Hey Virgo, welcome back to House of Virgo. This is Kelly. This is going to be part two from yesterday's reading. Yesterday was part one. Today's part two. If you missed yesterday's part one reading, there's a link in the description box below that you will be able to go back and listen to that. Just be mindful. Some of the messages may or may not resonate for you. Do put a like and a share or subscribe to the channel if you enjoy my readings. And as always, this is a channel for Virgo, Sun, Moon, Rising, and Venus. But all signs are welcome. Okay. So Virgo, yesterday what we saw was you had the King of Cups show up in the reading that also had the Queen of Swords and yourself. So it looked to me what the messages I received yesterday is that the Queen of Swords is trying to talk to the King of Cups who has his back towards her while looking at you. And you, your energy has been, you're not even focusing on that. Okay? You are, it looks like you're not seeing anything going on with anyone except maybe looking at your own reflection and self-reflecting, self-care, healing, that kind of energy. So I wanted to get into part two to kind of see what the energy was here going on because I feel like the King of Cups, Virgo, is and you know, as usual, generally the King of Cups will be a water sign, but it could be any sign for you who your person is. They just are showing up, no matter what sign they are, someone who is very supportive and kind. Okay? And I felt like the King of Cups has his back towards the Queen of Swords. It's not interested in engaging much more attracted to your energy at this time. Okay. So let's get into, okay, the rest of the reading here. All right. So we're starting with the Queen of Swords. Again, I felt like the Queen of Swords is trying to get information, talk to this King of Cups. There's a lot of stress that is happening now in part two. Okay. The Queen of Swords is sitting with a tremendous amount of anxiety, stress and nightmares, frustration, trying to get her anxiety relieved by the King of Cups who, like I said, has their back towards the Queen of Swords. And the King of Cups basically was telling the Queen of Swords, I'm sorry, I'm just not interested. Or I am focused on somebody new who is Virgo. This could be somebody that you meet, you haven't met, you meet them, and maybe they have another person and they've changed your mind and they're more intrigued or attracted to you or was someone that you already knew. Either way, Queen of Swords is like really like struggling here. It's a complete nightmare. There's a lot of grief on the Queen of Swords side because as I said, you can see the King of Cups is not even paying any mind or attention to that person. Let's see what else is going on with this Queen of Swords. Okay. So she's going to give up. That's what I feel like. She's going to distance herself and just get back into a peaceful place and try to resolve her anxiety as well. I feel like she's not going to pursue this King of Cups going forward. And I feel like she's going to withdraw or go silent or give the silent treatment to the King of Cups or just block the King of Cups. Okay. And move on. Now the King of Cups is, wow, Virgo, very sincere towards you. Very. Like I said, male and female energy, it just depends on your situation. That's Six of Cups. Some of you, this is like a reunion. They want a reunion with you. I'm very nostalgic about you. I want to bring some type of a gift. It's innocence, sincerity and kindness with this King of Cups here towards you.  What are you talking about? What are you talking about, Willis? Virgo's like not even paying attention. So let's see how things are going to play out between the King of Cups and the Queen of Pentacles. Okay. Virgo, it's like, I feel like for you, your thing is, yeah, you're going to have to change.  And what we saw in yesterday was the King of Cups had that Knight of Swords and was going back to the Queen of Swords and saying, I'm changing. So I feel like Virgo, your only communication to have going forward with this King of Cups is, don't bring me flowers. Don't bring me candy. Don't bring me wine. Don't give me promises. I want to see action and change. That's what the Knight of Swords is. Basically, here's a knife. You're going to have to cut your leg off below the knee and hand it to me and be willing to limp for the rest of my life for me. Now, Virgo, is that a little demanding and vain? Some may say, but I feel like there's a purpose behind this because for some reason, Virgo's just like not taking apologies without action, really. And Virgo, it just feels more like you're action oriented as far as going forward in your life, right? And if somebody comes in and apologizes to you, it's kind of like, all right, thanks. But if they don't try to make amends, they go hand in hand because apologies are just without action or without amends are just manipulative words. So I don't know why people think and apology. Oh, if you just say, you're sorry, everything's fine. No, it's not. That's only half the game, half the battle. Apologies have to come with amends and making change. Otherwise, it's lip service. So if you don't know that, now you know. And if you don't believe me or you don't agree with that, click off. Because that's how that works, right? There's consequences. If you steal from the store, sure, you can say you're sorry, but you're going to have to pay restitution and give that money back. Or you're going to have to do some community service. You're going to have to do something. That's just the law. It's an ethical thing as well. But that's just the law. And I feel like Virgo, the law of life, it's a moral thing as well. And I feel like Virgo, that's your, that's your thing. I don't see it being angry. It's just kind of like, you know, that's, you're drawing a line in the sand here. That's it. So somebody could come in with all the sweetness and all that, you know, gifts and, you know, want to reconnect in sincerity. And does that mean that person doesn't mean well? I'm not seeing anything manipulative here. But I feel like, you know, Virgo, you're more along the lines of like, you know, your actions are going to speak louder than your words. But they also have to align. Okay. So let's see what the King of Cups is going to do here. Because Virgo, I feel like the King of Cups is definitely going to try to come in and reconcile and try like to come in with kindness, maybe an apology of some sort. And I feel like you're, you're basically, I don't know, this, you know, the Knight of Swords can be a fast talking card. So you might just be real quick to nip it in the bud and have some type of communication with this person and say, I'm sorry, you're going to have to keep that gift or you'll take it and say thanks. But, you know, hey, you're definitely going to communicate this to this person when there is some type of reconciliation here. Okay. Here's the lovers again. I hate that. I have to cover this card up. You know, if somebody I'm going to, I'm going to ask that somebody email me some cute little bikini stickers or maybe some fig leaves. And I'll download it, cut them out and put them on there. Okay. Hold on. All right. So Virgo, the King of Cups is definitely coming in to make things right. Have a win-win solution with you and love. You know, you have two sixes and here sixes are about harmony, harmony, harmonization, harmony, appreciation, self-sacrifice, loyalty and responsibility, taking responsibility. Justice, you're given justice and this justice is all about consequences as well. You know, this is like going before the courts. Basically, Virgo's like, you're going to have to come in, you're going to have to be tried in the court of love. This is my court and I am the judge, the judge and the jury. And Virgo, I just kind of feel like your energy is like, you're, this is what you want. Nothing less than that. And I feel like this person will sit in front of the judge, you know, and do whatever they can to, this the law, this is karma as well. Justice is karma. So it's definitely like being honest and open, taking the stand and giving the truth, the whole truth and nothing but the truth to you, especially about how they feel about the lovers. They had the lovers yesterday, you being the one and making a sacrifice to whatever you want. Virgo, just speak up for yourself and just say it. You know, there's no guarantee. Just be mindful when I say the message may or may not be for you. It just basically means that there's no guarantee right now as you're listening to this, that this is going to go your way. There is a chance though. I think the most important message for you, Virgo, is to stand your ground and do not let yourself be a key word here. Do not let yourself be lulled to sleep or cast under some type of spell. Because nobody can take advantage of you unless you allow that. So that requires you to have your wits about you. But justice is good, especially with the lovers coming from the King of Cups. And I feel like the King of Cups, you know what you guys? Even if this is somebody new and you don't know this person, and that's going to be a small percentage of you. The story, the message still rings true here for you that if you're going to meet someone new, Virgo, part and what we saw yesterday was you have been leveling up and respecting yourself and setting boundaries for yourself. You don't have to set boundaries for other people. You just have to set them for yourself and then they'll get the message. You don't have to worry about setting a particular boundary for person A and then a particular boundary for person B. No, you just set boundaries for yourself, what you will and will not do. Then that just covers the whole thing. Then you don't have to worry about every single person. You just have to be clear with yourself. So you have the six of cups, six, six and six. This is crazy. So you have the six of pentacles here. So Virgo, you're definitely someone who is a very generous person, very giving, and you're more than willing to help offer advice or show the way. But I just feel you haven't lost that hospitality aspect of yourself, which is very attractive about you. You're willing to help people who are in need of help, but you're not willing to help people who are able to do things on their own. In that case, you might just point the way. But I see you here being very generous and fair going forward and to yourself. I feel like your message here for yourself, I feel like you're telling yourself is, I will love you, of course, but I love myself first unconditionally with a romantic partner. In a romantic partnership, your unconditional love should always be for yourself and of course your children. But I don't believe for a romantic partner that either love should be unconditional for each other because it's not the same type of love that you have for a baby or a child. You know what I mean? You don't want to have that kind of love for a grown person because they're not a child. They're not a baby. In a romantic partnership, we come together as a power couple. We work and we help each other be better, but we don't put ourselves in a position to