what's going on infinite fam welcome back to another video it's your girl jenice if you guys are new to the channel make sure you hit that subscribe button and join the family now today i am going to be surprising azea with an early christmas gift now azea is a type of person that doesn't like to spend money on himself he doesn't like to splurge or anything like that well i decided to get him a little something that he's wanted to get for a really really long time but like i'm telling you he's wanted this for years and years probably like two or three years but never pulled a trigger on it and honestly he's such a good boyfriend i just want to bless him and what better day to do it than around christmas time guys right now i say it is in the shower i don't know if you can hear the water running but i've been hiding this gift from him for some time now and i'm the type of person that i cannot wait to give people their gifts i'm just so excited and i feel like he is going to really love it and the reason why i'm giving this to him early is because i feel like he's really going to need it these next few days so i'm going to show you guys what i got him i'm not going to tell you guys what it is i'm just going to show you guys the box and yeah so this is the box it's a huge huge christmas gift i had it hiding back here in this hole i just knocked out all my shoes my closet is so freaking messy right now but yeah i had it hiding back there so we are going to just wait for isael to get out the shower and i'm gonna surprise him with the gift so let me know if you guys are excited out of the shower it's the gift hi babe what are you doing hiding why what are you up to bro all i see is your arm coming out the door with a camera what are you doing why are you hiding behind the door what are you doing because i have something for you what if i don't want it you're scaring me what surprise what the hell is that here grab it how's this it's a early christmas gift you're scaring me babe it's not a prank i promise what's in it i obviously i'm not going to tell you it's a gift all right so wait before you open it before you open it i'm just going to tell you like i told the infinite fam this is something that you've wanted for a really long time but you know i know that you don't splurge on yourself for anything and you know you're very hardworking and you're such a great boyfriend and i just wanted to bless you and i wanted to give you something that you'd enjoy for a really really long time and you'd love and appreciate so look at how you like that and um yeah okay that's all i have to say you is working hard for that ring you is working hard for that ring i'll tell you big turbo kit uh no babe do you really think a big turbo kit is gonna be that light everything it has nothing to do with cars okay nothing to do with cars no and i wanted it for a really long time yes don't sound so disappointed i'm not disappointed i'm not at all i'm just trying to guess it's something i wanted for a long time scratching the box let me prepare for this hold on get comfortable get comfortable all right wait it's not christmas okay so why are you giving me this now i just said it's an early christmas bro is it a prank no it's not a prank i promise so are you giving it to me early oh my goodness because i just wanted to i don't know you scare me i feel like you wouldn't need it bro i can't i need it sooner than later which is why i like allowed you to open it early oh my goodness your rib gets so slow so slow i'm trying to see if i can get some type of hint of what it is security seal bro that would security sales only come with expensive stuff what are you up to what do you mean expensive stuff what are you up to i'm not up to anything i'm zooming in i want to see your face right now i just ripped the security seal what are you doing open it break the freaking box goodness gracious do you see i can't open it figure it out babe hurry up it better not be a box inside of a box it's not i promise oh i think i got it booty it's all my dark room it's such a mess babe would you get me but um open it moncler is expensive babe it's okay you deserve it no what you get me oh my gosh just open it hey i'm assuming it's a coat i don't know you have to look and see okay the packaging oh my gosh okay i feel like a little kid oh okay yeah undo the buttons yo the packaging on this period so nice it feels like a coat it definitely feels like a coat it does it definitely feels like a coat okay baby you got me moncler coat stop this is expensive stop worrying about if it's expensive or not put it on did you get the right size what size did you get i hope i got the right size what size did you get i think it's at a size two which is the medium the hell is it two that's like a medium i don't speak expensive i just know small medium large how's it feel i feel like a million bucks oh my goodness oh that's so good on you yo babe you got i did i got you on moncler hold on hold on i gotta i gotta zip it up i gotta damn so cute yo you got me i'm just happy i'm so excited babe y'all how much did you spend on this we not talking about that wait can i zoom in on it though babe how much did you spend on this look at that guys i wanted this coat no lie for like two years i always wanted them on claire coat but they're really expensive and you guys know we're not the type to like buy stuff like this we have a few like designer things and they're nice to have and all that but it doesn't make you who you are or whatever but like this is something i always wanted and you bought it you're crazy i'm not crazy you guys it's always okay to like bless yourself and bless the people that you love with things that they really want and this is something that you really wanted for a really long time so look at the cartoon what is this cartoon i think that's like information on how to wash it and stuff like that yo what it's the small details man they got to dumb it down for people like me that's why they colored they color coded it they got the cartoon in it no this is fire do you love it yes i'm not sleeping it oh my gosh and you really would too what's the problem give me the camera give me the camera give me the camera give me the camera give me the camera yo she really just got me on my claire coat guys when did you order this um actually i ordered it like a couple days ago bro this is insane like oh my god yo this is yo she is working hard for that ring oh my goodness i'm not doing this for a ring i'm not but i would appreciate one don't worry baby that could that day going to come that they're gonna come i already got told you all the time i tell them all the time i got it locked in i'm just i'm waiting and there's a reason why i'm waiting but don't worry about that we're not going to get into that but yo how much did you spend on this here take the camera yo so pretty much like i was saying i told you that i wanted you to open this gift now because i don't know if you guys know what the weather is like in florida right now but it is going to be 40 degrees tomorrow and it's super cold and we threw out our coats when we moved to jerries when we moved from jersey so i'm like yeah baby needs a coat i mean i'll be fine i have like a trench coat so um oh he is filling him so yo you got me up on clear this is crazy so it's a 10 out of 10 yes oh wait what is this oh my god guarantee authentic that's right her no no fugey this the straight monk please oh my gosh no seriously thank you thank you i love you thank you so much i love you all right guys let me talk to you in a second when he comes down all right infinite fam so that is it for today's video i hope you guys enjoyed if you guys did don't forget to smash that like button comment down below team junece for being the best girlfriend ever it is now time for today's post notification shout out today's post notification shout out goes out to daisy lariah thank you so much for your love and support izan i love you if you want to post notification shout out all you gotta do is like comment share and subscribe turn on those post notification bells you know whenever we post a new video we'll catch you in the next vlog miss video we'll have you infinite fam


a hoi-hoi and welcome to today's video. It's going to be a little shorter. I'm just going to do a reading from the SCP wiki, but this one in particular is a song, well song parody somebody wrote, of it's beginning to look a lot like Christmas. Let's see here. This was written by Calibri Bold on the, or I should say the parody was written by Calibri Bold, the original it's beginning to look a lot like Christmas. I believe, let's just, let's Wikipedia this, was written in 1951 by Meredith Wilson. That's important to remember because it is still under copyrights. You'll unfortunately have to deal with an acapella version, where I read off the SCP wiki here, but let's give it a try and see how well we can do. It's beginning to look a lot like Eldritch, everywhere they grow. There's tentacles in your head, squirming there once again, where torture pain and going insane won't go. It's beginning to look a lot like Eldritch. You can't take much more, but the evilest sight to see is an undead corpse from me at your own front door. Flash-crafting books from their ancestral roots, it's the wish in the sarcastic said. Striders that walk and will turn you to rock, they'll bring you to Corbinic Dead, and nine-tailed fox can hardly wait to contain them all again. It's beginning to look a lot like Eldritch, everywhere you go. There's a god we summon from hell, he'll be contained as well, and the SRAs will never let him go. It's beginning to look a lot like Eldritch.  Yes, it's beginning to look a lot like Eldritch, bright forevermore. But the scariest sight to see is mtfi13 at your own front door. Hey, I got it in one. Thank you very much for watching. If you enjoyed the video, which I have no idea why you would have, but if you enjoyed the video, scroll down and hit the subscribe button, and then hit the notification bell when you're done with that so that you will be updated and notified when I upload new videos. And if you'd really like to support the channel, head on over to patreon.com forward slash D Sumerian, and pledge at any level like everybody here on the screen already has. It's one of the easiest ways to ensure that I can continue to make content for this channel. I've dipped, I said this last month too, I dipped a little bit under 400 then and then somebody pledged in the middle of the month so it went over, but I'm back under 400 again, which is always worrying because you know this is actually my primary method of earning money right now. But I appreciate anything that you can pledge on the Patreon page like Dr. Jay Redacted, who has pledged a hundred dollars, Sinjeriki, who has also pledged a hundred dollars, and Morgan, who has pledged 40 dollars. Frankly, it's nice to know that I'm not alone out there, and I will see you all again on Tuesday.


