Big video today guys. Hey guys, what is up welcome it to my channel I'm gonna skip the intro and all of that nady gritty because we are going to be ranking All of the eyeshadow palettes that I tried in 2020 and if you're new to my channel I love eyeshadow palettes. They are my favorite part of makeup. They're what I purchased the most They're what I reviewed the most they're what I believe you guys like to watch the most as far as my eyeshadow videos Reviews explanations all of that. So that's why there's so many I'm gonna be ranking 109 eyeshadow palettes that I tried this year if I'm missing any don't tell me You will stress me out. This is it. This is good. 109 is Enough now I'm not going to be ranking these in the traditional way that I normally rank these in my videos instead of going numerically I just feel like a hundred is too much. It would stress me out too much and not only that I feel like there's just too many it loses the value it loses the meaning when there's a hundred different places So instead I'm going to be ranking on a tier system I created 10 categories from the worst to the best so I put give or take 10 eyeshadow palettes in each category That way I think it makes it not only more meaningful But a little bit easier to get through as well. So let me talk about the tiers that we have So this is from worst to best. There are 10 tiers. The worst is no just no Then we have one step up, which is not what I was hoping for then we have Then we have I just didn't reach for it The next category up kind of morning the next one Worth a try then we have good but not great up higher the third highest level Great, but there's better even better, but not the best and then of course finally best now I actually won't be talking about my best category So I guess I'm talking about less than a hundred palettes because I already have the best eyeshadow palettes of 2020 in a whole separate video that I've done So if you want to see the number one top top top category those will be there, but anyways, let's just get into it Okay, so the first category is going to be No, just no these products didn't work for me if you love these good for you These just didn't work for me. I didn't like the quality of color. Sorry, whatever reason let's get into it in each category They are not in a particular order So just keep that in mind But these are kind of the bottom of the barrel palettes for me So the first one I talked about my worst makeup. We have the Urban Decay stoned vibes palette I just this did not work for me at all. The colors are really inconsistent I didn't get much pigmentation and the mattes didn't blend. I didn't like it at all the next one We have physicians formula rose all play eyeshadow palette I always say I don't like drugstore eyeshadows and I had high expectations for this well Maybe not high expectations, but my hopes were high. I was definitely hopeful and these looked like hood of beauty to me It's just a powdery hot mess the shimmers aren't even detectable on the eyes did not like that one the Shantikai Safari collection eye trio. This has gotten a lot of mentions on my channel lately about how much I don't like it It's just really overpriced this fell out in one video You can see my horrible press job that I did and I hate pressing eyeshadow palettes I feel like it's never the same afterwards, but anyways for me not worth the money This was like 70 bucks for some sheer eyeshadows. This is one that I've reviewed recently We have the color pop hello kitty and friends eyeshadow palette now. This isn't horrible. Okay, but at the end of the day I just I'm not vibing with the color story. There's nothing special about it I would only suggest you pick up this eyeshadow palette if you love hello kitty like me So I don't regret it, but I don't recommend it unless you like hello kitty It just it wasn't that good to me, but I will say the look I created I did like but it wasn't a very good Palette the next one this you've seen before it was in my worst makeup of 2020. We have the no bad cosmetics Tokyo palette I wanted to love this. It's adorable, but I just can't get down with no man's formula I have one more palette that I need to try and that will kind of close the grave if I don't like that one But I felt like all the colors they just disappeared off of my eye completely The fading was just a little bit too much than I could handle we have another nomad cosmetics eyeshadow palette This is the studio 54 quad again, like how could I not purchase this? It has these amazingly gorgeous glitters, but they weren't as pigmented on the eye I just couldn't get them to pick up like they look in the pan and just knowing the other quality that we can get From other indie brands. This just wasn't it for me the pure and raw beauty Christie collab I did not like really anything about this palette personally the color story I thought was kind of ugly if I'm being honest, you know what color story is a personal thing But the quality also wasn't very good to me. I couldn't blend the shadows It took forever to create a look even on the neutral side So I thought not only was the color story kind of not my thing But the quality was also not very good these I recently reviewed like two days ago and they've made my way into the no just no category These are the Dior Trio bleak eyeshadow palette that's come out for their spring 2021 collection check out that review if you want to see me completely Trash these because you get zero pigmentation from this a bottom color. I mean it's ridiculous You spend 63 dollars for the color not to show up and the top two colors They are okay. They're a little bit more on the sheer and subtle side, which is fine But I don't I don't get much out of this palette I spent a hundred twenty six dollars for these little guys and they're worth about five bucks each honestly They're only pretty then we have the Odin's eye free Hadeva palette now. This is not that bad Honestly, I tried a lot of good eyeshadow palettes and this one is just one that I didn't want to reach for This has to do more so with the color story It is so extremely warm and I feel like the colors don't pool how they look in the pan So it's even 20 times more warm on the eyelid than they look in the pan because this looks like at least a little bit More of a warm brown it pulls off almost red on the eye So the quality in here is not bad, but it's way too warm for me and the colors Don't look in the pan how they do on the eyelid and then the last one and this was more so Disappointment not necessarily terrible quality, but this is the ABH Norvina volume up for palette I did not like the quality in here I just felt like in the first three original palettes the quality was really good these colors I feel like weren't as pigmented. It was a little bit more difficult to work with so more so disappointed because I know what ABH can do So so that sums up the no just no category. We are moving up to a little bit better This is the not what I was hoping for I can make these palettes work absolutely But my expectations were a little bit higher and I was let down so we will start off with this Milani palette I tried this in the beginning of the last year. This is outlaw olive It's really a gorgeous color story not one that the drugstore comes out with very frequently And I just don't like the texture of these. I feel like they blend away a little bit too quickly But you can definitely make this work It's just I don't like drugstore palettes and I feel like this is pretty drugstore quality We have the visi art from bois palettes I just feel like this is the least unique palette visi art could have come out with because a lot of their other palettes Also have this color story. The shades are a little bit light. It's just an underwhelming palette The quality isn't bad, but it's very under the radar. Nothing. I really would consider grabbing for too often I have three palettes here. They're all from the same line. These are the Odin's eye Alva mini ocean palettes and while overall I feel like these Individually have some very beautiful shades They too take a lot to work with if you are creating entire looks with these So this one is the mini forest palette This one is the mini sky and this palette in particular. I had the hardest time working with I felt like these shades were Really chunky and then we have the mini ocean, which is probably my favorite of the three But again, some of the shades are just a little bit more inconsistent than others This is a these are palettes that I would grab for to do lead toppers because these middle shades are really stunning But as a whole they aren't quite as cohesive as I would like for them to be and they're a bit inconsistent in the formulas Next we have the Pat McGrath labs celestial divinity at luxe quad and Fleur fantasia this came out in her new holiday collection I just was expecting a little bit more now the look is really really pretty don't get me wrong But I feel like it pulls just so extremely light on the eyelids It's more of a spring color as opposed to coming out in a winter collection like it did It's just not my favorite quality from Pat McGrath It's not what I envisioned when I first saw this online It does not look like on the island like I thought it would look we also have the Sigma enchanted palette and the corda rosa palette. I did talk about this briefly in the worst makeup of 2020 I'm just not a fan of Sigma's formula I find the mats a little bit more difficult to blend and while I do love the glittery shimmer shades that they have going on it still takes a little bit of work to make sure that it stays a long time you need to use a glitter glue and Sigma kills it on the color stories I will give them that and I'm not gonna say I'm never gonna buy a Sigma palette again because I really do love their color stories And I can most certainly make the palettes work But I just am not impressed with the quality. We have the elf bite size eyeshadow palette in the shade hot Jalapeno now I love other palettes from this line, but these matte shades are quite atrocious Honestly, it's good for six dollars. Don't get me wrong. So this isn't even an insult to this palette It's just a little bit harder to work with. We have the Ophra summer edit pro eyeshadow palette This is really appealing to me, but I never ever ever reach for it I don't think Ophra has the best eyeshadow formula in the world. So this palette is really pretty honestly I think I'm just gonna end up putting it in my bridal kit because I will get more use out of it because I've set This on my desk and I've never grabbed for it. It's been on my desk for over a month And so the fact that I haven't grabbed for it It's been one of three that I kept on my desk says a lot. It's just boring and not the best formula We have the color pop sailor moon pretty guardian eyeshadow palette Obviously, I bought this more for the novelty of it as opposed to the actual Eyeshadows and color story. It's just not my favorite formula from color pop if I'm being honest And it's not really a color story that I found myself reaching for while you can create pretty looks for it The idea behind it was greater than the performance. We have the Natasha Denona mini Zendo palette I don't know you guys. I just was not feeling this palette The quality is not bad in here at all actually this quality is bad for this shade right here But I don't