harm ourselves in order to keep this person safe in the sense from themselves. You know what I mean? So you do have that generosity about you, but I feel like you know where to draw the line with people. You're more than willing to help this person get themselves to a better place, but you're not going to do it for them and they're going to have to make these changes on their own. You're more than willing to assist, but they got to walk the walk, right? And I see you letting this person do that. So let me see. I feel like Virgo, okay, between the two of you, there's a lot of things to work out and to figure out yes, there will be a lot of contemplation and consideration is going to come about in as far as how much each of you are able to invest in this relationship. Okay, and that's going to have to be kind of figured out first before it's a go. Something, if this is a past person, something caused a breakup and that needs to be reconsidered and dealt with. If this is somebody new, this could be along the lines of having some obstacles that you're both going to have to figure out between the two of you. So whether that be distance, whether that be a financial thing, maybe you work together and there's an issue with that, or there could be many obstacles that are going to have to be like you both kind of sit down and take a look at whatever the issues are and say, okay, we're going to have to overcome this. How are we going to tackle this? How are we going to do this? So, but I feel like overall, that's mostly coming from you. Okay, mostly coming from you. You've got the six of pentacles to the seven, which tells me that in order for something to be successful, okay, or this relationship from your side to be successful, there has to be a giving and receiving and a sacrifice, of course. And I kind of feel like for you, Virgo, I think you would probably be more inclined to say, well, if this person wants to try to make something happen here with me, I'll consider it, but I just don't see you jumping the gun with anybody, really. I see you being very thoughtful with yourself and considering and being very discerning. Okay, so tell me more about this King of Cups here and the energy towards Virgo, because I feel like with the lovers, I mean, yeah, there's just this energy of this person. You know, they want to heal any discord or arguments. They have a lot of regrets about something in their life, whether that was something with you or just in general, if you don't know them. I feel like this person is really ready to come into union, like a secret union here, and if not fully ready, it is what they desire with you. Okay, because you did have engagement, remember?  Their passion is off the charts for you.  I mean, oh, Virgo, this is like being passionate and on fire for the truth for making things right as well. Let me see what else is going on with this person. There you go. You two are going to end up in a happily ever after. Yeah, yeah, Virgo. I feel like, hold on a minute, and I definitely feel like that's something you want to just get into this joyous, happy outcome here. This card flipped out real quick. Yeah, so it's fleeting again, another fleeting card, but I am going to recognize it. I feel like for you, it's like, okay, I see all your ideas and your plans, but you know, and they're tangible and attainable, but I need to see the actual work and action. Yeah, I feel like Virgo, your energy is, your intentions, your thought process in life is to other people in general, it's like your intentions are great, but until they're actually action-oriented or actualized, then they're just intentions and so they're just ideologies or dreams or thoughts. You know, there has to be more than just good intentions, right? Because we could all have good intentions. We could all not want to steal candy from a baby, but when Jones are for candy and we take it, our intentions don't mean, it doesn't mean anything. Do you know what I mean? So I feel like for you, it's more along the lines of like intentions. I don't care how good they are. I got to see some real, something tangible, something real, that kind of thing, Virgo. I don't know, I just kind of feel like you're just raising the bar here and that bar that you're raising with someone else is creating this stronger attraction to you because you're basically giving the message, I am high value and I am high worth and I see that in myself and I want people to also see that as well and not intentionally putting it out there, but just owning it and I feel like being in that position for yourself, Virgo, is drawing someone towards you here, definitely. In such a way that this person just kind of feels like, oh, this is, Virgo's the one. Virgo is the person for me, like everything I've been looking for, everything I wanted. Now if this is a past person and you were with them before and they didn't see that, Virgo, let me just give the message straight up to you, okay? And I'm not trying to victim blame or do anything like that here, Virgo, but if you weren't living in your best self, living your best, let me ask you a question. What's different now with you than before? Has your behavior changed? Have you become more positive?  Have you begun to respect yourself, okay? Because sometimes people don't respect us because we're not respecting ourselves. Sometimes people could be attracted to us physically but not want to get into a relationship with us because they don't feel comfortable or safe with us, right? Now if someone is doing that and they're saying, now I'm changing my mind, it's no different than you being with someone and saying, you know what? You're not respecting our relationship or me because you're not respecting yourself and even though I'm attracted to you, I don't see us being in a long-term thing because your behavior is not working, it just doesn't fit in alignment. So let's not sit in place, blame everywhere. I think it's really important.  Okay, it's not to do it with the intent to snag somebody. It's doing it with the intent to live your best life regardless of what other people say or do when you keep the focus on yourself and your self-respect that you're attracting and I feel like that's what's going on. So if you were with someone in the past and they made poor decisions, then don't own them. Those decisions belong to that person. If this person is beginning to see your abundance and your ability to turn to them and look at them and say, you know, I don't need you. I did before. I do need you in romantic ways, of course, but I don't need you to the sense where I can't survive. I want you because for whatever various reasons, but I don't need to have you and if you leave, I'll be sad, but I'll be okay. And that's the energy I see you projecting here, okay? And that's very attractive to someone because it says what you want. You don't need me to keep you in a house and riches and furs and gold and all that stuff. You're willing to put in the effort with yourself, then you would be willing to put in the effort with us, right? That's the energy I see you projecting and I see that person looking at you saying, wow, Virgo is different or Virgo has changed now or Virgo doesn't want me the way they used to or doesn't need me like that anymore. And that also causes this person to change, okay? Because the old days that we can't change other people, but when we change, other people change as well. And so if we change for the better, somebody that we're close with is going to change for the better or they're going to change and walk out the door because they can't do it. It's one of the other. There's no need to chase. Just chase yourself, okay? So I like what's happened in here. Perhaps this person, this king of cups, saw something in a queen of swords in the past that they thought was better than what they saw in you. And maybe at the time it was, possibly, okay? But it could have been superficial. Look, it's all swords with this person. So I think if anything, there was a lot of communication that might have gone back and forth between the king of cups and the queen of swords. But I don't see anything romantic coming. I feel like the king of cups is all about romance, okay? There's all the love, okay? And the passion and truth. And that's what the king of cups is all about.  It's just kind of bitchiness, male or female energy. And Virgo, of course, is all about show me the money and being very grounded. Look at all the pentacles, okay? And the sword there to be also about truth, you know? I feel like that's the difference between the king of cups and the queen of pentacles here is queen of pentacles is all about being grounded in reality and the king of cups is all about the love and the emotions, right? There's a little bit of a balancing out going on here, for sure. All right, let's see the outcome here, okay? So the king of cups has some mental insecurities or issues to deal with here, okay? Definitely. That could have been a past person that they were with that might have been very abusive that it's had an effect on them. The heart but on their head, okay? And Virgo, you're in tune and you've got the high priestess. I think you already know or you will pick up going forward that there's some type of isolation or mental restriction with this king of cups. The king of cups kind of sits here with a little bit of this victimhood stuff. They might have been a victim of something in their past and they might still be kind of, this is the issue that the king of cups needs to work through within themselves, okay? If they were mentally abused or, so the Aida swords is always a card of someone who's like a self-sabotage or they imprison themselves, they isolate and that can be energy of a water sign. They could be very moody, they go within, they withdraw a lot and that doesn't make them bad, it's just how they cope with things, okay? So there's a lot of like this king of cups tends to isolate themselves from others, okay? It doesn't mean they don't love those people but they might be a bit of an introvert as well. Virgo with you, you've got the high priestess, right? So your advice here is to really get into alignment with your intuition, okay? I feel like keep your cards close to your chest going forward, okay? That high priestess, you know, it's all about like trusting yourself. The secrets are the things that you understand intuitively that you don't share with others. Keep them as secret for yourself and begin to start trusting yourself going forward. I like this reading, it's very detailed but I do feel like, let me see, yeah, there you go for the outcome. I feel like you guys are going to come into balance with each other. I feel like the relationship going forward will be very reciprocal with the two of pentacles where there's going to be a lot of, one minute you're going to need the king of cups and then the next minute the king of cups is going to be needing you. There's going to be an equality and two is always about balance, duality, choices, partnership, diversity. There is some diversity between you and this person for sure. I don't know specifically what that would mean but there is, you guys, there's maybe some