I think maybe shortly after my son was born I realized wow this is way harder than I thought. It was very challenging being able to balance everything between my husband and I. We both worked full-time and we had to face a decision about whether one of us had to push pause because of the lack of childcare options. I'm so fortunate to be with someone who continued to encourage me to pursue my career. But this is not the case for so many women who have to decide between work and childcare because they don't have any other childcare options and this is not okay. Unfortunately, women are still disproportionately the ones who have to sacrifice their career and their educational goals to be able to meet their childcare needs. So we need to improve that and having a system where more mothers in BC are able to return to work is good for our young children, families and our economy. So I think that having a universal childcare and early learning system is not a nice to have. It's a necessity for us to have the kind of province where people can afford to live where they work and where it's not seeing a strange for a mother to return to work or a father to stay at home and care for their child. We need to reach to a point where men and women are equal members of the workforce and have the same opportunities. This is the BC I want for my child.


It's time for intergenerational dialogue and co-leadership. It's time for young governments. It's time for young women empowerment. We must work together and co-create with youth, especially young women. 60% of the current population is young people under 25. And it will double. And it will double. So that's why the conversation with us and with the new generations has to continue. We have the average age of African leadership at 66, and we have the average age of the population at 25. So we have at least 40 years of gap between generations in leadership because young people are the majority, but they cannot sit on a table and have a confidence to have a conversation or to feel they're listened to or to feel their voices heard or they're taken seriously. And so, you know, elders and other people in generations where they take youth serious are important in this conversation because they can enlarge the table and bring a young voice here, a young voice there and make that conversation inclusive and make it empowering for the youth. We need parliamentarians to come in and hold governments accountable that they have to allocate budgets for programs, fighting FGM or child marriage. The gender agenda is my agenda and it is young women agenda as well.  This is not an issue to do with the older generation or the grandmasters or the mothers generation. This practice is continuing with the current generation. And we noticed from the different surveys that even the younger generation, the young mothers are still confused about this practice and some of them, some of them, they are still defending it as a practice that is requested by the society. So that's why the conversation and the dialogue between the different generation has to continue. It has to continue. As our generation we have the responsibility to take all the knowledge we have about these issues to our youngest generation because they will take the lead with their generation, with their peers and with the coming generations and they can deliver the message better than us. That's my belief that they can deliver the message better. They can't change their narrative. Absolutely. So they can drive it themselves. So intergeneration dialogue, especially in Africa, is very important because of the generation gap and we need to close that gap and we need to talk about intergenerational co-leadership. It's not about us or the other generation, which one of us will lead, which one of us will take space. It's about us coming together to co-lead because we need to take our continent where she deserves to be. One, two, three!


As one school year closes, we turn our eyes to next year. The Bemidji girls hockey team was still looking for a head coach and today the school announced that they found their guy. Michael Johnson will be taking over the program. He played for the Lumber Jackson High School, graduating in 1984. After one year of juniors, he went to North Dakota State, playing on the team from 85 to 87. He has experienced coaching girls varsity, leading the Cap City Cougars of Madison, Wisconsin for eight years until 2015 when he moved back to Bemidji. He says he's fired up to be back home and leading his alma mater. I'm just really, really looking forward to again working with the high school kids here in Bemidji and again giving back to the area that I owe so much to and I think it's going to be a really good fit and we'll see if we can build some success. While he was a forward during his playing days, as a coach, he focuses on the defensive side, saying the offense will come naturally from there. If we play really good team defense, we'll create offensive opportunities by getting turnovers on the blue line in the neutral zone. So I'm a little bit more defensive oriented. Having said that, I do like to get some speed through the neutral zone and maybe open up the ice and have a fast transition game as well. If you enjoyed this segment of Lakeland news, please consider making a tax deductible contribution to Lakeland PBS.


Okay, maybe not worked. Okay. I think that's got it All right, so here we are how you doing everybody say hello welcome to the live stream say hello All right, I'll put this down here All right, welcome to the live stream. This is a impromptu live stream I'm just going live quickly because I'm starting and I'm using a new software. We're using a Be live to go live today. Oh, let me pop up a little bit So we use a be live to go live today and we got a couple of things in the carousel That we're gonna go through quickly only about 15 minutes Maybe but it's that time of year, you know pest control for the people Okay, I'm a licensed pest control technician in the New York State and hey, how you doing rising stars siren Welcome the purple army is here. Okay, awesome stuff. So we're talking about pest control for the people It's getting cold at night that triggers your rodents around your house to start stockpiling food start expanding their range They start finding their way into warm spaces Yeah, they're looking for warm spaces, but they're really looking for food to stockpile for the winter Because that's what drives them inside in the winter lack of food. Okay, because they're out there all went along believe me I'm out there all went along They're under the snow. They're out there and every five to eight weeks the babies are having babies These mice are unbelievable. Right. So what do you do? What do you do?  Oh, we're gonna mute that And we're gonna pause that we're gonna share it. You see this is new software So it's gonna take me a minute to figure this out. So Screen share window boom. Okay, that's not too bad. So this is them there and okay They have a little key you open it up and you're gonna put the bait in block in there And it's big enough only for a mouse to get in there. Alright, so the mouse would crawl in there eat the bait now The bait that I'm using there is also in the carousel. It's a intercoagulant It's a really good bait. It's a vitamin K deficiency causes them to bleed to death intercoagulant, right and But their natural instinct when they when they start to feel ill is to Search for water, you know, watch their stomach, you know, they're gonna go search for water.     They're out there chipmunks, you know Rats, I don't want to mention it. But yeah, you know, whatever they're out there, right? Um, and your house is totally under pressure all the time. They're looking and they chew Okay, and they're always looking for ways to get into warm places. They're always looking for ways to find food Right, so you can got to control the population on the outside So you take these bait stations to protect the landscaping stations they look like rocks they about this big I'm going to show you them in a second and You put them on the four sides of your house and you put the bait inside them and They established those as food sources and they go there and Instead of coming into your house. I'm trying to find the putting the rock stations together I'm gonna just show you a picture of a rock station with the bait in it Let's see if I share I know Know that didn't work. All right, hold on a minute That's cuz I didn't do that All right, hold on a second Here we go. There you are. All right, so that's the station open This is the top and these are the baits they put in there and it closes over I'm gonna show you I think it close it so you close it over and they go inside there So you put one of these on each side of your house, right? And I think I have other pictures of them, too Let me see if Hold on a second. Can you see that? I don't know So that's it as well. So and it looks like a rock and you got a key special key to open it up Hey, I'm using be live. This is my first time using be live You know, it seems pretty to pre pretty good. Actually what I would like to do what I'm not sure is going on here Is I don't see a chat Over here Maybe I didn't do something right. I'm supposed to be able to see the chat, but maybe I didn't do something, right? Um But yeah, it seems to be working just fine. What do you guys think? I mean, you're the judge, you know As far as the quality it seems about the same Sharing the screen is about the same Close that close that But the bait that I use is what I'm telling you is is the contract blocks, which is an inter coagulant I Was using first I started using Restream at the beginning and then I was using stream yard I still use stream yard and Probably will use stream yard Tuesday and Wednesday when I go live for prime day this this week because I'm still learning the new software You don't want to go live on the pressure situations with the new software. So that's what I'm doing with that, you know, so Yeah, so But be live is supposed to be really good And I must have done something wrong because I'm not getting the chat in the be live And I was supposed to be able to highlight the chat and put it up on the screen over here But maybe I needed to put my URL From Amazon into the somewhere and I didn't do that.  Oh, we got a follower Chrissy started following. Hey, thanks Chrissy Now usually I play music and I hang the curtains and I dance a little bit But I'm on the new software. I don't have that set up yet I don't even know if I could play music with this one. I have to download my own songs to it I probably that's what I'm gonna have to do but Chrissy. Thank you for following. I really appreciate that That's great. Are you looking for pest control?  