the asymptotic cycle. Well, I'm cheating just a little bit but morally everything is absolutely correct here and now I have to make a, I have to state a theorem saying what is the behavior in the most general situation. So take now any directional or, sorry, almost any directional filiation on almost any flat surface and consider the sequence of first return cycles. They behave exactly as iterations of some intergevector, well, sort of, they mimic the behavior or iterations of some intergevector under Pseudonosov map. So now we have many dimensions, not two as before, so we'll get this asymptotic direction which was already discussed but we'll also get further asymptotic direction so if we consider the projection of all these vectors to a screen which is orthogonal to the asymptotic one so if we do not want to see the asymptotic direction but only the deviations we'll see that all these points would get aligned and in the case of Pseudonosov map it is sort of obvious because we have the top eigenvector but we have next eigenvector and the corresponding plane would be spent by the top eigenvector and the second one and then if we consider the screen which is now orthogonal to the plane we'll get similar picture and so on. So it's like having decomposition of an image of a large iterate of a matrix with respect to eigenvectors. So here's the formal statement of the theorem there for almost any surface and there is well in the vertical direction there is a flag of subspaces which was presented on the previous slide with these properties and what is the essential part of this theorem is that the corresponding numbers which are on the previous slide which were responsible for this deviation so here this deviation is already the length power lambda 2 which is smaller than 1 this deviation is length of the cycle power lambda 3 which is smaller than lambda 2 and so on in the case of Pseudonosov map these numbers are just the logarithms of corresponding eigenvalues in most general situation that's the Lyapunov exponents of the Hodge bundle along the Tecmeler geodesic flow and the corresponding connected component of the stratum now Tecmeler geodesic flow I would not introduce Tecmeler metric or whatever we have seen that we have a group action on our families of flat surfaces so by definition for this lecture the Tecmeler flow is the diagonal one e power t e power minus t on the diagonal so this is the definition of the flow now and the Lyapunov yeah and this is a very important remark so the theorem got this form only after the work of Giovanni Forni who proved that the Lyapunov exponent number g for the stratum is positive and later work of Arthur Avela and Marcelo Viana who proved the spectrum is simple so before it was sort of conditional theorem where the inclusions were non strict and to have all the strict inclusions one has to have this property which are two difficult theorems proved later now what are the Lyapunov exponents in this particular situation so we have this families of flat surfaces and in each family of flat surface for each family of flat surface we have group action the action of SL2R and we have a flow which is the action of a diagonal subgroup excellent so we have a base space on which we have dynamics and ergodic flow now when you have a flat surface you can consider its homology and taking for every point of our family of flat surfaces the vector space of homology with for example complex coefficients we get a vector bundle on this space this vector bundle is not trivial it's not a direct product because when you follow some closed line in the family of flat surfaces you come back with some twist as we have seen we have considered closed oocycle or closed geodesic we get a deformorphism of the surface so when we come back our vector our bundle twists and we get a co-cycle so every time when you have a vector bundle endowed with a flat connection and a flow on the base you get a co-cycle which is the natural co-cycle which is responsible for the mean monogamy of the bundle of the connection in direction of the flow so the Lyapunov exponents of this vector bundle excellent now I arrived to I still have two or three minutes and I arrived to the conference part of the talk so I want to report several recent results one of them is that there is a formula for the sums of Lyapunov exponents and it contains two parts so one is just this multiplicities of conical singularities the integer numbers which are responsible for geometry of the flat surface so this is really the sort of a trivial part of the formula and the other ingredient is Siegelwitz constant which is one and the same for every family of flat surfaces and what is great is that one can compute this for example for strata one can compute the Siegelwitz constant in terms of volumes of the strata so this is the two theorems are difficult theorems so this one took us about 15 years of work to prove so I use it as a black box and now the most striking result of last years is the theorem of Eskin-Mirzachane and Muhammadi which is sort of analog of Ratner's theorem in this situation I should say why it is so striking so genus the space of flat torre which appeared in Jens and Andreas series so this for genus one the corresponding modular space is a homogeneous space but starting from genus two the corresponding families of flat surfaces are not homogeneous space it is known that these spaces are not homogeneous so a priori there was no reason to expect that such a miracle as the fact that all orbit closures and all invariant measures are extremely nice exactly in