neurodiversity going on in this relationship. Someone could be on the autism spectrum. It could be you, it could be that person, it could be both of you or one of you. So there's going to be a need to understand and balance things out in this relationship. I do feel though that you two will figure it out. You'll be able to keep the balls juggling up in the air and you'll be able to take turns and understand each other and figure things out between the two of you which is very promising. But as far as like that engagement and everything that we saw yesterday, that is definitely the start of securing the relationship even more. And I feel like if, you know, I don't really know if you're, I did feel like that Queen of Swords was going to reach out to you as well, possibly in yesterday's reading and I implore you to go back and listen to that. I want to tell you that when I told you about the Queen of Swords when I started this relationship, when we started this reading that this Queen of Swords was going to give up on this relationship and if the Queen of Swords reaches out to you to try to get clarity, I felt like you really weren't going to give anything or say anything. You're keeping your cards held close to your chest. You're keeping things private and in secret. Your relationship with other people is nobody else's business. So, yeah, I feel like as far as any kind of communication or confrontation, I feel like this Queen of Swords is going to back off, okay? If you don't give them anything, if you don't get in the ring, a boxer, if there's a boxer in the ring and you don't get in to that ring as an opponent, that boxer is going to be doing nothing but shadow boxing and eventually they're going to quit because there's no contender there. So, I feel like for you, not getting involved, not saying anything, just keeping it to yourself and not giving this person any, feeding it any energy, they're going to just dip away. They're going to just dip away. What's most important to you going forward is forgiving and learning healing, releasing and healing the past so you can have more love in your present moments. I do not necessarily know. I feel like if you want to... All right, so I'm going to say this for both of you.  I'm going to tell you that, okay? But you both have to want it. As far as somebody new coming in, I feel like Virgo for you, the overall message is set your boundaries for yourself. Do things differently than you used to do. Whatever got you in a situation up to this point where something didn't work out, I feel like you need to do the opposite of what you were doing before. And I can't tell you exactly what that is, because I really don't know what you were doing before. That's up to you to look at and say to yourself, okay, well, I did A, B and C and how did that... What were the results of that and how did that work? If it didn't work, you need to try something else for yourself that is going to work. If you've done everything in the alphabet, A through Z, and it still didn't work, then that's time to let it go and let someone else start sorting that stuff out, okay? All right, so let me see what else you have here. Trust and true love, beautiful. Situation is calling for you to have faith, and this is the romance of a lifetime, Virgo. Wow, I keep getting all these ding messages, and then you've got to give your relationship a chance here. Work on the partnership. Okay, Virgo, and that's advice, and of course, as always, you can take advice, or you don't have to take advice. You do whatever fits you the best. Hold on, you guys. Okay, so let's get into a couple other things here. I'm probably not going to go for an hour long. Let's see what your messages of love are here. And so I'm uploading this Tuesday evening for Wednesday. On Thursday, I am just going to keep the format of doing general messages. So your everyday stuff, health, career, your everyday things apart from romance, but if romance shows up, then, yeah, we will definitely be taking a look at that as well, okay? But the focus will be the dailies, the everyday stuff, not love, okay? So let's see, messages of love for Virgo. So you have release, radical acceptance, and surrender can help you let go. And that's what I was talking about, doing the opposite of what you always did.  Sometimes the most radical thing you can do is release and let go, oh my gosh, all these notifications. Sometimes the most radical thing you can do is release.  And that might be opposite of what someone here is used to doing, okay? I know not all of you.   While others of you are like, no, it's gonna take me a while, okay? Strictly sexual. This connection is passionate, but not enduring, okay? Yeah, so you might have had a relationship with someone in the past. It could be this person or somebody else where that was the relationship. That was the gist of it. Passion is not sustainable long term. It does need breaks, okay? If you were in a relationship like that in the past, ask yourself how did you get into that, right? Because unless you were actually forced into that, that's a different reading. But most likely you agreed to it, and it might not have been what was the best decision at the time for you, but maybe at the time you thought it was. But perhaps it didn't pan out into something. So now, Virgo, some of you might be looking at yourselves and saying, you know what, I'm not doing those kinds of relationships, because that doesn't fulfill my long term end goal here, is to have someone a companion for my life.  And if that's something that you were okay with, then, yeah, fine. You're okay with that. Some of you weren't. And some of you, maybe it ran its course for you, and some of you might just be deciding, you know, that's what that was. And it doesn't mean somebody didn't care for me, but it's not going to work for me anymore. Don't worry so much about the feelings. Worry about more so what's going to work for you in the long run. So you've got stability. So this relationship can stand the test of time, and Virgo, this new person or this person showing back up in your life, yeah, there's a huge potential for that. Like I said, if this was a strictly sexual relationship in the past, sometimes, you know, it's totally fine. It's never too late to raise your standards and your desires. Sometimes people change their mind about things. And if that's the case, that's what the relationship was in the past. And then you, some person was like, you know, I think I actually want more out of this. And another person might be like, no, it's not what I want. Well, okay, that's okay. Separate and move forward. Okay, but perhaps for some of you, this King of Cups is coming to realize with that justice card that they do want something, a relationship of stability and something that can go the distance. And I do kind of feel like that's what that is, whether that's a past person or new. You just had marriage down here, right? So that's definitely the energy that was playing up in yesterday's reading. The relationship is more moving towards a secret union. And the King of Cups is going to move it in that direction. And all you have to do is, you know, keep your, set your boundaries for yourself. That's all you have to do here. Okay, yeah, look, King Cups feels regret. They feel remorse, regret for the way they treated you.  I don't know. They might have regret because of what they were dealing with in the past. If they went back to that person, baby mama, baby daddy, or if they, you know, decided they wanted to date someone else while you were in separation and that didn't pan out and kind of made you look like gold. Or if you never met this person and they were dating someone and that person was just giving nothing, nothing but grief and anxiety and that kind of energy or giving them the silent treatment or being kind of nasty mouth. I feel like the King of Cups regrets that they wasted so much time. And I don't necessarily feel that they feel that way until they meet you. It's like, where have you been on my life? I wish I met you when we were younger. I've wasted so much time on the wrong people and here's Virgo who's like loving themselves. Here's Virgo who's caring for themselves, who's independent again with that nine of pentacles, you know, self-sufficient, solo, enjoying their lives and really happy and joyous and free. That is attractive. Okay Virgo. So I definitely feel like you've got something good coming toward you in April and I think my friend Lucy told me that April 21st is the start of the Mercury Retrograde. So the pre-shadow period, perhaps like a week before that, so the week after, the week of Easter, some stuff is starting, we'll be starting to brew here, Virgo, okay? So I just feel like as far as your reading goes, I'm going to leave this here because I'm going to just skip the dice and the names and randoms and the charms because you had that yesterday, I should have probably saved that for today but it's okay, I'm going to leave it. I feel like if you are in that position where, you know, there's a third party or somebody else, I feel like that situation has ended or is ending and if that's a concern for you, yeah, that third party, they have somebody else or they will have somebody else very soon or they already do that they're making plans but they just could not, the thing is, and I just want to leave this one caveat, much like a Taurus, the King of Cups doesn't really budge if they're being forced to do something or if someone is trying to force something out of them. The King of Cups is either not going to budge or they're going to find a way to get that person who's trying to make them do something soften and ease up off of them. That's how the King of Cups works, right? Because they're very knowledgeable about themselves because water signs are often turned inwards, they're often self-effacing and a lot of times people can think that they're extremely selfish which they are, which is not necessarily a bad thing because to be selfish means putting self first, okay? So in a lot of ways I feel like the King of Cups is basically like you can't get blood from a stone, okay? That's the energy of the King of Cups and the King of Cups, which I can't pick my card up, oh, the King of Cups sits on that stone there, okay? With all that water, tumultuous water which, you know, the cup of blood, you know, the King of Cups just doesn't budge unless the King of Cups feels something. They have to feel it in order to move it and to move something forward and the King of Cups is not feeling anything with this past person. I just... I just have to tell you that. That's the message I'm getting, okay? So we're gonna leave that there for you and I'm just looking to see the full moon is on April 6th. Okay, so obviously Thursday I'm gonna do your... Thursday I'm gonna do your weekend reading but I'm gonna put in a full moon reading in there, I guess. One of the days, so expect it one day this week before the weekend and then we also have the Easter reading and yeah, so I'm gonna leave that there for you guys. I hope that you enjoyed this part two extended and I will see you all back here tomorrow. All right, take care. Bye.


