Välkomna Laddingtons!  Sämre frågor än detta Och när vi menar Europa, när folk säger Europa De brukar fokus på Västern Europe, så Great Britain, Frans Germany, Sverige, etc. Men vi har hela Västern Europe också Och jag ska inte säga att allt är så bra i Västern Europe att det är så baserat och allt är så bra i Västern Jag skulle säga att i Sverige det är bättre att leva här än i många platser i Västern Europe för att de har mycket problem unless you're a woman of course If you're a woman I probably would say that many Eastern European countries are better than Sweden in certain situations Men i Västern Europe de har plenty of problems of their own It's not been that long since the Soviet Union collapsed and basically all Eastern European countries they were in quite a mess when the Soviet Union fell when the iron curtain was lifted But if we look at where things are headed many Eastern European countries especially the Visigrad countries can take Poland for example they have made tremendous leaps in the right direction whereas Sweden for example is going in the completely opposite direction so if I say now yeah Sweden is perhaps better to live in or Great Britain is better to live in than some Eastern European countries if you ask me again in five years I don't know if I will say the same thing if you ask me in ten years I will probably say that most Eastern European countries are better especially in terms of safety for women etc. That being said I don't think Western Europe I don't think Sweden is lost at all this is just getting started and we shouldn't be so quick to just focus on the bad news but in the worst case scenario if nothing changes in Western Europe what can we do to save some parts of Europe and either when I think about these things in very grandiose scale I envision Western Europe as the Western Roman Empire and Eastern Europe then as the Eastern Roman Empire and as you all know the Byzantine Empire the Eastern Roman Empire they survived for much longer now what I hope at least is that Eastern Europe will continue to develop will continue to get economically stronger hopefully with the help of the EU ironically enough it might just be the EU that saves us Who knows I might be overly white-pilled on this but if Eastern Europe can continue to modernize in the good sense you know get rid of corruption get rid of a lot of the bad things continue to build themselves up to be strong and at the same time and this is my wish for all Eastern European guys I know I have you know a lot of subscribers from Poland and Romania some from Hungary as well Czech Republic please do emulate countries such as Sweden in certain regards but also tell as many people as you know every time you talk about Western Europe tell them the truth about mass immigration from the third world and you know keep firm stay strong in this regard you can just point to Western Europe you can just point to Sweden or France and say this is what we must not do so you can say we will continue to implement a lot of the good things you can see in Sweden but we will also avoid the mistakes Sweden has made so on a personal note what do I do I have my two heroic and glorious companies Legiugloria for clothing and training gear and eutonym nutrition for supplements and I'm proud to say that I make business transactions with the following countries Estonia, Latvia, Lithuania Poland, Hungary Czech Republic and of course internally in Sweden as well so in my view I want to drive as much business to Eastern Europe as possible at the expense of China you know all of my clothes all of our clothes etc are made in Europe in these different countries primarily in Poland also on a personal note if something happens here in Sweden I know I can send my family to Eastern Europe and continue the good fight here now I don't think anything of the sort will happen I think that's wishful thinking for some guys just don't want to engage in a marathon they want to have a quick decisive battle but you know I don't think it will come to that so we need to adapt ourselves to reality and not fantasize about some heroic crusade or anything might come later in a few decades who knows but as for now dispel any thoughts of violence because if you do violent acts the establishment will jump with joy because then they can implement more restrictive measures against any pro-european groups or individuals so basically my own vision and I put my theory into practice by my you know my business endeavors you know stimulate Eastern Europe bring as much strength and wealth to Eastern Europe as possible have them develop as rapidly and as good as possible then we have both a base for you know a strong support to retake Western Europe if it should come to that and also if we're talking geopolitically I'd rather have a strong Eastern Europe so we can have a strong Europe as a whole because you know what at this stage in time we can't have this sort of petty nationalism that each country in Europe is hating its neighbor because of a conflict two hundred years ago now understand that there's a lot of bad blood between you know a lot of Hungarian guys I know they don't really like their neighbors I understand but at the same time you know Hungary today on its own is weak just Sweden on its own is weak that's why I hesitate to call myself a nationalist now I might as well call myself a an imperialist and I say this and I'm one hundred percent on the side of you know my Swedish brothers but I also realize the time when Sweden could be