this situation of homogeneous spaces it was absolutely over-optimistic and still it is true and finally it is proved it is very complicated theorem and the proof is really difficult but it immediately produces wonderful results and well there are effective methods of construction of orbit closures which are under consideration and also one of the colors of this theorem is that for any given flat surface almost all vertical directions define a Lyapunov generic point in the orbit closure and this shows how one can sort of approach in a sense at least conceptually all kind of wind trees as soon as all the angles are rational and as soon as my unfolding construction produces a surface of finite genus so first you start, you pass from a billiard to a flat surface then you take a magic wand of Askin, Mirzahari and Muhammadi you touch your flat surface you apply the action of vessel to R something happens and you get a complicated complex manifold of some dimension orbit fold but which is actually extremely nice and this is the miraculous part of the story it is under construction there is no classification of SL2R invariant manifolds yet but in some cases it is possible to find the orbit closure excellent then you compute the corresponding Lyapunov exponents and in a sense you are done in the case of wind tree or in the case of this generalized wind tree which I suggested everything can be reduced to genus once so the formula for the sum of Lyapunov exponents is reduced in a sense to a formula for a single Lyapunov exponents which is computable because we know the volumes of corresponding moduli spaces so all this is sort of quite a technology but still it is at least conceptually it is clear technology how to produce things and I want to finish with several challenges and open directions and now when we have some fantastic theorem new horizons appear so one major question is to study all GL invariant suborbials all these flat surfaces this question is solved only in genus 2 by Kurt McMullen in 12 years ago and there are indications that the life is really interesting there are known examples which are sort of miraculous there is no understanding what is the geometric origin of these examples now one more question is basically differential geometry and it goes sort of backwards this dynamics and Lyapunov exponents some universal properties with respect to some universal estimates for the curvature of holomorphic sub bundles of the Hodge bundle and on the one hand and on the other hand it gives a possibility not to compute exactly but at least to estimate Lyapunov exponents this is also wonderful now one more very interesting direction is to study what happens when genus tends to infinity and there are many indications telling that many quantities sort of stabilize and do not depend on particular shape of the flat surface or particular collection of conical singularities as soon as your surface is sufficiently complicated and not too symmetric it seems like all of them are in a sense the same and maybe this is an approach to Win-Tree Billiard with irrational angle and so on it would be very interesting to relate this Siegelwitz constant to intersection theory and to find the answer for the expression for it in terms of complex algebraic geometry and also here I considered families of Riemann surfaces but one can try to consider families of complex surfaces of complex three manifolds and they are corresponding moduli spaces there is a group action so some things can be done and it would be very interesting to recognize and the corresponding dynamical systems renormalization systems of something which is analogous to billiards so this is to my mind a very cute question thank you very much for your attention


Everybody forced here ripping it 22 23 contenders optic Basketball hobby 20 bucks full case here. This is 20 bucks full case number one. There are the random results Let's go ahead and get started Mike D How you doing Mike? Happy Friday, dude All right Let's get to it Teach and let's go teach and grabbing a spot and contenders optic guys the next one is at seven now Is that seven only seven? Whoa This guy's going crazy dude kind of sit. What kind of says is your favorite? I'll bring one. 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I guess you got to respect any NFL team like it's it really is any given someday Pink Donovan for the cabs that's out of 75 Silver Larry marketing silver Palo uniformity for the magic on the inserts And a red ice hoop dreams been the bucket Ben Mather and Pacers First auto goes to the Hawks. It's variation green a rookie ticket auto AJ Griffin That's pretty nice 44 of 75 good start for a second tier Hawks And they give him Poisson that doesn't even make sense Anthony. That's not even an attempt You're just you're just being you're just attacking Sean Yeah, he does Yeah, Christian Wood You got silver Bronson for the next Silver hoop dreams is Tari eason Nice one there. I got blue ice sweet shots Steph Curry to 75.  I think they got Franz Wagner out of a 99 magic season ticket color match on the blue and Jazz out of 25 John Stockton on that is stew. That is sick. That's a great car design That the throwback 85 contenders. Yep. 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It's supposed to feel like the dolphins That's what it was for a while Don't move the ball because Pat's D was good, but I'm not even mark. That's crazy rookie ticket red Christian Brown for the nugs Silver Hardy for the Mavericks Uniformity Silver Step Curry Warriors Red ice Damian Lillard for the Blazers unless we've seen Zach Wilson do something. He's never done before And Paolo Ben caro rookie up-and-coming autograph for Orlando There we go up and coming five of 75 Jersey numbered Orlando magic I Am excited this season is coming up. I Am excited. Let's go magic Big hit there. That's going into a mag of course. 