Everybody forced here ripping a select football hobby 2021 select football hobby three bucks number 20 random teams. I'll leave the results on screen. You can always find them a latent sports cards calm There's a link in chat. Thanks for joining everybody. Good luck. There is now just 11 to go on that football mixer Just 11 to go Breaks in just a little bit Tray Lance got a white premier level Kellan mom to start us off Vikings 535 concourse over Herbert's Blue field level Justin Jefferson Vikings again 10 of 49 Nice Vikings break premier level silver Nico Collins rookie Texans Hey guys just 11 to go in the mixer break it and just a bit Sensations Brady concourse over Tyson Williams rookie for the Ravens try color club level for the football team here, D'yamy Brown 18 and 149 and Buccaneers rookie Otto Kyle Trask 37 of 49 Jalen waddle 93 and 99 rookie Jersey card for the Dolphins here guys I'm not gonna recap this break at the end silver premier level rookie ETN junior for the Jags were pretty far behind Anish the three boxer I'll try to catch up a bit Tyler Boyd 25 is 75 bangles Jersey Otto got an ETN premier level silver sensations Tray Lance Premier level silver Broncos Jerry Judy Yes, so I mean the Vikings are still having a good break. I don't know what you're going at Selections Elijah more rookie for the Jets club level silver Elijah more Jets rookie Joe Mixon. Oh, yeah, he did who do you get picked up by? Was it the Browns? Premier level light blue. Sorry. I misread your message 2099 for the Raiders, Brian Edwards club level silver Das News and Bears Eagles that's interesting. Oh, yeah, the Eagles cut their rookie Carson strong. I guess Najee Harris club level premier level tri-color Davis Mills 83 of 199 Houston Silverfield level Jalen Darden Mack Jones premier level base CTN silver concourse to Vonte Williams rookie Broncos Fletcher Cox XRC Jameson Williams Lions XRC nice one there. Yeah, good stuff there Is that a slight dimple on the top of this card Very small Summer concourse turn your own with Dallas Penny Sewell Send a box wrapper to the lion spot Not sure Penine will replace them for that small of a ding but We're the shot with an XRC for sure That's a good question. Where is an issue got tricolor concourse Ravens Tyson Williams 228 to 249 Carnel Lake Steelers auto 10 to 49 Johnson Chase base Fields phenomenon Redemption Trevor Lawrence rookie signatures memorabilia blue prism Big redemption for Jacksonville congrats on the hit Daniel C Heck yeah, Daniel Sick blue is the 75 or less. I'm pretty sure it might be lower because it's a Because it's Lawrence big card at Reeve base Silver snapshots Derek Henry for the Titans a silver field level rookie ETN for the Jags as well here Silver sensations Trevor Lawrence Jagged club level silver Giants Michael Strayham Steven Sandler's Mills Kyle Pitts concourse rookie white Falcons 9 of 35 Concourse rookie silver Rashan Slater Chargers Silver concourse juju for the Steelers. There's a no a gray field level rookie tricolor rookie Eagles Devonte Smith Number 96 to 249 a silver premier level Darden for the Bucks Premier level blue for the Colts Jonathan Taylor number 97 to 149 silver premier level Aaron Donald over the Rams Silver concourse rookie Zach Wilson Jets Jets Field level blue Falcons Deion Sanders. He's Deon three of 49 concourse silver David Johnson Texans Suburban premier level terrace Marshall Jr. Rookie Panthers Kid Darius Tony field level rookie base He ate a 49 Robert Woods Jersey card Andre Johnson would have been concourse rookie silver Titans that's with Patrick. I think if he had a good quarterback his whole career There's no doubt he would have been pushing maybe best all time with health and a good quarterback Like a lot of things had to break but he was he was that talented. She's good at everything man She's good at everything nine to go on the mixer now nine to go Who was Andre Johnson's best quarterback Matt Schaub Silver concourse DK for Seattle Mac Jones tricolor Jamar Chase rookie concourse 200 of 249 There you go Cincinnati nice Chase Silver Camara premier level Davis Mills premier level rookie base Gonzalez or Gates I go Gonzalez Silver concourse St. Brown rookie the Lions Blue concourse rookie Vikings Kellan Mons other Mon there 98 to 199 silver field level Matt Ryan Falcons Evan Silver sensations Devonte Adams silver premier level rookie D'Amie Brown Washington Premier level Smith Marcette rookie tricolor Vikings 71 to 199 club level silver Mahomes Chiefs XRC Muhammad that is Falcons Titans Nice trailing Berks Titans XRC Nice one there Two first-round receiver XRC's concourse silver Carson Wentz Trace Sermon on the field level Fields concourse got a Jonathan Taylor 50 of 99 Jersey card or five of 99 Jersey card Colts Mixed in silver premier level tricolor club level Packers and Mari Rogers One 12 of 149 concourse over Rondale Moore blue premier level Trevor Lawrence Jags have crushed the break congrats Jags 39 of 149 Geez had the blue prism Jersey auto as well big card big card here. It's Titans gold auto Julio Jones Six of ten Nice hit Titans Vante Smith premier level rookie silver selections Rashad Bateman silver club level Dwayne Eskridge Tien there's a Lawrence selections. We've got Luke Keekley Jersey auto one of 35 nice hit Carolina Good last box here, man. Really good last box Andromedis that is gonna do it. Thanks again everybody. We've got another three bucks are coming up


we conducted a comprehensive literature review to identify potential prognostic biomarker candidates for hepatocellular carcinoma, HCC. We then validated these candidates in three different data sets, including RNA sequencing, microarrays, and gene chip data. We found that 180 out of 318 genes remained significant in Asian patients after multiple testing correction, while none remained significant in Caucasian patients. However, we did find that stage in expression of certain genes such as CEMPH and CDK4 were significantly associated with overall survival. Additionally, we identified KIF18A as the only gene that predicted vascular invasion in both the aphometrics and Illumina data sets. This study helps to eliminate biomarkers with limited capability to predict overall survival or vascular invasion, which can be used to further a fine prognostic biomarker candidates for HCC. This article was authored by Otelia Menihart, Adam Nagy, and Bawlish GYNFFY. We are article.tv, links in the description below.


