That's why I woke up this morning with my mind stayed on him because he is real, real, real to me. Hallelujah. There's no life without Jesus. Hallelujah. Lord, we thank you this morning, Lord God. Father, we praise you. We glorify. We magnify your name, Lord God. We come this morning. Thank you for your mercy, Lord God. Thank you for your grace, Lord God. Your loving kindness, Lord God. Lord you say your mercy, Lord God. It's renewed, Lord God. Every morning, Lord God. Every morning, Lord God. Out you, Lord God. Hallelujah, Lord God. Our hope, Lord God. It's in you, Lord God. In the name of Jesus, Lord God. Hallelujah, Lord God.  We thank you because you're faithful in all your ways. Father, we pray speak to our hearts today. Give us a word. We bless you for what you've already done, what we've already heard. We pray anoint us today, anoint your servant today in Jesus' name. Amen. Amen. We give honor to our bishop. That's right.  He's worthy of praise. He's worthy of glory. He's worthy of honor. Again, we give our bishop honor today, our fabulous first lady, the ministers of saints. I want to thank Brother Richard for that wonderful song. That was the first song that I sang out at a big to-do, the Lord's Prayer. And that is a wonderful, wonderful thing. Is anybody here going through any tests or trials today? Is anybody going through anything today? Has anybody been tested today? The Lord tell me, don't worry.  He's already made a solution. He said, don't worry about it. We don't have to worry. I was thinking the other day that God is a master or the master producer. God is always trying to produce some things in us, some qualities and characteristics that make us like himself. He wants us to grow into the very image of Jesus Christ. And what he produces in us will always be a blessing to us. So I am convinced that unfortunately though, unfortunately part of the process is usually afflictions. We don't like afflictions, but usually part of the process is afflictions. So I'm convinced it's important that we understand afflictions. Paul put it like this, for our light afflictions. I like he said called our light affliction, which is just for a moment. In other words, trouble won't last your way. Tell you never, troubles, they don't last your way. Sooner or later, God's gonna turn it around just to you. He's gonna bring you out. There may be problems, there may be trials, there may be tests, but God's gonna bring you out. You'll be like the six million dollar man. You gonna be better, faster, stronger than when you went in and said these afflictions are just for a moment. He says, but it works for us or it produces for us a far more exceeding and eternal weight of glory. One translation said these light afflictions will result in God's richest blessings upon your life forever. So the word that the Lord gave me today is there was purpose in my afflictions. There was purpose in my afflictions. I didn't like it. I didn't care for it when I was going through it, but when I got to the other side, I found out that there was a blessing in what the Lord allowed me to go through because it took me to another level in him. There was purpose in my afflictions. Psalms 119 verse 71 is where we're going to take our texts, but it's interesting. There's some things that we can't understand until we get to the other side of the situation. Now, we as OGs, that's old men and old guys and old ladies, we have an advantage because we live long enough to see the other side of some situations. We live long enough to see that God is faithful regardless to what I'm going through. He's able not only just to bring me out, but he'll bring me out with a victory. He'll bring me out with a testimony. He'll bring me out better than what I was when I went in. Psalms 119 verse 71 says it is good for me that I had been afflicted. It was good for me that I had to endure some hardships. It was good for me that I had to suffer some adversity. It was good for me that I had some hard times in my life. It was a good thing. Now, at first glance, at first glance, this is truly one of the great oxymorons in life to opposite ideas, to opposing points of view. How can you use the word good and affliction in the same sentence? To say such a thing to a man or a woman that is intelligent just wouldn't make good sense. Turn to Isaiah chapter 55. Isaiah chapter 55 verse 89, Isaiah chapter 55. So the Lord is speaking through the prophet Isaiah and he says, my thoughts are not your thoughts. Neither are my ways. Your ways, said the Lord said, as the heaven is higher than the earth, so are my ways higher than your ways and my thoughts than your thoughts. The Lord is saying, I don't think like you think. I don't do things the way that you do. In fact, I operate on a whole another level that you can't conceive. He says, when I deal with an issue, I'm looking at it from an eternal perspective. So what makes sense to the mind of God usually doesn't make sense to the corner mind of men.  But let's go back to Isaiah. I mean, yes, Psalms 119 verse 71. It finished that verse. It said, it was good for me. Bishop, but not to me. It was good for me, but not to me. It was good for me, but I didn't like it. It was good for me, but I didn't understand it. It was good for me, but I didn't want to be there. But it was good for me. And here's the reason why that I might learn that I might understand that I might conceive the will of God and the way of God. I didn't understand it going in, but when I came out, I had an understanding. This was a large doing and he did it well and he brought me out with victory and joy and hope in the Holy Ghost. It was his doing. He said that I might learn your statues, that I might learn and understand your words, that I might learn your ways. See when we begin to get a heavenly perspective on life and a heavenly perspective on adversity, we'll begin to view things the way that God views things. And we will value the heavenly perspective over our own corneral thinking. See, I think I'm pretty sharp. I think I got it going on. But every time I think like that and I go to the Word of God, I say, brother, you need to come up. There is a way that seems right to men, but the Word of God will give you a way that will change your life. It was good that I was afflicted. I didn't like it. I didn't understand it, but it was, it was good. It was good for me that I was afflicted. Romans 1133 says, Oh, the depth of the riches of both the wisdom and the knowledge of God. See, those are the true riches. A lot of times we think, if I could just get a million dollars and have in the bank, if I could just have a six or seven houses sitting on a hill, if I could ride in the Mercedes like Deacon Sykes, I'd be all right. But if we will get the wisdom and the knowledge of God, he said, there's no good thing that I withhold from you. So you could have a house up on the hill. You could ride around in the Mercedes being, you can do whatever you want to do, but get my wisdom and my understanding. See, my wisdom and my understanding. See, some of us, we get a nice car and we won't pick up nobody. When we had a whole hoopty, we take everybody home from church, but now we got a nice ride and we hear, let me give you some money. You can catch the bus home. You feel more comfortable. But when I have the mind of God, when I have the love of God, you know, I was talking to my wife yesterday, we were just just kind of talking and I was remembering some of the adversities that we went through in our past. And she said, you know, it was hard. It was tough. And then she says, but it taught us some things. It's taught us I can trust in God. It taught me I can depend on God. It taught me I can rest in God. It taught me compassion for my brother and my sister. See, but some things you don't learn. You only learn in adversity. And so when I get to the place that I don't understand what God is doing in my life, when I get to that place and I must trust that God is working all things, all things, all things together for my good and for his glory. See, when God does it, I get blessed and I give him the glory. I tell the world, I didn't do this. God did it. Nebuchadnezzar, he walked around with his chest poked up and he crawled on his belly like an animal and said, and he came to himself and he said, the heavens do rule.  It was good that I was afflicted. It taught me some things. It learned me some things all the old folks would say. It learned me some things. And even when I don't understand, I trust him that he's working all things together for my good and for his glory. And see, that's where faith comes in. The Bible said, the just shall live by faith. The just shall walk by faith. The just shall prosper by faith in God. We used to sing a song, said we are standing on the promises of God. Not standing on, and old wives tell them we're not standing on Grimm's fairy tale. We're not standing on what mama and them said, because mama said, oh, I know I'm right. But we are standing on the promises of God. We want to look at the life, the story of Joseph. But what I want to do, I want to go to the end first and then go back to the beginning. Turn to Genesis chapter 50. Genesis chapter 50 is going to be on the same page or the next page from Exodus chapter one, because that's the last book in the Bible. If you go to Genesis 51, you've gone too far. 51, let's look at verse 20.  But God, hey, anytime you see a but God, that means a situation is about to change. It was bad, but God, I didn't know what to do. But God, we were seeking but God said, but God meant it unto good to bring to pass as it is this day to save much people alive. And so let me paraphrase it. It says, you meant it for evil, but God meant it for good. I couldn't see it at the time, but you meant it for evil. I didn't understand it at the time. I thought you were just doing it, but I found out that God allowed this to happen because he was going to bring some good out of my life. Hey, so let's go now back to chapter 37, verse one. I'm going to tell you after Bishop got up, I kind of didn't want to get up and preach. I thought we ought to take up the offering and go home. Come on Bishop, don't do a brother like that. Yes ma'am. Genesis chapter 37, verse one. It said, in Jacob dwelt in the land where his father was a stranger in the land of Canaan.  See, we can't despise the youth. Sometimes God will give young people a vision that he didn't give to you. He didn't give it to you because you thought you would think maybe it's impossible, or he didn't give it to you because you just tired and didn't feel like doing it anyway, but he gave it to a young man. Joseph is 17 years old. He was feeding the flock with his brother and the lad was with the sons of Bilhah and the sons of Zilpha, his father's wives.  Because you know snitches get what? Come on now. Don't be no tattletale like that. He told me, if he told me, I said, Dad, don't get me out there.  So now Israel love Joseph more than all the children because he was a son of his old age. And he made him a coat of many colors. And I was not going to talk about that. But that is foolishness. Do not prefer one child over another child because they will see that. I know a young man, he is almost borderline retarded. And his father treated him worse than he treated the other children. And that young man, even though he was retarded, he grew up and wrote his father a letter. He said, you did me wrong. You mistreated me.  