I don't know if you all have had that happen to you, I get it a lot, matter of fact so much so I'm actually going to share with you some comments that I've gotten because if you happen to say something about someone, maybe this person agrees with this person because if I agree with person A and you've called out person A, well then you're calling me out and so because of that it's not because I'm wrong, it's not because I don't know my words, it's not because I'm not safe, it's because you're not safe, it's because you don't have the Holy Spirit. Yeah I think maybe some of the stuff I see what you're talking about but you got them twisted because the reason why you're reading it from your head, you got to read it from your spirit, you don't have spiritual eyes, I've been told that before, as a matter of fact what I want to do, I'm going to pull up some of these comments and I want to do this, now I'm going to have my picture over so you don't necessarily, I'm trying not to embarrass the person, now you might see their name or what I'm not sure but I'll do my best to try to keep it where you can't see them but the point is the reason why you have this as a problem, the reason why this is important is because this is the issue with the body, the body does not seem like have a strong regard for the word of God, as a matter of fact what I want to do is let me first, let me play something to kind of illustrate this, this is from my new favorite, my new favorite false teacher, Prophetess, that would be the Miss Prophetess, I don't know if she's a doctor yet, the Miss Prophetess, Tiffany Montgomery, now I don't know if you caught that, she says what we're gonna do for me, we're gonna pray in tongues, pray in the Holy Ghost, we've covered that before, I won't touch that but she says I know, I know, I know, don't send me any emails or what have you about how we're not supposed to do it in public, you know, the Bible says the Bible, since we're gonna go by the Bible, the Bible says that regardless of what you think about tongues, if you have my view, if you have a different view, regardless what you think it does say clearly that if you're gonna do it in this gathering that there must be no more than one at a time, two or three at the most but she's saying no we're gonna do it all together and you've seen this in some of these churches where they just do this and it's like well have you read the scriptures, the scriptures say even if you disagree with my take on it with it however you whatever you think tongues are you can't do it the way you're doing it and so she's decides we're gonna do it in round when it comes to scriptures. When you go to the gym and you go and work out with a whole bunch of people does the workout edify the people that you're with no it just helps you out right but there's a reason maybe you're motivated to go to the gym instead of working out at home because half of us find out that if we work out by ourselves we're gonna do five push-ups instead of 20 we find out that we're gonna do two and a half sit-ups instead of 10 we're going to run around for 12 minutes instead of 30 but when you're working out with other people statistics show that you're actually more motivated to get the workout done than you are if you did it by yourself so so so if we all speak in tongues at the same time it's gonna motivate us to do it even more so even when we're by ourselves so forget what the word says let's just I've got a new a better way this is why this person is a heritage this is why this person needs to be disregarded this is why this person needs to really just sit down somewhere and just in the in the words of John MacArthur just go home just go home you have no business with a mic in your hand before the people one because you've been ordered but two even the heretical things that you're saying now I want to look at some of these comments I want to look at some of these comments because I think they're just they're interesting and what I'd like to do is whenever someone in this truth guys whenever someone gives me even a negative comment I think and say to myself huh maybe let me think about this for a second so let's look at some of the comments some comments say since you don't believe the gifts are for today it's weird you're even talking about this that well first of all I've never said I don't believe the gifts for the day I have never once said that the spiritual gifts are not for today I just said I'm a biblical continuation is meaning when I see one and it looks like the Bible like it's done in the Bible with an amen praise God but if I don't see that then guess what I'm not gonna do I'm not gonna just go off of that because you said it's the spirit but I want to look at some more of these let's see let me just find some more of these now some of these are good some of these you know these are smart folks some of these folks are just not and let me find a few I should have went and highlighted some of these you all forgive me let's see this person says my biggest problem you've got is that I see the title of your video pigeonhole yourself as a as a Calvinist again not a Calvinist can we can we say it just one last time I can I just get the people to understand I am not a Calvinist but anyway let's see and I already know what the answer your question is before watching it but what does the Bible say about a person answering something before they hear it now he goes on and say I guess it's a he Jason something he goes on to say a little bit more but let's see what does he say none of this is about pleasing the Lord if one traces the Greek and Hebrew words for profit the answer is rather plain and uncontroversy which I literally cover that in the in this in that video but I'm quite certain wouldn't land at the same Calvinist position you likely hold you like we hold what you find out anyways let's leave him alone there are some more matter of fact I'm not gonna run through these because you know why they can be annoying sometimes they really are but here's a problem the problem is everyone who disagrees now I shouldn't say I shouldn't say that that'd be wrong but there's a lot of people who disagree who would say that it's just you're relying too much on head now you rely too much on the Greek