So last time we heard, I told you how to construct iteratively left sheds elements using this Kronekalemma. So last time we had this Kronekalemma, which helped us to, essentially what it gave us is that bias pairing property, that the bias pairing property, so the non-degeneracy or the non-degeneracy of the Kronekalemma pairing at ideals implied somehow in dimension one implied the existence of left sheds elements. And then, well okay, so to close the loop we have to discuss how in general this bias pairing property, bias pairing is implied maybe by left sheds and could I mention one again. And this closes the loop and this is what I will, some of the first half will be spent on explaining that. Essentially what we discussed last time was how we get, some of our last time, last lecture, we discussed the bias pairing property for ideals of the form i sigma delta, where delta for sigma is sphere and delta I could I mention one sphere. And I want to essentially now go to general delta in a few small steps and then I want to argue how these things then really fit together, sum it up. And this somehow to give you at least a feeling for how the proof of the left sheds here works using this inductive principle. And then towards the end, so in the second half and tomorrow, I will go over another proof using transcendental theory, which does not really rely on an inductive principle but really just exploits some nice residue formulas. So let's go over this case of these ideals i sigma delta again. So last time we discussed this case i sigma delta, where delta was a could I mention one sphere. Today I want to verify this bias pairing property. And in the case where delta is general, I will argue that it's enough to consider the case where I consider a pair i sigma delta, which is just to remind you, this is the kernel of a map a of sigma to a of e. And here for sigma, and we will look at this in the case where sigma is 2k minus one sphere, over r field k, and e is a could I mention one manifold in sigma. And I will assume that such that e is k minus one acyclic over k, meaning that the homology of e vanishes up to dimension k minus one, and here is homology with k coefficients. And this is the case that I wanted to look at. And let me just remind you what am I trying to do. Well, I'm trying to show that if I look at the Poincare pairing in sigma to k, and I restrict it to this ideal, so this is the perfect pairing, and I want to restrict it to this ideal i sigma delta, then this pairing here should still be non-degenerate. So we want this to be non-degenerate. Well, you only have it in the middle, right? It's k minus one. Yeah. So it's strictly less than k minus one. Wait a second, so let me get to an example again. If k is two, then I'm looking at a three-dimensional sphere. The manifold is of dimension two, and I'm saying, ah, okay, you're right, up to dimension k minus two. Let me say it like that. Yeah, yeah. Yes, thank you, yes. Otherwise, it's just a sphere again. Is this connected or is it k zero? So h zero, it doesn't have to be connected, but I mean the case where we have a one-dimensional sphere is kind of trivial anyway. So the case that we're interested in is really going from k equals two upwards. So three-dimensional spheres are higher. So we don't have to assume connected, but with these assumptions, it will automatically be connected once we hit the interesting cases. But yeah, let's ignore the case of one sphere for now. All right, so we want to show that this is non-degenerate. Okay. And now let me first discuss how the criterion to show that this pairing doesn't degenerate. And once again, what we can look at is the case where, well, we can look at i sigma e, but now let us observe again that, so now it kind of, seems to be a mark becomes a little important. Just a second. This is a k sigma. Okay, so this is, so it's a middle pairing, the middle boundary. Yeah, yeah, so this is to a to k of sigma, which is just k. So e separates sigma into two components, into two components, right? So I have a component m and I have a component m bar. And again, I have that i, that these two components, they span my ideal and they are obviously orthogonal on each other because if I have a monomial supported here and a monomial supported here, then they multiply to zero because, well, they line different components, so they don't form a face with each other, all right? So they're obviously orthogonal on each other. So if I want to prove this, I restrict to proving it for, somehow we can restrict to showing non-degeneracy, non-degeneracy for i sigma m bar, okay? And here's the theorem. So i sigma bar, i sigma m bar satisfies the bias pairing property if and only if. Okay, so now what I do is I look at a k of e. Now, a k of e, this is, okay, so this is just the quotient corresponding to my hyper surface. Now, if you remember in this, when we discussed the partition complex, we remarked that there is a map of the cosmology into a k of e. So specifically we had a map from hk minus 1 of e with k coefficients to the d choose k to this ring, all right? So this came from the discussion of the partition complex in the context of Pankeriduality. And now, well, this here is, okay, this e is a sub-manifold of m, so I can write a map from hk of m, sorry, from hk minus 1 of m with k coefficients to, I can write down a map from hk minus 1 of m to hk minus 1 of e. And again, I carry with me this tensor product that is coming from the castle complex.  If and only if, let me call this map here star, and I can say star is an isomorphism. All right, the proof for this fact, it turns out to be just a little diagram chasing in the end. So what I do is, okay, so I write down a short exact sequence like this. So let me write down, okay, let me start with 0 up here and then try to make the arrows short enough so that I don't waste too much space. So I have ikme, which is defined for me as the kernel of a of ak of m to ak of e. All right, and this is a sub-complex, this is a surjection, that's all good and nice. And then into ak of m, well, again, I have my partition complex. So I have the map from hk minus 1 of m with k coefficients. Let me just not write down the k coefficients to the d choose k. All right, and what is the core kernel? Well, I mean, let me not describe it for now for the moment. Let me just call it b of m, bk of m for the moment. And now notice that our mystery map, right, this mystery map also appears again here. All right, so it remains to understand this map. So let's understand this a little. So this here I can think of as mapping, so this is included into i sigma m bar, which is isomorphic to a sigma m bar, which we discussed last time. And then, OK, so what is b of m, bk of m, let us discuss this. So b of m, this is a of m modulo. Well, now I look at the image of the direct sum over vertices in m, a star of the vertex in m. All right, so this was our partitioning map. In particular, this is just the kernel under the partitioning map. All right, so sorry, this is just the image under the partitioning map. In particular, this is the same as looking at a of sigma and modding out the annihilator of i sigma m bar. But as we discussed last time, having an injection from here to here, all right, this injection, all right, so now this is an injection from an ideal to the punctuality algebra to the string modulo, the annihilator of the ideal. So this is, the injection here is equivalent to the bias pairing property, bias pairing property. And in fact, it's an injection if and only if it is an isomorphism because, well, these spaces are Poincare duals. So this is an isomorphism if only if it's an injection. So now let's look at this. So this map here is an isomorphism. Okay, so if and only, okay, so this map here is an injection if and only if this map here is. But then these two spaces are of the same dimension. So if this map is also a surjection, this here will be an isomorphism. So these two spaces will be the same. In particular, I get this here is an isomorphism if and only if this here is an isomorphism if and only if this is an isomorphism. Which is exactly what we wanted, right? So isomorphism here is exactly the isomorphism here, which is exactly the isomorphism here. It's really just diagram chasing. So diagram chasing gives us a result, gives the isomorphism. All right, and that's it. So it's really just a diagram chasing. All right, okay, so now, okay, so now I want to, okay, so now I went to, I discussed hypersurfaces. And now I want to discuss general complexes, i delta, i. So how do we prove how to prove that i sigma delta satisfies the bias pairing property in general? So how do I do that? So I'm again in this case, okay, so I'm looking at a simplicial complex, right? A simplicial sphere and a subcomplex in it. So we are considering, so sigma is of dimension 2k minus 1 and delta subcomplex. So really, I can assume that delta is of dimension k minus 1. Why? Because I'm caring about this ideal, which is, all right, I'm caring about this ideal in degree k. So I'm really only caring about, I'm really only caring about the monomials up to degree k, I'm only caring about the faces of the simplicial complex up to cardinality k, which means that I'm only caring about the simplicial complex up to dimension k minus 1. So the idea now is construct hypersurface e containing delta such that a of sigma, sorry, a of e is isomorphic to a of delta and in particular that i sigma e is isomorphic to i sigma delta. And we will cheat a little, we will not achieve this in sigma itself, but we will achieve this in a subdivision of sigma, but we will see that this doesn't matter, that we do it in a subdivision. Okay, so let me explain that and I will mostly restrict to the case where sigma is a PL sphere just because I want to get to the transcendentality argument. So I will sketch the argument and this construction and why this is enough. And then I will kind of, this will be the, then I will summarize the, what we did and summarize the proof and then I will go to the transcendental proof. And so what, how do you do that? Ah, okay, so. Are you going to construct e and such that in the cycle? Yes, yes, I will, okay, so I will construct e such that it is a cyclic. So this here will be k minus 2 a cyclic, thank you. I will, yes, thank you. So, okay, so let me go over this construction. First of all, let me, let me note that the decomposition theorem implies something nice about some invariance property of this, of this, of the bias pairing property. Of the invariance property of the bias pairing property, there's too many properties. So an invariance of the bias pairing property under subdivisions. And for simplicity we will mostly assume, assume that sigma is actually a PL sphere. Because, well the construction is a little simpler in that case. Okay, so first observation, subdivisions of sigma outside that do not affect that do not affect delta preserve the bias pairing property. This, so subdivisions here, for us we will take stellar subdivisions. But it's essentially every, what you can take is essentially every simplicity map that preserves the fundamental class. I will give an example in a second. So for us, for us here stellar subdivisions, for us stellar subdivisions survives because I restrict it to the PL sphere case and I will just indicate what other marvelous things you can do if you want more general. And let me make a picture to illustrate this. So let's say we have our sum potential complex delta. All right, so this is delta and it sits inside our sphere sigma. This is delta and it sits inside sigma. And then, now perhaps for some, for some gods forsaken reason, we don't like sigma so much. For some reason it looks ugly somewhere and we want to refine it. So let's say here there's a triangle somewhere and we want to really look at the blow up. And so we want to take stellar subdivision here. So we have this and we want to replace this triangle by its stellar subdivision. All right? In any case, whenever you have a subdivision of simplicity complexes, you can induce a map. So from the complex before the subdivision to the sphere after the subdivision.  