really like the color story. I don't feel like the colors are cohesive enough for the type of looks that I like To create it's just not something I like to grab for and this color I get zero pigmentation from we have the Wayne goss the luxury eye palette in a moon stone So I love love love the look I can create from this but I can really only create one look Because these matte grays I find to be extremely patchy on me And I just think it's unacceptable for the price that you're paying you can absolutely make this palette work But it's a luxury eyeshadow palette so I was more so let down with the formula But this celestial shade incredible but not a fan of those matte grays the last category in mind Not what I was hoping for a category is the Pat McGrath labs mother ship sublime a golden opulence So this came out for the Chinese Lunar New Year Collection at the beginning of 2020 and the quality is just fine I just haven't reached for it really since I've done my review the colors I think are quite boring honestly and just kind of nondescript So it hasn't been one that I've been interested in reaching for and when I think of Pat McGrath I want something bold that's gonna catch my attention that I want to wear. I don't feel like reaching for this It's kind of yawn for me. All right. No, we're moving up a tier. We are in the seventh tier here so this is the Category these palettes were just To me now we're getting to the point though where these palettes are not bad The quality has now stepped up a notch more so a lot of these are just color stories that I don't grab for as much So we'll start off with the two that I just recently reviewed These are from the Charlotte Tilbury Hollywood flawless eye filter collection So this first one here is diva lights and then the other one here is star aura I just think Charlotte Tilbury has a much better formula while these are pretty these are not original color stories The price program is so inflated. It's $18 and so for that reason I had to put these here I will continue to use these but they're just we also have the melt cosmetics rust palette This my mom actually purchased for herself. So this wasn't even sitting in my collection, but I did try it a couple times I wasn't in love with this formula. I feel like melt is just so hit or miss and this was kind of a miss for me It wasn't the worst palette that I tried from melt Actually millennial pink was the worst that would should have been in the worst category, but I returned that so technically 110 palettes Anyways, yeah, I felt like the shadows just kind of blended away They weren't the easiest to work with but you could definitely get a look out of it We have both of the kaleidos Lunar lavender and then Sashimi city the looks you can get so pretty I just recently dug into them. This is Lunar lavender and then we have Sashimi city I will say though I feel like the formula of these aren't as good as the regular kaleidos formula the mats were a little bit harder to blend I had to keep reapplying because I felt like I wasn't getting the pigmentation that I wanted the shimmers Absolutely beautiful lid toppers But they didn't have as much pigmentation as I was hoping for so both of these palettes did require a little bit of extra Love to get a look that I wanted. We have the Tom Ford arabesque palette I was just extremely underwhelmed by this palette. It is very pretty, but I feel like the lid topper shades are too Lid topper II needed more pigmentation from these so it means a lot of times if I am wearing this palette I do have to dig into another palette. So while the shadows do blend really great They aren't a complete palette. We have the busy art spritz edit palette This one just kind of fell under the radar for me to be honest. I like other busy art palettes more It's just more so of a color story kind of things and I feel like the shadows in here Aren't quite as pigmented or as punchy as I would like for them to be so the color story is just kind of Antony we have the BK of beauty true beauty palette. This is one of those palettes. Like I said, it's very good It's just a soft palette and the formulation is a little bit more powdery than I would prefer You still can work with it. The shadows are easy to blend. I don't have a problem with the quality It's just not the typical formula that I prefer and while I've been at home all year I've been wanting to create crazy colorful kind of looks So this isn't one that I've been tempted to reach for quite as much We have the hood of beauty sand haze palette I just didn't think the formulations of these were the greatest They were a little bit harder to blend and on top of that I just didn't like this color story the other two palettes in this range They have a much better color story so they ranked a little bit higher But for this one, I feel like the color story. It wasn't that great neither was the formula I know for a fact this one is no longer sold. This is the morphe 35 I ice fantasy palette And it was a really great introduction to pastels for me I don't own a lot of pastel palettes and I actually really like this There are some things with the quality that I don't love but for the most part This was a great introduction to pastels to me now are they the greatest pastels? No, have I reached for this very often? No, so that's why it is in this category We have the elf earth and ocean palettes I picked this guy up for a sponsorship that I was doing with elf and this palette has some hits and misses in here So some of the shadows in here. I think are really great and I love the color story for this But like this shade is kind of a mess and then a few of the mats I have a little bit of trouble with so it's just a little bit of Inconsistency in here, but it's not a bad palette same thing kind of goes for the elf and j kiss a palette I do think that this is a great if you're looking to add some rainbows in your collection But I did notice a little bit of inconsistencies and it's just not the best rainbow palette that I have I like it. I have some really great looks that I pulled out of this But I also feel like the actual shades of the colors are a little bit more intense than I would also prefer They're just too vibrant for me. So not necessarily a more wearable tone of the bright colors So just not the colors that I would choose for myself. All right So now we're gonna move on to the fourth category, which is going to be ranking at number six at a time so these are Just didn't reach for for whatever reason I wasn't as tempted to reach for So we'll start off with the hood of beauty since I've already kind of mentioned them We have the khaki haze and the purple haze What I love most about these is the color story, but the quality is what is really kind of compromised with these So the looks I create are very very gorgeous But just know it does take a little bit of a journey to get there Why is my water bottle right there? It is a little bit of a journey to get there, but I love the color story So that's kind of what's pushing them up. We have the Odin's eye soul main eye shadow palette and this is a Absolutely gorgeous eye shadow palette the textures the dimension in here are really really stunning I just didn't reach for them because I have so many palettes in my collection So yeah, that's just kind of where it fell We have the makeup by Mario master mattes palette this one I wasn't in love with the quality just because I have better I have formulas like vizier art So I think for the average consumer and this is a great But it's not something that I reach for a lot just because I know I have better quality of these colors We have the vizier art shock a lot Pet it for this is beautiful. I like the formula of it It is a little bit more sheer than I would prefer But it's just a bit too warm for the colors that I tend to grab for so that's why this one is ranking here We have the Pat McGrath labs mother ship. What is this rose decadence? So I have the real deal as far as rose decadence and by the time that this came out I was a little bit tired of these pinky tones So this one was kind of run down for me. I was tired of these kind of colors. It's a pretty palette I feel like the formula wasn't as good as her normal Formulations these shadows feel a little bit more dry than what I'm used to from Pat McGrath as well We have the beauty by stony remedy eyeshadow palette So this is one that I tried fairly recently and I think the quality is pretty good The only thing is this isn't my color story. So this isn't a color story that screams out to me They actually have a new palette that they just sent me which I'm much more in love with that color story So I'll talk about that later on in another video But anyways not a color story that I'm really that obsessed with but it's a very unique color story We have the BH cosmetics mimosa eyeshadow palette and this is really really pretty But I just really wasn't tempted to reach for this guy this year I feel like the colors pull a little bit brighter and a little bit more different than they look on the pan It's just not my favorite formulation or color story from BH cosmetics And then the last one in my just didn't reach for category is the fade into hue from color problem Now this is fairly new and I really recommended if you're looking to add more color to your collection because it's a Relatively affordable way to get these colors. This is the palette that I'm wearing now I'm mostly wearing at this purple row right here and this palette is great a little bit harder to work with you Have to take your time. I think it's great considering how much it costs and the kind of colors that you have They're just harder to formulate So it's really great for that reason just know if you're going to work with this palette You do need to take your time and of course, it's not really a palette I'm gonna reach for a lot, but it works, you know, it gets a job done We're now coming up to the better category So at this point, this is really at the turning point of where I can say I really do like these eyeshadow palettes There's just a lot of competition. So this is the kind of boring category So palettes that I like but not ones that scream out to me. So we'll start off with this one This is Charlotte Tilbury the Fisticate now I totally bought this to complete my collection And I think you can see why this falls in the kind of boring category. The formulation is really nice It's just I bought it to complete my collection not because I needed those colors We have Tom Ford Noir fume now. This is really pretty I feel like this would bring differently if the year had turned out differently because I actually really love these cooler tones I'm a bit underwhelmed by the quality of the shimmers here because I know Tom Ford has better Quality but kind of boring and same thing goes for Della creme from Tom Ford again Kind of a boring palette probably would have ranked higher had circumstances been different this year And I actually went into work Again shimmers a little bit lackluster, but still a really great beautiful blendable palette Just didn't fit my needs this year. We have the Marc Jacobs Terrific eyeshadow palette. So this one I would say more so just not my color story. I think the quality in here was really good I think there was one shadow in here that I was kind of about the quality it but regardless It's a really gorgeous palette