strong on her own long past the time when Germany could be strong on their own long past same thing would basically all European countries and I'm not saying that all Europeans are the same I'm not saying that we should be one country but I say we could be a union because there is more that connects us we have more in common than we do not so you know I don't want to be a fan boy of the EU because the EU is very corrupt and very keen on replacing Europe's native population but I do want some sort of union at least I want some sort of corporation because let's not be let's not be libertarians in the sense that we believe oh if I leave others alone they will leave me alone I will live here peacefully ever after there will always be threats there will always be powers that seek to invade Europe as has always been the case so therefore we must be united you know what a single European country can't stand against these forces but united there might be a chance so basically my vision for Europe for the future what can be done is we hold our heads up high in western Europe we do try to enlighten and convert as many people as possible to understand what's going on then we also have to say you know what it's far from over it's just begun Europe is not lost and especially since we have large parts of Eastern Europe who are in a good position and that position is getting better we also have to understand that a strong Eastern Europe will mean a strong Europe as a whole so for me again personally and in my business dealings I want to direct as much business as possible to Eastern Europe make them strong instead of making China strong so think about that when you purchase something as well look where it's made look whose economy you are supporting look who you are making strong do you want to make Eastern Europe strong do you want to make Lithuania strong, do you want to make Poland strong or do you want to make Bangladesh and China strong it's up to you how you purchase what sort of clothes and products you buy anyway this is not a video where I talk about how good my business ethics are but I'm just saying that I do try to live in accord with my vision for the future so I'm not just going around here talking about how bad things are I'm just also trying to steer at least my little part in a better direction for the future so to conclude making Eastern Europe as strong as possible and also I wish I could reach out to more Eastern European guys and say you know what you need to be prepared because the EU as it is right now they will continue to try to bully you into accepting migrants and I just have to say stay strong do not accept the bullying they can't really do much you know they say money is power but power is power if you and I talk to you you know the Visigrad countries here Hungary, Poland etc if you say no there isn't much EU can do sure they can withdraw some funds etc but they don't have any real military power to actually force you to do it so you can say now we don't want to do it it's not good for our people we put our people first that's an excellent start and now I know some Hungarians don't really like Viktor Orban but you know I will say you're a bit spoiled at least yeah he might not be perfect I'm sure he is perhaps a bit corrupt who knows who cares at least he's not trying to replace his own population as we have in Sweden or Germany etc so you know what you have a good leader even though you might find fault with him same thing with Poland I understand your government isn't 100% perfect in every regard but again you don't have a regime that is actively trying to replace you in your own homes so yeah be a bit humble in that regard and for all this in European guys just if you want to talk about politics ask everyone to talk to you to just look at Sweden for example going from the best country in the world to not so great anymore especially in terms of you know if you have women you care about if you are for women you are against mass immigration from the third world simple as that if someone is for mass migration from the third world they are anti-female they probably don't even like women so keep that in mind and yeah stay strong heads up we don't chin up we don't need to be all the fittest this thing has just started Europe is not lost Eastern Europe is definitely not lost and you know we have a long way to go but nothing is over until it's over so thank you for watching XXO BOOM


And we will, yeah, our next speaker is Portia Anderson. I hope I'm saying that right. Who will tell us about it's warping symmetries of Knudsen-Tau puzzles with hexagonal and parallelogram shaped boundary. Thank you. Oh, and I guess so we should, yeah. Wait until you have your slides here. Yeah. Good. Okay. So you can see that. Yes. Please go ahead. Okay. Okay. It's actually Portia, but that's okay. Sorry. Yes, no problem. Okay. So this is going to be about this interplay between this combinatorial puzzle method of doing Schubert-Catholus and the geometric perspective. So Schubert-Catholus is about computing structure constants and the cohomology ring of the grass monion and true to the origins of this area of math. One way is to look at transverse intersections, triple intersections and count the points. So that's one way. And then we also have puzzles, which compute the same constants. And so sometimes an observation or symmetry that's clear on one side, like the puzzle side can show us something we didn't expect on the geometry side and vice versa. And so we'll see some good