It depends on which point to take, which point of view you take. But if you think about it on the level of converse polynomial functions, it is obvious, right? Because a converse polynomial function before the sub-division will be converse polynomial after. Right? Somehow you just have more, right? You just have more space, but somehow you just don't have break points at the point of the subdivision. That's fine. Okay, and then if you think about it, what you can show is that, well, that the subdivision here is really the image under this map. It includes the converse polynomials before and after. So this is just the pullback map plus a component that really comes from the subdivision. So this is really the image of the geese in. So let me just call it G, not G, G. And this here is orthogonal under, this decomposition is orthogonal under the Poincare repairing. All right? And now let's think about it. Let's look at i sigma delta and its subdivision. So i sigma prime delta and i sigma delta. So i sigma delta really consists of all the monomials that, somehow this here, all monomials not in delta. What does this mean? Well, these are all the monomials outside of delta. And now if I, okay, so this year all the monomials in sigma prime outside of delta. So really, I have the same decomposition here. So why does this mean, okay, so again, so I have this decomposition, this is orthogonal. Why does this mean that the bias pairing property, so the non-degeneracy of the pairing is true before, if and only if it is true after? Well, it's kind of clear, right? Somehow this, the splitting here is orthogonal. So I really only have to, so here I have it, okay, so if I have it before, then I have it here. This bearing, this here is the image of the geese and on which I have Poincare reduality anyway, right? Because I have it, right? Otherwise, I would already fail here. So I have Poincare reduality here. In particular, I have it here. The other direction, right? If it fails here, it must fail on one of these components. And that's it. There's nothing fancy about it. So this subdivision really preserves the pairing. Now let me note that really there was my, so now I did stellar subdivisions. I could have been more fancy. I could have said this triangle is really subdivided into something more fancy like attaching a torus. So this looks like a little minion now to this, but I mean, so really I could have been more fancy and more general, but let's sum up. So for the purposes of PL spheres, I really don't have to go into any fancy kind of subdivisions. But this is really, this is a more general principle. Whenever you can define this map on the convex polynomials, right? Whenever you have a nice substantial map, then you can define the map on the convex polynomials. Then you have to make sure that this map preserves the fundamental class, so the Poincare pairing is preserved. And then you get this decomposition. All good and nice. So now the next step is something that is kind of very nice and old in the anthropology is this folklore fact that if I have delta, a k minus one dimensional complex, and delta lives inside sigma, a 2k minus one sphere, then delta embeds into the boundary of its regular neighborhood and regular neighborhood of delta. All right? So example, if I have a graph in the three-sphere and then I take the neighborhood of the graph, then I can just by just by basic general position, I can move this graph into the boundary of the regular neighborhood. In particular, there exists a refinement, a refinement sigma tilde of sigma such that, well, okay, so I should say refinement sigma tilde of sigma not affecting, not affecting delta such that delta lies in partial of the boundary, somehow delta is a sub-complex of the boundary of the regular neighborhood, which is again a sub-complex of sigma tilde. All right? Okay, so how does this help me? Maybe I should move this up because that's better. So now, okay, so now I have delta sub-complex of some closed hypersurface, right? Delta boundary n. So this is what we have arrived at now. Now we are, I mean, careful. Now we are in sigma tilde. But as we observed there, the Poincare, but we can look at i sigma tilde instead of i sigma tilde delta instead of i sigma delta. So we are fine. Okay, and how do you move delta inside the boundary of the regular neighborhood? The thing is that you got the delta on the boundary. Yes. That is because, okay, so what you can do is take the regular neighborhood, all right? Let me try to, I mean, I cannot draw an example. I'm a very two-dimensional drawer, so I cannot really draw it well in dimension three, but here's the gist of it, right? So here's your regular neighborhood, all right? And now push your delta into general position, all right? Push it into general position. So if you do this in dimension three and not two, what you will have will not intersect the original delta. Push it into general position. Let me be like this, all right? So the issue is here now, of course, yes, yes, you do just do a radial projection to the boundary. All right, all right, all right. Where was I? Ah, yes. Okay, so we have delta in the boundary of n, okay? So we have, let me draw delta and hyper-surface. So we have delta and then it lives inside this hyper-surface here, all right? The issue is, of course, so what do I want? So I wanted that A of E, so the quotient of the phasoring, the quotient of the phasoring of sigma corresponding to E was actually isomorphic to A of delta. At least I need this in degree k, all right? And again I should write only in white. But in general, at this stage I only have a surjection, all right? I have a larger complex. So I can only say that I have a surjection from this larger manifold to delta. So what do I do? Well, if this is larger, then there is some monomial outside of, then there is some other, then there is an element of, then there is an element of A boundary n in degree k that lies in the kernel of this restriction map. In particular, there is a monomial, right? Some of this is generated by some monomials. And there is a monomial in this, there's a monomial supported outside of delta that generates this element in the kernel, all right? So there is a monomial somewhere, perhaps here, that generates an element in the kernel. So what do I do? Well, what I can do is I can just remove this phase and its neighborhood, all right?  And I can do this and repeat this several times until I have a surface with holes such that this is an isomorphism. So kill elements in the kernel, in kernel, by introducing holes, okay? Now, so far so good. Now the issue is, okay, so it's not so hard to see that we can make this surface sufficiently connected, right? If there were two different components, all right, so if boundary n had two different components, then what, just because maybe our graph had two different components, right? If delta had two different components, then what we can do is we can just attach a handle inside Sigma and make them connected. So the connectivity is not an issue. And us drilling holes does not affect the connectivity, but of course now we have a surface with boundary, all right? So e is now not closed. Problem, e is not closed. And so now what we have, so we have an e that is sufficiently connected that in degree k precisely encodes delta, but it's not closed. But now it's really simple. There's a simple trick that we can use to actually produce a closed hypersurface that does a trick for us that encodes a bias pairing property nicely. So let me use this backboard to finish at least that part. So we have e k minus 1 acyclic, but it has bound, it's now ak of e is isomorphic to ak of delta, but e has boundary. And now basically what we are doing is just we double e. So we look at e solution, so solution, well take e, double it, compactify and consider the resulting ideal. So let me explain what I mean. Okay, I will explain what I mean by compactify. So let's look at e on its own, all right? So e is my surface with boundary, all right? And then, okay, so as Pierre already says, okay, I can just double and obtain, all right? I take e, another copy and identify them at the boundary, the canonical way, all right? I have another sheet on the bottom, like this. So there is no compactification here, you're all right, but e lives somewhere, all right? So I should have said this construction that we did is obviously orientable the way that we did it. So orientable. So now, okay, but so I have, how do I deal with sigma outside of, right? So this is here outside, I have sigma tilde, all right? So really there are some phases here intersecting, so phases of sigma outside of e. And now I have an orientation and I have an upper part and a lower part. I have simplices intersecting my e from the top and simplices intersecting my e from the bottom, all right? And when I, okay, so now I want to see this in a new, somehow I want to look at a compactification of sigma tilde without my surface e as amount, right? So I want to compactify this such that the boundary compactify, such that the boundary of the compactification is exactly the double of e, all right? And really what I do is I really just encode what phases do I intersect from the top, what from the bottom, and that's it, right? I get a new superficial complex. So I get, all right, so this is my sigma, sigma hat. And now I'm done. Now I have a closed hyper surface in a new sphere, so sigma in a sphere sigma hat. And what remains, and this is somehow, okay, this is a diagram chasing, I will skip, is to show that i sigma tilde e satisfies the bias pairing property if and only if i k sigma hat double of e does, all right? And now I can go back to this theorem that I did for closed hyper surfaces and apply it here, and I'm done. All right, okay, so let me summarize. There is one more caveat that I have to go over, and then after the summary. If I identify this two copies, I get back my homology, yeah. I mean, so to fill it in here, fill the resulting antibody in, this will be our homology sphere again, yeah. All right, so for the summary, so what do we have? We have the Konek Alema, and what did the Konek Alema give us specifically, right? It told us we can construct left shits elements, provided we can prove the non-degeneracy of pairing, not, let me say