You didn't treat me like I was one of your childs. And it hurt me. And what are you going to do about it? See, don't don't grow up and let your children have to come back and say, you did me wrong. And you know what? He let me read the letter before he sent it to his father. And when I got through my mouth was hanging open. And I'm thinking you wrote this. See, but that was foolishness. He made him a coat, a designer coat. He didn't get it at a good will. He went down to Macy's and picked up some designer material, made this boy a coat. And when his brother and saw that their father loved him more than all his brother, they hated him and could not speak peaceably unto him. And this is where I want to get assessed. And Joseph dreamed a dream. And Joseph dreamed a dream. And this dream came from the Lord. And Joseph dreamed a dream. But Joseph was this arrogant young man.  So Joseph, who is hated by his brothers, he tells him about this dream that came from the Lord. And it said, and Joseph's brothers hated him even the more.  I remember as a teenager, I was excited and I had a dream and I went and told the saints and when they got through I felt so foolish. See, but we have to get to the place that we have wisdom enough when God gives us a dream, when he tells us, this is what I want you to do. Take some time to get along with God and find out, Lord, how do I do what you want me to do? And when do I want, see, sometimes we run out too fast. We get ahead of God and we make a mess out of the dream. See, sometimes given a, given the dream to people is like giving a baby a oozy. He gonna kill himself and everybody around him. But Joseph dreamed a dream and then he went and told his brothers and it said that they hated him even the more. And so in verse nine to 10, you can read it on your own, but he has another dream and he tells his father about the dream that he's the son and the stars are rotating around him and bowing down to him and his father, the one that loved him more than his brothers, told Joseph, Joseph, you need to slow your roll.  Who do you think you are? They had a song about that Mr. Big Stuff. Who do you think you are?  I'm God. And so his father says, you need to check yourself. And then Joseph's father says, Joseph's out into this fields to spy on his brothers. We already talked about snitches, what happens to snitches every time.  Tell somebody that ain't right. That ain't right. So his brothers decided to sell him for a slave to the Midianites. And they go back to their father and they take Joseph's coat of many colors that is now drenched in blood and ripped up and torn and tattered. And they tell their father, a wild beast has killed your son. The one that you love, the golden child, he is now dead and the father is devastated. And as tragic as that part of the story is, the father has to take some responsibility because he created the animosity between the brothers. And so they tell the father that, the son that you love, the one that you love, he's dead. Now I did tell you that Joseph had a dream and his dream came from God. He wasn't dreaming by itself, but the dream came from God. And now Joseph is in trouble.  But I have discovered when God gives you a dream, immediately he says, take him to the pits. I want to see what he's going to do. I want to see how bad he wants to make it happen. I want to see, is he going to fight to make the dream come to pass? And so the father is devastated and the Midianites, they sell Joseph for a slave to an Egyptian officer, a military man named Potiphar. But I want you to notice that down in this affliction with what Joseph is going through, still the hand of God, the favor of God was on his life.  Joseph was down in Potiphar's house. And before you know it, Joseph was running things at Potiphar's house. And then into Potiphar's Hoochie Mama, Cougarish, fast-tail wife. It was so bad they didn't even tell us her name. But all of us know some people like that, amen? Come on now. And it ain't just women. It's some fast brothers out there. Before I got saved, thank God I got saved. Oh Lord, we better move on. And his wife sees Joseph, young Joseph, handsome Joseph, six feet tall with a six-pack Joseph. And she begins to hit on Joseph. I can just imagine, she said, Joe baby, Joe baby, come lie with me. Let me make you my boy toy. Now, if I didn't get it right, all you Cougars out there, you pull my chain and let me know if I was something I should have shared that I didn't say. Just talk to me after serving. We'll be inconspicuous, but I'll take some notes. Cougars. But Joseph says no.  That's what you call integrity when nobody can see you and you still do the right thing. Integrity. And so Joseph ends up in prison. And again, God's favor is still on Joseph's life. His hand is still on Joseph because before you know it, Joseph is running things in the prison. And there's more to the story, but it says, and 10 years later, say 10 years later, say 10 years later. See, that's a long time to be in the same situation. That's a long time to be in the test. That's a long time to be in a trial, but 10 years later, but seeing those seniors, God was working on them. In the 10 years, God was teaching them some things. In 10 years, God was showing him some things. See, that's what I call a tribulation. See, a tribulation is a trial that goes on for a long, long time.   But after 10 years, Joseph is called to interpret a dream for Pharaoh. And he did. Joseph interprets the dream, and not only that, but he lays out a plan for seven years of plenty and seven years of famine. See, God will give you divine wisdom to know how to prosper in the years of plenty and how to prosper in the years of famine. And we're right at the edge of those years of famine, but you don't have to be afraid. But God said, I'm going to take care of my people, just like down when they had the people down in Egypt. He said over in Egypt, there was famine and death, but not in the land of Goshen where God's people are. We're living in the land of Goshen. We're dwelling in the land of Goshen. God blesses his people over in Goshen. And so he lays out a plan. And so Joseph is set free. And Pharaoh makes Joseph second in command over all of the land of Egypt. So let's go back to chapter 50. Let's look at that. Chapter 50 verses 18 through 20 says, we had the end of the story says, and his brothers, the ones that treated him so badly, also went and fell down before his face and said, behold, we be thy servants.  I'm not in the place of God. I'm not going to judge you. I'm not going to get tit for tat. See, when God works on us and he saves us and he puts us in that what I call the contrite machine, we go and proud and we put on that machine and it beats us up and we come out humble on the other side. He says, I'm not God.  He said, but as for you, you thought evil against me. You meant me no good. But hey, God, and that's what I like. Every time I see a but God means a situation with bad, but God intervene in everything has got to change. Everything has got to get fixed. Everything has got to turn around. He said, you meant it. Yet you meant it evil against me, but God meant it unto good to bring this, to pass this moment right here. So, Joseph's brothers had those who had afflicted him.  See, delay wasn't denied. See, in our life sometimes God will promise us something and because we don't see it right away, we think God has changed his mind. But I've discovered that many times delay is not denied. Delay means you not ready for it right now, but you just hold on because it's on the way. I got a blessing for you and what God has to me. It's for me. They hired a wicked prophet to curse the people of God. And he tried to curse and I've had some people try to curse me. But that wicked prophet says, how can I curse whom God has blessed? So you need to understand, I don't care who comes against you. I don't care who talks about you.  How can they curse who God has blessed? See, I belong to God. I'm a child of the king and no weapon, no weapon, no weapon that's formed against me shall prosper. It won't work. And so delay was not denied. But Joseph's afflictions helped to prepare him for the promise.    But God, I said, but God, but God made it for your good. It won't always be like this. The Lord will perfect, fix those things concerning you. Sooner or later, see, I've been going through some tests, I've been going through some trials, but I had to tell myself sooner or later, this thing's going to turn in my favor. He's going to turn it around for me, for me. She tells somebody, I've been going through a chance, but it's all good. See what I've been through, it's all good. Every tear that I shed, it's all good. The sauce that I had to give up, it's all good. It's all good. It's all good because the blessing hop is on the way. I declare to you today, every test, every trial, every affliction is just preparing you for the blessing, preparing you for your destiny. See, God has called you to do some things that nobody else can do but you. We have divine destiny. God has not called us to do the same thing, but God has called every one of us to do something. What has God called you to do? And see, sometimes we look at numbers and we try to decide if the hand of God was on my life, it would be big and it would blow up, but no one preached a hundred years and all he did was save his family and all of mankind. My divine destiny and every test and every trial that you've been going through, it's preparing you for the blessing. I said, it's all good. It's all good. It's all good. We've got the victory.  You're going to come out with peace. Tell somebody I'm coming out. I'm coming out. I'm coming out. I'm not staying here. I'm not staying here. So hold on. Your blessing is coming. Hold on. Your healing is coming. Hold on. Your breakthrough is coming. Your miracle is on the way. What I have been believing for, I believe that it's on the way. The Bible said weeping may endure for night. But joy, hey, but joy, but joy, but joy will come in the morning. The joy of the Lord is my strength. So I can rejoice. I can celebrate. I can get my praise on them. Even when things look bad, I can get my praise on them. Even when I don't know what to do, I can get my praise on them. Because faithful is he who has promise. He said God is not a man that he should lie. Nor the son of man that he should repent. If God said it, if he said he's going to bless you, he's going to do it. If God said he's going to heal you, he's going to heal you.  