Okay, James P. Madonna of Progressive Discussions. MegaLife21, I am now going to vote. I am here at the Hilltop School in Lodi, New Jersey. I am going to vote now. This is Election Day 2018. Yes, Tuesday, early afternoon, Election Day 2018. Probably one of the most important Election Days in modern-day American history. All right? And as you well know, I am definitely not going to vote for any conservatives, no Republicans.  But unfortunately, I am forced, because I don't have the options, I am forced to vote for the lesser of two evils once again, because I have a choice. Either vote for the frying pan or the furnace. And here in New Jersey, we are severely lacking well-backed, well-known, true progressive candidates here in the state of New Jersey. So I will enter right now until, you know, until our revolution or the Democratic Socialists of America or the Green Party or whatever the progressives actually are well-known in New Jersey. They exist in New Jersey, but they are not well established yet. Until then, I have to settle for the two evils.  You know, he put up a big argument about the fact that there are signs in other languages. He made a big stink about it. This redneck guy, this right-wing redneck, he really put up a fight, an argument, not a fight, but an argument, a dispute about this fact. Unbelievable. All right. Now I will enter. I will probably see my long-time friend and former co-worker from the 1990s here, Mr. Robert Gennaro. And now I enter. Okay, here I am, James P. Madonna. Progressive discussions ready to vote. November 2018 election, Tuesday on a rainy day, but the rain does not stop me. No way, no how. I am here to do the right thing. Here we go. I'm not even going to pay attention to these demons. To these demons. I'm not paying attention. Of course, I do not know any of these independent candidates. Unfortunately. Unfortunately, I do not know them. And this is the problem we have here in New Jersey. The independents are just not being backed full force by organizations like Our Revolution. I hope in the very near future they will be. I hope they will be backed. Okay, here I am voting for the lesser of two evils once again. Okay. Got it? The lesser of two evils is our public question. Securing our children's future bond act. I say it sounds good to me. It sounds progressive to me. I'll vote yes. What do we got here? Local crap, official school election ballot. I guess Board of Education.  I don't really get involved in local politics, really.  Might as well. Okay. Okay, America, this is how we do it in New Jersey. It's controversial. Some people say the paper ballot is best. But any form of voting can be corrupted and rigged. Corrupted and rigged. There's a lot of noise here. I'm sorry. Cast. Vote. Done deal. See you on the next Election Day. James P. Madonna, progressive discussions. Megalife 21. Okay. This is James P. Madonna, progressive discussions. Megalife 21. I have voted. I have cast out the demons. Well, we hope we're going to cast out the demons today. Tuesday, Election Day, November 2018. We hope we're going to cast the demons, at least out of the House of Representatives. And oh, lovely. I smell carbon monoxide fumes as I'm leaving. But anyway, that's it. I voted. I did my duty. And let's just see if the voter turnout was large enough. This Election Day, November 2018. Okay. I want to give a greeting to the only true progressive in the state of New Jersey, progressive candidate Lisa McCormick. And I also want to give a shout out to her partner, Mr. Jim Devine. And please, our revolution, American Socialists of America, if that's the correct title. I don't see the Green Party out here, but definitely get behind the progressives before the next election here in the Garden State, because many of the politicians, unfortunately, especially two-party system politicians, are corporate whores, typical two-party system corporate whores. They are corporatists. And if they pass away, they're so crooked, you will have to screw them into the ground like a corkscrew. Unfortunately, but that's it. Bye-bye.


Welcome back to Tennis Talk, my name's Cam Williams and we have the final tournament of the year for the regular season for the WTA. The lint's open. It's happening this week and we've got some big names playing in this event to finish up the season. We've got some big names also out so let's go look at the withdrawal list. Taking a look at the players that are out of this event and it's a long list. Begu, she's pulled out, Garcia, Golovic, Kinepi, Kostjuk, Li, Martic, Osorio, Tomjanovic and Zhang have all pulled out of this event. Taking a look at the top of the drawer and we have eight seeds in this one with a buy for the top four seeds. And Rado Kanu, she is the number one seed for the first time in her career in a tournament. She comes in as a wild card but she is the number one seed. She'll play a qualifier in the second round no matter what because we've got two qualifiers going at it in the first round. Then we have Kornay versus the wild card, Grabria. Winner of that match takes on either Barthul or the number eight seed risk in the second round. So the player that you've got to watch out for in this one is obviously Rado Kanu. Let's see how she does because the last couple of losses that she's had have been pretty one-sided. So let's see if she can get back into some form and let's see if she can deal with the pressure of being the favorite to win a tournament. Second part of the draw now and we've got Collins, the number three seed gets a buy in the first round. She awaits the winner of Minnan versus Dodden who are going at it in the first round. Then we've got Van Udvang versus the wild card Kraus. Winner of that match takes on either a qualifier or the number five seed Alexandrova in the second round. The two players that I'm watching out for in this one, you've got to watch out for Collins and also Alexandrova. Just coming off a win with Russia at the Billie Jean King Cup. So she'll be full of confidence, interested to see how she does this week. And don't forget this part of the draw, take on the Rado Kanu part of the draw. So this is all the top section of the draw, which means the semi-finalist is going to be maybe Rado Kanu versus either Collins or Alexandrova if we're going by the rankings. But of course, WTA is not as simple as that. Bottom half of the draw now and we've got the number six seed Castella. She's taking on the Russian Rakan Mova in the first round. Winner of that match takes on either a qualifier or Pedersen in the second round. Then we've got Kalanina versus Trevesen. Winner of that match takes on the number four seed Kudamatova and you've got to watch out for Kudamatova. She's coming off the Fed Cup win along with Alexandrova. So she'll be full of confidence in this event and she's had a pretty good 2021 as well. So she'll be looking to finish strong and she'll be a very tough opponent with a lot of confidence going into this week. And the bottom part of the draw now, we've got the number seven seed Paolini. She takes on Yastremska in the first round. Winner of that match takes on either Boral or Zhang in the second round. Then we've got Ferro versus Sastavic. Winner of that match takes on the number two seed Simona Halop, who comes in as a wild card. So we're going to be watching Halop this week, of course. Got to keep an eye on her. Did really well last week. Got back into some form at the Transylvania Open. She lost in the final, but she got some wins under her belt and she got to get to the final there. So interested to see how she does this week. And are we going to finally get that Raducanu matchup that everyone wants? The Raducanu versus Halop matchup. Don't forget this part of the draw takes on the Kudamatova part of the draw as well. So Halop will like her chances in the bottom half of the draw. It feels like it's a little bit easier. But then again, you've got dangerous players like Sastavic, who's been in some really good form of late. And that could be the second round opponent for Halop. So we'll see how it all plays out, but it's going to be a fun tournament to end the season. So there you have it. There is the lint's open and the defending champion is actually Savalenka. She's playing the WTA finals, of course, but got some big names still playing. You've got Kudamatova, who's in great form at the moment. Obviously, you've got Halop and Raducanu as well as the number one and two seeds. But let me know down in the comments below. Who do you think is going to win this event? Taking a look at the draw here and I'm going to go with Alexandrova to win this one. I feel like she's got a pretty good draw. And after winning the Fed Cup with Team Russia, I think she might be able to get through this. And she's at the top half, which is very dangerous. She's got players like Collins and Raducanu at the top. I just think Alexandrova, you know, these tournaments always have random champions. She's the number five seed. So I'm going to go with her this week to take out the last event of the season. Again, let me know down in the comments below. Who do you think is going to win? The lint's open, the final event of the season before the WTA finals. Who's going to finish with a win?