just not one that I was very tempted to reach for this year We have the Wayne Goss Imperial Topaz eyeshadow palette this I liked a lot more than his pearl moon stone palette It really is a stunning palette. It's functional I don't have much bad things to say about this. I think it was a little bit overpriced weird year for me Didn't really grab for it, but it's a nice palette. We have the Ophra and Leora palette if you don't know my friend Leora She curated this really really fun palette and this really introduced me to Ophra's eyeshadow formula And there's something about this eyeshadow formula that I feel like is really nice I was surprised and I was surprised by the versatility of this palette again I just had so many palettes this smaller one kind of fell to the back, but it's really nice We have the Nars Extreme Effects eyeshadow palette I really like this one and I know a lot of you guys really liked this one as well The only thing is I felt like the shadows swatched better than how they actually applied So I think I was almost a little bit disappointed by this palette The quality is still really nice and you can create really good looks with it I just felt like they swatched so good. We have another Nars palette This is one that I just recently reviewed This is the Euphoria face palette and you can check out my video to see what I think about it The colors are just kind of black. You know the quality is really really nice I can't deny the quality so I feel like this is a palette that I will actually grab for When the year continues on because the quality is good But there definitely are some things in my opinion that could be changed about this palette We have the Melt Cosmetics She's in Parties palette Now this is by far the best quality that I have tried from Melt. I really loved it I just don't love the color story I just feel like these dark shades right here are too close to each other and not only that They're not tones that I reach for I love this first beginning half of the palette But it got a little bit repetitive towards the end here and it's just a bit too dark for me We have the One Size eyeshadow palette that came out for Patrick Star's brand. I really like this I feel like it didn't get as much love as it deserved. Now is it one that I reach for a lot since I purchased it? No, because it is boring. I just didn't reach for it as much But I do like the quality of this you can get it on sale still I believe and I think it's most definitely worth the value and it's just really great basic colors and then Finally the last one in this category is the Jacqueline Hill Volume 2 palette with Morphe I thought the quality in here was really really nice What I'm not so in love with is the color story it definitely runs way too warm and colorful for my preferences and Everything I feel like has a warm undertone to where it looks like really hot fiery on the eyes Which I'm not necessarily the biggest fan of so that's more so personal preference But I will say the quality in here is really good. I know a lot of people are gonna disagree with me But if she came out with another palette, I would totally pick it up because the quality I was very very impressed by have I used it not much maybe one or two times since I've reviewed it So we're gonna move up into the next category, which is worth a try So these eyeshadow palettes I really like I might not be in love with them But I really like them and they're definitely worth a try if you're interested in the color story So we have a couple from Charlotte Tilbury so be prepared We have the super blue eyeshadow formula and at the release of this I really feel like there was a shift in her formula. I just don't grab for blues a lot I don't know. I just didn't grab for this palette a lot But it is really really pretty then we have the desert haze quad The formulation in here is absolutely incredible. It's just a little small quad of mats So I didn't find myself reaching for it a lot this year But something about this formula is so buttery smooth delicious And then we have the last one this also came out with the super blue quad from Charlotte Tilbury So the formula is spectacular It just wasn't a color story that I reached for a ton But it's definitely beautiful if you like the color story next we have from busy art the petite four lilas I would love this a lot more if it had a little bit more depth to it I feel like it's definitely one shade on your eye not a lot of depth the look is pretty, but it's just missing one color So next we have elf bite-size eyeshadow palette so that I do really very much enjoy I feel like the quality of these are really good, especially for $3 now Are they gonna be in Natasha Denona quality? No, but for $3 the mats are workable for sure And the shimmers are quite pigmented and look very nice on the eyes This was the year that I Discovered at nabla cosmetics and there is no running back for me. This is the dreamy palette This is the first palette that I ever tried from nabla this shade is the only dot in here And of course it was one of the first that I use but everything else is buttery smooth I love the tones of this palette I just haven't had enough time to give it love because a lot of palettes came out at once We have the makeup by mario master metals So this one is that not weird, but it's where you only got five But you paid a lot of money, but you're supposed to do it with a mixing medium And it's really metallic on the eyes. That is definitely the case It's really really cool, but I don't have time to do all that mixing and whatnot Then we have the hood of beauty at naughty a new palette I really like this the colors in here were very very nice And I like the way that everything blended the only thing is I already had all of these colors In my other hood of beauty palettes So for that reason this one falls in this category, but if you love the color story definitely Worth a try We have the color pop at forest site palette. This is the collaboration with raw beauty christie Now I will say I'm not the most experienced with this. I've only used this palette Twice and both books. I didn't use a ton of colors. So that's why I'm not as well versed in this palette I had two different experiences the first one when I reached for the more neutral colors I was like wow this palette is incredible the second experience when I played with these colors I mixed them with one of the new denessa myriks twin flames A little bit more difficult to blend, but it's also is a more affordable palette So I like it so far then we have the elf retro paradise palette I picked this up way after it released, but I was very very impressed by the formulation of this Of course as always you might need to give it a little bit of extra love But the colors in here I think are harder to formulate and they did a really nice job with it And I really enjoy it the color story here And then the last one that we have to talk about in this category is the odens eye alva eyeshadow palette This one again is really beautiful. There's a lot of dimension in here If you like these pinky kind of mobby tones I really think you are going to enjoy this formula the colors are kind of Nondescript in a way that I didn't grab for it a ton, but every time I use this palette, I thoroughly enjoyed it All right, you guys we're making it. We are on to Good but not great. So these palettes are really good. I can't say anything too bad about them So the first one that we have is the natasha denona mini retro palette I thought that this was a beautiful spring palette The only thing is this color pulled much more gray on me than I had wanted But this is overall a very beautiful palette the next one that we have is from makeup by mario This is the master metallics. This is definitely my favorite palette that he came out with I love the italian formula that is in here. I love the extra little dimension that these shimmers have And it's been really great to kind of pull this with other palettes to just top off my eyelid We have the d or sprints palette. This is definitely a surprise to me I did not expect to love this as much as I did This was part of their summer collection in 2020 And I just really feel like the shadows were very soft But in a good way and it really was a fun way to play with color while still feeling very Comfortable with the fact that I did have color on my eyelid. We have the visi art petit four praline palette This one was my favorite from these little petit fours. I just feel like it's a really great color story for me It is one that I grab for a lot now This is fairly new so I haven't reached for it a lot But I know I really am going to enjoy this well into 2021 We have this quad from nabla cosmetics. So this is their glitter quad It's the miami lights glitter quad and I have to say they have one of my favorite pressed glitter formulas I don't use pressed glitter very often. I am wearing some loose glitter from sydney grays today But it's very far in few between that I do do that And I really enjoy the glitter formula that nabla has we have the tom ford first frost eye shadow quad This was by far my favorite quad from tom ford that I've tried this year It is a little bit lighter, but I really love the quality the eyeshadow looks so Sophisticated on the eye and I just I love this color story the next one that we have is from sydney grace This is the enduring love palette If you enjoy cool tones, you will really love this palette Now I didn't dig into this palette as much as I would like so I do plan on using this some more in 2021 I think I've only created two looks with this palette But the quality is impeccable and I'm looking forward to continue trying this this year We have the muse a beauty impressionism palette. I think it's a really fun play of colors I love the colors that they chose. I love the way that the colors are laid out It's not normally a color story that I would go for but I do have to say I do feel inspired by this palette And you can create some pretty looks I think it was this color that I wasn't that crazy about but other than that It was a really great palette and I also created a really beautiful neutral look with this palette So it does have some versatility there The next one that we have is the maybelline nudes of new york palette Probably the best drugstore palette that I've tried like truly from the drugstore that you can get from cbs The quality is so good given the price The mattes are really blendable. The shimmers are great. The only downfall of this I would say is probably wear time But other than that it's the best quality at drugstore palette that I tried this year And the last palette in this category is the abhm resi palette. Now this palette did grow on me I didn't use it a lot because at first the color story just didn't speak to me at all But the longer I've had it the more I've stared at it the more I realized that I actually really do enjoy this palette a lot I feel like if you break them up into quads, it makes a lot more sense And I really enjoy the colors that I'm resi chose I love the quality that this palette has to offer. It's really beautiful and glam So it was an underdog for me that kind of swooped up towards the end of the year that I realized Hey, I actually really do like this palette. So the next category that we have is great But there's even better meaning there's more categories going up