examples of that with this project I've been working on. Okay. So here is our space that we'll be working in, of course, is the grass monion of T-planes in Cn. And so that is a smooth complex manifold and projective variety. And we have these Schubert varieties. We have these Schubert varieties and we index them, especially for puzzles with binary strings with K ones and N minus K zeros. And we can think of those interchangeably with partitions that fit in a K by N minus K rectangle. So, and then have opposite Schubert varieties which give us a dual basis. Oh, sorry. I didn't even say, okay. Schubert varieties give us an integral basis for the cohomology of the grass monion. And then the opposite Schubert varieties give us a dual basis for that. And we have with this perfect pairing, which is taking the integral of product of two of them, which I will say what that integral is talking about more soon. So, okay, let's define a puzzle. A, okay. So let's let this triangle lambda mu nu denote this boundary shown here on the right. So, we take this equilateral triangle and then we label the edges with strings of ones and zeros. And they go, they're right in the orientation shown. And, okay. So then the first one is going to be the first one. And then the second one is going to be the first one. Okay. So then those are lambda mu nu. And then a puzzle is a filling of this boundary with pieces in this allowed set here. And the way you fill them in is just as you would expect like a jigsaw puzzle where you have to align the any touching edges of the pieces such that they have the same number. Okay. And then at the bottom here is just some random examples of what these things look like. Okay. So then we have this theorem by my advisor, Alan Knudsen and Terry Tao. And it says that puzzles do Schuber calculus basically. So puzzles compute the structure constants for the homology of the cross-monion in the Schuber variety basis. So what that means is for these product expansions of two Schuber varieties, you expand that out in the Schuber variety basis and the coefficients you get are these constants. And so the C lambda mu nu equals the number of C lambda, or sorry, of lambda mu nu puzzles. And this, and then here is the like geometry version of this. And if that's unclear what this integral is, which it was to me for a long time, the integral, you can think of it as pushed forward to a point. And then how we turn this into like an intersection theory kind of problem is you can, if your varieties intersect transversely, the product of their classes is the class of their intersection. And so we can like squish them into one class with their intersection in there. And we wanna turn it into a triple transverse intersection. We have to perturb at least one of these with something in GLN to make it transverse. And then the integral counts the intersection points if it's a finite intersection and it returns zero otherwise. Okay, and then here's just an example of puzzles computing some structure constants. So what if we want to expand out the product X0101 squared. So then we put those 0101 strings or lambda and mu on the sides of a boundary. And then we fill in many puzzles that we can find. And then what ends up coming out is just two. And then we have one with the new equals 011 0 and then new equals 1 0 0 1. So then the C lambda mu news for those news is one and then everything else is zero. So then that tells us that this expands out to this sum here. All right, now we're moving on to think about some puzzles with different boundary shapes which is pretty unorthodox I guess because normally they're always triangles. So we're gonna do parallelograms and the next slide we're going to talk about how to take this and complete it to a triangle so we can think about it as a normal Schubert calculus problem. Okay, so let's take a parallelogram boundary shape and we're gonna label it with strings of zeros and ones going clockwise. And we demand that the two opposite labels have to have the same content which means the same number of zeros and ones and same for alpha and beta. And that's what this sort notation is saying. So sort means take all the zeros in the string and put them ahead of all the ones. So this is just saying alpha and beta have the same content and same with lambda and mu. And then what we do is we can complete this thing to a triangle in a trivial way. So we glue on these gray triangles where, so with this, these labels turns out that if you have a string of zeros and ones and then you have another label on the other side say beta in this case then that forces this other edge to match beta. So gluing these on just forces these to be the same and it preserves this inner parallelogram so we can just glue those on and chop them off as much as we want. And so that gives a bijection between these puzzles and ones like these. So, and then we have a geometric way of thinking of this as a ordinary Shubert-Taco's problem. Okay, and then the next thing to note is that if you rotate this parallelogram puzzle 180 degrees then you get another one of the same sort. So that says that this rotation gives us a bijection from parallelogram puzzles with labels going around this way and just the rotated version. So that is silly from the puzzle perspective but we wanna ask about the geometric meaning. So let's put those two bijections we just saw together. So we have this bijection with gluing on these triangles so we follow that up. Here we can rotate the parallelogram 180 degrees