you rely too much on the Hebrew you rely too much on the English you rely too much on head now it's well all I've got is the Bible now I think the way I feel is the Holy Spirit you think the way you feel is the Holy Spirit two Christians can have the Holy Spirit and disagree well what do we do then when two Christians who both have the Holy Spirit disagree do we have a fight an arm wrestling match to see who whose Holy Spirit the strongest and we know it happens because Paul addresses it have believers can disagree he's he writes a letter in 1st Corinthians and his whole point is that there be unity no divisions he and Barnabas also had a disagreement two people with the Holy Spirit same thing a disagreement it can happen as a matter of fact it happens a lot and so absent anyone else coming in and settling the score what do we have to rely on you have the Holy Spirit at the Holy Spirit your Holy Spirit is no more greater than my Holy Spirit so what we have the one thing that we have in common which is going to be the word of God so let's look at the word of God rather than following after our heart after our feelings instead of leaning onto our own understanding now the Bible is clear that if a person how do I put it if you have all the knowledge in the world and and there is no love or by proxy or by extension no Holy Spirit then what are you as matter of fact let's go to 1st Corinthians 13 if I have to get the prophecy verse 2 and know all mysteries and all knowledge and if I have all faith so as removed mountains but do not have love I am nothing so clearly knowledge by itself isn't enough obviously but the thing is though they they present this false dichotomy where you have either one or the other you have one of it so if you have some knowledge then that's a bad thing now it's only a bad thing knowledge is only a bad thing if your knowledge disagrees with my understanding because I can promise you the very same people the very same people that would accuse me or you of not having the Holy Spirit and relying on head knowledge if we go to the actual Bible itself which we should if we delve into the Hebrew Greek they wouldn't have a problem with the Hebrew the Greek or the scriptures themselves if what we if our conclusion was the same as theirs I've seen them go to the Greek go to the Hebrew now there's times they do so and they're wrong but still there should never be a problem with go if a person ever says has a problem with you go into the Bible that person it makes you wonder if that person is saved or at least not as safe as they claim to be I would put it this way if you have a problem with the person being in the word that much you're one of two things either not a Christian or not a very good one that's the only solution you're either not a Christian or not a very good one now because you're not a very good one you can change you can grow you can mature but you cannot rely on this ignorant statement of saying that you don't have the Holy Spirit you don't you're not you're not singing it because what that necessarily means that I'm right they will accuse you and I of being proud or arrogant you don't knowledge puffs up but what are they saying they're saying you don't have the spirit like I do because you don't see it like I do so you must not be safe that's really the height of arrogance all we're doing is going off of the scriptures a matter of fact the Bible tell here's what here's what makes it so bad if you're going to say that a person needs to rely on the Holy Spirit and not so much the Bible can I tell you and I'm not trying to be mean anyone that would make that statement however this is how utterly foolish this is it's nonsensical saying a person relying is relying on the Bible more than the Holy Spirit well where does the Holy Spirit come from huh if only the Bible will tell us where the where these words come from if only there was a passage telling us what's the source of these scriptures oh it does all scripture according to Paul all scripture is inspired now this word all I mean inspired is they are God new style which is breathe so breathe out by God all scripture is inspired or God breathe by God and profitable for teaching for teaching for a proof for correction for training in righteousness why so that the man of God man and women because it's Anthropos may be adequate equipped for every good work so the very thing that you're harping on that you're saying and I'm using thing and I don't mean call all these things but the very thing that you're harping on saying that we don't have we're literally using the scriptures are from him the scriptures come from the Holy Spirit and so to deny that or to say that someone didn't have the Holy Spirit and they're relying on head knowledge you ma'am you sir are acting silly you are being contrary to the scriptures what you just said is antithetical now we know the importance of the word because matter of fact let's go to Psalm he says in Psalm 119 119 says your word have I treasured in my heart that I might not sit against you so what does what does the word do it causes you to not sin this word right here treasure this word right here means to kind of keep to guard to hide okay and we're going to come back to that in just a second but it's very important that we recognize that by the way Anseltine for all thank you so much for the super chat or super sticker thank you so much but the word is ridiculously important I didn't sound right for me saying that the word is ridiculously important but it is without it you are going to stumble without it you go matter of fact here's my question how many of you have faith without the word of God is it possible now think about this for a second is it possible to have faith without the word of God or I'll put it this way how much faith can you have without the word of God how much faith could you possibly have without the word of God I when you think about it your your level of faith is dependent on the word of God where do you get that from Cory it's almost like it's almost like you're making that up Cory well I'm not Hebrews 11 here's what it says now faith I have an Holy Spirit and so forth I think produces this now faith is the assurance of things hope for it the conviction that things not seen for by