Okay, here's one All right, hold on let me share the screen Do-do. I know it's hot too many buttons to push All right, so there it is All right with my cell phone. So this is the the side of the the house of your siding that's the channel runs up the corner They run right up that to the attic, okay? Now you can see all the feces up here You can see them chew through the vinyl right up in here They chewed all up in there and now they're in the attic they ran up through now They're in the attic so you're gonna go down to the bottom and you're gonna stuff the bottom right down here I'm gonna show you my hand right up in there. You're gonna stuff that with the copper mesh or Foam we also have we also have a foam great stuff foam expandable foam You can do that and you run around all underneath here. It's as loose. They can get right under there They can get in around the hose they can get around the windows And you're gonna go around you're gonna look for any holes And you're gonna seal them up with the copper mesh with any caulking or I got Expandable foam already and watch this you just leak a slide right up there in the corner like that Because there's no channel that's supposed to be locked in on the bottom and there again the corner is wide open They run right up that corner and getting to your attic easy peasy now look at this now This is your utilities coming in okay now that one doesn't look too bad as a mess But look at this one's wide open now that they're gonna come right in through there You gotta cut you know stuff that up with copper mesh or just fill it with foam You know this is the time here you got to get out there and do that There's a rock station on the side of the house right and they established those as food sources and I'm gonna show you I got some pictures of a Actual mice in the rock stations, but I opened them up to fill them up and there's mice inside All right. Hold on. All right Share this one And there he is. He's just hanging out right um And you can see the nibbled on the bait debates the square so they nibble around the edges and you know that's him I opened it up he was building a nest in there and They established those as food sources they start going there and they have it's a pheromone trail so they pee and they poop everywhere they go and that's how they follow each other around so they start establishing those things as food sources on the outside of your house they start going there and You start controlling the population around the outside of the house once you get control of around the house You have the least likelihood of having any mice in your house now, of course, you know mother nature Peaks and valleys there's going to be Times where there's an increase in the population just from natural maybe acorns drop or a predator dies or something happens and Or they get pushed out because of construction similar else that you might see some activity But but as long as you keep those filled with bait on the outside every month checking I'm filling them up on a regular basis It'll get control of the population before it gets out of control and but you know It's something you have to keep vigilant and do it every month to keep up with it every every month, okay? Hey Boston butterflies here. How you doing? Welcome to the stream. We're talking about pest control for the people.  The other thing I got is is these these T-Rex rat traps Hopefully you don't need them They'll work on squirrels flying squirrels Mice rats Anything like that.  Oh Sorry about that and Yep, everybody's having fun in the chat. Thank you guys. I really wish I could highlight your chats I'm very upset about this. I didn't do something right with the with the The be live software I'm supposed to be able to do that and to your comments and bring them on but I don't I don't see them anywhere And I don't know how to do it. So I suppose I probably should have put my Amazon URL for the stream into the software that might have been what I needed to do So that's everything that I have in the in the car cells just a quick Overview of how to control the mice population around your house So you're gonna use the black boxes on the inside of your house. You're gonna put some in the basement Under the kitchen sink in the attic in the garage, right? Then you're gonna use the next item in the carousel is the rock stations on the outside of the house one on each side There's box.  Okay, make sure you seal everything up because first of all, they chew second of all things freeze and crack and and it just you know things settle and Things were always opening up is what I'm saying. So every once in a while you got to go back outside and check Okay, you can use use the expandable foam as well as another thing. I'm highlighting. There's the expandable foam Again Be vigilant. Okay, it's not hard. It's not rocket science, but they're out there I would tell you that every house has mice in it Okay, you just don't know it because they're not encroaching on them on your living space but they're in your attic they're in your basement and places where you don't know about and If you don't get it on the control, it'll get out of control eventually I have customers that'll tell me I've been here 15 years and I've never had a mouse now you do Now you do and that's just the way it is Okay, and then you got the pest control rat traps. Okay, so that's it quickly. I'm gonna review what what I did with the be live I'm gonna review what I did wrong with pulling in the the comments because that's something I'm very excited to do and Yeah, I'm very heavy. Thanks for stopping by. I'm gonna go catch the rest of the mech game I think they're losing and I'm very sad about that. All right. So how do I stop this? Okay, have a good day and I'll see you remember Tuesday 8 o'clock guys Tuesday 8 o'clock Tuesday


Today we are going to improve your compositional abilities in photography by looking at nine practical ways of crafting photographs by using the examples in the work of Steve McCurry. At the end of this video you can have a better understanding of how to recognise and use these self-same techniques in your own photos. How's it how's it? Learning composition in photography is really is a vital skill in improving your images but especially for the beginner photographer it can be extremely confusing. To keep things simple we're going to leave all the talk about figrids and fennabachi spirals to the textbooks who have their almost sterile images which seem to be created solely to illustrate their points but we're going to turn instead to that vibrant and gorgeous work that Steve McCurry has created throughout his career. This is going to inspire you to start seeing compositions which are all around you so long as you know what to look for. Fill the frame. This is probably the simplest compositional idea and it's just to really get as close to the subjects as you possibly can and fill the frame with them make it as explicitly clear to the viewer as to what exactly this image is all about especially with portraits it gives your images a real sense of impact. You can either do this by walking closer to the subject or possibly by using a longer lens. When you do this in your photography when you fill that frame in this composition then you are forcing the viewer to address the thing that it is that you're photographing. There is something about patterns that has an almost supernatural urge to draw us in. Anything that repeats or has a flow that more or less fills the frame holds our attention and when you do that you can really you know keep the viewer glued to the image because the eye is constantly seeking around looking for things and then once the eye thinks oh i'm quite happy and i'm quite quite safe here and this is this is nice break up the pattern break it with something unexpected and see how that elevates the image into something really powerful this is what Seth Gordon called a purple cow so try and find these purple cows in your photography something that leaps out of the viewer that they go wow i didn't expect to see that this is what patterns do repetition they lead us into a almost a sense of security and then we can use that purple cow to snap the person out of it and probably one of the first art techniques that you were taught when you were learning to draw was was leading lines you know those lines that draw the eyes that lead the eye to a place in the image they can give an image a sense of depth they can draw the the eye towards ideas and subjects that you want to you know to highlight they're all over the place in nature they can be man-made they can be natural they wherever you look you can find leading lines don't be afraid to use them none of these techniques are the hallmark of a of a of a novice or a noob nobody's going to laugh at you for using what are actually quite basic elements everybody uses them just embrace it see what you can come up with when you look at finding natural frames within a scene then you can start elevating your images into something a little bit more than just a snapshot for example it's a great way of highlighting again elements with this scene that you want to bring to the viewer's attention in a way that is not quite the same as when you get close and you fill the frame because then you can give the images or the subject a sense of context a sense of place you know you can you can bring some some flavor into the image and you can use frames within frames within frames the more that you learn to recognize and the more depth you can be at creating these little vignettes almost throughout all of your images that that the eye can be drawn to can you see how all of these things sort of build up that you can use them either individually or within like a context that you are layering ideas on top of each other and makes your photograph so much richer diagonals are much like leading lines but they have a lot more dynamic motion to them that they they really make the scene come alive in a way that often leading lines don't necessarily do because leading lines can be quite flat occasionally but diagonals start giving you so much more dynamic motion and they can also help walk the eye through a scene where you don't have necessarily this this sort of stereotypical sense of leading lines so look at this photograph here there isn't a you know there isn't really like a traditional leading line but the the diagonals draw your eyes through they're walking you through this alleyway and of course making you explore the image a lot more than you would if it was just kind of just a flat a flat scene now Steve McCurry does this a lot and it's what makes his portraits extremely direct that is it's almost almost i