But these palettes are just great. So this first one is from the charlotte towberry holiday collection this year This is the diamond Dude dude. Hello This is Follow follow. Okay. This is the dazzling diamonds luxury palette of pops This is by far the best palette that I've tried in the way of palette of pops from charlotte towberry I love topping off an eye look with these released like beautiful Glitteries all over the eye and they don't fall out That was my big problem with some previous formulas that she had was that they like fell out everywhere We have the charlotte towberry bejeweled eyes to hypnotize This is one that I feel sad that I didn't get more used out of this year stunning color story You get a lot of creative looks with this. I love the variety that you have with this and the quality is great I don't know why I just didn't reach for it as much as it deserves But it's a really great palette. We have a couple guys from pat mcgrap So we have the interstellar icon quad now I think this one is my favorite of the quads that came out because the color story is really unique I do not have a quad that looks like this. I don't think I have a lot of palettes With these colors anyways to begin with so this palette is really unique It's been really great and grungy for this time of year The next palette that we have is the pat mcgrap labs risqué rose. So this one in particular I loved for the quality, but these are repetitive colors within the pat mcgrap line But the formula is really beautiful and I love the color story It's just not as unique as interstellar icon, but it is still a wonderful palette Next up we have the tiny marvels palette from sydney grace This was done in collaboration with my youtube friend meld tobson She kicked booty with this palette. I love this color story This is a palette that I plan on keeping out on my desk so that I can use it more because I really love the color story I love the light pastels. I love how wearable it is, but there's also some pops in here and sydney grace Such a good formula so wearable so pretty the next palette that I have is from bh cosmetics This is the avocado toast palette So many people went crazy over this palette and for good reason We loved greens this year green was a very popular color BH really killed it with their formula this year. I feel like for their price their formula is Unmatched now this isn't technically a drugstore brand, but it's definitely the best affordable brand as far as quality Goes I love this palette the next one that we have is from busy art This is the itan do violet palette or violet itan do excuse me and it's a gorgeous purple palette I feel like they really laid out the purples in here very well You can get a more wearable look, but you can also get a little bit more of an obnoxious purple kind of look Now this isn't my favorite from busy art I feel like the original purple palette the liaison that they came out with was a little bit better It was a little bit creamier, but nonetheless busy art formula is still so good that this is so really nice And then the last palette that we have in this category is the kaleidos What is this the escape pod palette? I think color story alone is the reason why this palette is ranking So high the quality of these shades right here were really really stunning And so shiny and glimmery the math weren't the best But just how creative this palette may may fail is the reason why it's ranking so high It's such a gorgeous unique beautiful color story. Okay, you guys we are on to the second best category This is the last category that i'm talking about today. This is even better, but not the best So these are palettes that I wish I could have put in my best eyeshadow as a 2020 But they didn't quite make the cut so the first ones that we have are from rare beauty These were the most surprising palettes to me of 2020 I honestly was expecting these palettes to be very bad. They were pretty affordable for a high end brand, but I Love the formula here. I love the color stories. I I prefer this palette a little bit more This is confident energy and this one is magnetic spirit But I feel like even though these are all shimmer. They look beautiful all over the eye They look beautiful layered. They have such a unique color story. It's really great to create looks with I just was so shocked how much I loved these palettes this year. We of course we of course Had to put in fire roses was so so so close to making it to the best of the best It just didn't because It really isn't the kind of color story that i'm in love with but the formula This is the best formula that miss charlotte tulberry has ever ever ever ever ever Come out with it is so good this Shade right here. This is a super pop Sorry if you didn't get a hold of it Sucks and then the other palette that I tried this year from charlotte tulberry that had an amazing formula was queen of glow So this actually belongs to my mom But I do need to eventually pick this one up for myself because the formula here is extra extra creamy and just a beautiful Overall everyday wearable color. I love this quad this next palette I haven't gotten the chance to talk about yet too much on my channel. It will be in an upcoming ranking palettes I've tried this month video, but we have the two faced pumpkin spice palette now. It's not amazing quality I've had a few shades that were a little bit more difficult to blend, but I Love this color story.  We also have the color pop and moulin Collaboration palette. I really feel like color pop stepped it up with this formula this color story It's a great neutral color story, but you still get a lot of dimension a lot of different textures I wish all palettes that they did were as good as this one. So stunning We also have the natasha denona Trio chrome palette. I butchered the name every single time I said it this year But I'm more so in love with the mattes I was actually a little bit underwhelmed by the trio chrome formula But the matte shades were really inspiring to me and I just love creating Different color combinations with this palette and then this one it hurt me Now to put this in the best of the best because quite frankly It was probably one of my most used. This is the bronze palette Now the only reason why it didn't get in the best of the best was because I really did kind of feel like Every look I created with this palette looked the same But I loved every single look. Oh, I just gave away the next one I loved every single look with this palette. The quality was phenomenal if you're into bronzy tones. I highly recommend that one and The last palette that I'm going to talk about today is the color pop stone cold fox palette This one blew me away. Again one of color pop's best this year and uh most of that has to do with the amazing color story if you are A regular follower of mine Cool toned queen over here. I love all tones But cool tones have a special place in my heart So I was elated to see color pop come out with this and the quality is just fine You know, it's not natasha denona quality, but it's on the upper end of color pop quality. I would say So that is not the end. We still have the best category But if you watch my channel, you've already watched that video But if you haven't I'm gonna link that video down below to send you over there So the stone cold fox was the last palette that I have to talk about. Oh, so there we go guys That was me ranking every single eyeshadow palette almost every single I don't know if I was as accurate if I could hopefully every single one and it was really fun I'm sorry if you wanted it numbered by number, but I really just feel like I would have been randomly placing some palettes anyways, so Yeah, I am looking forward of course to trying all the new eyeshadow palettes that come out in 2021 I already have some palettes on backup that I am behind on getting a chance to try But I have said this in a previous video I'm really looking forward to seeing what kind of eyeshadow palette content I can create This year. I've had so much fun creating eyeshadow palette videos And you guys have really seemed to enjoy them. So this year I really want to focus on creating some unique eyeshadow content. So that is all I have for today's video I hope you guys enjoyed it. If you aren't subscribed to my channel already I would love it if you would consider taking the time to do so and I will see you guys in the next one Bye guys. Have a good one


With a better understanding of your team and how to communicate with them, it's now time to plan the project. You might be the type of person who likes to make it up as they go along, but when it comes to group work, planning is essential. Planning the project is important because it helps you stay on track and establishes the processes you will follow. Your professor isn't just concerned about the finished product. They want you to learn from the process. If you've planned properly with your team, then you will have a successful process and a superior finished product. First, your group needs to work together to review the assignment or project your professor has given you. What is the purpose of the assignment and what is it asking you to do? What criteria will you be marked on? Remember, your professors and TAs are available as resources if anything is unclear in the assignment. They often have office hours where you can ask questions about the assignment or seek guidance throughout the planning process. Next, you need to identify the main tasks your group needs to accomplish to complete the project. Your assignment instructions often provide information about key tasks and other guidelines. You will need to identify roles and talk about who will do what task. Sometimes a role may be carried out by an individual or sometimes it'll need more than one person. But planning in advance helps ensure accountability. Examples of roles include facilitator, note taker, timekeeper, devil's advocate and more. When working out roles, it's a good idea to be mindful of each member's preferences and comfort levels discussed in understanding the team. Your group should also build in flexibility around roles to manage absences or unexpected hiccups. It's really important for your group to record the key tasks, deadlines and roles you agree to do so that everyone can refer to them and keep group members accountable. This could take the form of a to-do list or group charter that keeps you on track. Remember to build in some flexibility as some things may shift as you move along. This makes regular check-ins especially important to track your progress as a group. York University offers many tools and technologies to help you engage in planning effectively and the web offers free calendaring and planning tools for teams as well. Remember that you are going to need to engage in tasks that require not just subject knowledge but also academic skills like research and writing. In particular, it is critical that your group practice academic integrity by tracking and documenting the sources you use from the outset. Such as journal articles, websites, books, etc. And by following assignment guidelines on which parts of the work need to be carried out individually and which ones collaboratively. All this work has a payoff. By pulling together to plan your project, you'll both receive amazing results and acquire the skills to collaborate for success.