it the men of old gain approval now you'll find people that come against me for you will oftentimes come to this particular passage for various reasons by faith look what it says though here's here it is the important part verse three by faith we understand that the worlds were prepared by the word of God so that what is seen was not made that I'm sorry was not made out other things which are visible by faith able offered to God a better sacrifice now here's the question how do we know this how do we know any of this how do we know by faith that the worlds were created by God is there any way how do you know it raise your hand if you were there raise your hand if you absolutely know good witness can testify yeah I know exactly what happened when God created the heavens in the earth I know exactly how you don't know except for the word Romans 50 matter of fact let me put Romans 15 for I didn't put it up there but I should have put it up there because there's a reason for this is the bible Romans 15 for for whatever was written in earlier times was written for our instruction so that through perseverance and the encouragement of scriptures we might have hope so this this ability to have hope comes from the scriptures you have to have the scriptures if you don't have the scriptures then all you have is blind faith and those people who claim to have faith but don't read the bible are soon to be former Christians not that they were ever Christians to begin with but they are soon to be former why is that because your faith was based off of a thought and not anything tangible someone told you but you never you never saw or investigated for yourself and so therein lies your problem so someone can tell you Islam is true someone can tell you that part of the bible is messed up that there's errors they can give you all sorts of things but because you have not investigated yourself well then you're subject to whatever they tell you now there are those that will tell us that too much of this reading is a problem as a matter of fact they'll try to convince you and I and others really others not us because we know better when someone doesn't read the bible tries to convince people who do read the bible and i mean read the bible as a pattern of life they don't go very far as a matter of fact oftentimes you're going to find that those people who say they rely so much on the spirit when they're they don't want to talk to someone who reads the bible a lot why because they recognize that they might be at a disadvantage and so what they want to do is they want to they want to impugn you in the eyes of other people i'm gonna put up a passage i'm gonna put a passage not a passage but a clip that we've covered before and i want because i think i think it's it's germane to what we're talking about it's like a lot of these guys they're making their brain do mental gymnastics and binge watching videos and reading all the time because they're really they have not confronted the stuff that's really inside of them and it's a it's an escape now how he knows is i don't know that when people do that it's what they're doing is they're they're covering up there there's something hidden it's an escape from something it's an escape from something how do you know that i how do you know that you you don't know that you don't know that as a matter of fact let me put this up he says the theology has become a bad word uh oh i moved i lost it there it is has become a bad word uh to those with no understanding yes as a matter of fact good thing that you put that up because we're going to talk about someone who's going to say that there's too much theology can be a bad thing but let's go back to him listen to what he says about us so it's like okay well i do drugs but you do you know like you snort books yeah you know how's that come out of your mouth you snort books does that does that i do drugs but you snort books you're you're reading too much you're reading the bible way too much my friend that's too get out of that bible and go do something with yourself you know it's like you know like you're you're just as much a distracted with your stuff it's it's just addiction and another flavor you know what i mean um you don't really even care about god you don't have a relationship with god you're just you're just obsessed with the things of god you know because it's a great distraction from your own trauma okay a couple things this is this is this is bob this is funny he says you don't even care about the things of god you're just it's a distraction you don't why would you say that who wants to read the bible now the things of god are literally his words literally his words those are the things of god so to turn around and say that we're being distracted by the how do you make make that make sense it it really doesn't what it what it really is and oh by the way oh by the way oh by the way this is a pastor please let that sink in this is a pastor who would make that statement that you don't even care about the things of god you just into the word of god huh you don't care about the things of god you're just into the word of god this this i'm gonna need some help with that now what's funny what's interesting is that uh his partner is is sitting there and just eating it up i want i want to play that again i want to play that again let me let me fast forward to that part again i think this this is just it's interesting but it's all so sad because the amount of influence that these two gentlemen have as well as others that are like them is amazing people are listening to them and they're not paying attention what they're saying if your leader or the person that you like or the person that you follow says anything like this i might i would say listen you you one you don't have a biblical leader you might want to find someone else that there's here's what's funny you would at least hope that they could find something in the back of what they're saying in the scriptures but what's going to be absent from what they're saying is a bible verse it's like a lot of these guys they're making it's just addiction and another flavor you know what i mean you don't really even care about god you don't have a relationship with god you're just you're just obsessed with the