suppose you could call like a signature approach with portraits the eyes of his subjects connect with you even now when you're looking at them i want to probably a quite tiny screen they connect and they draw you in to the image they make you feel that this person is looking at you not the camera that they're really engaging and that that that's what really gives these these images a sense of power and especially when you consider going back to the first point about filling the frame that the people within them fill the frame there's nothing else that is distracting us from this this intense gaze that they're holding you with symmetry is one of those quirky little aspects of composition that often we're sort of told to avoid if you look at your television screen or films or what have you it's very rare that the subject is centered in the frame often it's off to one side and or third and all the other aspects that we've looked at in this video so far but don't be afraid to put your subject center to put them in a symmetrical environment because that way again you're drawing the eye and that's the whole point with composition is to draw the eye into aspects of the frame that you want them to look at symmetry isn't just a very static way of looking at things but it can also be extremely dynamic when you employ things like soft diagonal lines within it so long as they are symmetrical this for me seems to be the one aspect of these videos that i think you can really really get to grips with and and explore deeply that brings us to the rule of thirds it was going to show up eventually wasn't it you know this is probably the one that everybody knows that everybody uses it and and i think you know a lot of people kind of dismiss it because it is so basic is like the the go-to and it's really just you know about sort of dividing up the screen into various lines and putting points of interest on those intersections and using the lines within that rule of thirds to place important elements like the horizon and so on and so forth just because it is basic like all of these things that we have talked about don't dismiss it don't just say oh well i can't do it because it's a basic thing use it but don't overly rely on it don't fall back to it the whole time and think well that's enough because i've done a rule of thirds see how you can add in some of the other elements that we've talked about in this video the compositional photography tools that you're learning to elevate it to something a bit more i know you can do it because you know it's there there are no real rules by all means you know take some of the examples that we've looked at today you know employ them in your own photography see what you can learn from them and take those lessons you know put them into your own photography but feel free to experiment these are not hard and fast rules go run with it see where it takes you see what feels right for you this is what photography is all about and that's probably the best compositional tip that i can give you is that does this feel right for you a great way of practicing composition in photography is to actually indulge in minimalist photography for a little bit and i put a video together talking about just this very point here which i'm sure that you'll find absolutely interesting thank you ever so much for watching and i will see you again soon


So last time we discussed some relations between braid varieties and maybe Detroit varieties and being homologous. So I tried to outline the general picture. I didn't really give concrete examples except for a very easy one for two-strand braids. So here is like one concrete non-trivial example, which I really like and especially this top picture keeps me up at night for less, I don't know, 10 years. So there is a lot of information here. So this picture is from a paper of Dunfield Gugman-Rasmussen who started this Kavanaugh-Prasantz-Kentripi gradient homology.  So rather they predicted based on various tools and methods what this homology should look like. In this particular case, this prediction was certainly confirmed. And there is a lot of stuff going on in this picture. So let me try to explain some pieces. So this picture represents the triply graded homology of 3, 4 torus not. So it's triply graded. So you have a degree, which is represented by the height, by the vertical direction. So this is a degree 0, this is a degree 1, and this is a degree 2. And then for zones of direction, it's supposed to represent the q degree, whatever it is. And then these numbers written there are the t-degree or homological degree, maybe in some slightly weird conventions that I won't really discuss. But in any case, you have three gradings and the whole homology is 11 dimensional. So there are 11 dots in this picture. There are five dots in a degree 0, there are five dots in a degree 1, and there is one dot in a degree 2. And so in particular, as I said many times, we're interested mostly in a degree 0. So these are these five dots here, which happens to be a cutland number, but that may be not so important right now. What is more important is like, what is the corresponding algebraic variety? What is the braid variety that I talked about last time? And that was also studied by a separate group of people, most notably Thomas Lam and his collaborators in the world of cluster algebras and cluster varieties. So they considered the variety corresponding to cluster algebra of type E6. And I will explain why E6 probably next time, but I mean, it doesn't matter really. So you have E6 in kind of diagram, you can build a very specific cluster algebra for it and then the result break procedure, how to build cluster variety if you know what it is. If you don't know what it is, it is just essentially up to some torus, it's a positroid variety. So this is this open stratum in grass mining 37, which I denoted by P37 last time. So some miners again represent every point in the grass mining 37 by three by seven metrics. And you require that cyclically consecutive three by three miners are all known zero. And up to some torus, this is this variety. And so these people also computed the homology of this variety, even before understanding the relation to homology. And so in particular, they found out that the homology is five dimensional. It's concentrated in even degrees, H0, H2, H4 and H6. So H0 is one dimensional, H2 is one dimensional, H6 is one dimensional, but H4 is interesting in two dimensional. And this perfectly matches this bottom row here because they have zero, two, four, four, six and there's exactly the same numbers. And again, like maybe I didn't say it clearly last time, but it's not true that this variety has a fine paving or anything like this. It's pretty complicated, non-compact variety. So it's not completely clear, as we speak, why this homology is all given. And another feature of this homology, which was phrased differently in these two worlds. So in this world, in the original paper done for Google and Erasmuson, which goes back to 2005 or 2006, they observed that there is a perfect symmetry of this picture. So there is a vertical axis of symmetry. And if you flip it, of course, you get the same picture up to some regretting which they wrote down explicitly. But it's clear that all these dots are symmetric. In this picture, it's less symmetric, but what they observed is so-called curious hard leftist property which was also observed in many different settings which are closely related. So there is a unique class in H2. So there is a unique algebraic two form here. And you can look at powers of these two form and this would give this class in H4 and this class in H6. And what they observed is that this multiplication by this class in H2 actually gives an action of SL2. But it's centered slightly weirdly. So these two rows that are written here correspond to the weight filtration and co-homology. And so this action would preserve the weight or rather the difference between the weight and homological degree. And so there is an SL2 representation here which is four-dimensional and there is a separate SL2 representation here which is just one-dimensional and trivial. But in general, you always have this action of SL2. And if you center things properly you would see that the picture is symmetric. And so this table 46 is actually by graded again by homological degree and the weight filtration. And if you do a proper change of gradings you get this picture which is clearly symmetric but also there is an SL2 action here and it's natural to assume that there is an SL2 action here. And so I think like again, I personally think that like these two pictures are very nice and illustrated like the first non-trivial example that people can compute and study in all possible details in both languages in many settings. And the air computation of homology is completely independent of not homology just uses some recursions in cluster world and some laboratory sequences and things like this. But here we got completely different computation and matches. So I think it's really cool and nice. And another thing is that like, where is this SL2 action? So for them, one property of cluster right is that there is a pretty canonical two form which is very, very important for building this cluster structure, whatever it is. And so these two form is not random. And so I guess what I'll try to explain how to build this two form and not homology and how to relate it to various structures here and how to actually prove this symmetry at least what is the idea, okay? So this is kind of illustration for the previous lecture but it's also, I think, announcement and advertisement for what will follow because that would be a bit more abstract but I'll try to keep it easy. And again, please ask questions if you have any. So any questions about these two pictures? So I just borrowed them from the picture of Dunfield Book of Rasmussen and LaMageberg. Okay, so if there are no questions let's discuss about different structures. So I wanna talk about not only the SL tool which will appear but more generally like what kind of homological operations can we construct or what would we expect in the homology? And so just very roughly what happened in previous lectures. So we start from this polynomial ring R there's a polynomials in N variables which was think as kind of associated to strands of our braid. And then if you have a braid and N strands we construct a complex of RR by modules which is denoted by T beta. And these are equivalently modules over C of X1 through XN, X1 prime through XN prime where this X is index prime is going to left and right action of R. And so natural question, what can we say about this complex of modules or by modules like from the viewpoint