The political transfer season, like football, is open, and as expected, many names and faces are popping up. There's realigning of interest and highly irrelevant lots who I call the lepers of politics are stepping into the game. They have one goal, cost chaos, ranker, and heightened uncertainties. They know they cannot win an election, even in their homes, their villages, but they are determined to make themselves relevant by saying whatever comes to mind. Very soon, social media will be a gurg with photo ops of politicians buying Akara beans and bread by the Rottai, even buns. Parties are looking for the strongest yet vulnerable link in other parties who they can poach. Politicians have started discrediting themselves and by July, full-scale tribal and religious affronts will have kicked in. Politicians amongst themselves know who the next favorite or most qualified candidate is. Yet, they do all this just to remain relevant and earn a sit-down peace talk which will lead to a political appointment after elections 2023. In all this, all you see are politicians attempting to make a way where there has been no way. Politicians put their best foot forward, swimming against the tides. Where are the citizens who are affected by the decisions of these politicians? We do not appear anywhere. We wait for the most emotionally relevant news, reports and lash on it. We walk around town accusing every politician. Yet, we are not ready to recommend a politician of choice. Neither are we willing to join a party. Until December, the song on our lips was direct primaries, direct primaries. A call that made politicians laugh because for direct primaries to be effective, you need to join a party. And as you already know, we are unwilling to join any party. They are all evil. The people have the right to demand for a particular candidate. But we are silent on this. We prefer to ridicule every single candidate that is presented. Politicians have a plan for 2023. They fight now and decamp by November. We, the people, do we have a plan? We're comfortable following the same politicians that we call corrupt, insincere and tribal bigots. We're comfortable in our one corner, one corner. Saying, Inogo better for this country. When in reality, what we are saying is, Inogo better for you and I, the individuals in it. Nigerians succeed in business because we understand that the outcome of every business endeavor has a direct effect on us. If only we understand that the outcome of any political activity has a bigger direct impact on us, we would be more participatory in elections. For 2023, we need to study and understand politics. Join a party. Get your PVC. Look beyond tribe and religion. Call out your family member and tribesman that is working against the nation. The more disunited you are, the weaker you are. The weaker you are, the stronger the politicians become. You shout, you don't want old Sikh leaders. Yet, our weakness is equivalent to the weakness of the Sikh leaders we are calling out. God bless the federal republic of Nigeria. God bless the people of Nigeria. Oh wow, I think I'm actually trying to think of what I know what to say. My mind is going to the fact that what you're saying directly actually links to what I was trying to get at. Even though I started from a point of motivation or like for me and maybe because I'm a lawyer, I'm so detailed that I'm breaking it down.   So I do a systemic break it down. Now, how do we as a people follow these instructions? Plan it, get our PVC, when we don't even know exactly where we want to be. When we're bowed down, we thought of a person who is helplessly helpless. No, that's the way we are now. They turn not to take a job, but the truth is that a lot of us really feel helpless. We're too broken to see what things are happening. The number one was change your mindset. If we're going to actually be functional, effective, or create any change politically or otherwise, we have got to be resilient and strong. That's what the politicians know. There's a strategy to the madness. Every politician that is doing something and succeeding, I admire their guts. I admire the drive. I've seen and I look at myself that even if you do anything successfully, good or bad, it takes a lot. And so maybe that is something that we as citizens need to adopt and see. What you're saying about the politicians. I also program on radio and behind the scenes talks are very detailed. There are things that you should record and play. Now, one of the politicians told me something. He said, listen, Nigeria has never suffered. That's it. He said, because until we get to a stage where it is unbearable, things will not change. You know why? He said, because right now, he as a person sees people who are qualified citizens and is encouraging them, coming to the fold so he can help people like us who want to make an impact. And they tell you, no, I don't have interest in politics. He said, it's because they still have something to eat. When it becomes when it hits so hard that, you know, you don't have a choice, you will know that you need to get into politics for you to survive. And you see, we, there's so much again, like you say, and I blame those that call themselves the elites, not necessarily because of their status, but how enlightening they are because, you see, we mislead the general public, the people under the bridge, like you rightly said, why do I say that? Now, there is this talk about constitution, throw the constitution away. But the truth of the matter is that the constitution cannot be thrown away. What you do generally, globally, is to amend. You can get into parliament and decide that A, B, C, D, E, F, G, we don't want it. And that is what you call the people having a say in the constitution. Right now, constitution amendment is ongoing. Nobody is talking about it, no single person. Then we come back in 2024, because from now to 2023, forget anything about removing the constitution. Nobody is going to talk about that. We come back 2024 and we start shouting. Whoever is, of course, Bari won't be there. The next person, again, we're victim. Then we go back to the same palace. He has to go, constitution, throw it away. That's it. I mean, I'm just processing everything everyone is saying. And in my head, of course, it just comes to my mind, it's fact that we've lied to ourselves a lot. There was that misconception in the past that I can live without politics. I can do well without being there.  I don't need politics. Well, I think in the past years, we've come to understand that your politicians matter a lot. The quality of people that are brought forward to the presidential elections, gobernatorial, senatorial, all of those categories of people, they matter a lot. I was having this conversation with someone the other time. You realize that the people that eventually rise up to say they want to become candidates for whatever party or whatever position are people that have been in this game for a long time. Very true. And they probably started as dropouts. In fact, if we look at it, look at the trend, they're mostly dropouts. Because our constitution allows that with a high school certificate, I can context for any election. So it means that they actually, or they just need to prove that they went to a secondary school.  