In this question, it's given that the sum of two angles of a triangle is equal to its third angle and We have been asked to determine the measure of the third angle. Okay, so let's first draw the diagram So, let us say This is my triangle. Okay, so triangle is a BC let's say, okay, what is given guys the sum of two angles of a triangle is equal to its third angle, so a Solution right so angle a plus angle B. Let's say is equal to angle C Right determine the measure of the third angle. So we have to measure the Find the value of this angle angle C now by angle some property by angle some Property of a tank triangle Property of a triangle, what do we know? Angle a plus angle B plus angle C is 180 degrees Right, but angle a plus B is angle C. So this is equal to angle C from here So that means I can write angle C plus angle C is 180 degrees that means twice angle C is 180 degrees That means angle C is equal to 180 degrees by 2 Which is 90 degrees now don't go by the figure which I have drawn So while we were solving the question, we didn't know the value So hence clearly angle C doesn't look like a 90 degrees, but then that was just a representative diagram So hence the solution is angle C is 90 degree Correct in the same problem, let's say if they say that you know angle a is 30 degrees Let's say if they say that angle a is 30 degrees if If let's say if this information is given then you can find out angle B as well. So angle B will be how much? 90 minus 30 that is 60 degrees Correct, isn't it? So angle B will be 60 degrees if angle a is 30 Why because a plus B was 90 which was equal to angle C, right? So this is the solution to this problem Now in this question, it's given that of the three angles of a triangle One is twice the smallest and another is twice times three times the smallest We have to find the angles. So let's first draw a representative diagram. So let's say this is my triangle a, b, c, a b C right and let's say one of the three angles of a triangle One of the three angles of a triangle one is twice the smallest. Let's say C is The smallest one C angle C is smallest one and let us Assume it to be x degrees. Okay. So one is twice the smallest. That means let's say angle a is equal to twice angle C that is 2x and The other is twice the smallest. So angle B is equal to 3. This is the smallest So 3x. So what is the assumption assuming? Assuming what assuming angle C to be the smallest To be the smallest angle, right? Anyone you could have taken as smallest. There is no problem. So I have taken a C just Just like that. So there is no a problem if you consider a or B as smallest the problem will not change now So, what do I know now then clearly angle a plus angle B plus angle C is 180 degrees. Why is this because of angle? some property Property of a Triangle right now some of three angles in a triangle is 180 degrees. Isn't it? So angle a was 2x angle B was 3x and angle C was x and This total is 180 degrees So that means if you see 2x plus 3x plus x which is 6x is equal to 180 degrees So clearly x will be 1 8 0 divide by 6 which is 30 degrees. Isn't it guys? So what do we observe that the smallest angle was x which is 30 degrees, right? So hence smallest angle is smallest angle Is equal to 30 degrees. This is a solution


Take the next ride on to Indiana 2 East. Finally, it has been a long, long haul. Tea, coke, rice, and transmissions. We've been driving for, man, I think almost 13 hours. 12 hours, 12 and a half hours. It's been a long ride. And it has rained almost the entire way up here. Non-stop raining is still raining. But we are about to be here, guys. It says we're about a mile away. As you can see, Indiana, pretty much famous for cornfields. That's about all we've been seeing up here. Cornfields, windmills, and cornfields. We was about halfway up here. Kentucky, one of the trailer tires on my brand new trailer. Brand new tire and wheel on a brand new trailer. The valve stem was leaking. We noticed it was almost flat. So we had to swap tires on the trailer. And we're going to have to get a new valve stem to put on the tire. But other than that, we didn't really have any issues. Made it up here pretty fair. Just a long drive. That's for damn sure. I absolutely wore out my wipers today. They've ran non-stop. They sound like total trash on my windshield now. So we're going to get some wipers. Take the next right onto Harding Drive. Then your destination will be up. Let's see. I don't want us to turn. This street came up quick. I didn't even think it was a street. But it looked almost like a back alley. It didn't look like the street to me. This is CRT, guys. Holy moly. Ready to get out of this truck. CRT. There's John's Dooley. Which also has a 46 RE swap. Manual valve body. Hell yeah. We made it. Let's go in here and find old John. Cold up here. We got out of the truck the last time about a few hours ago. And it was like 60 degrees. Temperature much colder up here. Top of Indiana. Cross bite on the trailer. Made it up here alright. It's time to get some dirty work done. Some serious go fast parts. There's the man right there. Look at him. Look at him doing his thing. He's in there building transmissions. Building transmissions. Hey, we're open for business. We're open for business over here. We made it, guys. Got an old gas pump out here. That's awesome. Let's go in here and see where the magic happens. He made it. There's the man right there with the plan. 12. Oh my goodness. It was 11. It was a long time. Almost 12 and a half. Look at his little play shop right here. What's it really? He's got transmissions going on right here in the front door. I was going to call but I was like, I'll just leave him concentrating and driving. You know. There we go. Well, no incidents. No flat tires in the trailer. Actually, we did have a flat. You're shitting me. Really? Yeah. Brand new trailer, brand new wheel. The valve stem was leaking. We got about halfway up here in Kentucky. Notice the tire was flat at the gas station. But swapped on to spare. We're all good. It didn't run all flat, thankfully. We made it. As you can see. John Cope over here. I'm going to let you guys go. Stay tuned. We got some serious cool stuff going down here soon. And it's all I can say. It's going to be badass. John said something about lining up old Mr. Haney and Frostbite and playing with him a little bit. So stay tuned, guys. We'll see you on the next one.


Ah, OK, all right, fantastic. Yeah, so I will talk about the homology of the quote scheme, which I called last time QDGRN. So this was the space parametrizing quotients of CN of rank N minus R in degree D. So then the kernel as rank R in degree minus D. And this is on P1. And I'll point out today when this P1 assumption is or is not necessary. So in any case, what we discussed, we talked a bit about this space. It compactifies. So this compactifies the space of maps of degree D from P1 to the Grassmanian. In case, this is when R is less than N. OK, so when R is less than N, the generic quotient here is actually locally free. OK, so it's a vector bundle. Obviously, when R is equal to N, then we're parametrizing just torsion quotients. So it depends. Right, so I'll comment on this as we go on. So we know that there is a universal sequence on the quote scheme. And so we've established various things that this is a smooth projective variety. In this case, over P1 is a smooth projective variety. There's this universal sequence. And we saw it be our diagonal arguments that we talked about, I guess, last time that, in fact, the churn classes of the universal bundle generate the cohomology multiplicatively. So if we write, well, maybe I'll take the dual here. This is the universal sub-sheep. So the churn classes live in the product space. So if we use a quenet decomposition fix a basis for the cohomology of P1, a point, and the fundamental class, then we can decompose this. And so this is, if you want, point on P1. And so then what we know from our previous discussion is that the class is ai and fi. And this is generate the cohomology. So yeah, we also, so this is one thing that we understand. And last time I had written the Poincaré polynomial. So let me actually write it again. So here, if we also see the Poincaré polynomial. So if we, so this is here, we sum over the Poincaré polynomials of over degrees, basically. So this is, OK, and then we saw that this is a beautiful expression, the following formula. So here we have the, this is the Poincaré polynomial of the grass man in itself. I'm not going to write it. And then you have a product of two types of series. So these are, OK, so this i goes from 1 to r. And so this was calculated by, I guess, by Linda Chen about 20 years ago. And the calculation involves localizing the code scheme. So and just it's using the cellular decomposition that gives, that this localization yield. So in fact, you can do it with isolated fixed points in this case. And so it's, yeah, so it's a calculation one can do. I should say that Linda actually calculated for every variety of flags. So the so-called general, the so-called hyperclose schemes parameterizing not just the not just the subchef of the grass man, but then a flag of a certain type. The most interesting examples are the full flag or the grass man, kind of the more one for representation theory, maybe, and the other for geometry more. But in any case, there's a formula for any type of flag. OK, so what we saw is that so we're going to focus a little bit on explaining this Poincaré polynomial today. So one thing that we noticed last time was that what we have seen is that the homology stabilizes. So we're interested in this. So in other words, we saw that the betting numbers start to stabilize for d greater than or equal to k. OK. And so also then this was basically you read it off from the formula, essentially. We went over this last time. And I also explained, or rather, let me mention briefly at the end of the lecture, what it stabilizes to. How do you calculate it? And in fact, we have that the stable homology. So this is the stable value. You put them in a series. In fact, given, and one can check this easily by multiplying the Poincaré series by 1 minus t and evaluated at 1. There's a simple polar one. And so then this is just of the theory. So this is in fact given by, let me write this. So is this pre-factor, the Poincaré polynomial of the grassmine in itself. And then it's just this expression. And so we commented where we exactly left off. So it's been a few weeks, I guess, but where we actually stopped last time was in commenting that this is actually the polynomial algebra. So we, right. So let me just, so we specialize. So in fact, I will do this today. And I'll explain why and what the general difficulties are. But we saw very quickly. So what's very, very easy to observe is that in the kind of extreme case when r is equal to n. So when r is equal to n. So in this case, so this is this QD, is this QDG rr, if you want. So this just parameterizes sub-sheeps of rank r of CR. And then the quotient is just purely torsion, yeah? So this is a zero-dimensional sheaf. So it's a rank, it's a, so it's a rank zero sheaf one. It's just a torsion sheaf on the curve, you know? So in this case, in fact, we commented that this polynomial is, so the stable polynomial is actually the Poincare polynomial of just the polynomial algebra in the ATIA bot generators in these A and F classes. And so moreover, also, the Poincare polynomial has a very nice form in