tomorrow, maybe in sigma, we can show the non-degeneracy of pairing. What pairing did we look at? Well, we looked at, I mean, not the pairing, what spaces, what ideals did we look at? Well, we looked at the kernels of these generic linear combinations of X v, right? But this is not tomorrow, it's not so important for the moment. What currents we looked at? We looked at the kernel of the already constructed map, right? Our candidate for the left shits map that came close in A of link of the vertex w in sigma. All right, remember somehow there was this, okay, so we wanted to apply this basic representation theory of the Konek Alema. All right, so we noticed that we have to show that the kernel of the old map mapped under this model, the perturbation, the map that we want to add, the divisor that we want to add. And we want to say that this does not intersect the image, all right? And then we observed that this is equivalent to just saying, okay, the pairing on either one of them does not degenerate, and then we were done. But so this we have to show, all right? So this here, all right? So this is, notice that this is in co-dimension one. So we really gained something here in terms of induction. And now then the second ingredient was bias pairing property, right? So it's about non-degeneracy of the pairing at ideals at, okay, so at i sigma delta, well, is implied by left sheds properties at hypersurfaces for hypersurfaces, all right? So this was exactly, we identified, we looked at A of E, all right? So why was this a left sheds property? So what we looked at here, all right, was we looked at A of E. And this was, okay, so this was a co-dimension one manifold inside sigma. So sigma was of dimension 2k minus one. This E is of dimension 2k minus two. So the co-dimension of sigma of K of sigma is one higher than the co-dimension of K of E. So we took out this additional element and we wanted the kernel under this element to be prescribed. All right? So now we wanted to have the kernel and the co-kernel under this element, under this multiplication to be prescribed, which is a left sheds property. All right? So this here, somehow, this was K of E, right? So we took out some of our linear system of parameters, theta for coming from A, coming from sigma. But this was really, somehow, it was one too long. So we think of this as really of K of E modding out linear system of parameters that was one shorter.  All right? And this last element acting, this is really something saying about middle isomorphism, which is saying something about the left sheds. All right? So this is a left sheds property. All right? Okay. So the non-degeneracy of the pairing at ideals of this is applied by the left sheds properties for hyper surfaces. And so we see that we actually, we gain in dimension in both steps, all right? Seemingly, somehow, we are in a very nice position. The only issue here, this is why somehow I said there's a caveat and we have to want to explain a little how to get around that caveat, is that these ideals here and these ideals, okay, so let me just say it like this. So these ideals here are not in general of this form, all right? So these are not in general monomial ideals. So let me explain briefly how to get around that issue. And then after the break, then we will do a break and then we will go to the transcendental theory proof. So issue, these ideals here are not monomial in general, not monomial in general, or orthogonal complement to monomial. All right? So the issue is not, so if I just looked at this here, this would be the orthogonal complement to a monomial ideal, that would be fine, but I'm pulling it back to the link of a vertex, which I'm intersecting then to ideals, and that is not so easy to say that this is a game monomial. So I have to say something. So intersection of monomial ideals in general is not monomial. So how do I deal with this? So let me make some space somewhere. Let me make some space here, all right? So maybe it's not so clear that this is not in general monomial ideal. That's the camera. It's in general just the orthogonal complement of the monomial ideal, but I could just as well look at the orthogonal complement, which would be the image of this generically in your combination, and this would be clearly a monomial ideal, but the image, if I already satisfy the transfer of the prime property, is just the span of the images of the individual maps. But this is just generated by the monomials of the individual v and the index at w. Okay? So how do I remedy this? Well, somehow, let me say that this here is the orthogonal complement. So let me repeat. It is orthogonal complement to a monomial ideal if the union of the vertices star v and sigma, where I go over v in my index at w, and I take the union of the vertices again, star v sigma, with the star of the new vertex star w and sigma. If these two objects here are sub-manifolds, sub-manifolds, okay? So if this is the case, then these are nice ideals that I can deal with in a nice algebraic way. So how do I then... Well, let me give you a situation where this doesn't happen, and then let me describe how I get around it. Yeah, it's really just the neighborhood. I take the union of the stars, right? So I take all the faces that intersect my vertex set, and I draw it in a larger closure, right? The issue is that tomorrow... I could have some picture that looks like this, and then maybe another vertex here, and this could have some nasty singularities here. So this here could be... Let me draw it nicer. So this here could be another one, another vertex in my index area, it's somewhat degenerate, it's not a manifold, so I'm not in a good business, okay? So that's a good position. It's not a manifold. So how do I get around that? So let me sketch off how is this circumvented? So, yeah, sketch of proof. So, conventing the caveat. So here's the trick. So let's say we want to prove... we want left sheds for, let's say, for explicitness, let me say, for sigma 2k-sphere, 2k-dimensional sphere. And let's say it doesn't have an order on the vertices that satisfies this property. So you can show that most spheres that you will be looking at in daily life, they will not have an exhaustion by vertex of the vertices in some order such that at every intermediate step you have a manifold. So let's say this is one of these nasty spheres where you don't have an order that is nice. So let's say we just fix an arbitrary order. So arbitrary order on vertices. And now the trick is the following. So we look at sigma, and this is a sub-manifold. It's a co-dimension once here. In this here. Yes, yes, yes, exactly. It's a weaker property than shellability here, but still you can find many examples where you cannot get it. That's right. So this is a weak form of shellability. Weaker than shellability, but still much too strong. But what we do now, so sigma we can think of as a co-dimension one sphere in a bigger sphere 2k, in a bigger sphere 2 sigma bar. So this is then a 2k plus 1 sphere. And then we saw that observe left sheds for sigma. This is what we discussed on Monday. This is equivalent to saying that the ideal i sigma bar sigma satisfies the bias peri property. And now this is equivalent. Well, now I can again reduce to the k-skeleton. So this is in degree k plus 1. So this is equivalent to saying that i sigma bar, and I take the k-skeleton of sigma satisfies the BPP, the bias peri property. All right. So now let me call this an empty composition because everything intermediate is a manifold. Maybe m is not so nice, but an empty composition. So there's an order on the vertices such that every intermediate step you are a manifold. So now what I do is I find a refinement sigma hat of sigma bar such that there exists a hyper surface with boundary such that... Well, what do I want? Such that a k plus 1 of e is isomorphic to a k plus 1 of this k-skeleton of sigma. In particular, the hidden motive here is that the ideals are the same. Sigma hat e isomorphic to i sigma hat e, sigma. All right, that's the motivation, the hidden one in degree k plus 1. So I find this and such that e has an empty composition. So it turns out that in this code I mentioned, you can actually construct e in such a way. You can construct many hyper surfaces with the first property.  So now... So now what do we have? Well, then the bias pairing property. Then the bias pairing property for i sigma hat e in degree k plus 1 is equivalent to a left shift property for e. All right, this is what we observed. But now this e is nicely decomposable and we can actually imply the induction. So now the e is nicely decomposable. But to construct the left shift elements here, well, we apply the conical lemma, but now the kernels here, they are nice. In e, they are nice. They are nice and orthogonal complements to monomials. So let me... I have this somewhat bad habit of if there is not enough space and I will just squeeze it, so let me not do that. But in e, the kernel of this direct sum of an initial segment v initial in the order under the stars of vertices in sigma pulled back to a link of w in e is monomial and so we can complete the induction now because now we have a monomial ideal and we have gained the dimension. So e is of the same dimension at the starting sigma. So the dimension of e is the same as the dimension of the sigma that we started with when we wanted to prove the left shift. But now we actually gain a dimension because we are looking at the link here. So now we have the dimension. This is the dimension of the link of the vertex in e plus one. But now we gain a dimension and therefore we can complete the induction. And that is the overview of this argument. All right. And this is somehow where I want to end with this argument. Then after a 10 minute break, I will go to the transcendental theory argument. All right. So for the final section, so what we will do is we will actually, we will talk about slightly more general objects and we will give a new proof based on some very nice residue formulas. So beyond positivity again, the cycles and transcendentality. Not in kind of a new age way, but in the sense of transcendental extensions. Okay. We have what is our object? We consider mu as a simplicial cycle, k-cycle, which for me is a pair of a simplicial complex, mu. So this is a simplicial complex, mu in these brackets, a simplicial complex, which I will call the underlying complex or support or underlying complex. Maybe let me specify the dimension here for simplicity of dimension d-1 of dimension d-1. And I have mu of the underlying complex and this is also of dimension d-1. And then mu is an element in the homology dimension d-1 with k coefficients of this underlying set, of this complex. And this is what I call for me a simplicial cycle. All right. Now let me consider a of the underlying complex. This is always in a hidden way. There's also always a linear system of parameters here. And this is just the phase ring again. So this is really just, again, k of the polynomial ring modulo i of. All right. And now what I will think of, well, is, well, I already, when we discussed this last time, I explained that there is a, that there is a canonical isomorphism between hd-1, this underlying complex, and