But because there's that room for it, and we, the people that are quote-unquote educated and enlightened, we've given them that room. We've succeeded in business, like you mentioned, but we've not succeeded in politics. We've left politics to people that really don't have the capacity. If you bring these people to business, they're going to run down the business. We know that. So we don't even employ them as entry level. You don't even make them entry level. They don't qualify. So, and we know that because they cannot process the kind of things, thoughts, and knowledge that is required to be the business. So why then have we left them to run our country? But before I come in here, I don't know if this is that. You'll be shocked. Look at, okay, let's take it. A state governor is a successful businessman. Thank you. Ogun State. Makinwa. Makinwa. No. A lot of the government. A lot of them are successful businessmen, former MBA lawyer and chief, whatever, MBA, Nijambal station, and all those people. Yet they get there, and everything is messed up. Exactly. They don't have the team. Look at people around them. You and I, we should be the team. We are busy in tech, and we are busy there saying, no, this government is useless. And we've been saying that for years. That's what I was saying. We've been saying the same thing for years. So it's not about one party. Because it's the system that we've created. And it doesn't take one person to build the country that we want. We need more than, and we need to stop this our messiah mentality. That's it. So we can build a better country. You know what I mean? I was just here sitting, thinking of a counter job tutor at Nijambal station. The truth is, this is my personal music, random stuff. I just feel so sad that it's almost like we're letting nature take its course. In the next two decades, or even less than that, some of these old guys, will be gone by virtue of life. They're going to die.  It's almost like we are watching waiting for that to happen before we take it. I told someone that Buhari plan, cool. Lots of people did all of this. You know, they seized power, and I'm not saying we'll do that. I'm just saying that. You need to be active. What are we doing beyond tweeting and all of that? They're even using us as a strategy to tweet. So I think it's saddening to me that we're all just intellectually talking. We're just going to wait for another 10 years where they die off and we're going to take over. Nobody wants to take that lead. What are we doing? That's the problem. But it was scared. Because blood has to go on the line for that. You see, beyond blood, what is happening, I think here is this. I think that we do not realize that we had this systemic brain failure. You see, for 20 years or more, let's look out for one one primary school and secondary school. We had a period of time, so all those in their 40s and 50s now, went through a system of education that gave them academic skills, but actually not give them enough thinking skills, planning skills, life skills, to be able to say, okay, where are you going to? We naturally went through school and system of life in Nigeria. And maybe the most of Africa to say, go to school, get an education, get a job. So now we're not taught how to fight for life. How to fight for things. The last fight in Nigeria was independence. No, no, no, no. I remember my friend. We could go on and on and on the fight. Or answer us. Or do Nadeco... No, that answer is a different one. You see, different born came. If you want to go through the Nadeco era of the late 80s and 90s and 90s and stuff, then you know what it means to fight. But Ruth is next. Stay with us.


The federal government has launched a national development plan 2021-2025, presenting the plan to President Mohammad Ibrahim during a brief ceremony ahead of the week's virtual federal executive council meeting. The Minister of Finance, Budget and National Planning, Zainab Ahmed, said the NDP 2021-2025 is a successor plan to both economic recovery and growth plan and the Vision 2020-20. According to the Minister, the plan will focus investment on massive infrastructure and share economic stability, enhance the investment environment, improve on social indicators and living conditions of Nigerians, among other targets. Ahmed further revealed that the plan is a pointer to the type of Nigeria that all Nigerians desire, adding that it will encourage the use of science, technology and innovation to drive the growth being targeted.


When I hear somebody saying one support a time, I wish to tell you a story. It happened like this.  It's our urge to make sense of our existence and our lives. Stories are very important for cultures. People want stories. We're fundamentally narrative creatures. I think that we are formed by stories. Stories to engage us, entertain us, stimulate our imagination. It tries to point out the wonder of life as lived and it makes us look beyond ourselves. India has always had stories being told or rather retold in different ways. Mainly through performances. And if you look at our music or dance, in fact many of them do have either direct narrations of stories or at least suggestions of stories or certain contexts in our epics. So I think our whole tradition of art and literature is linked to the telling of stories and the ways the stories are told. You have Pandun Ki Kade in Rajasthan, which again is really chronicles of Pandavas. And these are Muslim singers. And these Muslim singers are really community singers. Antiquity of our tradition is not as remarkable as its continuity. For us mythology is not something dead, not something that belongs to the past, it is something that surrounds us, wherever you go. In the Indian context, most traditions of storytelling relate to a particular community.  Bawls are mostly the poets, the travelling poets who is searching for the spiritual fulfilment or the ananda through songs, dance and music poetry stories. We tell these stories and we try to touch that moment. So in this way, Bawl is a storyteller. As a Bawl, we say that the song must become our body. We must realise it in our body and then only we can give because otherwise it will not touch you. We need to talk also about how the same material is told in 50 different ways. There is Kathakali which is of course, it is what one can say it is a kind of mime but then there is the song in the background which narrates the story which is acted out following very much of the instructions of Bharatha Aindranathesh Aasthara. Pav Kathakali is also one of the examples how the small stories are taken into this globe puppetry which Kathakali also is performing. We are four puppeteers, one drummer, one singer and one stage assistant and we create exact illusion of Kathakali in a short time. You have a story telling in performing arts. You have a story telling in literature. You have also story telling in pictures. A singular story will not do justice either to the human diversity of existence nor of the plurality of human imagination.