this case. This pre-factor disappears. So this guy disappears. So there's just one. And in fact, the full Poincare polynomial. So this is the, so there's the stable polynomial and the full polynomial is P of z, I'm sorry, z and t is equal to this. So I will, for the rest of the lecture, I will actually focus on understanding the homology of this particular space. So this quote scheme of torsion quotients of CR, OK? It's just, it's very accessible. So we can say a number of things about it. And I also comment on the general case as well, how, you know, what the difficulties might be. So where the analogies kind of break down. OK, so I should also say that, you know, in, so this is obviously, so the Poincare, these are the Poincare polynomials, but I should say for this punctual, sorry. OK, so for, so from now on, so assume, so we're going to assume from now on, assume, we have always this R, R is equal to n, OK? So this is, this is, and QD will always mean this now. Yeah, so QD, QD denotes just the scheme of quotient, of quotients of length D of CN, OK? So, well, OK, so let's, so I should say that, yeah, so there's a, there's the Poincare polynomial here, of course, and in fact, you know, one can, one can, one can generalize this and there are, recently, there are calculations of the, of the motif of this quote scheme. Yeah, so, so, so sort of kind of along the same line. So the motif, so in this case, the motif of QD is calculated, OK? So, so these are recent papers of my recall fee. Juan Jaro Fanteci, Peroni, so, so, so, so it's a, yeah, so in some, in some ways, this exploits the, the Taurus action and the Biennese-Kibirula decomposition and also a few, a few other sort of a classic multi-dig techniques, but in any case, this is, this is understood. And what I want to kind of, to do today, my point is to go a bit beyond, I mean, to understand what these formulas actually mean. So I'm not going to, why do we, why do we, why do we have this Poincare polynomial? And here, you know, I wrote both the stable polynomial and the, the, just the total polynomial and we can go in two directions, right? I mean, we can, we know that you can approach the study of the homology in, in, in two, in two ways. So one, one thing is that, you know, we have the idea about generators and to, we, we could try to understand something about the, the shape of the ideal of relations, yeah? So we know that these generators become relation free as D, as D is large, gets large. So it's kind of a, it's, it's an interesting, it's an interesting direction. So I can tell you right, and we expect these relations to be natural, geometric. So I'll just say one thing about this direction because I'll actually focus in on a different one. So let's, let's observe the following, yeah? So, so on QD now, because we have the universal sequence, so on QD, so we have this universal sequence, which in this case, actually I will denote by, I will denote the torsion byte, just to the, the quotient byte to E to indicate it's a torsion sheet. So this is on QD cross P1. So if we project here, we can, so we have the complexes. I'll just write something obvious in some sense. So I can consider the complexes, which I, here, I'm gonna call E sub n, which are the push forwards of the universal sub sheaf twisted by just the, just ample line bundle from P1, coming from P1. So for all n, yeah, you can do this. And so a feature of this is that if I look at the Segre class, so you notice that if I look at, I'm looking at the, you can look at the trend classes, but I'm actually interested in looking at the Segre class of the m, this is zero whenever k is greater than D. And the reason is that this Segre class is actually calculates the trend class of the associated sheaf, which comes from the torsion. So you obtain in the same way by push forward. And this is just locally free of degree D. So then the trend class is vanished past D. I'm sorry, locally free of rank D. So the trend class, the trend class is just vanished past D. And so one thing that one can ask is, do these vanishings, do the Segre vanishing generate the ideal relation, the ideal of relations, RQD. And when we take the generators to be these at the abogen, obviously they can be expressed. These are polynomial expressions in the abogen. There's by the Riemann-Roch formula. So you can say, well, so you can ask this question. And then it's somewhat supported by the fact that indeed there are no relations. We know that there are no relations in a homological degrees lower than D. So these relations are all, these would be relations in higher homological degree as they should. So we expect this. So I'll leave this a bit open, because I want to say a bit more about the other direction. But this is a question that sort of, it's natural and it's kind of forced on us by the fact that we know what the stable, what the stable cohomology looks like. So we understand when we know the stable cohomology want an explanation for it. So this is something that one might pursue and I'll defer answering this question, okay, for the moment. But this is a type of geometric relation that you might expect to give you all relations, essentially. This is kind of a Mumford type relation in the homology of this code scheme. Okay, so now in fact, let me go in a different direction now, which is, you see, which is looking at the Poincaré polynomial here. Let me highlight it and I wrote it here. Yeah, which is to sort of understand, produce not a, instead of starting so to speak the general shape of the relation ideal, you may attempt to produce some sort of canonical additive basis for the homology, since the Poincaré polynomial looks very nice. And this can be done just by, I guess, a system of Hecke correspondences. So this is what I would like to explain next. So our goal for the remainder of the lecture is to produce, so we leave this a bit open and here we want to produce additive basis for the homology of the code schemes, okay? So this would be an Akajima type basis, okay? So let's see what we can do. So it's actually straightforward in this case and Poincaré polynomial kind of tells you what to do. So we're gonna consider the nested code scheme. So we can consider for any given D. So here this sits, so Etilda is a locally free sheaf of degree by one less. So you study the sort of nested phenomenon. So the quotient is certainly just a skyscraper sheaf at a point on the curve. So yeah, so there's a, so what is this then? So how do you constitute this? Well, you know, it just sits as a projective bundle over QD times P1. So in fact, so we have that myomorphism. Yeah, so then, so this, right? So what you need to do in order to generate the nested code scheme starting with, so starting with the basic setup of E and CR. So a point in QD here. Oh, you fix a point where you want to have this Hecke modification. And then you're looking at just a surjective morphism from E to E to just the skyscraper sheaf at this point. Okay, so this is essentially, this is the fiber of E dual, okay? So, and of course you look projectively inside the code scheme. So therefore, overall one can show this easily that this is, I mean, this is completely straightforward. This is just a projectivization of the universal sub bundle, okay? So then there are these natural maps. So you can, sorry, you can map to QD here and well, you can also forget about E and just map to the deeper sub sheaf, so to speak. So that maps you to QD plus one in here. And this side you can check is a generically finite of degree D plus one. So it's a D plus one, one map, okay? All right, so it's just kind of a basic geometric setup we have and then on, so this comes of course, there are two basic exact sequences. One is as the exact sequence, the universal sequence. So this would be what is a torsion and this is precisely, I'll explain in a second. Or maybe I, oh no, it's actually okay. Okay, so now this, I'll explain immediately what this is. I mean, here of course this maps to QD cross P one and this diagonal, you see there's a divisor here which is just a pullback of the diagonal on P one times P one. So this figures here, so this parameterizes the elementary modification at the point, yeah? And then there's this L which is just a hyperplane of the, so this is a line bundle which is the hyperplane bundle of this projective bundle, okay? So this is a fundamental universal sequence which connects the two sides on QDD plus one. And so then of course, B is just the exact sequence of this projective bundle. So how, so this is on, so there's a quotient Q but this is just on QDD plus one, okay? And this is pullback from QD via this, yeah, this is pullback from the base from here. From, yeah, it's pullback from here. This is just the exact sequence of the projective point. This is the universal. So then, so this analysis, it's easier, I mean, okay, this setup has been much studied, for example, in the context of just the Hilbert scheme of points on a surface, no? So in fact, by Lotar, so it's nothing, it's nothing very surprising, you know? I mean, it's kind of a standard way to move between, inductively between understanding the Comolge of QD, understanding of Comolge of QD plus one. So also, so therefore, you know, but these exact sequences allow us to relate easily the trend classes, which is something that we want to do. Well, let me just, yeah, so here I have to be, right, so I will call the hyperplane in this P1, I'll call this H0, okay? So then, whereas, so we have two P1s, yeah? I mean, on this, so we have to be a bit so, and on the basic, oh no, I'm sorry, I'm sorry, I'm saying it exactly the way I don't want it, okay? I apologize, so I'm gonna call this H, okay? And when I write a universal sequence in the context when this would be H0, this H0 will disappear, so this one is H0, yeah? So we have two P1s, they each has a, in the picture here, each has a hyperplane, so we have to keep them apart. There's H in the basic one that comes into the projective bundle and H0. So then, if we, so from the universal exact sequence gives, gives the following, well, if you, if we write the trun classes, we sort of, we do this kinetic composition, so we can, now we have two sets of classes for E and for this E tilde. So then you just, well, I'm not gonna go into the details of the calculation, but it's completely standard, but gives, so setting these, so with this notation, okay, we have, we have that, and now I'm just gonna write a full series, so to speak, so this is absolutely standard calculations, okay? So you might not even, it is kind of important, but as one, right, so it's something that you definitely want to understand. We also have that, so these are both classes, so this is on QD, D plus one, yeah? So we're looking, so we decompose, we use universal sequence and the kinetic composition on the second P1, so to speak, in the problem, and this is what we have. So here, just to, right, so I should also say that for the, I'm interested in the trun class of the quotient on the projective bundle, and this is in fact, well, again, this is immediate. I remind you in a second of the classes if this is a bit confusing. So in any case, we have, and this is something that is useful and it's a beautiful formula, in fact, that when everything is said and done, the two universal classes differ by just the trun class of the quotient bundle here, okay? And again, I don't write it by, I mean, this is a multi-degree formula, but okay, okay. So, but let me also