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Let's go man and Struce On the heat auto here Struce To 75 Yeah, guys, you're just if you're just hopping in had to change the schedule up We're kind of breaking what fills We had a lot of breaks with spots left and somewhere down the line that were full So that's why we're doing this one right now Chargers they're they're they're a cursed team They are a cursed team the chargers They have been since the San Diego days Is the only way to rationalize it at this point is a non-rational way 13 to go on the bowman mixer with bonus boxes Get them John Yep two bonus bowman chrome hobby boxes Five gold packs five silver packs in that mixer Keldon Johnson. There is a blue rookie Yovitch It's a 99 for Miami red kairi for the Dallas Hoop dreams red ice is christy for the Lakers and numbers game yannis number to 75 bucks blue ice I'll agree jets are a cursed franchise too. I'll agree with that as soon as as soon as rosters got hurt. 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To 75 lottery ticket for houston and kaminga to 99 on the sophomore contenders auto for golden states His name is alblades I don't know what team he played for but his name is alblades jr There's two alblades in the world Wait, oh duke and or dame play this weekend interesting My name is 13 national titles Why did they win a national title? Silver scotty barns are the raptors red.  We need a big auto. We had a palo Yeah, i'm the same wave on I don't I don't really watch preseason if it's on the tv at work. I will see it but Never will I ever go out of my way to watch a preseason game of any of any kind? all right, silver jammy grand blazers orange terry rozier to 49 for the hornets and I feel like preseason ticket prices are like just as much as like a lot of regular season ticket prices for a lot of sports Anthony Edwards optic illusion for the wolves The sweet shots and bead sixers depending on the franchise some franchises that might be cheaper, but That's a big one. 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I think Yeah to 75 Yeah, after the oracles clinched I mean the oracles and rays are gonna be the two teams that are just like letting off the gas for the this this weekend rj barrett or they should be a little bit Especially for the for the pitching Certain stars should not be playing as much either get that rest mobily out of a 99 for the calves Got a whole other month of play for you if you play it, right? silver jalen brown silver hoop dreams of j dubb for o kasey And he can marry rookie uniformity to 75 kings no lambier to 125 With the pistons auto All right, paul george. That is pink to 75 kuzma for the wizards Red kevin herder for the kings hoop dreams grizzlies david roddy Sweet shots red ice anthony Edwards tea wolves And bison daniels variation blue rookie ticket auto for the pelicans 49 Yeah, king's finally on the board Yeah, give me tough to hang the tenders out to guys not a ton of hits the gold coming out I think we've got a gold coming up. There's a viscel base paint to 75 reddish for the blazers red step curry silver numbers game zubak for the clips It's like a gold Nice luke donchich optic illusion number to 10 Congrats, dallas very nice gold luke Mavericks is thomas c That's pretty sweet. I like that a lot and watson variation of 49 blue rookie ticket auto for denver. I think they'll be pretty good Blake But I don't know the west is the west is so stacked that I don't know if they'll place as high, you know what I mean West is west is stacked. I mean the blazers are the blazers are out of it now officially silver rookie johnny davis wizards Blew out of 99 siddique bay hawks red ice sweet shots luke donch it's dallas And a first class ticket for dallas of jade and hardy It's 125 karan butler all-time contenders for the wizards Yeah, what was great Bobby quit Yeah, they could still place pretty highly in the west and not winning fewer games theoretically Because if the the suns are going to be better The lakers are going to be better for a full season than they were last year You obviously got the nuggets still Thunder should be good I mean, there's a lot of good teams the last we can just keep going dude. Dallas is going to be pretty solid. I think Clippers could be good.  Let's see The the the bowman mixture down to three The bowman mixer is down to three nine on contenders optic number 10 Okay All right All righty then Come on doug peterson get the jags going good here Are they over three? They're one and two What their losses are against like houston and like all the cheese But the offense doesn't look good at all neither has a defense like they just haven't looked good. They beat the colt's week one How was their win? What's up spencer? Oh jaker mayfield's here Hello jaker mayfield Yeah, oh, I mean Just keep talking about it. The young guys don't learn cursive I learned cursive. I don't even use it. That's an interesting Um to 99 mobily sidique bay dyson daniels kevin dirant nickle the yovitch jordan pool Franz vogner it's 75 down to the michel 75 is yannis embied reddish kuzma Steph curry Uh jalen brown and keegan murray to 50 a scary terry 25 kevin dirant christian wood Zach levine lebron james Lebron autos coran butler ladel isaam mobily jake laravia bill lambier nick van exel Ty Ty washington jr. Herbert jones jonathan kaminga dylan terry a j griffin max streus james worthy blake westley Dyson daniels paton watson and john stockton We had gold optic illusion of luke donchich for dallas palo band caro rookie variation season ticket green redemption for the magic Up and coming jersey numbered five of 75 rookie auto palo band caro And a gold rookie ticket auto of jalen williams For the fund, uh, that'll do it. Thanks. Get everybody more basketball coming up