There's been a lot of talk about the Roman Empire online recently, and we clearly are not in Rome. So we thought we'd ask students at the Edmondville Library what their Roman Empire is. What's your Roman Empire? My Roman Empire is football. The beach. I think about the beach all the time, for no reason. Anything by Jane Austen. Like just Jane Austen in general. Birds. The birds. The birds aren't real. The birds aren't real. You're so right. That is my Roman Empire. I think about that all the time. Currently it's Travis Kelsey and Taylor Swift. You and the rest of the world. Do you think it's real? Truthfully no, but I'm gonna say yes. Construction engineering technology. Global health. Actually the Roman Empire. Egyptian Empire. One direction covering teenage dirtbag. Red Bull is a racing team. Thrice. I don't know. So sad, but so real. Plants. Do you have a favorite planet? Plants. Sorry. The Disney version of Hercules. I think I can quote that entire movie from memory. It's a great movie.


I'll be talking about, well, obviously there'll be some amount of introductory material, but the new work is joint with several co-authors, so there's Mark Gross at Cambridge, Sean Keele at UT Austin, Konsevich at IHES, and Bernd Siebert in Hamburg. So those are my collaborators for the joint work.  So these were originally defined in a very algebraic fashion, and by Fomin and Zelovinsky. And later, Fok and Goncharov gave a more geometric description of them. But right from the outset of the subject, there was a conjecture that a cluster algebra had a canonical basis. So it's some algebra over a field, say the complex numbers, and what they were conjecturing is some way to give this algebra a canonical basis as a complex vector space, some infinite set, which is a basis of this algebra. And they were motivated in part by some examples which they discovered, which they called finite type cluster algebras, where there was a distinguished set of elements called the cluster monomials, which were already a basis. However, there are some more complicated cluster algebras where these cluster monomials are linearly independent, but they don't span the whole algebra. So they were sort of looking for some additional elements to add to this set to form a basis. And sort of one of the reasons they expected this would be true was some work of Lustig. So in representation theory, Lustig had described something called canonical basis for representations of algebraic groups. And so they were partly motivated by that work to introduce the notion of a cluster algebra in the first place. So the idea that there should be a canonical basis around was also inspired by this work of Lustig. So the new insight which led us to work on this problem is that this should be regarded as an instance of mirror symmetry. So cluster algebra is the ring of global functions on some variety, a cluster variety. And that variety is, in some sense, a Kalabiyaw. And it's not a compact Kalabiyaw, it's a non-compact Kalabiyaw. And if you apply mirror symmetry to this variety, then this canonical basis arises naturally from a construction, a symplectic construction on the mirror. So that's a very rough overview of what I want to talk about. But this is at least to motivate what I'll do today is I'll talk about what is the correct notion of a non-compact Kalabiyaw variety. That's what's called a so-called log Kalabiyaw variety in our context. And how do cluster algebra fit into that framework? So that's what I'm going to talk about today, log Kalabiyaw varieties. So let's have a definition. So what will it be? Well, it'll be a quasi-projective variety U, and it will have the property that's given by, so we take a smooth projective variety X. Let's work over the complex numbers. And I remove a divisor D. So D will be a normal crossing divisor. So locally it just looks like a union of coordinate hyperplanes in Cn. Let's also fix a bit of notation, let's say the dimension of X is n. And it should have the following property. So KX plus D should be 0. That's the log Kalabiyaw property. And so equivalently, it seems that we have a holomorphic n form that was called a holomorphic volume form on U. So what do I mean by that? So that's just an n form, top form, which is nowhere 0. And it has simple poles along the boundary. So with simple poles, D is a sum of irreducible divisors DR. That's the notion of a log Kalabiyaw variety. And I'm not going to, that's right, effective reduced no multiplicities. That's right, that's right, yeah. So for various reasons, this is regarded to be the correct notion for correct generalization of Kalabiyaw's this open non-compact setting. So one reason is related to the Stromanjiyaw's as low conjecture that if you want to have this so-called special Lagrangian torus vibration on U, it forces the form to only have simple poles along this boundary divisor. So for various reasons, this seems to be the correct generalization to non-compact setting of Kalabiyaw varieties. Okay, so a couple more definitions. We'll say it has maximal boundary if this divisor D has a zero strata. In other words, a point cut out by n branches, analytic branches of D. So in other words, symbols a point in D, so locally isomorphic to just the origin in the union of coordinate hyperplanes in CN. And one more definition we'll say it's positive, this variety U is affine, or more or less equivalently D supports an ample divisor. So for instance, the case that Alessio and Al have been talking about, if you have a far-known variety and you have such a divisor in the anti-canonical linear system, that's an ample divisor, so this will be a positive example. Yeah, so I say a log-Kalabiyaw has a maximal boundary. If when I look at this boundary divisor, there's a point, so just sort of a boundary has a zero-dimensional strata. Is it clear? I'm sorry. Ah, good question, yes. It's independent of the choice of compactification, actually. So that's not, that's a theorem, but yeah, so that's right, that's right, yeah. Let's just give an example. So, you know, so here's an example. I can take P2 with a smooth elliptic curve, or I could take P2 with a rational nodal curve. This guy's maximal. Was there another question? It is a property of U, yeah, I won't prove it. It's not too hard. Basically what you have to do is show that this property is invariant under blowing up. Yeah, I'll probably abuse notation a little bit and talk about U or the pair X of D. But I will allow, yeah, so it's a, yeah, so I'll allow myself to pass between different choices of X, D of the same U fairly frequently. So I think really one should think about U as the object. Okay, or it's roughly equivalent to say that D sports, certainly if D sports an ample divisor then U is a fine, yeah. And the reason I mention this is often that, you know, you'll, how do you construct such a thing?  