quickly remind you how, what this class is, okay, so I didn't say that this, okay, so this lambda is actually this hyperplane class, yeah? So it's one of the, okay? So in other words, if we think of this being just a, so this has a structure of a projective bundle over QD, right, as we, so let me remind quickly here what the classes are. So this H is the hyperplane on P1, lambda is, well, there's a hyperplane line bundle here, so this lambda is C1 of L, and in fact, the entire cohomology of this nested Hilbert scheme then is written over the cohomology of QD, where you have these two generators, you have H and you have lambda, and then, you know, of course the relations are that way squared is zero. And also there's a basic relation on, just on any, on the projective bundle, and this is the top churn class of E dual L is equal to zero. Now it's just a standard relation on any projective bundle. So that's why it's no mystery that this A tilde and F tilde then can be written in terms of precisely these classes up here, AF, which generate, which live here, they generate the cohomology of QD, and then, you know, you have these extra two classes, H and lambda to contend with, one coming from P1, the other one is just the hyperplane class. Okay, so any questions so far? Yeah, so then, so then we're in a position to define these Hecke and Akajima type operators in, right, so we have this, so our setup allows us to define, it's the collection of two R operators. So, right, so we, let's, let's remind ourselves as the basic notation here, this maps to QD by phi, and to UD plus one by psi, and while psi is a finite map, is a generically finite map, this phi is not, it has, the fiber has dimension R, because this sits as a projective bundle over QD cross P1 here. Okay, so it factors like this. Okay, so let's, so we have these operators. They will be indexed by, yeah, by, we have two indices here. So what, so this goes from the homology of QD, the homology of QD plus one, and let's say you take a class alpha, or you pull it back to the nested quote scheme, and then you intersect with this canonical class, which is the class with the term classes of the canonical quotient of the projective bundle, and then also with this hyperplane class on P1. So here I ranges from zero to R minus one, this Q is a vector bundle of rank R minus one, and of course A is either zero or one. Yeah, so, right, so if you, if we're looking at what this does, so basically, for example, A zero zero is just the nested Hilbert scheme itself. Yeah, you don't, and for, to generate a rest of the operators, you don't use just the correspondence, which is the obvious one, the incidence correspondence, this nested quote scheme, but you also use these natural cycles on it, yeah, which are these classes. I found it more convenient to work with the term classes of Q as opposed to powers of the hyperplane class, but essentially it's the same thing, no, I mean, so you can say, well, I could use the lambda to the I here, but I think this is better to see some, to calculate with it, in fact. Yeah, so this operator, then how does, what does it do to us? So if to a cycle of a certain dimension, well, it raises by, oh, I'm sorry, minus, I'm sorry, you can, you can see that this is obvious, you can just, okay. So essentially they raise dimension, no, in the case when, well, when the maximal case, when I look at i is equal to r minus one and h, then it's actually, it preserves dimension, but in general, it will raise dimension. But I'm also, of course, interested in looking at the transpose operator, the one that goes from, so if I look at the transpose operator, so given by the same cycle, but I view this as going from qd plus one, qd, okay, then this, the dimension goes down in this way, okay? So this lowers dimension, okay? So one raises the other one, lowers dimension. Okay, so then, well, so the, there are a couple of basic propositions here, the first one is that these operators commute, okay? So, of course, to see this, you look into, so you have to compose, no, you have to compose the operators and they're raising, they're both raising, so you, so you, I'm not gonna show the argument, but I will just, you have to look into, I'm sorry, you have to look at the double nested quote scheme, and then, of course, there's a projection to the nested quote scheme where the length of the quotient is actually one scheme sits in the other with the quotient of length two, okay? So this is, so this here, you look at, I'm using F for the one that will disappear that you project out of, yeah? So then this, so this is where the cycle, the correspondence, the cycle will be supported that the composition of these two operators will be supported on this quote scheme. And then, you know, you just, this, and if I'm just looking at this inclusion, then the torsion here has length two, I'm sorry, the quotient has length two, it's a torsion shift of length two. So then, you know, you just use the basic geometric cell, this is generically two-to-one map, so you can, yeah, so you can argue the commutation, okay? But so, so basically looking at the Poincare polynomial, you would like to say, well, you know, you just act with these operators on the vacuum, on Q zero, so on, there's a, and then they just sort of freely generate the, the, the, the comology of the, of the, of these torsion quote schemes. And that is indeed the case. So, but you do have to, you're interested also in establishing the commutations with the, with the transposed operators, right? The lowering operator, so to speak. So, so we also wish to calculate, right? This type of commutator. So, yeah. So then again, you know, a calculation is needed here. So, you, so, so I'll just say a few words in the remaining time about what this is and how, so maybe I should kind of be a bit organized here and say that, let's say that you want to look at this composition, of course, as a correspondence. So this is a cycle supported on the following scheme. So this is, I'm going to call this Z minus. So this proceeds not, you know, not unlike the calculation of the Nakajima commutation relations for the Heisenberg algebra on the Hilbert scheme. No, I mean, it's, it's this type of, it's kind of this type of analysis. So this Z minus is just, you're looking at, well, so, so three subsheves here, yeah? So, and the main point is that they're contained in the same, I call this F, I'll explain in a second. Okay, so it's a cycle supported like this. So you have three subsheves of CR, but in this relation, so two of them are actually contained in this F minus. And of course, they're contained with quotients, which is a skyscraper sheaf, no? So then of course, then you want to project this, yeah? So you take, so you, so then you take the, well, well, so in fact, you're, how do I, yeah, so let me, maybe, let me see how I organize this best and it's very little time left or so. Yeah, so let me say also about the other one, right? So if I'm looking at A, maybe plus composed with A, I, A, this is a cycle supported on, let me call this Z plus. And the Z plus is the, why it goes. So again, you have three subsheves and the ones you care, so you have E at this E and D tilde. This I think in this course, Kim, this is E tilde and this is F plus, yeah? And they're in this relation. So these E and D tilde are related by the fact that they actually contain a large subshift is F plus. They both contain it. Yeah, so then these correspondences, both this Z minus and Z plus both have dimension, well, the dimension of the quote scheme QD plus R. Now, they play differently, which is why the commutator in general will not be, will not be zero, yeah? So in fact, the Z minus is smooth irreducible. So in fact, you can say what it is. I mean, it's simpler to describe in some sense. So this is over QD minus one is a product of two projective bundles. So here, you have two, you have a universal E one on the first copy of P one and P two. So this is simpler, but this is not the case about Z plus. Z plus is not irreducible. So the plus is not irreducible. And in fact, it has one component which is smooth, one component. Well, if you look at this, I can look at a component when E and E tilde are the same is given by E in CR being equal as a point in the quote scheme to the tilde in CR. And then of course, you have a, you pick a subchief, okay? So this is one component of it. And in fact, so this component, let's call it C, it's kind of a distinguished component plays a major role here. So C is just isomorphic to the projective bundle over. So you see it has the right dimension, it's a big, it's a big locus in fact, in this Z plus. But in any case, so what you want to do is ultimately project to, so you have this, let me, okay, it's okay, I won't discuss too much. So let me just, because let me write this to comparing. So of course you want to project to QD cross QD where you want to compare these two compositions, no? So here, so let me write them side by side. So this is C plus here. And again, you want to project to QD cross QD. So our operators are cycles supported. So what these compositions are cycles supported on Z minus and Z plus respectively, which you then project down to QD times QD. And you hope then you calculate and calculate the difference, no? So, but largely these calculations will overlap. So these, so basically, yeah, let's think a little bit about this setup, I'll just explain because geometrically it's very simple what goes on. So if I'm on this side, where in fact, I'm looking at sub sheaves, which have a common. I'm looking at pairs, E and D tilde, which have a common. Okay, which are, yeah, I'm sorry, I'm sorry. Just a second, I want to do this right. Which are both contained in a common sub sheaf F minus with some quotients. So unless, you see, unless these maps are the same, so this quotient, unless P is equal to Q, and in fact these two maps are exactly the same, this phi and phi tilde. In other words, if E and D tilde coincide as sub sheaves, unless that happens, there's a pullback here, which maps you to the other side. No, so which is basically will give you a point in the correspondency plus. And conversely here, if I start with a sub sheaf of both, unless again, E and D tilde are identical, there is a push out, no? Which they will be sub sheafs of a common sub sheaf minus of the correct degree, degree by one, higher by one, okay? So, and so this is a, so basically this establishes so there's a, there's an isomorphism between an open sub scheme of Z minus and then open sub scheme of Z plus, okay? So that under which, under this isomorphism, I don't have time to explain this, I'm just kind of giving a flavor of this, of this calculation, the classes, the cycles that are supported precisely match on both sides, okay? So then you have to, you have to see where you don't have, where you don't have this picture, where you don't have this isomorphism, and that is exactly over the diagonal in QD times QD. So you have, you always this diagram works perfectly unlike unless the two pairs E and D tilde are actually the same. So then, yeah, so let me write this as a lemma. So there is an isomorphism. So as I said, this is not at all unlike the calculation of the Nakajima commutators on the Hilbert scheme where this type of analysis is also done, okay? Which basically interchanges, which matches the cycle supported on the, on these two sub schemes on Z minus and Z plus, okay? So then you have to deal with what's left