God did it. Your blessing is on the way. But I like what Peter said. He said now, after you suffered a while, after you've been through affliction for a while, after you've cried for a while, say he's going to make you perfect. He's going to mature you. Says he's going to establish you. He's going to bring order into your life. Then it says that he's going to strengthen you. He's going to make you strong in the Spirit. The Bible said be strong in the Lord and the power of his might even when I don't feel like being strong. I've discovered that I can stand in the strength of the Lord and the enemy has to flee. The enemy has to run. The enemy has to turn back because I'm standing in the strength of the Lord. It's all good. It's all good. It's all good. He said, and then I will settle you. He says, I will give you peace. See sometimes we think we need stuff. We need peace. He says, I'll give you peace that will pass even your own understanding. He says, I'll give you all things that pertain to life and to godliness. In other words, I'll give you everything that you need. He told Moses, tell Pharaoh I am that I am. In other words, I'll be whatever you need me to be. I'll be a bridge over troubled water. I'll be a way out of no way. I'll be whatever you need me to be. So it was good that I was afflicted. It taught me to trust him. It was good that I was afflicted. It taught me to rely on him. It was good that I was afflicted that it taught me that he was able to do. The Bible said, now unto him that is able to do. And not only is he's able, but he's able and he's willing to do it just for you. So you got to know this promise is unto me and my children and my children's children and all them little rashes will fall. She got me a promise to my grandfather, Arling Alexander and that promise transferred to his daughter, Ruth Alexander. And that promise transferred to me, Kevin Ross Alexander. And that promise is extended for my children because my children are blessed. But it was good for me that I was afflicted because it taught me how to love him. It taught me how to trust him. Tears so sweet to trust in Jesus. It said it's so sweet to trust in Jesus just to take him at his word. And this is the part of life. He said, just to rest where my heart is broken. I can rest when I don't know what to do. I can rest when I don't see no way. I can just rest. I'm not going to lay up all night and toss and turn away, but I'm going to rest him. Oh, yeah, yeah, yeah, yeah, yeah, yeah, to rest him. It says all for grace to trust him more. I've learned I can trust him. I've learned I can depend on him. Even in my day of affliction, in the fires of affliction, his hand is still on me for good. Job said, though he slay me, yet will I trust him. I don't understand what he's doing, but yet will I trust him. And see, when you trust God like that, he'll move on your behalf. And the end of the story God gave Job, double for his trouble, because he learned to rest. I don't understand it, but I'm going to rest. I've been in this affliction for a long time, but I'm going to rest. I don't know what to do, but I'm going to rest. So as I look back over what I have learned in the fires of affliction, I can truly say it was good that I had been afflicted, because there was purpose in my afflictions. And at the end of the story, God blessed me. And so understand, I don't care what you're going through. I don't care what the test.  God's going to bring you out. God's going to deliver you. God's going to make a way for you. You're going to learn to trust him. You're going to learn to love him.  God bless you. God has spoken. God has spoken in this place today.  You know, sometimes we've been made promises by some folks, and that perhaps their intention was good, but they couldn't follow through. But then there were some promises given to us, and the person who gave the promise just told a lie, didn't intend to do what he promised. But I thank God our Father in heaven has given us a promise, and he won't go back on it. Hallelujah. Thank God for the promise. Somebody says, it was a song that they used to sing, every promise in the book is mine. And somebody kind of broke that song down and said, that's not what I sing. Not every promise in the book is mine.  Come on, somebody. But I'm not claiming that. I'm standing on the promises of Christ Jesus, the one who saved me, the one who raised me. Hallelujah. I'm standing on the promises that he's given his children. Hallelujah. I'd like to sing that song. Anybody know that song? Standing on the promises is not a way from it. All right, I'm going to get it back, because I must sing this song. Standing on the promises. The devil is a lie. You got the verses? All right, let's go together. I know some of it, and we'll get it together. Standing on the promises of Christ my King, glory turn away, just let his praise history, glory in the highest I will shout and sing, standing on the promises of God that can't end out, fears are safe. I can't breathe my Savior as mine. Couldn't help but going back to Psalm 37. He said, though he shall not, he shall not be utterly cast.   He said, the Lord upholds him. He said, the Lord, when you fall, the Lord reaches down with his hand. Come on somebody. He doesn't call for anybody to come do it for him. He said, I'll take you myself. I'll raise you up and I'll keep you. What a promise. What a promise. What a promise. Hallelujah. Hallelujah. I understand. I want you to raise your hands, raise your hands in submission, in humility. Lord, we thank you. Come on, cry out. Lord, we thank you for your promises. Thank you for that that you've spoken into our hearts. Lord, help us to understand and to depend on you completely because you're our father. You're our savior and we submit our will. Lord, we submit our will. Lord, we have desires of our own but we cast that down and we say, not my will, oh God, but your will be done in me, oh God. And I thank you for the promise that you've given me. Hallelujah. Hallelujah. Thank you, Lord. We praise you, oh God. We magnify you. Thank you for your compassion. Thank you for your mercy. Your mercy that extends forever and ever because we belong to you. Thank you, Jesus.  We thank you for everything that you are to us, oh God. We thank you for the road that we've traveled, oh God. We thank you for giving us the full knowing that you came. Hallelujah. You're going to bring us out, oh God. Just give us strength, oh God, we pray. We'll run every step of the way. We'll run every step of the way, oh God, until we see your face in peace, oh God. We thank you. We magnify you. Hallelujah. We can't stop praising your name, oh God. Hallelujah. Hallelujah. We thank you and praise you. We give you glory and honor. In Jesus' name we pray. Let the church say amen. Amen. Amen. I'm glad I was here today. I said, I'm glad I was here today. Hallelujah. Hallelujah. Hallelujah. Hallelujah. Hallelujah. Thank you, Jesus. Everybody here today look like they want a hiss. So we're not making an altar call for anybody here, but if you're in and you don't have a promise from God that you're satisfied with, it's because it promises for everybody from God, whether you're in the church or out of the church. If you don't belong to Jesus, you need to know Jesus. It's good to know Jesus. Hallelujah. He'll take your sins away and He'll claim you as His own. That's the great thing. And I hear some naysayers say, well the church people are doing the same thing the world is doing. I'm going to help you out with that. The scripture said, if a church person falls, that's a promise. We can do the same thing and if we repent and ask for forgiveness, the Lord will pick us up and hold on to us like we never failed. You don't have that luxury. You got to make it right. You got to get it right with Jesus. You need to be baptized in water in Jesus' name for the forgiveness of your sins and the scriptures and He promised the gift of the Holy Spirit, the Holy Ghost, that will come and live down on the inside of you. Oh, what a wonderful thing it is, elder. While you're going through the test, as you said, He gives you peace. Hallelujah. All right. I'm through. I can go on. Elder Ross, this is good. Thank you. Thank you. Thank you. But you know the Lord, I have to just put all this stuff away because it keeps coming. You all can sit down. I'll just say this. Then I'll turn this, turn it over to Sister Marilyn Alexander. For my early, early years in the church, probably when I was about four or five, if you can imagine, if your mind can go back there. There were two brothers in the church. One of them was a deacon and the other was a preacher. And the deacon was legally blind. He played a guitar and the elder, he was an older man. They both had a guitar and they played and they sang a song. And the song went something like this. My mother doesn't like me here. She won't take me in. And just because I'm saved, living free from sin, why should I fear when thou are near, thou caret, Lord, for me?  The Lord cares for us, no matter who don't like us. Thou cares, Lord, for me. That's all that matters in my mind, as long as he cares. Thank God for his mercy. I'm gonna get out of here. I'm gonna sit down. I can't stop praising his name. Hallelujah. Hallelujah. If you're glad, God saved you. Come on, it's offering time. Saints are the most I got. Come on and bring your offering. Bring your blessing. Bring it with a chill for a spirit. It's wonderful. Hallelujah. Hallelujah. The Lord loves us. Hallelujah. And he's feeding us from on high and we bless his name. It is so good to know that we can depend on Jesus. We can depend on God. He will sustain us. He'll pick us up. Hallelujah. And put us back in our rightful places. Lord, we thank you. Lord, we thank you. Lord, we thank you. The great and precious promises of the Lord. Amen. God bless you. God bless you. Amen. Has everyone been served? Hallelujah. Hallelujah. Hallelujah. Father, we thank you for the offering. God, we thank you for those that gave. And we thank you, Lord, for even those that had not to give. We give your name. Praise and glory. Teach us how to give, oh God. Your word says that if we give, it shall be given unto us. Press down. Shaken together. Shaken together. Yes, man. Running over. Running over. Running over. Hallelujah. And he'll cause men. He'll use men to be a blessing to us. Father, we thank you for that. We give you. We're standing on your promises. Hallelujah. Thank you, Lord, for teaching us how to give. God bless it for this house in the name of Jesus. Make it meet and let it meet every need. In Jesus' name we pray. Amen and amen. Aren't you glad you were here today? Aren't you glad how the Lord encouraged our hearts? Hallelujah. And I think I can run on. I think I can run on. Hallelujah. With joy. Hallelujah. And with great confidence. Hallelujah. In the Lord that I serve. Come on. We're determined. Hallelujah. Thank you, shine now. Hallelujah. Protect us and keep us until we come.