and then we come back down and complete that again. So now we have this triangle here with alpha and beta on there and then here they're swapped and same with lambda and mu. And so this is making this equality is claiming something about something geometric. And so the first question I wanted to answer was why is that? So how can we understand this geometrically or prove it? So when we did that, we ended up finding a stronger symmetry actually which is a combinatorial statement which we don't understand combinatorially. So that says that we can just swap two of the opposite edge labels instead of both pairs at the same time which would be the rotation. So if we just swap alpha and beta then we get the same number of puzzles still. And then this picture kind of is suggesting how this was done which was that we had to sneak through this triangle version in order to prove that equality and we don't know how to just directly combinatorially prove this. And then by the way, this also is true for if we allow some of these different special pieces which compute different homology theories. Okay, so we can also swap lambda and mu. So here's just a glimpse of a cute little puzzle-based way to do this which is where you commute sides of the triangles because puzzles have commutativity because the homology classes commute. So you commute these sides triangle, it turns into this and then you can fix this edge down here and then again you can commute the sides of this triangle so you get the same number and then you just follow it around. That's, yeah, so that's just a brief glimpse of that and it doesn't work for equivariant because that piece has no rotations. So, okay, and then I'm just gonna probably place through this really fast. So it turns out that this yellow bit, this yellow parallelogram can only have at most one filling and then it does have a filling. If and only if these fixed labels down here, eta and theta fulfill a certain condition in relation to each other. So we sum and that is, you know, the theta choice has to be unique to eta so we call them eta and eta prime and we sum over choices of eta and then it turns out this number of these is sum of products of Littlewood Richardson numbers where one of them is the number of puzzles in this green triangle and the other is the number of puzzles in this. So it's kind of nice that we can write as a product of Littlewood Richardson numbers and then here's the just a glance of the geometric version of that. Okay, and then we have this stronger thing coming up where, so we have these puzzles as I mentioned that with this special piece that computes the structure of constants and the equivariant homology, T equivariant homology of the cross monion. And so those constants live in this polynomial ring in n variables over z and it's just, it's a richer version of ordinary homology. So if you prove something, we prove this here then it's also proved with the ordinary. And these compute these polynomial structure constants while each of these special pieces contributes a weight, yj minus yi, which is given by where it's located in the puzzle. And then to get the weight of the whole puzzle you take the product of all the weights of the pieces and then the structure constant is the sum of all the weights of all the puzzles that you can get with those edge labels. Okay, and then here is this result about swapping edges of equivariant parallelograms. So if we take the same exact setup and now we're allowing the special equivariant on this pieces, then what happens to the structure constant if we swap alpha and beta. So it turns out that... I've minutes left. Okay, it turns out that what that does is you, it's a permutation of the variables in the structure constant. And so for, let's look at the alpha and beta case. So if you are swapping alpha and beta you look at, here's this permutation that will act on the structure constant. So what it does is reverses the first A variables, the first Y1 through YA and leaves the others alone. And then, yeah, so that is this here is saying that if you swap alpha and beta you effect on the structure constant is just you're reversing Y1 through YA and then leaving the rest of them alone. And then swapping lambda and U is something very similar but you're reversing the last, quarter of the last variables. Okay, and then this is just a very brief flash of what the gene of the group is like where it's you pull the problem back to this product of smaller gross monions. So, and then we also have this take it a little bit further and we find that the actually the number of these equivalent parallelogram puzzles is the same as when you swap. And that is not automatic. So when these equivalent puzzles sometimes you can have even the same structure constant but you will have a different number of puzzles say for a commuted version and then they just add up to the same polynomial. So this is not a given. So the proof of that is simple though. And anyway, so this bijection is definitely not understood combinatorially at this point. And then this is just a brief flash of like an example. So this is some of these puzzles, the set of all of them and then you swap lambda and then you get these. So still glimpse and finally hexagons. So we can ask, we can generalize some of this stuff to questions about hexagons because a parallelogram is a degenerate hexagon for one thing. So we can do this thing where we complete that to a triangle in a very similar way. And then we can, we also asked this question