essentially and on one side, so you have to examine and this plays differently on the two sides somehow, which is why you get a non-trivial commutator sometimes, which I'll tell you what it is, okay? And so, yeah, so in here there's a big, there's this, this is actually this component C that I described, which sits as a, so this is just a, so in, okay? So this C sitting inside QD cross QD plus one cross QD is in fact isomorphic to, this is just the, let me write it here, write it here. So this is just a projectivization of the universal bundle on the, over the diagonal, yeah? So over the diagonal in QD times QD cross P1, okay? That's how you should view it. So, so then this leads to the following conclusion in some sense, so we have the following proposition which sort of sums up if one does the calculation in detail, so it's, the point is that this commutator, viewed as a correspondence in QD times QD in the homology of QD times QD is always supported on the diagonal. And so, so in particular, concretely we have, well, A, I mean, maybe I should say one here, that this commutator is zero whenever I plus J plus A plus B is less than R, just for dimension reasons, it cannot be supported on the diagonal, it is for dimension reasons. And two, if you have a match, then the commutator is the identity for complementary pairs of operators. So whenever I plus J is equal to R minus one and A plus B is equal to one, okay? And so it's zero otherwise. Now, it does appear that this algebra does not close in the sense that, so you would say, well, you know, this would be, what happens if you exceed this number? So, so more precisely if I plus, well, then you put it like this, it's okay. So it seems that the commutator is calculated by a cycle which can be determined precisely by a tautological cycle on the diagonal. But in any case with this, so the conclusion in, I'm sorry, I went over it, I just, I'll be done in a couple of minutes, but it's kind of hard to interrupt at this particular moment. So, yeah, so what this means though, is that, let's write this maybe as a theorem, is that the homology of the code schemes in this case has a canonical basis given by applying these operators, the operators IA to a basic state zero, say to the zero where, where this just spans is just the homology of q zero, q zero is just a point.  And then the Poincare polynomial does the rest, the Poincare polynomial basically strongly suggests this picture, and so, then in fact, they, you know, they, they span the homology as well, just because the Poincare polynomial tells you so, but this computing the commutator with the, with the, with the lowering operator, with the transpose is important for understanding that they're actually linearly independent, so you generate linearly independent classes. Yeah, so you, so you have this picture, and yeah, so, you know, so basically you can say, well, you have these, then these basis vectors are labeled by a, by a multi-plat, NIA, you know, which corresponds to applying each operator a suitable number of times, you know, so, and zero one and so on, A i minus one, zero, these are the basis vectors, and these are sort of canonical, there are no choices involved in, in come, in producing these basis. So it's, it's kind of an Akajima basis, I don't know. So, yeah, so I should say also as a remark that it does not matter at all for this, that C is P one can take C an arbitrary smooth projective curve, and also it does not matter at all that you start with a trivial sheath, it could be, it could be anything, so, so it could be quotients of a fixed sheath is zero, but of course, torsion quotients, yeah, so we're in zero quotients on some arbitrary curve, see nothing changes in the analysis that of course on a curvy of odd homology. So the zero and one that I have here, this second index, of course, is there to model the homology of P one, yeah, so you have the class of a point and the fundamental class for an arbitrary curve, you have to include the odd generators, yeah, so then, so then, so then you have a larger family of operators, but in some sense, you know, the analysis is not, it's not, or difficult, and of course, let me maybe stop sharing because I'm here. Yeah, so, so the analysis basically, I would say, I should say that I knew about this, you know, of course I knew about this Poincaré polymer, but his operators for a long time, but giving this lecture is an opportunity for me to sort of understand how to, you know, present this best and yeah, so, and somehow calculate carefully, and you could say, well, what does this have to do with anything, so I just wanted to, because we started with these lecture SL2s, and of course, the idea is that, you know, how do you calculate, you know, we wrote this in the Chao of an abelian variety, and you know, if say, you want to express this on the code scheme, then having such a basis might help, so you can, either of the two approaches I outlined, one is to, you know, I would produce this canonical basis, so then it might be easy to write the lecture SL2 in terms of these operators, and the other approach too, if you have a good system of generators, when you control a bit the ideal of relations, you understand the ideal of relations, you might also be able to just define the operators on the generators, so on just, and then see that they're compatible with the relations, if the relations are geometric, so basically, it's kind of sets the stage for this type of, any calculation on the homology, for example, of the lecture SL2s, but other, you know, other calculations too, so it's kind of, it's good to have, in some sense, this type of description, so yeah, I apologize, you know, I don't know how I always run over, so I run over a lot of time, so if, I don't know if there are any questions here. Thank you very much. I'm actually gonna approve the statements you gave us today, because I've never seen them before, so I was a bit lost. Sorry? I'm gonna approve the things you discussed today, in the end, like the revision or the theorem. Am I gonna approve the- The last part, like when you give the description of Z minus Z plus? To the commentators in, I try to sketch the, how you approach the, it's a calculation, right? So it's actually kind of, yeah, so I try to sketch a bit how you calculate this commentator, basically there's a large overlapping part in the two sides, when you, in the two compositions, and then you have to somehow come to terms with the difference, and so, yeah, this is, I guess, as I said, I'm sort of working them, I worked them out, I mean, I knew for a long time they were there, but I worked them out basically for the purpose of this lecture, and then, yeah, so there isn't, they should be written carefully, but in some sense, I did indicate a little bit, how you know, how you go about it, but I mean, you mean to have this in all, in full detail, and just the calculation of the commentators. Yeah, yeah, maybe, I don't know, maybe I have to go through the details, please. No, no, I mean, it's, for example, I didn't think, you know, for example, the fact that they commute, I, you know, yes, there are arguments to be completely, to be written carefully, you know, to be given carefully, and of course I could not in one hour do it, you know, but yeah, it's just an intersection theoretic calculations now in successive nested quote schemes that give you these commentators, you know. So, but of course, maybe there is an optimal way to, which still maybe I haven't, you know, to present this, because as I said, if you look at, it's not, so maybe you would hope for a kind of a finite Heisenberg algebra, and this doesn't seem to be that because you do have these, some of the commentators will not, yeah, I mean, you would be properly supported on the diagonal for dimensional reasons, so you know, so you'll have some cycle on the diagonal, but I, so I'll have to think a little bit about that. So the algebra doesn't seem to close completely, you see, but it's enough what already the propositions that I wrote, I mean, the fact that they commute and the fact that in there is, so to speak a, for each creation operator is a matching annihilation operator that, so the committer is the identity is enough to provide this basis, so this basis is given, it's enough to show that these are inject, this is, they're linearly independent. Now, this, if you keep applying the creation operator, the vacuum you generate linearly independent vector, so what I, you know what, the propositions I, the commentators I actually calculate are enough for this purpose, but yes, for an optimal presentation of the algebra, I agree there's still, one has to, one has to work maybe a bit more, so thanks for the question, yeah. Actually, I have one comment to say, but just to, which is that this doesn't, you know, this picture, which is fairly simple, you know, it doesn't, doesn't work for, unless for the case of a non-degenerative grass mania, so when R is less than N, so you look at a code scheme in that case, it's more subtle, I mean, you know, the Pankarebo-Lehmann looks also beautiful, but these hecke type correspondences do not tell the whole story in that case, okay, so one has to work harder. Just as I would say it's, you know, in the modular theory of sheaves on surfaces, it's the same situation for rank one, for the Hilbert scheme, the hecke correspondences capture the cohomology, but it's not the case for higher end sheaves, yeah, so for, so it's kind of an interesting one, yeah, so we've been interesting thing to study basically, the case when R is less than N, so here I only focused on the case of torsion quotients, but yeah, anyway, so just, yeah, so I, yeah, I apologize, I ran over time and yeah, and so. Is there a senior infrastructure for all of the coasting? Was there any general theory about this point is how to work with the crazy as an algebra? But here the coasting is for sheaves on Q1, for general understanding, you know, I'm sorry, I have to say that somehow the acoustic, there's something about, I can't hear, I heard a little bit, but not quite completely. So, I mean, here the algebra structure is for cold scheme of sheaves over Q1, is that a similar structure for other cold scheme like sheaves on hygienist curves? Yeah, you know, so for this particular cold scheme, I ended up discussing of torsion when the quotients are torsion, this is smooth for any curve, but in general the cold scheme on a curve is can be quite singular. So then, you know, you have to write, so the structure is much messier. So you can, yeah, so you have to sort of work virtually, but then, you know, how, yeah, so it's a more difficult picture to make sense of. I'm not saying it's impossible, but in principle, lots of things do not go as you, it's not smooth, it can be quite badly behaved, you know, for higher, for higher genus curve, but and an arbitrary gross mind, you know, I mean, in this case, in this kind of, why I won't say the general, but kind of limit case when you're looking at torsion quotients, it's okay. So that this goes through, as I remarked, this is basically, it's the same picture, but in general, no. So another thing that you could, one could look at is the code scheme of zero-dimensional quotients on a surface. Yeah, so sort of generalizing the Hilbert scheme. So you look at quotients, not of C, but of some higher rank trivial sheaf on a surface. So that's also not smooth. But it's, I don't know, it's a little thing. Thank you. Okay. Thank you.