The speaker of the morning, Larry Gooth, who will speak about Lipschitz's constant and degree of. OK, thanks for having me. I was Tom's grad student 15 or 20 years ago, and he was a great advisor. And I wanted to come and say thank you. The winner of my first year, I started doing a reading class with Tom. And that winter, my mom needed an operation. And I was kind of nervous, and I didn't do my reading. And I came to talk with Tom about it. And he was very down-to-earth and real. He shared some of his experiences. And he made me feel comfortable to talk to him about anything. And in that spring, I joined his group. And I was part of Tom's student reading group. And that was my, oh, by the way, my mom did well. She's well, and she'll probably watch the video with this. So then I joined Tom's reading group, which was my most important learning experience in grad school. And it's kind of delicate to run a student reading group, because we're all learning to explain math.  And the advisor has to kind of help everybody understand and help us learn to explain better. And at the same time, to be kind of careful of everybody's feelings in the group. And he did a really good job. He somehow created a sense that we're all trying to understand the math together. And so I have tried to carry this sense, now that I'm an advisor and I have my reading seminar. So to carry this sense, and also when I teach and when I give talks and things like that. Okay, which maybe is a good moment to mention that this talk is in a little bit different area than most of the talks in the conference. And I hope it will be clear to everybody. And so please stop me any time if you have questions. The notation isn't clear. If the main idea isn't clear, or whatever. So as I talk about the relationship between some geometry and some topology. And the geometry is the Lipschitz constant. So I have to start by reminding everybody, what is the Lipschitz constant of a map? Okay, so suppose I have two Romanian manifolds. M and N are Romanian manifolds. And I have a map between them. Then the Lipschitz constant of this map is the supremum over all of the curves in the domain. So I wrote gamma one, because gamma is a one dimensional curve. Of the length of the image divided by the length of the curve. The Lipschitz constant is like a stretch factor. It measures how much this mapping stretches curves. Here are a couple facts about it. It's also related to how much it stretches other things. So the infinitesimal version of this is that if you look at the length of Df of a tangent vector, that is at most the Lipschitz constant times the length of the tangent vector. And if you look at what f does to surfaces of higher dimensions, then you can say that if you have a k dimensional surface in your manifold, then the k dimensional volume of the image is at most L to the k times the k dimensional volume of the surface. So hopefully that gives some sense of what is a Lipschitz constant. And what the talk is about is how does this geometric property of a map relate to its topology? And there are fancier topological questions, but we're just gonna talk about the degree of the map. How does the Lipschitz constant relate to the degree? Okay, now there's an old sort of fundamental proposition about this. It says that if m and n are closed and orientable, so we can talk about the degree, and f is a map between them, then the norm of the degree of f is at most the volume of the domain over the volume of the range times the Lipschitz constant to the power n. What is n is the dimension of these manifolds. Okay, and I'm gonna sketch for you a proof of this. There are several reasonable proofs, but the one I'm gonna sketch for you uses differential forms, because differential forms are going to be one of our main characters going forward. Okay, so let me say a little bit about differential forms. So suppose I have a k form on a manifold, then I'm gonna talk about a notion of how big it is, which is called its L infinity norm. So the L infinity norm of the k form is the supremum over any point in the manifold and any k tangent vectors. Of what happens if I take my form and I evaluate it on those tangent vectors, and I normalize it by dividing by the norms of the tangent vectors. Or if you like, I could insist that I take unit tangent vectors and I ask how big the form could be. So that's the L infinity norm of a differential form. Okay, and this L infinity norm, it also behaves nicely in terms of mappings and Lipschitz constants. So I'll put that on my list of facts. So if I have any omega, which is a k form on the target, and I pull it back, here's what happens. The L infinity norm of the pullback is bounded by L to the k times the L infinity norm of the form. So this one is on N and this one is on N. Okay, so this isn't hard to prove. I won't do the proof, but if you take this fact about tangent vectors and you plug it into the definition of L infinity, it pops out. And another way to think about it is that k forms are sort of dual to k manifolds, and this fact is dual to this fact. Let's see, there's one more basic fact about differential forms. Suppose I integrate a differential form over a k-chain or a k manifold. The norm of that is at most the volume of the k manifold times the L infinity norm of the form. Questions or comments so far? Yeah, good. Why it's not what? Oh, it is, yeah, good question. Okay, it is also, create, the L infinity norm of the differential of the map. Yeah, thank you. And so this is like the infinitesimal version of this. This is the soup of how much the mapping stretches tangent vectors, and that's the same as the soup of how much it stretches macroscopic curves. Yeah, good, any other questions? Okay, so now we have the tools to prove this fundamental proposition. So here's the proof. I let omega be a normalized volume form on the target. So it's one over the volume of the target times d volume of the target, and so it's normalized, so it's integral is one. And also I can think about its norm, its L infinity norm, and the L infinity norm of the volume form would just be one, but because of the normalization, we divide by the volume of N. So now how does this relate to the degree? The norm of the degree of F is the norm of the integral over the domain of the pullback of omega, okay? So now that's now bounded by the volume of the domain times the L infinity norm of the pullback form. Then this norm is controlled by that. So that's bounded by the volume of the manifold times the Lipschitz constant of the map to the N times the L infinity norm of the original form. And then we plug in what that is. That's one over the volume of the target, and that's the proof. Okay, so the question that I wanna talk with you about today is how accurate, how sharp is this basic upper bound on the degree? And this question was raised by Gromov in the 70s, and he worked out a few interesting examples, and then later he came up with some questions. So for example, what happens if we're thinking about maps from SN to itself? So let's say these are the unit N sphere. How accurate is this bound? So these have the same volume, so the volume factors just cancel, and the proposition says the degree is at most Lipschitz constant to the N. And that's basically, that's accurate up to a constant factor. So for every L at least one, there exists a map F with the following properties. It's Lipschitz constant is at most L, and its degree is at most some dimensional constant times L to the N. Here's what the maps look like. So these were discovered by Gromov in the 70s. So here's the domain, and here's the target, and we're going to try to build this map F.  So those balls, they are right over here. So we have BJ, our 10 over L balls in the domain. Okay, how many are there? How many 10 over L balls can I fit in here? Well, they have volume, something like one over L to the N, and the total volume is one. So the number of these balls is around L to the N. Okay, now what do I do? What does my mapping do? Well, for each ball, it maps BJ comma boundary of BJ to SN comma base point degree one. So let's draw a base point on this SN. So over there is the base point. And actually, let me put it at the bottom. Pretend it's at the south pole. Okay, so what happens? I take one of these balls. It's small, so it's basically Euclidean. So it's basically a Euclidean ball of radius 10 over L. And now I'm allowed to stretch it by a factor of L. So after I do that, I have basically a Euclidean ball of radius 10. And I take that Euclidean ball of radius 10, and I wrap it over this sphere, and I put the whole boundary at the south pole to make a degree one map. So that's the mapping on each of these balls. And notice that it takes the whole bound, all of the boundaries, they all go to the base point. And that makes it easy to extend the map. So in between these balls, there's this like blue interstitial region. And what F does is it maps the rest of SN colored in blue and it just sends that all to the base point. So it has Lipschitz constant L because each of these had Lipschitz constant L. And this has Lipschitz constant zero. And it has the desired degree because the degree, there's one degree for each ball. The number of balls is around L to the N. This one should be equals. Yeah, this one could be equals. Yeah, you're right, I did. Any other questions or comments? So for spheres, this proposition is sharp up to a constant factor. And you do actually need the constant factor. So Gromov showed that you can't have degree exactly L to the N. And you can't have degree 0.99 L to the N. You need some modest constant. Okay. And then he raised the question, how often does it happen that this proposition is sharp up to a constant factor? So here are some examples. So proposition one is sharp up to a constant factor. Okay, so we just saw it's true for maps from the sphere to itself. It's true for maps from a product of several spheres to itself just by taking the products of the maps we've constructed. It's sharp for maps from CPN to itself. Takes a little more work.  But there are many examples that were not so easy to understand. And about 20 years later, Gromov proposed a problem, a particular manifold that he couldn't figure out what the answer was, and he thought it could be interesting. And what it is is a connect sum of two of these things. More generally, I'm gonna have XK be S2 cross S2, connect sum to the K. And this problem maybe is in the Russian tradition of problem solving, where the idea is to give the simplest problem that goes beyond current ideas and points towards maybe a general theory. Okay, so the problem Gromov asked about is what if you take a self map of one of these guys? So let me give it a name. D sub K of L. So that's gonna be the maximum of the degree of F, where F is a self map of X to the K and the Lipschitz constant of F is at most L. Okay, so Proposition 1 says that this is at most L to the fourth. And the question Gromov asked is, is it approximately L to the fourth, like for these other manifolds, or is it actually much smaller than that? And trying just a little bit to build examples, the examples had much smaller degree, maybe like L cubed. Okay, so that question was open for 20 some years and my graduate student, Sasha Bairdnikov, made a cool construction for S2 cross S2, connect sum with itself. He proved, so Bairdnikov, for K equals two, D sub K of L is at least a constant times L to the fourth. And his construction also works when K is three. And he described this to me in my office and then he said, but I think K equals four will be different and he was right and he worked together with Fegermanine and they proved it. So theorem, Bairdnikov and Manin. So for any K which is at least four, D sub K of L is little O of L to the fourth. So remember that little O means that it's less than any small constant times L to the fourth, asymptotically as L goes to infinity. Cool, so there's a phase transition between three and four. Okay, and then we worked together on understanding better, more quantitatively, what this little O is like. And we pinned it down decently. The three of us together proved that D K of L is in some range. So it's less than a big constant L to the fourth over log of L to the half and it is more than a little constant L to