of pure commutative algebra or homological algebra. And so the first observation which is very easy is that for any semantic function F in N variables the left action and the right action on this complex is actually the same. And this is true for a single crossing and this is true for any braid in fact. And this is just again if you have seen Zorgil by modules this is true for any complex of Zorgil by modules just by definition. So this is just true. And more abstractly you can say, well consider this algebra B which is the quotient of polynomials in N variables and N variables with primes and quotient by the ideal generated by F of X1 through XN minus F of X1 prime up to XN prime for all possible semantic functions F. So of course it's a vision to take I don't know elementary semantic functions if you want. So this is clearly an algebra and the action of my polynomial algebra in X is in X primes on TB on every term of TB if you want factors through this algebra B because the left and the right action of semantic functions are the same. And for people who have seen Zorgil by modules this B is usually called BW naught. So this is an decomposable Zorgil by module for the longest element of SN but if you don't care about it, it doesn't matter. What matters is that if you have a random complex where semantic functions on the left are not equal to semantic functions on the right it never comes from any break or anything resembling a break. So this is one series restriction. The second series restriction is that the action of X size on the left is actually homotopic to the action of X primes on the right with the exception that you need to twist the action on the right. So any braid corresponds to the permutation in SN and the left action of XI is actually homotopic to the action of XW of I prime where W is the permutation corresponding to beta. So here is one concrete example, some random braid I don't know what it is. So how do you get the permutation? You just start here, label it by one and then trace your strand until you get here. So this permutation would send one to two, two to one and then three goes back to itself. And so what this fact and this property says is that X1 is actually homotopic to X2 prime and X2 is homotopic to X1 prime and X3 is homotopic to X3 prime. And this is true again for any braid and in general, it's not true for random Zorgi band module but this is pretty special for braids. And somehow way to remember this as people in not homology community like to say is that this action of variable slides through the crossings. So you can imagine that you put some mark point here and the action of X corresponds to this mark point. We'll draw this mark point, let's try it at least. And so you can slide this mark point over here and you get another action of X in here and they're homotopic, they're not the same. And then you slide it again and you get here. And so one way to phrase it is that the action for different mark point is not the same but it's at least homotopic. So in homology, it's all the same. And then after you close the braid, this means that you identify X1 in the X1 prime. And X2 and X2 prime and X3 and X3 prime. And so before X1 was homotopic to X2 prime but now after closure, X2 prime is the same as X2. And X1 prime is the same as X1. There's a one here, but whatever. And X3 is homotopic to X3 prime which is the same as X3. And so if we close the braid, we actually have an action of sampling of an algebra except that some variables are identified. And so I think some people asked this question in the first lecture, like, do we still have an action of R? So in some sense, yes, but you need to be careful. So a better way to phrase it is that you really have this mark points and what you can say is that the action of polynomial variables correspond to the choice of this mark points. And then if you slide this mark point around, the actions are homotopic. And if you close the braid, the action of the mark point on the top gets identified with the mark point on the bottom. And so in particular in this example, if we ignore all this homotopia and stuff, what we can say. So the action of X1 is the same as the action of X2. The action of X2 is the same as the action of X1. And the action of X3 is separated. We don't know anything about it. And so effectively we have an action of two variables, polynomial ring in the homology, in X1 and X3. And so if we phrase it like this, so you can say that if you have the closure of beta, it's some link with R components. And triply graded homology of this link is naturally a module over a polynomial ring. And you have one variable, one X variable per component of this link. And of course the components of the link correspond to cycles in the permutation W corresponding to beta. So in this case, the permutation is actually just the transposition one two. So there is one cycle connecting one and two. There is three separately. And so you have two in three different cycles and you have two interesting variables. Set differently, you have equivalence relation on axis given by the cycles and given by the strands of the break. And so this is quite well known. And so you have an interesting polynomial algebra action. And again, I didn't write it here, but if you have a node, all axes are the same. So maybe let me write it here. So for nodes are equal to one and all X1 and X2 and Xn, that's the same. And so effectively you have just one variable action. And it's actually free over this one variable polynomial ring. And you can ignore it and that gives you finite dimensional vector space, which we discussed below, this is free. But if you have a link, even with two components, usually it's never free. And you can study the module structure over this polynomial ring and you get a lot of interesting stuff. And we will discuss it very soon, okay? And again, this is just very, very general property which happens here. And we'll see these actions in lots of different settings later tomorrow and on Friday, especially on Friday. Now you can do a little bit more of homological algebra and try to study homotopy between axes and X primes more closely. So we don't just say that they're a homotopy, we need to say like, what do we know about this homotopy and what kind of homological operations we can build from this homotopy. So let me just try to spell out what does it mean that to have a homotopy between X1 and X1 prime and if we're going to put prime here. So there is some operator X1 and the differential of X1 is X1 minus X2 prime. That is the definition of chain homotopy between X1 and X2 prime. Similarly, there is an operator X2 and the differential of X2 is X2 minus X1 prime. And there is an operator X3 and the differential is X3 minus X2 prime. And now we close the braid. So what happens? So differential of X1, so we identify prime variables with non-prime variables with the same letter. So differential of X1 is X1 minus X2. Differential of X2 is X2 minus X1. And differential of X3 is X3 minus X3. So this is actually zero. And differential of X1 plus X2 is X1 minus X2 plus X2 minus X1, so it's also zero. And so we see that X1 plus X2 and X3 actually give you closed operators on this triplicated homology. And so these are non-trivial homological operations of degree minus one, which are interesting. So in a sense they're kind of monodromes of these marked points. We're saying that you have a marked point here. We use X1 to move it here and then we identify this point with this point with the closure, which I don't want to draw. And then you slide it here and then you go back to here. So X1 plus X2 is the monodroming which we use to come back. And this monodroming turns out to be closed. And this gives you interesting homological operations in lean homology. And for each cycle in a permutation corresponding to a braid, you would have such a monodroming. And so you have one of these monodromes per component of the link. So one monodroming. And well, so it could be maybe not so interesting, but like one abstract thing which you can do, which is if you want to say it in hydro with some kind of causal duality, but like you can just do it and you can use this Xi variables. So this monodrom is to deform the differential on the chain complex. So you start from your chain complex Tbeta which is associated to a braid. And you tensor it with polynomial rank and in zero variables, Y1 through Yn. And then on this thing, you deform the differential by taking the old differential and adding Xi, I, Y, I, the sum of this where Xi's monodrom is from above and Y's are some formal variables. And again, you can check that at least when you close the braid, this query is equal to zero. And maybe I didn't say this, yeah, I didn't say this but this is a Lincoln variant. So theorem that we proved was not Hogan-Comp that this deformed homology is a Lincoln variant. So it satisfies all the market promotes it satisfies all the braid relations and you can define this deformed homology. And again, you have one monodrom per component and effectively this means that you would have one Y variable per component after all this identifications. So the number of deformation parameters is again the number of components in your link. And so here is one concrete example which might be interesting to some of you. So you have this hope link with two components so you expect two deformation parameters and two X variables which are alive. So the old complex is this complex from R to R to R with differentials given by zero and X one minus X two. And that we discussed in lecture one how to get this complex by first considering the complex of Zorgil bimodals and then taking home from R. And then the Xi, there is only one Xi or like took signs up to a sign which goes from the middle R to the left R and this is this map and this is a chain map because if I apply this guy and then differential I get zero or if I apply differential then there is no Xi so I get zero as well. So this is honestly a chain map chain and the morphism closed in the morphism of this complex. And then we can deform it by this rule so that you just put Y one minus Y two times this sign this red arrow. And again in this deformed complex this square is equal to zero. That's easy to see. And you can actually compute the homology. So this is R of Y which is just polynomials and X one, X two, Y one, Y two. You have two generators Z and W which leave in this degree and in that degree. And there is one relation that Z times Y one minus Y two is equal to W times X one minus X two because this, the differential of this guy in the middle is precisely W X one minus X two minus Z times Y one minus Y two. Maybe I'm off with some signs but it doesn't matter. And so this is something