So if you happen to know D is an ample divisor then, okay. So let's see, what about some more examples, more sort of general examples. So the first example really to be example zero is that U is just a torus, algebraic torus. So here, my holomorphic volume form is just coordinates in this way, DZ1 over Z1, wedge, et cetera, et cetera, wedge DZN over ZN, where Z1 up to ZN are the characters of the coordinates on this torus. And so here, you know, XD, what can I take? Well, any torus X is any smooth, projective torus variety containing, you know, with torus C star to the N. And D is just the so-called toric boundary to complement the torus. You know, if you want to be completely specific, you could take X to be projective space and D to be just the union of the coordinate hyperplanes. So that's somehow the sort of base case which is rubber easy. Maybe I'll just say immediately, so I said that, you know, these varieties in some generality tend to have a canonical basis of global functions. So in this case, one can see it immediately. Look at the global functions on this variety. Well, those are just given by the canonical basis, just given by the characters torus, so all the mononials coordinate ZI. So this is a canonical basis determined up to scale. So why is it canonical? It's just exactly the units in the ring. So that's the characterization of the spaces. Of course, this isn't a very interesting example, but we'll sort of see in more complicated examples. This basis can be very interesting. So example one, let's suppose I have a log club. Yeah, I want to construct another. Here's one way to do it. So let's pick a subset of co-dimension two and I want this to be contained in a unique component of D. And so I am assuming I want everything to be smooth and normal crossing. So I should say Z is smooth and intersects all the boundary, intersects the remaining components transversely. Then let's blow up Z. So define X tilde, D tilde, B the blow up Z and D tilde will be the strict transform. I'll write D prime, strict transform of D. So then check that this is again log club. Yeah, so KX plus D tilde is zero. So let's just give the picture at least dimension two at least looks something like this. Here's my X D I choose. So in dimension two, I'm just choosing a point contained in a unique boundary divisor. I blow up that point. There'll be an exceptional divisor E over the point P and D tilde is the strict transform of the boundary. So E is not contained in D tilde. The higher dimensional case is very similar. That's right.  Yeah, so maybe I should just say, why did I write these two conditions down right at the beginning? So our results will be the strongest under both of these assumptions. So in case you have maximal boundary and positivity. Questions so far? That's right. Yeah, that's actually that's my next comment. So yeah, so I'll refer to this as a non-toric blow-up. Yeah, so let me make the following remark.  So this contains U. So pi, this blow-up, it's called pi the blow-up, is an isomorphism over U. So just the inverse of pi includes U and U tilde. But if we look at the complements, what's that? That's this exceptional divisor E intersected with U tilde. So this is a line bundle over Z. So we've added something to the interior. And so there's some sort of, in symplectic geometry, this is some sort of surgery. So sometimes called a handle attachment in symplectic geometry. So Weinstein. Well, I had a P1 bundle and I removed the zero section. Oh, maybe you're right. Okay, maybe I should say an affine. What do you call that? Okay, that was probably worse than it was, right? Thank you. Okay. Yeah, so I just wanted to go back to Sasha's remark, is of course we could but also blow up a stratum of D of any dimension and define, so this will be X tilde goes to X, and then D tilde should be the full inverse image. So we include the exceptional divisor in D tilde. So then, of course, U tilde hasn't changed. So we regard this as sort of a, from our point of view, the most important thing is the interior. So we regard this as some sort of trivial operation that we should be allowed to do whenever it's convenient.  We'll do this whenever it's convenient. Because, as I said, U is the main thing and U is unchanged. Okay, so now if you know a little bit about rational geometry of surfaces, here comes the first exercise. So first of all, we say a toric model of a local R-V-L-X-D is a diagram. Start with R-X-D. We do a sequence of toric blow ups, let's call that X tilde D tilde.  In other words, we have some X bar D bar and some non-toric blow ups here, sequence of non-toric blow ups. Such that this eventual variety X bar D bar is a toric pair. Toric variety with its toric boundary. Oh, you can imagine there's some kind of roofs. To be honest, yeah, well, yeah, no, it's not a silly question. I mean, so this is a big theorem if you just ignore the boundary by who is it, and that if you have any birational transformation between smooth projective varieties, it can be factored into a sequence of blow ups and blow downs. That is a theorem, I think. Yes, yes, right, right. Yeah, but you know, all I'm saying is that's a non-trivial theorem and I haven't even thought about whether that's sufficient to, yeah. But let's, I mean, so this is some kind of definition which may or may not be correct. Yes, yes, right, right. So I'm, as I said, maybe this needs slightly modifying, but this is the essence of it. And then the lemma I want to mention, just to give you some idea, is that if we're in dimension two with maximal boundary, then there is a toric model. And so, of course, in dimension two, birational geometry is very explicit. Any birational map can be factored into a sequence of blow ups and blow downs of points. That's a very classical result using that it's not hard to prove this lemma. How to say? So one can always blur up the bound, you know, do toric blurts on the boundary. Oh, I'm sorry. Yeah, so no, there's no uniqueness. Right, that's a good point. Yeah, this is highly non-unique in general. Yeah, so sorry, thanks for the comment. Let me just give an example. So for instance, yeah, so maybe the best example would be something like this.  And so that is an instance of a toric model. But there'll be lots and lots of minus one curves on XD that have absolutely nothing to do with the original exceptional curves. So there'll be all sorts of different ways of blurring this down to P2. So, you know, so for instance, as everybody's aware, you know, if you have a cubic surface, that's obtained from P2 by blurring up six points. But there are 27 minus one curves on the cubic surface, there are many sets of six curves inside that 27 that you can choose as the exceptional curves that will blow down to P2. And so, you know, that's an example, a very classical example, you know, which sort of, if you sort of study that, it's related to the value of V6 and so on and so forth. So somehow the point is that there can be many, many different toric models and it's a choice to choose one. Yes, sorry, I wanted to take a rational nodal curve here. Yes, sorry. The point is, oh, sorry, sorry, so that's not a toric model, sorry. I'm sorry, I'm sorry. Yes, so let me call it D bar or something. And I'm sorry, that was my fault. Yes, so blur for a bunch of points here. I'm sorry, this is supposed to be an example of a toric model. What I'm saying is it's not unique because there are lots of minus one curves here which can be blown down instead. Yeah, thanks for the correction. Sorry, okay. Yeah, so let's just do a quick example. So here's something where, you know, the toric model is not completely obvious, but it's not hard if you think about it. So for this picture, actually maybe I'll slightly license here. Yeah, I mean, actually I'm always going to assume that D is non-empty. Yeah, but another example would be something like P2 of an elliptic curve. So somehow that's never going to have a toric model because