the fourth over log of L to the fourth. So it gives the, is that legible from? Okay, great. So it gives the general shape that this thing behaves like L to the fourth divided by some power of log L and we don't really know what the right power is. So my goal for today is to try to give some intuition why there is a transition between three and four to describe something about what these maps are like and something about why these are different. Any questions or comments about the plan? Right, right. So the question is, is there any connection with simplicial volume? There might potentially be some connection but there's not any very direct connection with simplicial volume. Right, thanks for this question. Okay, so the question is basically, what is the metric on X K? So to define the Lipschitz constant I need to have a reminding manifold with a metric. And for all these phases there were canonical metrics which is what I've had in mind. But for this space there is no super canonical metric. So we pick a metric. So G is some metric. Now, this also depends on G. So okay, I'll stick a little G over here. But these results are independent of G up to the constants. So if I stick a little G over here the only thing that changes in this result is that that constant depends on little G. And the only thing that changes in this result is that those constants depend on little G. Oh, is there any hope that the constant doesn't depend on the metric? No, I don't think so. But conversely, it's a nice feature of, if you ask about the exact value of something like this it depends on the metric almost always. But if you ask about the asymptotics up to constant factors then it's independent of the metric. So in some sense it's a question about the topology of this space. And FEDGA has developed this into a fairly systematic theory of how to compute these things for many different manifolds. And it's based on the rational homotopy properties of the manifold, which I don't know super well so I'm gonna focus on this example. So now we're ready to look at this picture that I pre-drew. This is a picture of the manifold S2 cross S2 connects some with itself. So on the left here I have a copy of S2 cross S2. This is like the connect some part and here's a second copy of S2 cross S2. So this picture corresponds to K equals two but you can imagine bigger case. Okay, and I drew some stuff on it. The main characters in the story are going to be the differential forms on these manifolds. And so I drew something that hopefully helps suggest kind of standard basis of differential forms. So what is for example A1? So the AIs and BIs are two forms. It is Poincare dual to an S2. So actually each of these forms is Poincare dual to an S2. So A1 is this, so okay. So this here is an S2 in my S2 cross S2. And then I tried to draw like a blue neighborhood of it. And A1 is a differential form that lives on this blue neighborhood and it is Poincare dual to this S2. And so on for all the other letters. Okay, so I'm gonna go over here I think and I'm gonna explain our plan. Okay, so I'll start with an observation. If I take the integral on XK of A1 wedge B1, what do I get? Well this was Poincare dual to an S2. That was Poincare dual to another S2. So the wedge product is Poincare dual to their intersection which is a point. So this integral is one. And therefore if I have a map from anybody to XK, then the degree of the map is the integral over the domain of the pullback of A1 wedge the pullback of B1. So this is the, we're gonna try to study that thing. Okay, now our main lemma is that if I just have a map from a ball, that this is a standard Euclidean ball to XK, and if K is at least four, then when I look at the integral over the ball of the pullback of A1 wedge the pullback of B1, then that is smaller than L to the fourth over log of L to the half. And this implies the upper bound down there pretty easily. So if I have actually any manifold mapping to XK, I cover it with charts which are balls. And if I restrict the map to each of these balls, then I can apply this lemma to bound this thing.  So everything can take place on a single chart which is a ball and this is the main thing we have to understand. And on the other hand, if K is two or three, this inequality is false. This integral could be as big as L to the fourth, constant times L to the fourth if K is two or three. So here's our plan to try to understand this. So I'm gonna suppose that the integral pullback of A1 wedge pullback of B1 is around L to the fourth. And if K is two or three, that can happen. And we're gonna try to sort of visualize how it happens. In particular, the thing we're gonna try to visualize are the pullback forms. We're gonna try to visualize the pullback of A1 and so on. Okay, and if K is at least four, then in our visualization, we're gonna see something that starts to look like it can't happen. So we're gonna try to move towards getting a contradiction. And that would sort of, so if we really got a contradiction here, this would be a proof by contradiction that says this cannot be as big as L to the fourth.  All right, so let's begin with the case K equals one. So here is my four ball. And inside of there, I'm gonna try to visualize the pullback of A1 and the pullback of B1. And it might look as follows. The pullback of A1 might look like some parallel two planes. And then the pullback of B1 might look like some parallel two planes going in a transverse direction. Okay, so let me do some writing to go with this picture. The pullback of A1 is supported on the horizontal slabs. So it lives here. And what are they? They're thick planes and they go in the x1, x2 direction. And now, I don't wanna write anything else about that. Okay, and I guess one other thing we can say about them is we know how big they are. So the L infinity norm of this should be around L squared, which we thought about earlier. Okay, and then the pullback of B1, everything is the same, except it's on the neighborhood of these vertical planes. And let me also put a little scaling scale in this picture. So that distance there is one over L. And the thickness of one of these guys is also approximately one over L. So that's maybe one over 10. Is all the description of K equals one. So this two plane, which has thickness one over 10 L, it got stretched out. So it's thickness got increased by a factor of L to be as thick as this thing. And then each bit like that goes here and wraps around that two sphere. And so if you go down this two plane, it wraps around the two sphere many times. So that's what the map looks like. And then, all right, let me pause there. How is your image coming of these pullback forms? So then let's think about the integral of pullback of A1 wedge pullback of B1. So A1 is a two form. Yeah, right, that's right. The picture is confusing. So this is a two sphere and A1 is a two form. Okay, so what happens when we take the integral of the pullback of A1 wedge, the pullback of B1? Well, the pullback of A1 wedge, the pullback of B1 lives on these intersections here and here and here. And there are L to the fourth of them. You can count by thinking about it. This is like a four dimensional grid with spacing one over L. So there are L to the fourth of these intersections. And each one of them is just a pullback of this. And so when we integrate that one, it's like integrating that one, which is one. Okay, so this is around L to the fourth. So, okay, so we did the case K equals one. The case K equals one isn't a hard case. It's just S2 cross S2 map to itself. It's just the product of two of the mappings that we constructed earlier. Let's go to K equals two. K equals two, something tricky happens. So now we have to think about the pullback of all four of these forms. So maybe the pullback of A1 and B1 that might be as above as before. But then what do we do with the pullback of A2 and B2? Okay, here's a first attempt. Here's my B4. I'll draw it a little stretched out. And the pullback of A1 and B1 look just like before. And I think to myself, well, that worked well. So maybe I should make the pullback of A2 and B2 look similar and maybe I'll stick them over here. Okay, so this is not possible. And let me tell you what the problem is. So this attempt does not work. Here's what the issue is. A1 wedge B1 is the same cohomology class as A2 wedge B2. So these are equal in H2 Dram. And that means that there exists three forms, C, where DC is A1 wedge B1 minus A2 wedge B2. Let me draw C in the picture here. So geometrically, C lives on a neighborhood of a curve that goes from A1 wedge B1, which lives on this ball, to A2 wedge B2 that lives on that ball. Okay, this picture is now really confusing from the point of view of the dimension of everything that came up before. So that's a two-sphere and that's a curve. So let me try to label it a little bit. So C is a three-form on XK. And C is supported on, lives on, a three-disc cross a curve. And the curve goes from here to here. So if our pullbacks of A1 and B1 and A2 and B2 look like this, what would the pullback of C look like? Well, there would be one strand of it connecting each disc of A1, pullback A1 wedge, pullback B1 to each disc of pullback A2 wedge, pullback B2, have all of these strands. And the problem is that if we think about the geometry of these strands, there is not enough space to put them all in the four ball. So geometrically, the purple strands, thickness, radius, one over L, they have some length that I'll call A, which we don't know what it is, but in the picture, in attempt one, A would be one. And the number of purple strands is L to the fourth. Okay, let's digest each of those and see if they make sense. So why do they have radius one over L? Well, if I take a purple strand over here and I map it forward by F, it needs to cover this purple tube over here. And the thickness of the purple tube over here is some fixed number. It may not be huge, but maybe a 10th or a 100th. And this purple strand can only stretch by a factor of L. So its thickness needs to be at least around one over L so that it can cover the strand in the target. Okay, so that's why this is true. Its length is some number called A. That's just the definition of A, so we don't have to check it. And the number of strands is L to the fourth. That's because the degree of our map is L to the fourth. So there are L to the fourth, there are L to the fourth pre-images of this guy and there are L to the fourth pre-images of that guy. We have to connect them all. So we need L to the fourth strands. Okay, so what's the total volume of all that? The total volume is L to the fourth, one over L cubed times A, which is L times A. And that should be less than one because it fits in the unit ball. So the conclusion is that A is actually smaller than one over L. So these strands that connect a blue vertex to a red vertex, they can't have length one. They need to have a much smaller length scale, one over L, like that length scale. So let's try again, now that we know where A2 wedge B2, now that we know that we need to have these short purple strands. So attempt two of our four ball. We have pullback of A1 and pullback of B1, just like before. And we know that we should put the vertices, pullback of A2 wedge B2 close by so that they're close enough that we can connect