concrete which you can compute. And again, you see that the homologies even degrees but you can ask like why this is interesting and that I will explain in a second. Maybe let me actually explain this and then ask some questions. And so like one application which we found is the following. So you have N, K, N, torus link. So this is a link with N components. For example, this for K is equal to one you have T and N. So all components are actually on nodes and the link in number between different components is equal to K. And so we just computed was deformed and undefined triply graded homology for this link. And so how do you describe the answer? Again, like if you like combinatorics you can do it recursively as I sketched in the first lecture. So there is some really complicated recursive description and there are some more explicit combinatorial formulas but it doesn't give you, for example, the module structure for this axis and it doesn't give you some other interesting structures. And instead we deform the homology and say that the result is J to the key where J is the following ideal. So you have the polynomial ring in two N even variables X1 through XN and Y1 through YN and N odd variables theta one through theta N and you look at the ideal generated by XI minus XJ YI minus YJ and theta I minus theta J and you take intersection of such ideals for I not equal to J. So if you don't like these odd variables you can just ignore them. So again, if we restrict to degree zero part what we have is the ideal generated just by XI minus XJ YI minus YJ intersection of false such ideals for I not equal to J and this is just ideal in the polynomial ring in X1 through XN and Y1 through YN. And so this thing has a geometric meaning. So this thing because you can consider N points on C2 which is of course related to Hebrew scheme of N points on C2 but that we will see a bit later I guess on Friday but for now just consider N points on C2 with coordinates X1, Y1 and so on up to XN, YN. And then you look at the place where it is at the diagonal where some of the two points coincide. So where do I's point and J's point coincide? Well, this is a co-dimension to hyperplane which is given by equations XI is equal to XJ and YI is equal to YJ. And so this is the ideal of that hyperplane. So maybe I's point is equal to J's point. So this is precisely the equation that XI minus XJ is equal to zero and Y minus YJ is equal to zero and so if I have the union of all this co-dimension to hyperplanes they take the corresponding intersection of ideals and this is it. And so this is RG and this is kind of super analog of this which is not so important for now. And then we take the powers of this ideal which also make a lot of sense from the Hebrew scheme point of view and from the work of Mark Heyman especially but that we will discuss properly on Friday. But for now this is some ideal in this polynomial ring. And so the claim is that this ideal as a module over polynomials and X's and Y's is actually this deformed homology of Torus Link and KM. And to me I would say this is, I mean let's get a lot more structures. So maybe we don't know or like the combinatorics is hard to describe in the actual dimensions of graded pieces of this ideal but this ideal is clearly interesting and important and the fact that it's related to ring homology I think it's quite remarkable.  So you just quotient by maximum ideal in Y's and then a separate result of Mark Heyman plus a little bit of work if you have all these status says that J to the key is free of our Y variables. So actually you don't lose any information. And so it is kind of a paradox that you want to describe some module over X variables. So as I tried to explain in the beginning for any link with N components you expect an action of polynomial ring and N variables.  And this turns out to be the right answer because in this case it's flat over Y's. And it's interesting that it's much easier to describe this deformed homology before describing undeformed because somehow, I mean, I don't know any good commutative algebra description of this thing rather than J to the key and Y J to the key. And yeah, I mean, in examples you can see this but yeah, I think that's all what I want to say here. So any questions about this theorem about the deformation and about anything. Maybe it's a remark. So this is kind of one similar thing which is not exactly right in this setting but the motivating is that if you have a variety of store section and you want to describe its homology sometimes it's easier to describe a covariant homology by localization or anything and then just kill a query environment. So roughly speaking, that's what happens here. But it's not exactly the same thing. Okay, so let me still pause for questions. Any questions here? Comments, no questions. Okay, so anyway, so if you don't care about this example you might see it again on Friday but for now this is just an example. And then the next thing is really what I want to talk about today is this symmetry and SL2 action like how to get it from this X's and X's and for this I want to do some homological algebra. So I want to rephrase what we've said in parts one and two and three using slightly more advanced homological algebra language. So you have this algebra B. So these are polynomials in X1 through Xn and X1 prime through Xn prime with the relations that symmetric functions and X's are the same as symmetric functions in X primes. And you can resolve R. So R is naturally a module over this B and you can resolve R over B by three modules. And so you can denote this resolution by curly A. And so concretely this curly A is what? So you have B, you have the variables Psi1 through PsiN which we saw before. So the differential of PsiI is XI minus XI prime. And these are the homologues that I talked about. And you have additional variables U1 through Un and the differential of Uk is more complicated. So the differential of Uk is H PsiI times the complete symmetric function HK minus one of XI and XI prime. Maybe kind of good way to write it is HK minus one of XI, XI prime is actually XI to the K minus XI prime to the K over XI minus XI prime. That's kind of easier to digest. And then you have this thing, you multiply by XI and you sum over all I. And a very good exercise to understand what's going on is to show that D square is equal to zero. For example, if I take D of Uk, I get this thing and I apply the differential again, this will be zero. And this is not completely obvious but this is not hard to see. And slightly harder exercise is that this is indeed a resolution. So there are no more relations. And so you can think that like in R, you identify left action and right action, obviously. So you need this size, but then you want to describe seizures between sides and these are given by these things. And then there are no more interesting seizures after that. Anyway, so this is some complex and this is, you can think of it as a digital algebra if you want. And so the theorem that we've proved in a recent paper with math, a plug-in component on Mellet is that this digital algebra over here acts on any Rukia complex, T beta for any braid beta. So you have an action of all this size, which we saw before and the action of this use, which is more interesting on any complex of bimodials for any braid. And so let me try to sketch the argument for this. So the argument is the following. So first of all, you need to construct this action of size and use for every crossing. And size you construct explicitly, that's not so hard. Maybe I don't want to write it, but it's really, really easy. And like this is why the left action and the right action are correct. And the action of U is just zero because you have a homological degree too. There is no rule for it. And you still need to check that this relation is satisfied. So for example, if U is equal to zero, then this right-hand side is zero. But this is easier in this condition. So I don't want to explain this, but that's true. And then you want to extend it to the product of crossings, to arbitrary braid. And here is an interesting idea. So you, if you wanna extend the algebra action to the tensor product, what do you usually do? You build a coproduct on the algebra. And so you construct the coproduct delta on this algebra A. More precisely, this coproduct would go from A to A tensor A over R.  The second copy of A has X primes and X double primes. And the coproduct is given by these explicit formulas. So maybe it's not so important. I will comment on geometric meaning of this coproduct in a minute, but the coproduct of Xs is just Xs and X1. So Xs on the left, coproduct of Xs primes is one tensor Xs prime, which is Xs double prime. So this is kind of, if I Xs prime, it will be the right most. And then coproduct of Xs I is Xs I tensor one plus one tensor Xs I. That's not surprising. What's more surprising is that coproduct of Uki of this new guys is Uki tensor one plus one tensor Uki plus an extra correction term. And this correction term involves complete symmetric function in Xs I, Xs I prime and Xs I double prime times Xs I. So there is some formula. Maybe it's not so important. I just wanna say that if you don't put this correction term, it doesn't work. And this correction term actually is super important for what we'll follow. And so one of the exercises I think is to check that this is indeed a chain map from here to here. And I mean, you can explain it more abstractly why such a map should exist, but maybe it doesn't matter. And so if N, M and N are two modules over this algebra, you have an action of M, A on the left on M. You have an action of A on the right on N. And you just have an action of A tensor A on this thing. And then using co-product you can translate it to an action of A on the tensor product. And so because we constructed the action on single crossings and we can extend it using co-product to arbitrary products of crossings to arbitrary braids. And maybe let me say that like because of this correction term over here, it's really non-trivial. So although U's are equal to zero on single crossings, when we start multiplying things, these correction terms will accumulate and they become non-trivial. So even if you have two crossings where U is equal to zero, once you multiply them, these correction terms will give something non-trivial. And the action of U will become non-trivial pretty soon and interesting. So there is some question. Some question in chat, Eugene. I'm sorry. The product that bi-algebra hope for algebra. It is a bi-algebra, so co-product is an algebra of homomorphism if you want. So we defy it on generators and then we extend it to products. The only subtlety which I'm really swiping under the rock is that