Ευχαριστούμε και ευχαριστούμε! Είναι Vicky εδώ και σήμερα είμαστε συμβαίνοντας την 4η Ανυβερσαρία του Αλτιννού. Αυτό είναι ένα Βαρτι-Βλογκ-Ωπ. Ευχαριστούμε για πολλές εμπειρήσεις και πολλές δεξιότητες. Σήμερα θα δημιουργήσω ένα πολύ καλύτερο καρδί. Είμαι πρόκληρος τηςζυμή της εμπειρήσεις και θα είμαι πρόκληρος της μπανπούςπς από το Never Give Up Stumbset. Πρέπει να είχα αυτήν την εμπειρήσεις της εμπειρήσεις και νομίζω να λάψω ένα καρδί. Πανωθεί να αγκατεραστούω ένα καρδί. Πολύ θα έρθω ένα εμπειρήσεις εμπειρήσεις εσένα για όλα αυτά τα πράγματα και τα γερνά που έχετε στην ζωή που εξετάσουν δύο νίντια. νιντιάς. Αλτενιού έχει ένα νιντιάς που είναι called bamboo which is absolutely perfect for stamping these. Είμαι σκάφτας them twice to get a nice and crisp image and now I want to add some more dimension with the second color and for that I'm using caramel toffee now in the bamboo trees there are some bumps some nopes I don't know how you call those and they are always darker than the rest of the tree and that's exactly the spot where I want to have that orange color so just because my ink pad is quite big decided to apply the ink in the areas that I want to by using this fantastic tool this is quite spongy at the tip and it picks up ink and applies it on the stamp beautifully so it works great another way to do that is if you have a marker for example mementos would work great so I use the matching dies to cut out my bamboo trees and I'm just popping out all those little pieces I picked one of those ninjas and I'm going to stamp him with black ink and I'm actually using all the new permanent black and I'm doing all the stamping with my team holds stamping platform because I want to stamp this ninja a couple of times so that I end up having a nice bold and crisp image now for the body parts such as the hands and the face there are other little images that fit inside and I'm using blush ink by Altenew to stamp those to finish off my ninja I need to stamp the swords so I placed them where they need to go and also for that I was following the guide the back of the packaging I started by stamping them with light gray but this is where I decided that it would be awesome if I actually used embossing powder so I'm going to stamp them one more time with embossing ink this time and then I will apply platinum embossing powder by Altenew I'm using my heat can to melt the embossing powder and this is going to give a beautiful shine on the swords and a more realistic look and now I'm going to use the matching dies to cut out my ninja and here is a tip you might want to stamp the ninja cut him out and then stamp the swords the blades of your swords because this might be an issue if you don't align those blades correctly now it's time to assemble my card first I'm going to stamp the sentiment on a piece of white cardstock this is just a panel that is slightly smaller than a standard card I use one sentiment from the same stamp set that says because you are awesome and now I'm going to apply some texture at the background using the warp speed stencil this is by Altenew and it has just been released for the month of March I'm going to apply with my spatula some texture paste on top I'm not going to cover up completely the background I'm starting from the center of these of these rays and I'm moving towards the outside without covering up the whole thing I went with white just because I don't want to turn my background into being very busy after all I have lots of focal points to stick on top and now let's peel off the stencil and see how the background is looking I really love the white on white texture and I think it provides the perfect background for my ninja to make my card really fun I'm going to stick my ninja with one of those mini wobbles so this is actually a spring and I'm going to stick one part on top of my ninja at the back of my ninja and the other part on top of my card base I'm peeling off the other side and then I'm going to stick this ninja on my panel making sure that he is nicely centered at the texture that I already have there and the card is already looking awesome as it is but I will embarrass it a little bit more with those bamboo trees I'm just trying to decide where everything is going to go and then I will add some tape adhesive at the back and stick them down so the idea here is that this ninja is flying and is coming towards you and I'm going to stick those bamboos in two opposite corners which frame my ninja quite nicely I'm going to use my scissors to cut off all the excess and my panel is ready and to finish off my card I'm going to stick this panel on top of a standard card that's 4.25x5.5 I'm using some foam tape at the back to add some dimension and I picked the cardstock to match perfectly the bamboo trees all they knew is an awesome company they have amazing stamp sets and this is a card for them to celebrate their fourth birthday don't forget this is part of a blog hope with lots of inspiration and giveaways make sure to join me on my blog to learn all about or how to enter the giveaways thank you all so much for watching and have a lovely day