them. So what are the labels in this picture? So the blue stuff is the same as before. This red dot is the pullback of A2 wedge, the pullback of B2, and this purple thing is the pullback of C. Okay, but now we have a new issue, which is, I know I've picked what that should look like, but what does the pullback of A2 look like? And here's what's tricky about it. The support of the pullback of A2 is disjoint, the support of the pullback of A1 or B1, just because the support of A2 itself is disjoint from the support of A1 or B1. Okay, so I would sort of like to make the pullback of A2 to be some kind of two-plane like I did before, but that two-plane is not allowed to hit the blue two-plane, so that hopefully it looks sort of like this, and we'll have to take out our four-dimensional visualization skills. Think if this makes sense. I would like to have the pullback of A2 and the pullback of B2 be some parallel planes like before, but I want this plane there to somehow be going behind those planes there and not hit them. So here's how to do that. At this point, so we're gonna try to do some visualization in 4D, so in that sense, I'm in a good place. But at this point, I think it's best to describe in coordinates by algebra what all these things are. So let's remind ourselves. The pullback of A1 is supported on some thick, thickened two-planes in the x1, x2 direction, and the pullback of B1 similarly supported on some thickened two-planes in the x3, x4 direction. Now what are we gonna do with the pullback of A2? It's gonna be supported on some thickened two-planes in a different direction, say the x1, x3 direction, and then the pullback of B2 will be supported on some thickened two-planes in the x2, x4 direction. So what do I wanna know? I wanna know that this two-plane doesn't intersect either one of those. So let me explain why this two-plane doesn't intersect that two-plane. So I brought some two-planes with me, two-plane, two-plane. Okay, so one of them is going in the x1, x2 direction, so that's this two-plane, and one of them is going in the x1, x3 direction, that's that two-plane. I brought a stiffer one, and they appear to be intersecting along a line. But what I have right now, and what I'm holding up in front of you, are some two-planes in R3, and these two-planes live in R4. So if I want to get them to not intersect, all I have to do is take this two-plane in my right hand and move it a little bit in the fourth dimension, and then it will become disjoint from the two-plane in my left hand. So as long as I may put these sort of in general position, these ones won't intersect that one, and for the same reason these ones won't intersect that one, and for the same reason these ones won't intersect these two. But those two will intersect each other transversely, just like we'd like them to. So that's the trick of how to arrange all these things to build the map from a ball to X2 that has a really big pullback of this differential form. And secretly we have almost constructed the mapping. If you look at this picture, like various parts of this picture, I've told you should be mapped to various parts of X2. So I know who's mapped here, I know who's mapped there, I know who's mapped to C, and so on. And to finish the job, I just have to fill in the map to send the in-between stuff here, which admittedly looks like a mess, to the in-between stuff over here. And the in-between stuff over here is a four ball. It's not very topologically, it's trivial to map something to a four ball, and geometrically it's not that hard either. So that's how we do it for K equals two. Can think for a moment, but I don't have too much time. So this is for K equals two. Then if I had to go further and do it for K equals three, what would I do? Now I need to have a pullback of A3 and a pullback of B3. Okay, so I need another plane. And there's something I haven't done yet. I haven't yet done the X1, X4 plane. And this one is the X2, X3 plane. So that's how we map to X3. Okay, and then if I try to map to X4, I have a problem. So first of all, I have run out of coordinate two planes. There are not any more of them. And then I say, well, maybe I'll try a plane which is not a coordinate two plane. But if I take a plane that's not a coordinate two plane, some funny angle, it will intersect some of these two planes. So it will hit the things that I've already put there a lot. And even if I'm allowed to curve it or something, it will hit the things that I've already put there a lot. And that was the intuition that Sasha explained to me in my office like three, four years ago. Why he thought that the case K equals four should be different from all the other cases. Okay, now that's not a proof yet. The main issue is, you know, in this special example, I assume that the pullback of A1 was supported on a bunch of parallel planes. And that doesn't have to be the case. So we need to make this argument more general. And just briefly say, there's some analysis involved in doing this, especially in getting this quantitative bound, is based on doing Fourier analysis on these differential forms. We break them into low-frequency parts that are simple and those low-frequency parts look sort of like this. And we can tell a version of this story that why K equals four is different. And high-frequency parts, and Fourier analysis lets us prove some bounds that the high-frequency parts don't contribute very much to the degree.  Okay, thank you, Tom, and thank you all for having me. Questions? Yeah, that's correct. So the function that we defined, if you change the metric, it only affects this function, dK of L by a constant factor. Yes. That will also only change the constant by a constant factor. And that's because our question about degree is a homotopy question. So if two spaces are homotopy equivalent and the maps in the homotopy equivalents have some Lipschitz constant, then all the homotopy properties will agree up to a constant factor. If we were to ask about the Lipschitz constant of some diffeomorphisms or something like that, then maybe we could get maybe the smooth structure. Except I haven't come up with what is an interesting question to ask. I would involve that. Right, the question is have I thought about a more interesting format of fold, like a K3 surface? I personally haven't. Based on this, FEDJA has been trying to develop like a systematic classification. You're given any manifold of any dimension and you try to compute these things. Which is going pretty well, but I don't know if it's like completely and totally covers all the cases. And topologically what it involves is the rational homotopy theory, the like minimal, Sullivan minimal model about which I wish I knew more. Yeah, everything we said adapts to a connected sum in higher dimensions. Right, that's right. So the number three is special to S2 cross S2. There's a different number, which I think we've even computed what that transition number is in each dimension. For S n cross S n, it probably, it is that dimension, yeah. So Tom, building on David's comments at the beginning of his talk, Tom just answered Francesco's question. Sure, the answer is if you look at S n cross, if you have maps from S n cross n connect some K times to itself. The transitional K is you take the dimension of lambda n of R2n. And you divide it by two. Any more questions? Yeah, you get the same, you get the same kinds of answers. Yeah, yeah. So I think if we look at connect sum of S k cross S n minus K some number of times that we can say where the transition is and you could connect some with different with various Ks, it's a little bit messier. But in, I mean, I think in principle, we could reduce the question to some purely topological question. Any more questions? Okay, if not, let's thank Larry again.


So, um, she is... If you're new here, my name is Kyra, do not forget to like, comment, and subscribe. My coffee looking so yummy. But anyway, before we get into this vlog, I want to show you all some pieces I got from Boohoo. They are sponsoring these videos. Thank you so much. Boohoo are sponsoring. I've been wanting to work with them for a while now because, girl, I'm shopping at Boohoo. I'm looking for some of the pieces, okay? So, I'm about to show you all the things that I got. So, let's just start with these sweatpants. Um, this is a brand they have on their website. It's called OFCL Studio. And I really like how these sweatpants look. They do have a matching top to it, but it was sold out in my size. But these sweatpants look so fire. I cannot wait to style these. In my closet, okay. I always got to have a good pair of sweatpants. And these are in the size medium. So, I cannot wait to wear these. I love this color. So fire. And like I said, this is a brand they have on their website now. So, and I really like the stuff from this brand. So, make sure y'all check them out. The next item that I got are these Q another pair of sweatpants. That's how like they're gonna be good for when I like, you know, I'm wet or something like that. This is gonna soak it right up and make me dry. So, and y'all know this summer time is coming up. Um, we got a pool and I'm gonna probably wear these out by the pool. So, and like I said, I love this brand y'all. This is on their website, OFCL. And I like this color. I bring really into like the aqua blue. So, I had to grab these. Like I said, sweatpants are just an essential in my closet. So, then the next bottoms that I got are these leggings. Y'all know I've been into working out. So, I got these leggings.  Or white sports bra. It's just this cute like blue deep blue. It's like how the pattern looked. And um, as you can see these are size small.  I don't like when leggings fall on me. Um, I'm getting smaller y'all. So, I had to get a small because I don't like I said, I don't like when my leggings fall on me, especially when I'm working out. And these look like they're gonna compress me in all the right areas and just hold tight. And so, I don't have to worry about like so, I don't have to worry about keep pulling up my leggings when I'm working out. So, so yeah, I got a pair of these. So, yeah, they have a really good athletic wear selection. So, check that out. And lastly, I got some bathing suits because I've been really lacking in the bathing suit area y'all. And Buhu has real affordable um, pieces and they all hate. Okay, so I had to grab some swimsuits because I wanna so I got this two um, what is this called? The two pack triangle bikini in black and white, simple easy. So, and I really want the black one and it has padding in here so I got like a simple black one and then the simple white one as well.  