this is not associative. So maybe I should say this properly, although again, I guess nobody here would care. The delta is co-associative up to homotopy. And so it's not actually, oh, sorry. I'm sorry. That's a good time to ask questions. I don't know why it crashes every time I give a talk here. Usually it doesn't crash. I'm sorry. So I was saying that this is co-associative up to homotopy. So if you want to be really careful, you need to keep track of this homotopies and higher homotopies and they give you some kind of infinti structure, but you can actually avoid it in this case by some tricks. So we can write something about this higher stuff, but it's really, really nice to end. And instead, we just prove abstractly that this higher homotopism which give this homotopy the associativity and some kind of higher associativity relations, they exist on abstract rounds and then it doesn't matter how we choose them, the result will be the same. But in principle, there is something interesting. And again, it's like something really, really basic. It's all about this algebra B and resolution of R over B. There is nothing else. And again, maybe it's actually a good time to say that because it's resolution of R, you know that R tensor R over R is R. And so this is a resolution of R. This is in some sense resolution of R tensor R over R which is again R. And so you know that there should be a chain map which relates these resolutions and we just write it explicitly. And extend, yeah. So this is an algebra homomorphism. And then co-associativity of the homotopy, I mean, it's immediately follows from what I just said but there is a trick like how to prove it very clearly and that's what we use. Anyway, that is definitely too technical. But in particular, why this co-associativity is important because if you have a triple product of three things, so if you have, I don't know, M tensor N tensor K, a priority, there are two ways to define an action on this thing and they're different and they're honestly different but homotopy. So let me try to sum up things. So there is an action of this algebra on any read and there is an action of interest in operators size and use on any Rookier-Kohm books for any read but they're not closed. And so instead, so solution to this issue is you just deform homology as I did before. So in the form homology, you deform the differential and then you can also deform use by saying that instead of use, we consider use plus we gain some correction terms involving partial derivatives in Ys. So this is again some formal homological algebra thing but this gives in operators in this deformed homology. And so what we proved is the following thing. So this deformed use, so this FKs, they're actually closed under D. So this is easy to see because somehow the differential of U is not equal to zero. So these two terms don't commute and these two terms don't commute because Ys don't commute with partial derivatives. But somehow this is designed so that they cancel out each other and this commutator is zero. Now it's more or less clear that they commute between themselves FK and X sub I also commute that's clear. And then FK and Y sub I, they don't commute but you know how they commute with each other. So the commutator before closure is something like HK minus one of XI XI prime. The commutator after closure is just K XI K minus one. And so these Fs are what we call the pathological classes and they act in deformed homology of every link. And that is kind of the main result is that this F2 corresponding to U2 is an interesting operator and that satisfies this kind of hard left so curious hard left condition. And that operator actually leaves to an action of SL2 on deformed homology for any link L. And so as a corollary this deformed homology has an SL2 action. And so it is symmetric as this explains the symmetry that we saw in the bottom. And this resolves this conjectural look of Donald Rasmussen that for knots you always get a symmetry because for knots this deformed homology is essentially the same as under form. And so we get it like the bottom line if you don't care about the symmetry is that we have a lot of interest in commuting operators acting in link homology. And they're very new. You can ask tons of questions about them and how do they act? What are the other relations? Can you, how do you interact with this SL2? Like what is the structure of homology in known cases as a model over this operator? So they're all very interesting questions. And this is very, very, and so geometrically you can ask, well, okay. So we don't care about all this monological algebra. We like algebraic varieties. What happens there? What happens for this great varieties or this droid varieties or whatever. And so we need to find some interesting operators acting in co-homology of these varieties. And of course the natural guess would be that there are some kind of the astrological classes in co-homology. And we just multiply by this the astrological classes. And of course, this is the case. And that is actually one of the motivations for us to build this digital algebra story. And this goes back to the work of many people, Atya, Bot, Shulman in particular, and more recently I guess Lisa Jeffrey and others. So how to build the astrological classes on character varieties and more recently on braid varieties. And so this FK would correspond to some algebraic K form on this braid variety. So how can it be built? So we'll start from the group JLN and Bot and others give you a family of differential forms on the group and on some related spaces. So given any symmetric function of degree R in N variables, one can construct a characteristic class which I would call phi zero of Q in two R's homology of the class find space BG. So this is probably familiar to many people. If you have a degree R symmetric function you get a characteristic class. What is also familiar to many people is that you have another interesting class which is, I mean, really it's a form in this case. I mean, I should write a form instead of H. So this is an algebraic differential form on the group itself. So, and the form will be two R minus one form and this is closed. So it represents a class in H to R minus one. That's why I wrote it. And in fact, if you think about the respectful sequence for the universal, for the vibration EG over BG this form, this class would kill that class by a differential. You have seen this, but in any case for any symmetric function you can build an interesting co-homology class of degree two R minus one on the group itself. And this is also quite known. And what is less known is that you can continue this procedure. So you have a two R minus two form on G cross G. So this is just the product of G with itself. And these two form, two R minus two form, sorry it won't be closed, but it would satisfy some kind of co-cycle condition that the differential of phi two, the differential of this guy will be two R minus one form and it is related to two R minus one. Okay. Yeah, so what I was saying, that you have a G cross G and there is some differential form on this. It's not close, but you know the differential. Here is a concrete example of this. And now what you can do, so if you have two algebraic varieties X and Y with the map to the group or alternatively you can think either we have algebraic varieties with the map to the group or we have just matrices kind of depending on parameters which live in this algebraic varieties, X and Y, then you can build an interesting to an interesting form or two R minus two form on this.  So you have some form here and you have some form here. And the important point is that you can glue them together by taking the form on X, the form on Y and this correction term which is phi two of Q which appeared over here. And maybe that was too fast, but it doesn't matter. There is a procedure how to build interesting forms and product of varieties. If I have an interesting form here if I have an interesting form here I take the sum of these forms and just add this form correction term which lives on G cross G. And I claim that this is very similar to the co-product on my DG algebra that you have this kind of element or the action of U on the left, action of U on the right and then do you have these corrections or which glues them together. And so classically people have used this construction and this specific form phi two of Q to build a two form and subtract two form on character varieties. But Anton Manit, for example, proved this that one can use it to construct algebraic two form on this braid variety. And this two form on braid variety is constructed exactly like this. What is the braid variety? You have a bunch of matrices for each of them we put the two form to be zero and then you glue them by this procedure inductively. And this gives a non-trivial two form and this turns out to be almost symphactic up to some torus action up to seven things. And this two form is closed and it gives you a cohomology class and it does satisfy Q's hard left shift theorem for this braid variety exit beta. And so you can use this to build an action of SL2 on the braid variety and you can use this to prove this Q's hard left shifts by this machinery. So actually these people also observe this Q's hard left shifts in this case but they use very, very different machinery. But this is exactly the same two form which appears here and which represents the class in HF or two. And so maybe just to conclude I wanna say that in principle this works for any symmetric function. And so the fact that I had this F2F so this gives me an analog of F2 but I can build F3, F4, F5 and so on. It has lots of interesting things and this gives you, in this case it gives you lots of interesting forms and cohomology classes on X of beta by the same procedure you have something on matrices and then you keep building them. So it might be less explicit but you might not need it to be more explicit. And so for example, the next optological class would correspond to sum of XI cubed, I guess and this would leave here. So this would give you four forms. So if R is equal to three, where is R? So if R is equal to three it would give you six form on BG it would give you five form on the group which is there if the group is at least JL three and it gives you a class in H upper five of the group. Maybe I'll write it down actually so if R is equal to three, you have H six of BG and you have H five of G and you know that there is a generator of degree five in the group and this gives you a class it's not closed by the class it's in a four form on J cross G and then by some reasoning you can check that this four form has an interesting weight and this actually gives you a generator here. So