about 180 degree rotation yielding a bijection between these. And that was a weirder problem because this is, it can be a bigger triangle than this. So a different cross-monion and everything. So I did prove that for ordinary homology at least. And then we also find the similar edge swapping result where hexagons with like symmetric edge content. So there was these two cases that are nice where the first one is the opposite edges have the same number of ones and zeros. The other one is you have this like three ways, symmetry of the same content. And then for each of those, you can swap between those symmetric edges and end up with the same number of puzzles. So... Two minutes left. Okay. So yeah, this is the last slide. So this is just further questions that I'm interested in. So it's, we want to extend these results potentially to other homology theories. So do these, can we swap edges and get the same number of puzzles? Or what does that even mean? What is the right question to ask for some of these other ones? So like there's the K theory version and we have these puzzles with an upside, the right side up 10, 10, 10 that computes the structure sheaf basis and the upside down 10, 10, 10s that compute the ideal sheaf basis and the, yeah, the parallelogram symmetry kind of implies something interesting with that. I don't think I would like to understand the geometry of this better. And then, you know, echo variant hexagons, I haven't really felt much into that. I'm not sure what the right question to really ask would be right now. And then lastly, there's these segue shorts McPherson classes which have both the 10 right side up and upside down 10, 10, 10s. And so I'm, yeah, I'm interested in seeing what is the deal is with that. They have any symmetries and what that might mean. Yeah, so that's it. Thanks very, very nice talk. Thank you. And I guess there might be time for questions after all the talk.


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every villain has an origin story every villain doesn't start off evil well at least most of them don't when you check any type of story any type of literature any type of movie that has an antagonist usually the villain has a pretty cool upbringing or maybe their character is lighthearted a good person and then maybe their circumstances changes them maybe they get bullied maybe they lose a loved one and then they go rogue it happens all the time and in sometimes that villain like reveals why they know why they think the way they think why they act the way they act and you can sometimes empathize with them just a little bit you still know they ain't up to no good but for that split second of them explaining why they are what why they've become what they've become you care just a little bit that is what MJF did tonight and we got to talk about it what's good yeah it's your boy Ross back again with another video so I know I didn't get to stream AEW Dynamite tonight sorry y'all I've been swamped with videos for the main page and I'm trying to get some content ready for you guys on my personal page and I was just tired man I've just been running at a thousand miles an hour so I literally just needed some time to recharge my batteries but I woke up went to Twitter and I saw some of you guys talking about this amazing promo by MJF so I had to check it out on YouTube and one of the comments that's really struck me with me and it kind of goes into my intro in this video is usually every great villain doesn't start off as a villain they usually turn into that because of circumstances and MJF pretty much revealed why his character is what it is now why he is a heel and I love it I will say this MJF CM Punk feud is fan-tastic in a sense of they are telling stories with their promos alone that have you wanting to see their match and it's not just you know they add layers to it and this is why I can appreciate what they are doing right now because MJF won the first match and I'll use one loosely here he cheated as a heel should and they're setting up the second match to be even more intense she can't wait to check out but they threw a curveball here last week CM Punk showed a photo a real photo of MJF meeting him at like a fan meeting greet when he was a kid MJF comes out has nothing to say one of the I think the first time ever in AEW history he has nothing to say he comes out this week with a heartfelt message like he's breaking out in tears he's selling this he is selling this like no one else should be able to he's selling this promo segment and it comes off genuine it's like he breaks his gimmick his heel like gimmick to show you why he acts the way he acts why is he the way that he is and I love this because the crowd is trying to shit on obviously he's the heel they don't care but by the end of this whole promo segment he had a lot of people in their chair there was still some booze but he had a lot of people in there giving him a damn near standing ovation because of what he was saying in the ring and how CM Punk was truly his hero and you believe that talking about him joining the football team and getting picked on football team throwing quarters at him feel like he didn't belong feel like he didn't fit in I think all of us not all of us maybe some of you guys went to school y'all were the cool kids I wasn't necessarily a cool kid people knew who I was because of people I associated with but I wasn't just Mr. popular you know I'm saying I didn't just have all the girls back then you know so and you know I got made