you can't get rid of the elliptic curve by blowing down. What I'm saying is that, you know, if you have this toric model picture, there's going to be a relation between the two boundary devices. They're going to differ from each other by a bunch of P1s. So, yeah. Okay, so this is an example of a toric model. So I'll label the self-intersection numbers of the curve. So I've got a conic and a line in P2. So where's the toric model for this surface? Well, let me describe it in the following way. So first I'm going to blow up this point here. One of the intersection points doesn't matter which one. And now I want to blow up this point here, Q. Now this looks suspiciously like the surface F2. So in fact, there is a minus 1 curve like this and that can be contracted to the toric surface F2 with its toric boundary. So this is the F2, it's just a rational ruled surface with a negative section of self-intersection minus 2. And so where did this curve E come from? So if you sort of trace it back, what it was, if you take the tangent line, it's the conic at the point Q, that's what this curve is. It's the strict transform of that line. So after one blow up, that line is here. After one more, it's meeting the boundary at a smooth point. Oh, so I'm just drawing the self-intersection of the curve. So it's the, yeah. It's just a convenient way to keep track of, yes, sorry. And so here's my exercise. Exercise one, a find a toric model P2 with a rational nodal curve. Rational nodal cubic. And let's just one more exercise. It's no vibrational java cube smooth surfaces. It's fairly easy, but this is the case. So show that if B is smooth, Xd log Cy2, then that does not exist at all. Incidentally, perhaps I should have said, it's easy, but let me put it out. Note, by the adjunction formula, Xd log Cy2, Kd0, in other words, the arithmetic genus of the curve D1. And so D is either smooth elliptic, rational nodal, or a chain of a wheel of P1s, a cycle of P1s. So we're assuming, remember, it's nodal. So we don't allow, for instance, a cusp. So let's also give a sort of negative example in this direction. So let's have a log Cy3, which is irrational. So certainly if it's irrational, so it's not birational to P3, then it certainly doesn't have a toric model. So if you know a little bit about Farno-Freefolds, this isn't hard. Just by definition, I assume that D is a normal crossing divisor. And you might say, why don't you consider the cusp? So in fact, if I had a cuspiddle curve, and I tried to resolve the boundary to normal crossings, then the boundary would, if it stays anti-canonical, I would have to have a curve of higher multiplicity. So there would be a curve in the boundary where the volume form has a pole of higher order. And so that's kind of out of this realm where we understand how to do mirror symmetry. Yeah, sorry, I meant to add one thing, so this is with maximal boundary. Necessary condition for a toric model. Sorry, I'm saying that this is necessary to have a toric model. That's right, yeah. So what I'm saying is, oh, sorry, let me just continue. Yeah, if I did say it, I didn't mean to say it. I'm saying, no. No, no, because... Yeah, I'm not claiming any kind of if and only if here. I'm just saying that certainly a necessary condition, both of these things are conditioned, so both of these things are necessary to have a toric model. First of all, your variety should be rational, and secondly, the boundary should be maximal. So I'm just saying, of course, in dimension three, we know a lot about pharno-manifolds. There are some examples of irrational pharnos. So, for instance, I can take X, be a smooth quartic, in P4, that's irrational according to Iskosky and Manin, but I still need to cook up a maximal boundary divisor. So what's my D going to be? Well, it's a hyperplane section, so it's in the anti-canonical linear system, which is just O1. It's a hyperplane section. Okay, so this presents a problem. How are we going to make it have a zero stratum? Well, in fact, what we'll do is make a singular example here that can be resolved. So what I'll do is take D to be singular with what's called a cusp singularity. So let's be specific about this. So for instance, this is my quartic as an example, and D will be U equals zero and singularity, so the cusp singularity, of P and D. So I'm setting U equals zero and let's look in the affine piece where T is non-zero. So then this becomes zero to the four plus Y to the four, X, Y, Z zero inside C free. So that has minimal resolution, a cycle of P ones. That's an example of a so-called cusp singularity. Okay, and so now what you can do is take this pair, X, D, and do some blow-ups, so resolve to get this guy with maximal boundary. So the picture is in X itself, so X is smooth, but this guy has some kind of nasty singularity. When you blow-up, you do just one blow-up, you get this picture where you have two divisors that meet like this. You've got two divisors meeting along this triangle of P ones, and if you sort of do a couple more blow-ups, you get to this guy X to the D. You can kind of see, quite normal crossing, but here's, these guys are going to become zero strata here. Yeah, so it's not too hard to produce examples which you absolutely cannot have a toric model. Something similar will work, yeah. Yeah, yeah, I think basically the same thing will work. So let me just make this remark that, roughly speaking, a toric model or the existence of a toric model is the same as having a a torus, a full-dimensional inside U. So certainly if I have a toric model, I get this torus, so if I have X, I've got the torus in here, and that lists to here. So this map is an isomorphism over the torus, and so I get my torus in X. So certainly this direction is obvious. The other direction, well, this is sort of related to the discussion we had earlier. What do we know about birational geometry in higher dimensions? So you probably need some sort of minimal model program, and you need to think about singularities. But something like this is morally true. Okay, so now I can tell you what a cluster variety is. So two conditions. So the second one we've already discussed at length, cluster variety is a log collabial U, such that so first of all, there's a two-form, so sigma is going to be a non-degenerate holomorphic two-form on U with log poles along the boundary divisor. Again, so I'm writing U is X set minus D and again this condition will be independent of D, basically due to the lean. So what does it mean to have log poles? So I've got this sheaf on X of holomorphic one-forms with log poles along D. So this is just locally generated by so yes, at a point P so here I'm saying that P in D in X, where D has our branches so I'm allowed to have D log of Zi for i corresponding to a component of D and have the regular a DZJ for the other J so then my sigma should be in a global section of this wedge two of the sheaf. It can locally be expressed in terms of these differential forms at a point P. There should be a Turing model. So this condition one you should think of as some non-compact analog of a holomorphic symplectic or hypercaler variety so I like to call this log holomorphic symplectic and so of course an immediate example is that in dimension two this is nothing new so in dimension two cluster well the first condition is vacuous because you can just take sigma to be equal to omega n is equal to two and so cluster