Then these are the matching white bottoms. And then, before I even seen the two pack I ordered uh, basic black one but this one doesn't have padding in it so if you're a girl that doesn't care for padding, this one is for you but if you do like padding those two piece, the two pack triangle swimsuit will be for you if you like padding in here some suits. And then the last thing that I got are these cute bottoms which I'm gonna match with the white um, triangle top I just think this pattern is so pretty so cute the orange and yellow perfect for the summertime so, you know, when I go outside in tanning I'm gonna throw these on. So, yeah, I just like this color and also I really love orange in the summertime especially against my skin cause I'm more tan so I just really love those colors together so yeah so yeah girl that is all I got from boohoo and you can shop everything that I have all the links will be down below in the description box and yeah, thank you so much boohoo for sponsoring this belong so yeah, let's just hop right into the video Hey y'all, what's up? I'm back to another video that's your new hair, my name's Karim Karim anyway, I'm in Target earlier so my mom is back inside so I did all these things surprise cause we're about to paint the house and well, to her like me so I think I'm dirty please cause she's a O.T.D. so she sent me a list that I have to get and yeah, we're here in Target, I need some more too guys so I run out so we're here in Target and I'm gonna get some surprise I'm gonna just do a b-roll because oh I got my nails done too I don't know if I like them they're kind of big but whatever cause I told him take first where but I should have told him crossing I don't know, anyway um, so yeah we're here in Target we're gonna do an early sale it's white as fuck, I love it it's Monday morning and anyways y'all it's Monday morning and y'all know my girl birthday is on Wednesday Aliyah her birthday is on Wednesday and she's going out of town tomorrow but she has to drop her dog off at her mom's house so basically my girl booked her a trip to go out of town and um yeah, so she booked her trip came playing nothing for her cause she don't want to sit down so today before she leaves I don't know if y'all know this but she lost her luggage like a while ago like a mini, she lost everything in there woo woo woo and she's always carrying a carry on so like a little roller carrying on so I'm like and you know, sometimes you want to buy them but you like I ain't spending my money on that like I'll just wait until somebody to spend the money for my birthday or something but she never told me she wanted this but since she lost her luggage I was like fuck it I'm gonna just get her rolling Louis luggage so I'm on my way to go get that now me and Tasha are surprising her today we popping up at the crib so yeah I have to celebrate my girl before she leaves on her trip because that's my home girl like I know like I go on my way for her cause she would do it for me like that's like my girl like that's like my girl, she's so supportive I have never met anyone like her like her energy matches mine you know what I'm saying so this my girl, I'm gonna make her feel special on her birthday cash now on this grud and gonna spend a check so I'm about to meet D Charles I told him what bag I wanted, he isn't there yet it's his off day so he coming he coming in for your grud okay I'm gonna like get in him now and then so me and Tasha are gonna meet up at two I gotta get her her balloons and I'm gonna get that ahead of time so it's already done because I gotta finish up a vlog girl I should have been finished this vlog yesterday sometimes when I'm editing y'all you have to think, when you're editing you have to think about like where you want your clips and stuff like that and girl yesterday I don't know why I just couldn't think like I was so out of it then my home girl called me we was on the phone for a little bit and then I fell asleep so but yeah so I'm gonna go home and finish a vlog I can't even go over there so it's ready to go live and there's a party city right here so I'm gonna get her her balloons her gifts it's gonna be so cute and then Tasha's getting her a gift too and some flowers so I'll probably check in with y'all once I get two fips because I'm gonna listen to my music so when I think he didn't like I said I know I said that I was like wow I gotta like that and I just moved to Atlanta but yeah like I said we couldn't get her gifts I gotta come in here to get Chanel for a scene but then we headed to the we lost her luggage at the airport and she always bringing a little rolling one with her hold me down alright I need the where's that's at the rolling one just the people the ladies I got it's out of stock they don't have that lineage it's nothing here like this I'll tell you we got there's more of this than that so let me see let me see if anybody has that we need to buy today the whole this will work the 50 close place I got it since now the Orlando we ain't got nothing rolling that's ugly as shit what does Linux have they don't even pop up on here they don't have anything let me see if they have anything else should I just get her a duffel then that would make more sense because right now you won't be able to find it no I don't know but I know she got a little roller she already has a roller I'm checking all these different ones right now I need to go to a 50% that's still carrying I don't know but I don't know if she's going to want to carry that because she always carries a roller carrying they don't have this one if I ordered the one from Cincinnati how long would that take they're not going to ship it today they're going to ship it the next morning it will process tomorrow damn how can I tell her I got her this but not tell her that the shit won't delay because I don't have money today they have to accept it it won't just be Cincinnati accepting I'll just give her the receipt and be like that's the best shit to do do you want to check anything over here do you want to see if you got a lunch I'm going to come back if they don't just text me or let me know damn I'm mad what's up they have rolling things sometimes they don't work I was like you're lucky I won what about Fendi okay what's going on I know I wasn't here all the weekends don't work do you have any rolling luggage I know last time you had like two was it cute it was the one with the big cream she also like sure Dior she did I don't know y'all so we in Dior now trying to find a luggage trying to find a luggage but it's hard because everybody got money now and also I'm like who is really buying the luggage right now everybody everybody is traveling me outside I know I'm going to have to order you know what you wanted this that's what I was going to get picture before I do all that we are the last minute crunch look at my luggage it's our birthday today newton is sold out and I ain't getting no chameleon yeah it's like 8-9 you should know luggage is like 8-9 it's easy 8-9 easy very fashion girl this is the best one let me see the green one that's cute and this one so this would be like the it comes in a few colors and then it also comes in a few sizes this one is really cool I'm most lucky to carry on this for a woman too so this comes in a carry on as well okay do you have another color so what do you have on your offsite it's a luggage it's carry on and it has the blue oblique print on it like that oh like all damn I wish I had a pig I'm gonna go get it that was nice I'm gonna get you one too no it's okay is it this one yes but not this size it's a little smaller the removable ones are really so this is the small size which is like carry on it's and you have this I can get this here for today's memorial day so Wednesday I don't know if she's gonna like that that's very mainly thank you though you're welcome are you sure they don't have it at linux let me you know what I'm gonna ask Jeremiah to call his shop over here just to give a check look at the puppy oh my god I need to pet it oh my god can I pet him so cute no I want a chicken wing the curly poodle oh I know him the boy yeah damn they be like oh what about go y'all hi do you have a rolling luggage yeah I am why everybody was blogging my hair looks like this what's up what are you doing we looking for a rolling luggage so let me see let me see you know you know it's sold out too though really yeah but if not we're gonna go downstairs it might be the best option I was thinking gang gang gang I'm a pro he got the plug got the luggage they had it in red but y'all know this girl is a neutral girl I know my girl I know her and I know she's a neutral girl and the suitcase has just a little bit of red in it like on the like a little red patch but hopefully she likes it if not she can go ahead and exchange it she's gonna have to order it but at least I get to show up with something before she leaves you know hopefully she likes it it's cute it's giving black girl luxury okay and Tasha's gonna meet me later so it's gonna be a little cute back to my house because I gotta finish some edits on this video and also I have to give my mom her boo because she's a hangry girl so I'll probably check in with y'all later once I am on my way to her house to give her a gift I'm so excited I gotta get I gotta write out a card I'm gonna put I hope this replaces what was lost that's cute anyway, I'll check in with y'all hey guys y'all we waiting for Ilya to face we waiting for her to come she got like her flowers so cute she got balloons in the back yeah she got balloons she shouldn't avoid us but she like give me 30 minutes she know what the fuck going on she know I'm gonna stop playing but I'm gonna check with y'all once we um surprise her whatever you did it we thought you did though it's a lamp for me yeah they cute it's real cute wait this gotta come down some some glasses but she was like she don't wear it on vacation let me see let me see what's giving they're cute but I wouldn't pick them those are really cute she's a fashion girl please stop lying to me she's a fashion girl I'm so shook right now like uncomfortably we are too I don't want me yet I'm not gonna let you in the house no I'm saying it because we have a good time because we can talk about anything and everything because we're there for each other to lean on or listen it's not everyone that can be this kind of friend but you are so on your birthday I'd like to say thanks for being that kind of friend a true friend I don't know what I do with that you I love you take the day off and relax you work so hard and deserve it wow that was a good card did you just wear a lot of money did you just shoot for a month and there's one in perimeter so you have two locations one avalanze I just have to say the avalanze is bougie I'm not there thank you so much you're very thoughtful you had everything so I was like Kyra what did she want you do no for real he hasn't been to the spa I remember you said you wanted a spa day thank you so much okay the card is right there I like this card this is really big Kyra it's making me extremely uncomfortable I can't lie your future looks bright must be Otto's candle that's basic friend I hope this replaces what was lost love Kyra why this must be where should I put these where we have a mental symptom yeah in the corner there she goes over there too she's not a jumper well she be jumping on couches if you aren't leaving I'm comfortable yeah it's the day of a bed delivery day my mom got the second camera on while they were doing this but let me go on this side I'll just mention whatever I get on that side and whatever I got over there and then minus the 83 nah not really just go like you know if I got like 41 over there I can just I want 41 no because I have the night stand oh how big are the night stand 1 just 1 night stand 2 night stand 2 so it's 38 41 and a half 38 and a half it can go 38 and a half let me check it out let's round it up to 36 36 a little