ad of the simplicial complex. So this is a canonical isomorphism here. And what can I say then? Well, now what I can consider is a dual to some mu b. This is a dual to mu in the homology. All right. And what I can in particular do is I can think of mu b as a quotient, mu dual as a quotient of ad. Sorry. I don't get what you're doing. You do the slope on creditability because it seems it's good. It has come over. You don't have any class in commons. This is just a top degree. All right. It's just, I'm paring on the final. Okay. So it makes no sense. We have kind of like fundamental chain. Yes. But you don't have canonical commons to class. Maybe you can say that you have canonical chain. Well, I still have a paring, right? I can still just... Oh, maybe you're right. Because you have not only commons to class but canonical chain, which interprets as a quotient. Yes. Exactly. All right. And then what I can do is I can consider b of mu. And this will be the smallest quotient of a mu, such that b of mu in degree d is exactly this class in degree d. All right. Kill everything, yeah? Maybe get it lost. Yeah. You said your order doesn't... It's ad of mu. Yeah. Maps to... It's co-homologer. It can pair with this homologer class. It can make to your field. Yes. But not to m check. Yeah. Because... So I want to... I would just consider this as a one-dimensional... All right. No, it's nonsense. What is mu check? The mu is... You arrange your problem in... OK, OK. So... So mu check is... OK, so now I have a pairing of degree... of degree d to my field, right? To k, to my ground field. Yeah. You get a function of all your things too. Yeah. So I... OK, so now this pairing will give me a one-dimensional quotient of this top degree. All right? Of ad. Yeah, or you get one-dimensional quotient of co-homologer, yeah? Yeah. It's not a class in co-homologer. Yes, I get a one-dimensional quotient. Yes. It's not a class. You say that mu... Maybe you want to write with mu check. It's a dual to mu in h-dimensional. It's a quotient. Yes, yes, yes. That's what I mean. So it's a quotient of hd minus one. Therefore, I think of it as a quotient of ad. No, because literal and corrective. Like, you get elements here and... Ah, OK, so yeah, OK. OK, so quotient, OK. So it's quotient, yeah. OK, thank you. Of hd minus one. Yeah. All right? And what I get here is essentially the... The quotient of A of bracket mu with the fundamental class, right? With the prescribed fundamental class induced by mu. All right? So, and again, this still depends, right? So B of mu, still in a hidden way, depends on theta, right? So this is still theta inside. So I should write, right? There's still theta encoded inside. All right? And this is a Poincare duality algebra. It is a Poincare duality algebra. Duality algebra. The fundamental class in degree d. OK, so now I can ask again... OK, so I can ask again, does B mu have the left sheds property? And again, the theorem, maybe I should not start on the bottom of this to write the theorem. Yes, sir. Where do you stop? You have something, you have a manifold. You leave these inside the manifold. It doesn't have to be in the manifold. It's just a simplification complex. Yes. And then you have a homology class in the simplification complex. There's no manifold here anymore. OK? And so the theorem is as follows. This is going with Thomas Papadakis and Valisirica Petrotou. And for this, I will not just say a generic element. So I will be a little more specific. So mu d minus 1 cycle over K arbitrary. This is an arbitrary field. And now I take a field extension and what is the field extension I take? So while I have the matrix theta and its entries, and then I have the element L, the left sheds element L and its entries. And I see them all as some algebraically independent numbers. So each of the entries is a variable and I think of my new field, I give a field extension where each of the coordinates is an algebraically independent variable. OK? So I have all my entries here. And then basically I take the field extension and I take the field of rational numbers with respect to all those variables and this here will be K tilde. OK? So K tilde, rational field of rational functions of rational functions over K and I extend by the individual variables theta and L. And now I mean immediately a generic element. Then B of mu satisfies the left sheds property. So IE, so let me B mu K to B mu to D minus K L to the D minus 2 K is an isomorphism. It satisfies the whole amount of relations that is, right? So the hot dream on bilinear form QKL does not degenerate at monomial ideals. And then let me give one additional property that is very nice, know it in characteristic 2. So if the characteristic of the ground field is 2 then we have something even nicer and even more beautiful. This is that Q, the hot dream on bilinear form never degenerates. So QKL of U, U is not equal to 0 for all U in B mu K. And here I should say K should be less or equal to D half. So that's kind of the strongest form of not degenerating anywhere. This is somehow no ideal. I degenerate at no ideal. Surprising. But that's what happens here. All right. Okay. So now we are all over this field extension. Maybe I should say, right? This is now over this field extension K tilde. Okay. So why is this surprising? Go to the classical hot dream on relations. All right. You have the signature plus one and minus one. So definitely there will be one point where it will just be zero, the pairing with itself, just by intermediate value theorem. And you can show that tomorrow, I mean, if I pass to the, I mean, if I pass to the algebraic closure here, right, if I take K tilde but take the algebraic closure, then definitely there will be points where this is zero. So this is really something that only works in this. Okay. Yes. Yeah. I mean, you can also achieve this. I mean, okay. So let me open question. What happens in other characteristics, right? Other characteristics. But the point is this will never be true if you pass to the algebraic closure, right? So you take the extension of the complex numbers by these transcendental variables, then it might be true. But if you take the algebraic closure of K tilde, it will not be true. That's the point. Okay. Okay. So let me give like the first few indications of what we will need. I mean, we will end in like 10 minutes. So I think I will have to repeat anyway.  All right. So corollary, well, what are some of the things that we need to do? We need to do something. We need to do something. We need to do something. So corollary, well, what are nice cycles, right? So if sigma is a sphere, right, then you take just the fundamental class as a cycle. Then B of, or A of sigma is just B of the fundamental class. Similarly, if M is a manifold, an orientable closed manifold, then B of the fundamental class is really just, well, it is A of M modulo, well, the kernel of the partitioning map, right? The kernel of fM to the direct sum of the vertices in M, A star of the vertex in M. All right. This is just the kernel of the partitioning map. I mean, but somehow, what else is encoded by these? Well, for instance, pseudo-manifolds. If you have a pseudo-manifold, if P is a pseudo-manifold and it's orientable, then B of mu satisfies left sheds. So B of fundamental class satisfies left sheds. So these are nice examples, but these are, one has to be a little bit careful. So this B of mu here, you could look at its Betty numbers, right? You could look at the dimension of the K graded component and try to extract some combinatorial meaning. So for instance, for manifold sigma, or for manifolds of spheres, and you can also show that for pseudo-manifolds, the dimensions of the graded components are, in these cases, the dimensions of the graded components here of the Bi is independent of the linear system of parameters. So they have a combinatorial meaning, even though in this case, we don't have a closed formula, no closed formula. So in the case of cycles in general, one has to be careful in the sense that there is no combinatorial meaning immediately. So it's really somehow a rather interesting property to have left sheds for cycles. And perhaps if I'm done with this sketch early tomorrow, then I will give some applications of this. But really there seems to be no algebraic geometry analog. So this is kind of left sheds, but really there seems to be no good algebraic geometry interpretation of this left sheds. Because it should depend on the coefficients of the cycle. Yeah, yeah, it depends on the coefficients. It's kind of, yeah. Constructible function. Yes, yes. Yeah, but it's a kind of, it's a, it's small. It's, yeah, it's a left shed theorem that is very far from any kind of geometric interpretation. All right, so what will be, what will be the idea, what will be, what will we work with? Well, we will basically look at residues of the degree function and work with them. And so the main trick is, well, first let us, let's define the degree function, right? So the degree function is just the map, right? It's just a way of identifying B of mu with K, right? B mu on degree D with K. So how do I define this degree function? Well, here's a way to write it down explicitly for facets. So define, define this, or kind of, if you want, just normalize this by defining it on a facet. Define this by, okay, so I take a facet for F facet of the underlying complex. So this is again a D minus one-dimensional face, D minus one-dimensional, simplex. And then we can evaluate the degree of corresponding monomial XF in the following way. I consider the vertices ordered, and then I look at my cycle, right? And I look at the ordered, the oriented coefficient of my typical cycle. So remember, mu, this was an element in HD minus one of the underlying complex, right? And then I divide this by the determinant of theta restricted to the minor corresponding to F, okay? This is the determinant of this. Remember, theta was this matrix, right? I think of theta as a matrix with some entries and then face F cuts out a minor, and I compute this determinant. And I can prove actually that there is, well, actually, yeah, let's not do this today, right? So we will tomorrow prove that somehow we will see that this is consistent. Maybe I will give you just intuition why this is consistent. And then we will look at this function, and in particular, not for function for faces F, but we will look at faces. We will look at, we will try to evaluate this at monomials of the form X tau squared, tau is a face of cardinality, cardinality d half, or at most d half inside, yeah, of cardinality d half inside my complex. And then we will see that this actually, this function really has very nice poles, and then we will compute some, yeah, we will do some basic analysis with this, prove some nice identity, and then this anisotropy, this total anisotropy, the anisotropy everywhere will just fall off, and then we have to modify a little to do this in general characteristic, and we'll prove just the Hollermann relations there. All right, and I think somehow it doesn't make sense to start with the proof now, so let's finish here, and then, because I would have to anyway, repeat a lot tomorrow. All right, thank you.