is the same as the existitoric model which as we just saw is the same as maximal bound. But this is a non-trivial condition in higher dimensions the same as in the compact case yes it wouldn't be because the boundary is certainly not an empty you see yes so no it does not okay so there's a little lemma really clarifies this first condition so so what does it mean to have a Turing model so the blow-ups are very constrained a Turing model you know for cluster variety so remember we've got some picture like this X bar do some blow-ups and this is a Turing variety and so what I eventually want to get a form on X or equivalently on X tilde sigma but of course that will define a form sigma bar on X bar because outside co-dimension 2 I can just take it to be sigma and then this will extend over co-dimension 2 the usual sort of Hartog's property so if I have a form here I'll certainly have a form sigma bar on X bar but now we know this is a Turing variety so what it follows is that sigma bar is actually constant coefficients so this is just given by a skew matrix so this is a non-degenerate skew matrix so I want this to be have log forms along the boundary that's right so what I was trying to explain just then maybe it wasn't clear let me say it again if I have the form on X or equivalently X tilde I can get the form on X bar automatically because after all if I just remove the co-dimension 2 subset of X bar then that will be an open subset of X tilde I can just restrict the form no no no it's okay so this is a bunch of blow-ups so there's an open subset in here let's call it maybe V which maps isomorphically and this is complement is co-dimension 2 so I have the form on V it will automatically extend to just by this Hartog's property so what I'm saying is if I have sigma up here I certainly have this guy down here okay but the why is that a good thing is because on the torus we know that this let me just write it down if I take this log forms on the torus that's just a free module it's a trivial sheaf so it means that when you take global sections you're just going to have so this form just becomes a matrix so now what's the condition so now I say okay now start with this form and start blowing up what's the condition that this form lifts that's a non-trivial condition so let me tell you what it is oops where are my notes so what's the condition so I must blow up the following type of center so it's a co-dimension 2 center so C will be this center will be the intersection of some component the boundary with the hypersurf is given by a character of the torus so let me write this down and explain so here where so what's C so C inside D is a component of the boundary chi is a character of the torus so you know this is just a monomial lambda is just some scalar non-zero scalar and chi is determined by following condition so I have this form sigma bar I've got a divisor I can take its residue along that divisor so now I'll get a log 1 form on the boundary divisor that should be equal to the log chi up to a scalar so roughly speaking what I'm saying is if you take the two form on the toric variety the boundary divisor take its residue you'll get a 1 form so that defines some kind of foliation of that boundary divisor you must blow up a hypersurface contained in that foliation sorry this is just a scalar so yeah that's the condition there's no condition on lambda that's right yeah I mean the whole thing remember you have to remember that we're in this sort of situation we're on a torus this form is constant coefficients so I know that when I restrict to this boundary divisor I'm going to get something with constant coefficients that's right yeah there is some rationality involved yeah exactly so somehow a priori when you restrict this you just get a 1 form with arbitrary complex coefficients but I'm saying up to a scalar those have to be multiples of a rational vector yeah do you see what I'm saying there is a non-trivial condition here if this divisor when you restrict if you get something which is a tuple which is not a scalar multiple of a rational vector then you cannot blow up there's no character somehow if you think about in terms of foliation that doesn't close up and you don't get algebraic manifolds there's a non-trivial condition hiding here if this guy is not rational then you won't be able to do any blurb at all and so what I want to I'm out of time but let me just say so what's the picture so the picture of course is changing coordinates without loss of generality let's say that this component is given by z1 equals 0 and this chi is given by say z2 so then what are we doing so up to what we get it's just the dimension 2 picture across a trivial direction but remember dimension 2 is just this simple thing you've got a a toric boundary divisor you select a point and you just blow up and so what we're saying is that in the cluster picture just look at a single blow up it's just exactly this dimension 2 picture across a trivial direction of course that's locally so for a single blow up but globally of course I see I sort of like to draw this picture here's a picture of your toric variety of its toric boundary what you'll have is that these guys that you blow up they'll be now hypersurfaces in the boundary divisors but they won't line up so it won't be globally a product but it will be locally along this stratum a product so that's kind of the the condition that comes out of this log holomorphic symplectic condition under the assumption that you have a toric model so what I've presented here is a sort of algebraic geometry version of what the guys in cluster algebras were doing to begin with so they had you know they in their combinatorial description they had this skew matrix and then there was I'll describe more next time there was this process of mutation and the notion of seeds and so on but somehow this is the interpretation in algebraic geometry of the combinatorial data that goes into the cluster algebras the skew matrix corresponds to defining a holomorphic two form on your log collabi alve variety okay so I should stop here but I'll continue in this vein tomorrow um actually this afternoon yeah so thanks


We've been enough to go from Katanjali. Thank you my dear friends for joining me Here's another poem from Katanjali On the seashore of endless worlds children meet the infinite sky Is motionless overhead and the restless water Is boisterous on the seashore of endless worlds The children meet with shouts and dances They build their houses with sand And they play with empty shells With withered leaves They weave their boats And smilingly float them on the vast deep Children have their play on the seashore of worlds They know not how to swim, they know not how to cast nets Pearl fishers dive for pearls Merchants sail in their ships While children gather pebbles and scatter them again They seek not for hidden treasures, they know not how to cast nets The sea surges up with laughter And pale gleams the smile of the sea beach Death dealing waves Sing meaningless ballads to the children Even like a mother while rocking her baby's cradle The sea plays with children and pale gleams the smile of the sea beach On the seashore of endless worlds children meet Tempests roam in the pathless sky Ships get wrecked in the trackless water Death is abroad and children play On the seashore of endless worlds is the great meeting of children