more space yeah yep got it thank you guys because this is a bit too heavy to push yeah especially after you have yeah and then we still have 34 for the end tables and you still have the mattress yeah would you like it to set it down a little more do you want to have it that high what or do you want that mattress to be high oh how is it going to look I don't know how high it's going to be with the thing have an idea I guess after you're after let me see this goes away you got to open it this is going to go like I don't want it too low is that fine did you have a pretty much decent set on top of that yeah just flap it thank you it's not that it's tall oh I guess it's fine it's fine yeah but this one but I just wanted to see how difficult it is oh you want to hide all the black you have the remote for the yes so just lower it so you don't see the black thank you if we take that this one this is so cute thank you my bed look it's so cute oh my god I'm going to do a bedroom makeover but this is what it's so this is the night stand this has been here the night stand has been here already but one of them broke because the movers had dropped it and they broke the marble like seal it or something she's going to have a professional do it so but they look so cute they're going to look so cute next to my bed hey y'all so me and my mom are spending quality time today first stop is starbucks because obviously we need our starbucks what I'm not getting them you can't vlog in time I'm literally far from the car you who I'm far from the car yes literally so we have starbucks first stop and we're going to the movies to go see a quiet place too starbucks this is tara we're going to get started for you today hi tara we're going to be on our way to the nail shop sugarcoat midtown I like how they do my feet the best you need to go get a car thing because this right here is not making it cold we just got done seeing the quiet place I just need you to be quiet right now how did you like it it was good you really like your pedicare because your feet is all over here in the spotlight that tattoo though why is that tattoo there you won't tell us that though second toes moment but yeah we just got done seeing the quiet place some people we make this man on the elevator like the security man he was like y'all going to see that and that's like a 7 out of 10 but I was like yeah alright but it was really good but obviously it's going to be a part 3 for sure how it ended if you see it but we're getting I'm going to have some food hey y'all so it's the next day and we are in the process of re doing this room she want to put her little touch on this and um the carpet the rug in this room was not up to par y'all it was just so wrinkly and just not cute but um she is my my um found a regimen or something to like make the rug better or whatever though I'm about to take this stuff off the shelves y'all this weather is making me so sleepy I didn't get to show y'all the rest of what we were doing yesterday because my camera died but basically what we were doing was filling the holes in with these white little screw things so tedious like but oh here they're from Ikea obviously but it's these little white things that you just put in the hole so you won't be able to see anything obviously so it blends in and saw seamless or whatever but anyway so today Sunday and I am modeling for Jasmine's lip class I'm going to be her model so she's going to be demonstrating on me so I'm just waiting for her to come get me um I wish I could have showed y'all my baby hairs from yesterday that was kind of cute with this ponytail but yeah messed up so but anyways I'm I'm waiting for her to come y'all I cannot wait till my couch comes like I just can't wait but anyway I'm waiting for her to come and then I gotta come home I know she's gonna want to celebrate after I just know it but I have to come home and work like I have to like there's no answer but so but I'm excited like this is my first one of her this is my first time going to her class in Atlanta so I'm excited to see her in her element so and obviously I couldn't wear lashes but my skin looks so good like it looks it's baby soft it feels so good y'all if y'all watch my skincare routine girl tea period my skin is so smooth it's so soft it feels like a baby but I swear to God like I'm obsessed so so y'all this is a little behind the scenes of her class she's matching me for my foundation so she's the expert she know yeah this is perfect look hey hey to the vlog yeah yeah it's gonna have to come all the way down here see yes perfect that's my color I need to go get that yeah wait how do you feel about my skin is my skin popping or what no it looks good okay so we just had to stop at her shop because obviously behind the scenes she had to match me but we're gonna be on our way like I said this my first time going to her class in Atlanta she looking cute I need to go get those your slides this shit is too big look at these I should have got a small bitch you look good I was just like do I look skinny you looking real good bitch I'm gonna play this back for you okay yeah let me see cause I don't believe wait let me it was a real cool class it's okay for me so um I think they're closing up she's having like a little talk so she's on camera give her my place so I can be a little over is it giving crime what's the time 1-1-12 it's good some closet arrangements yesterday my mom had did it while I was away I was doing that make-up class for Jasmine and I really like this I'm not gonna lie she snapped so I love how she put all the YSL and the YSL the perfumes I was really iffy about my perfumes being spread out but um it makes sense so I'm not really mad at it at all the Dior now and then I'm gonna get another perfume tray to put on the side in the middle so it's just on display cute and then the Louis section Louis shoes the Hanifa flowers and I'm gonna get a Louis Duffel that's gonna go on top of there so so yeah that's what it's looking like so far in the perfumes that I um are waiting to get on display just at the bottom so but yeah let me know how y'all like this so far and make sure y'all shop my Amazon store that's where the um the perfume tower is at cause y'all are asking me where it's from so but yeah she's just my mom is just out here getting this rug put together for my beauty room, posties it was just wrinkly and ugly and I didn't like it and tomorrow my couch comes y'all I just got my rug delivered for my living room we're gonna set that out tomorrow before the couch comes and tomorrow um a man is gonna come and put this film on the windows cause my mom's so sure she don't even gotta do all that she like no I want you to to best out of y'all for me and girl I got neighbors across the street across the way that is looking in my way I can't be naked but out here they're gonna be able to see I'm gonna be able to see them but they can't see me so the man is gonna come here and do that because I don't really want to you know block my view you know what I'm saying I want to be able to see my view um and have my windows up in the background of my couch so yeah girl my mom a little extra you know let's go so but yeah I'm just gonna finish working oh my god my closet looks so cute from right here it's so cute she always coming to do something I swear but I'm about to start back working and I'm gonna check in with y'all later I just had to update y'all because I've been working all morning and I haven't been really vlogging so yeah I had to update y'all good morning y'all it is Tuesday June 8th and the man is out there putting I hope y'all can hear me the man is out there putting the stuff on the windows so people can not see me but I can see them touching I just got out the shower my couch is literally on the way they're outside but it's just getting pushed to the side because I still gotta lay out my rug and like iron it and stuff so it's nice and then I'm gonna push the furniture and align it how I want it so so I'm just saying that call me back and tell me they're outside so y'all so we just had to go to y'all's kitchen it was like a seven out of ten what would you rate it the chocolate cake was good though you better go get my chocolate cake I'm not fucking playing with you tell me what I can't eat she want two sweets in one day right now I want my cake we are gonna get my artwork I don't care you better take your day off today we're about to no we ain't about to do nothing no we're not gonna do all that why why you don't want to do it what is this black thing we can start doing stuff I'm gonna check in with y'all but I got some work to do so Gigi fans welcome back to another another show welcome today's episode will be cleaning my mom's store I'll be putting a few more up there for you I don't know why she don't got the cleaning gene it must have skipped her maybe my grandbabies will have it but this one help she don't know how to clean clean she pick it up but she don't know how to clean clean like clean clean okay KGG if you don't know what KGG means we're a girl gang let's get it together so this is the next day I just got back from my what did I get? capitation from my girl out here at opolar so now we're gonna get to tackling the living room area before we're about to tackle the living room area I'm gonna sweep so we're gonna sweep so let's do it we just got your toys on do you eat them at night? no they messed it up we like this this take over, Michael Jackson style oh lord wait from the pole little white couches we just got yesterday it's in the same family okay see if you have cardboard floors you will win so we had an order of dice and peas for your cardboard floors this from amazon? yes it's on the amazon store amazon in the cleaning section make sure y'all shop on my amazon store obviously I've been cleaning up her cause like I'm modeling her that way so you want to demonstrate how we're going to do this she just took out one took out one crease like half of this crease I'm gonna demonstrate how to do that so just in case y'all have a crease rug this is how you get it out take you back for another episode of cleaning at 101 with Angela aka Angel aka my name is she before you put the damp and I just washed this real quick in the washing machine and then after it's done took it out because I had already did the rug and the glam if you want it on steam and not leave it in cause it's on steam don't keep it in one area too long so we have to do this for the whole rug area so I'll tune back in which y'all want to see it done just later so now we're gonna vacuum it this attachment cause you don't want to vacuum it if it has a beater bar that's a no no only beater bar it's for carpets this is a rug and that's carpet okay it's a flat four rug only and I'm gonna have vacuum her own rug this is what my living room is looking like so far so now I'm gonna hunt for a coffee table a tv stand and a tv and um stay tuned for the next vlog I'm gonna be doing a bedroom makeover so stay tuned for that but let me know how y'all like my couch I love it the quality is amazing I've worked away and I just am so excited so I love y'all so much thank you so much for tuning in and this couch would not be here without y'all so I love y'all so much and I appreciate y'all so much so thank you and I'll see y'all next video