this is kind of the form which you can build from the cubic symmetric function and it's there. And in this case, you see that cohomology is a ring is actually generated by this class in H upper two and this class in H upper four and it's a very natural question which is as far as I know, we're wide open is it true that cohomology is generated by this theftological classes for torus nods? So maybe I will ask this question. Some problem slash conjecture. Is it true that for co-primes whatever braid variety is generated by the theftological classes or maybe equivalently as H H H is generated. You see that all this theftological classes they are in even degree and we just proved that cohomology is in even degrees. So maybe it's not so surprising. And one reason to believe is that in slightly different setting there is a space with very, very similar homology and their homology is generated by the theftological classes. And another question is whatever. So suppose that it is generated by the theftological classes. So can you describe this co-homology? So again, here co-homology would be a ring because it's the co-homology of a space. And you can describe this, can you describe this ring by generators and relations? And here co-homology is not really a ring. There is no ring structure, but it's a module over F2 and F3. And hopefully this module has one generator or one kind of co-generator. And again, what a, how to describe this module explicitly? What are the generators and relations? These are all wide open. And I think that's an excellent problem for someone to work on. So at least I try to convince that, like besides these combinatorics and the fact that we know the dimension of homology, this opens up lots and lots of questions. Like what are these theftological classes? How do they act? And what can we say about all this? And I think I'll stop here. So again, I'm very sorry for all these interruptions and technical issues. Any questions? Can you say a little bit about which symmetric function gives rise to this element that does this curious hard left shift? Yeah, sum of XI squared. So if you have sum of XI squared, this would give you, so again, you can write it down. So if R is equal to two, then you get two form on G cross G. And this is precisely that two form. So you'll get three form on G, but you'll get a two form on G cross G and this is these two form. And so the recipe is that you just take this. So for this particular queue, you take this correction term, this two form on G cross G, which is very explicit.  So you do it for the first matrix, the form is zero. For the second matrix, the form is zero. For the pair of matrices, you have this correction term. And then you have the third matrix. And then you add correction matrix, corrections are for the product of the first two matrices in the third one and so on. So this, you build inductively these two form.  And like whenever you close the braid, this gives you a class in H upper two. And this class in H upper two will give you hard left shift. And maybe I wanna add that, like Anton actually used this theorem over here to prove curious hard left shift for more general character varieties because they can be somehow stratified by this. So this is like really, really powerful thing. And like most of what I explained in the beginning was motivated in kind of trying to translate it to not tiering more algebraic language, more kind of homological algebra language. Yeah, now I can see, sorry, I crashed again. I don't know, maybe my network or my sum or something, I am very sorry. Were there any more questions? He didn't have any more questions. One more chance for a last minute question. Well, if not, let's thank you, Jean again.


And I suspect that we are going to face now a different situation, Mr. Camel in the Middle East, which has strong specificities, including environmental, geographic specificities in this major channel. And here we are at the right place to discuss these issues. Thank you. I'm not sure if the presentation is up and running, even without it. First, thank you all for still being with us. I know we stand between you and your breaks for the evening. That's indeed a very important topic, especially in the Middle East. The good news for the Middle East is that national food security has been an item for at least the last 30 years in the Middle East. While it's becoming a hot topic around the world, we've been dealing with it for a long time. But there has been major changes over the years, especially over the last almost eight years in terms of how we look at national food security. Let me highlight my contribution here is that I come from the private sector. However, the role or the projects we do is are in public-private partnerships with the government. For example, in Qatar, a couple years back, we flew in cows by plane first airlift of cows. We worked within one to two years to ensure 100% self-sufficiency of dairy products in Qatar. Today, as was mentioned, we are working in Egypt to again ensure self-sufficiency in sugar by doing land reclamation for a land as big as Singapore, desert land, while using disruptive or new technologies in this. But let me look at food security. Historically, in the region, it was easy. It was about availability, affordability, accessibility. And essentially, governments would import it because only 3% of our land overall are fit for agriculture. Almost 97% depending on the country that you are in does not have the soil or the water needed for you to do agriculture. Government had lots of money, so they can import and subsidize food programs around the region. For example, in Egypt, 70 million people receive some kind of food subsidy for bread, 70 million out of about 110, 120 million people. But this model is not sustainable. Oil prices will not always go up. We heard earlier today they will start maybe going down. Governments cannot continue to run budget deficit. And we had another complication really, which is the health crisis in the region. We have the highest, or the second highest, diabetes rates around the world. So what happened over the years is that diabetes, heart problems start coming early. We have them at the ages of in the early 40s versus 50s in other parts of the world. Again, government responded by spending money on healthcare, putting the hospitals, sending people first abroad, then putting hospitals in. And now they realize also it's not sustainable. It's neither sustainable to subsidize imported food, nor is it sustainable to continue spending money on treating sick people from chronic diseases. And that's now where food security has graduated into. It's about wellness, nutrition. It's not about now making, let's say, poor people or rich people just having food in their tummies. It's about making sure they have the right food in their tummies so that they do not get sick so that I don't have spent too much money on them buying insulin for their sugar problems. So when you have this change in food security approach in the region, which said, we need now to have good food. We cannot continue to import it expensive for many reasons. And we cannot continue to have people eating unhealthy food. So now we are working to ensure we have some kind of self-sufficiency or at least a reasonable domestic component production of agriculture. And that's what we are seeing things happening. What made this happen? Technology, pure and simple. Without disruptive technologies, we would not have been able to reclaim the desert. Without disruptive technologies, we wouldn't be able to optimize the production and minimizing the use of water. And without these technologies, we wouldn't be able to develop higher yields for cows, higher yields for sugar, higher yields in every area of the agriculture sector. Do we face challenges? Indeed, many. The public sector has different expectations from the private sector. Major mismatch between the public sector and the region and the private sector. Goals, expectations, timelines, return on investments. Add to it another component. We really need always R&D research and development and our regional companies, and I've run two of the larger ones in agriculture, do not have the funding to do significant R&D investment. So we need the third component in the public-private partnerships, which are universities. The universities in the region are well-funded. I was an ex-academic, but regretfully a lot of the academia are focused on having a higher research index than maybe having relevant research. We know in the industry, whenever we mention academia, they say, watch out, these people do research that prove that blind people do not drive. And we say, of course not. They have a relevant impact for research. We just need to tie them with the food security, the government, the policymaking, and the private sector. So where we stand today, it's the best of days because now food security is the end topic. No longer are investors chasing more buildings and more real estate, and people are looking at investing in agriculture, but also it's the toughest times. Indeed, as was mentioned, when you are looking at one healthy earth, one healthy water, one healthy soil, one healthy food, all of this will contribute to one healthy human, and we have several bottlenecks nowadays, especially when we have the regional geopolitical situation. The tensions that used to be here are coming back very fast, and they are driven increasingly by agri issues or water issues. For example, the water issues will impact the Turkey, Syria, Iraq food supply, and agriculture, as well as the Ethiopia, Sudan, Egypt water supply. Add to it all these countries use common underground water, and increasingly as we do land reclamation, we are using or maybe overusing the underground water, so we are ending up with a lot of issues there. Hopefully technology will come to the rescue. Instead of having lots of cows to get meat, we get our energy from alternative proteins so that we'll have less cows, less gas emissions, and we'll be able to have also healthier food in the process. So let me stop here and we'll wait for the Q&A, except to repeat that technology has been a saver for the region on food security and the region has been among the pioneers in promoting nutrition and healthy nutrition now as a direction, instead of just more food for people. Thanks very much, Mr. Camel, for all those messages and two of them, which I think we'll all remember, a message about health and the situation of diabetes and heart disease. By the way, something which is also taking place in Southern Africa, which is also experiencing this major change in health challenges, and also this call for technology and science as a driver for sustainable food production.