fun of as well I got picked on the bullied as well you know I'm saying a lot of kids go through that and he was telling us about that really diving into why he is the way he is and when CM Punk left wrestling that's when he was like no I don't want to be the best in the world like CM Punk I want to be the best in the world in spite of CM Punk because he left us when we needed him the most and I love that this is great bro this is so good because he's coming off as a baby facing his promo because he's opening up but he's still a piece of shit he'll don't get it twisted and CM Punk comes outside comes out to the ring confronts him no microphone which I like no microphone came out and was like is this true is this real this is how you really felt and I like that and you have to say nothing it's and it's it's giving me the vibes of him being able to open up about his how he really feels about CM Punk and why he is the way he is but it's a way to lure CM Punk in with his kindness with his empathy and I love that I love that I do think this is a nice little work I don't think he's turning face but I do think this is a way to storyline wise get CM Punk to let his guard down just a little bit but everything he said probably comes off storyline wise real and I love that this is how you set up a match another match with somebody there's a lot of times you see and I have to use WWE's example because they are they are really they they do this a lot we'll have a nice segment a lice lead up to two people in their very first match and you know you want to continue it you know you want to continue the feud it's usually never one and done thing so we get the second match but the feud has an intensified like you would think it should it's kind of formulaic it doesn't come off as a next level in their feud it just comes off leveled out the same way like their first match here we started off here at a pretty decent level and now we're at this point before the match and I'm we're not sure how how high this this feud this rivalry will go and in my opinion some of you may disagree even before this promo I did think MJF should still win this match but after this promo for sure he should definitely win this match and he should be at some point this year the AEW champion bro I know some people want Adam Cole to maybe the throne Adam hanging main page but I don't think it's time for Adam Cole Adam Cole will have his time for sure I just don't think it's time because of all the other stuff he has potentially going on with the you know young bucks his crew like there's a lot of a lot of decision-making he has to make so I don't think it's his time yet I don't I think if anyone drops the belt to anybody it gotta be MJF is this is his time this is his year and see him punk can definitely help him become that star in the company cuz I'm gonna be honest with you see him punk is not there to be the top guy he's not no if he was to come back to be the top guy it kind of he kind of plays into what he didn't like about WWE coming back to be the top guy to ultimately take someone's potential you know spot he wants to be able to enhance and elevate all these great talent and he's doing that with MJF we are new MJF is very talented on the mic and he's pretty good in the ring my opinion he has the wrestling psychology down he has a down pat but see him punk is here to help other talent get over so you can have your you know not even you're a new senior punk you can have your MJFs you can have your Darby Allens these household names in the industry where fans can get behind them just like they did with see him punk and I love that I think see him punk well she'll lose this match put him over and that would ultimately catapult storyline wise in my mind and in the fans eyes it should catapult him to be potentially being a number one contender in my opinion and see him punk and take a loss it's see him punk bro he's not out here getting squashed he can take a loss because people are gonna love the fact that see him punk is still wrestling in 2022 people didn't think he would ever come back to wrestling he's wrestling he will always be cheered he can always take a loss here and MJF did his thing bro MJF did his thing on this segment he is arguably the best heel in the business still think he's a heel and he is the best promo damn near in the business I ain't gonna lie to you bro I can listen to MJF being asshole all day because he knows he knows what he's doing he knows exactly what he's doing man so comment down below let me know did you guys watch this segment did you guys enjoy this segment did you guys actually start to feel just a little bit of sympathy from what MJF was saying just a little bit and are you guys looking forward to their future match and who do you guys think should win do you think it should be see him punk get his win back or do you think MJF should get a win over see him punk again and potentially be the new AEW champion sometime this year let me know man it's I don't talk about AEW outside of the streams and sometimes interaction with you guys on social media and I make make videos like this but when I make a video like this that's how you know I really enjoyed what I saw and I want you guys to be able to get my thoughts and opinions on what I saw so death best believe I'm looking forward to where they take this feud because this is this is some good some good entertaining shit man but I appreciate all love and support roll it to 70k I appreciate y'all kicking me see y'all next week