My dear viewer, welcome again to our series of 40 days of prayer. Today is day 18. We are glad to have you here. It's a Sabbath morning. Allow me to say happy Sabbath. I'm glad that you chose to be part of this ministry. Blessing our when we come before the Lord to see him in prayer. And before you go to church, if you are in tune this morning, perhaps could be watching it in the afternoon depending on your time on the evening. But before you go to church, just a few minutes, that we can commit ourselves to the Lord as we go to church. The text we are sharing today is from the book of Psalms number 73. And we read verse 25 and 26. The one of God says, Um hava'i in heaven b'atvee, And there is none upon earth that I desire beside thee. Verse number 26. My flesh and my heart faileth. But God is a strength of my heart. And my portion forever. I'll read one more time. Psalms number 73, number 25, verse 27 and 26. Um hava'i in heaven b'atvee, And there is none upon earth that I desire beside thee. My flesh and my heart faileth. But God is a strength of my heart. And my portion forever. Let's pray. Lord we thank you. As we speak from this text in the next few minutes. We invite you presence in Jerusalem we pray. Amen. There are times when we go through difficulties and challenges in life and we feel so much broken, so much rejected and rejected. We feel we are so lonely and even God himself has hid himself from us. Such experiences are very common. They are not new to us. They have been there before. The author of this particular song, who is a friend of David, Kondasaf, he went through a trying moment, a friction moment. And he lost it. He felt broken. He felt God was not with him. He felt rejected in fact. Before he comes to this verse number 25, when you read from verse number 21, it's very interesting. He says, in that situation my heart was grift and I was pricked in my reins.  Some version says I was senseless and ignorant.  Passing through trials in life and until you lose your senses. I mean, what this text is saying that there are difficulties that can come your life. Your challenges can come your way until you start even doubting God himself. Talk about death. At times death can be very devastating. Death is the last one. There's nothing that can explain how one feels when he has lost the last one. Imagine losing a last one and especially when you have trust in the Lord, you praying, but then God seems not to answer. At times we have death through accidents and expecting death and things just happen and you're fast struggling to understand where was God and why these things have to happen. So you've just been working well employed at your workplace and things have been well but boom, from nowhere, things are turning around and becoming shapeless and before you know there's a light on your desk and you shun the door and you don't even know where to start. At times your marriage could be doing well and you're good people, you're surfing church, you're nailed or deconized or committed in church, you're giving your title and offering and everything is okay. But boom, something happened, disrupts you and your marriage is going down the drain.  You wonder where is God?  You're praying but God seems not to answer. At times, when life becomes meaningless and appreciating God's power in your life becomes meaningless, mistakes of that situation and says, I was there and I could not understand how I became senseless, I became foolish and the ignorant, I was like a beast because of what I was going through but then he says in verse number 23, nevertheless I was continually with these.  You don't mean it, it's just a situation you are in and that's what's happening. But then this order says even so in that situation I never allowed my heart to wander away to doubt the presence of God and his love upon my life. And so in verse number 25 he says, Whom have I in heaven apart from you God? In my crisis, in my afflictions, in my trials, in my tribulations, whom do I have apart from you God? He says I have only you and there is none upon earth that I desire beside me. Think of Job who would lose everything, that twinkling of an eye has lost everything.   Who does this kind of thing? Who allows heaviest power such kind of things to happen? You have lost everything. I've lost the seed of my womb and you're still here talking about this God? What a God can do such kind of things? In fact, she's suggesting to Job just cast him, cast this God that we have nothing to do with him. You see, she didn't mean it. She didn't mean it. She was overwhelmed and so at that moment when you feel overwhelmed, the scripture is advising you never to allow your heart to wander from God. Yes, in this world we shall go through tribulation. Jesus says, be of gun shia because he has overcome. You can say like phase over here, um havae in heaven bati and there is none upon earth that I desire beside thee. Version number 26 is, my flesh and my heart faileth but my flesh in my heart may fail but here he says faileth presently this is the situation I am in my flesh and my strength are failed I am lost of ideas I'm lost of options I'm lost of anything I know that could be a source of my health I'm here hopelessly yet waiting upon the Lord says my heart and my flesh faileth but God is the strength of my heart love is in my portion and I hear Paul asking what can separate us from the love of God and goes ahead to talk about things that ordinarily ought to separate us tribulations, you know conflicts, promise unemployment struggles of every kind he says no even in these kind of things we are more than conquerors in Christ Jesus the love of God will never leave us my dear viewers in Sabbath morning I want to invite you as you prepare to go to worship or wherever you are whether in church or wherever you are God cares for you God loves you the God of the mountain is still the God in the valley he cares for you despite the pain you are feeling today he loves you despite what you see today and I want to encourage you that may we hold hand to the Lord may he be our refuge and our strength in the time of need is such a friend and I hear one musician saying what a friend we have in Jesus and in that text he says what needless pain we bear because we do not learn how to take everything to him in prayer God is waiting to address our emptiness our brokenness he is waiting to deal with our pain he is waiting he knows how to do it to fix our brokenness and we can trust him one more time you can trust him with that condition with that situation with that circumstance you can trust him and I'm here to pronounce today that weapon which he cast against you shall not prosper in Jesus name just hold on to your faith we are upon the Lord and we shall see the reverence that comes from God just walking to church on this sabbath and worship him in confidence and waiting upon him and we shall see the miracle he is going to perform join me as we pray gracious for the heaven indeed whom we have in heaven apart from you what we desire here on earth if not you our heart and our flesh may fail but you are strength and our portion forever yes indeed in this world we are going through tribulations but Lord when we imagine and think of the glory of the victory of a scene the glory of the new Jerusalem that we are preparing for us we admire and get committed for that the joy thereof and the good things thereof cannot compare with the momentary trial and afflictions we are going through we are willing Lord this money to say like job I know in whom I believe the Lord our faith will not waver because of the current situation we are faced with and I'm praying for my viewer Lord I know what they are going through it could be a family issue it could be a work related issue it could be anything health or anything but they are in their lowest moment when as it were they are losing their senses Lord we inviting you in these ugly desperate situations of our lives that you can come and gather those broken pieces and bind them together that your children may have hope one more time gracious father we don't have any other source of hope apart from when your children are crying to you your promise to hear even before they speak you are going to answer and Lord we are seeking you this morning that you may answer the needs of your children as many as they are presented Lord we pray in Jesus name thank you media viewer for this moment the Lord thank you for choosing every time to be here for this for the days of prayer this is the source of your blessings and thank you for being a blessing to too many people by sharing this message and if you have not been sharing and invited to share but also inviting you if you have not subscribed to our channel please just click that button that anytime we have such programs may come to you and you may continue being blessed remember to pray for the 7 member list that you have let's continue praying and praising let's continue pleading let's continue interceding let's continue asking in faith and indeed miracles will be vowed by God till tomorrow I'll be with you


When the great African kingdom of Dahome is under threat, only one group of people can save it. Strong female leads. It's about time. Let's talk about Wakanda Forever. For starters, this is not historically accurate. It's not based on a true story. It's set in an alternate setting, an alternate timeline where there was this elite women tribe that could take on anything, any obstacle that was in their way. And who is the greatest threat to these people? Well, the answer's obvious. The white man. I love Viola Davis in this. She's a beast. And let's just say overall this film, actually pretty decent. I like all these characters. They're not just one dimensional badasses. There's actually backstory. This feels like a Disney movie. It's very colorful. The people are all beautiful.  I mean, this is not what it would have been like. It is its own thing. Whether or not audiences are gonna go out and see this, that's questionable.