things of god that's a hard thing to say you really don't care about god or you just obsessed with okay that's obviously there's nothing difficult about it obviously there's nothing difficult about it let me just say this have you ever thought about how satan actually attacks think about what satan does now we've covered before satan doesn't have a whole lot of tricks up his sleeve he really doesn't have a whole lot of tricks up his sleeve and by the way cg i don't have a problem with this addiction i don't have a problem with this addiction at all one iota i am fine if you say that i'm addicted to the word of god no problem no problem uh giordnalli's thank you for super chat he says there is two there are too many passages that approves god's word like his equal 17 plus personally a testimony uh ever since i embrace god's word my life's a man a man and oh the last part thank you i appreciate that that i didn't put on i didn't put on the screen i'm sorry there it is i should have put on the screen thank you thank you and tyrone dunson thank you also for the super chat you know what i can't tell you how when i'm in the word how my life is better matter of fact even on the days just going day by day if i have the word of god in me in that day i'm better off i'm i'm so much better off by having read his word even if even if it's it's an extra minute literally an extra minute if i read an extra page now there's two types of readings you can do there's the reading for just your own uh comfort level your own edification almost like reading the reading a book for book sake i'm just enjoying this i'm not necessarily looking for it to prove a particular lesson i'm just reading it and just absorbing what it's what it's giving me and then the other way that you read it i'm i'm studying it i'm looking up a particular doctor i'm covering that's both are fine both are fine there's no there's there's always that you've heard the saying that you you you can always get too much of a good thing yeah but not this the person that said that didn't say that about the bible hadn't read the bible you could never get too much of it but now let's think about let's think about satan because who of all the people that you can think of of all the people that you can think of human beings or demonic who would have the greatest interest in you or me or anyone else not reading the word who would ever say let's have less less word less less is more now i was an architect student i was an architect student and there's a famous architectural phrase you've heard it didn't know it came from an architect that says less is more never heard that before less is more well less is more is is fine when you're doing when you're doing space and design less is more fine but not when it comes to scriptures but when it comes to satan in the scriptures less is more he's only got he's only got two tricks up his sleep and both of them pertain to the word now let's go to it as a matter of fact genesis three now the serpent was more crafty than any beast in the field which the lord god had made and he said the woman indeed has god said is that does the bible say so is that what the bible says are you sure about the bible are you correct are you are you sure that's what it says hmm well the woman says yeah it does say that okay good yeah it does say that it does say that but you're misunderstanding now what does he do what's he trying to do the bible says that the fruit that she saw it was pleasing to her eyes that it was desiring so he's going to twist it to kind of get you to do what you want to do he's not going to twist the scriptures for you or distort them to cause you to do what you don't want to do in your flesh in other words he's not going to twist something and then cause me to all of a sudden let's say you want to go i don't know do drugs i don't do drugs except for coffee and sugar and sweets which is sugar and some more coffee and sweets sweet coffee but other than that he can't twist he can't twist something to make me want to go do drugs listen if you if you said cori there's there's there's a ton of xyz drug over there okay i wouldn't care how much if every demon on the planet was whispering in my ear it it's not gonna work for me now cori you just said you're gonna cut back on sweets and your daughter just brought home this beautiful looking cake i'm hooked i'm hooked matter of fact that demon didn't have to whisper that much i'm there if it's sweets i'm dragging the demon to the sweets that's just that that's just me he is going to tempt you in the areas not that that's your weakest and the reason why you're weakest is because that's what you want and so when you see people out here doing these little ridiculous things these these exaggerated movements calling it the Holy Spirit it's because it fulfills a desire for them being great being seen as great and then those in the audience i want to be great too even if we're just great in our own mind that's why you have a world full of people who just simply make up things as they go call me this i i feel like going by this gender i'm this i'm that identify as this i well you're gonna have a hard time identifying as someone who's cool when you're in hell so god wants you to be serious about what is real like his word now we also have jesus who similarly who similarly was tempted by the devil the devil did the exact same thing the devil does the exact same thing that he does with e with jesus the exact same thing he goes at him with the word what is jesus's response jesus's response is in matthew 4 4 he says but he answered it is written man shall not live on bread alone but but by every word that proceeds out of the mouth of god where is this word established where do we where do we know this where where can we find this word today that proceeds forward out the mouth of god oh i know the bible so we should live off of that that's our source now if the devil will go after eve in the garden remember you and i have never been in a situation as ideal as the garden of eating no sin whatsoever at least yet no sin perfect you and your spouse are running around naked eating up eating fruit and so forth not having to care in the world none of that in them in the most ideal situation how to say to attack them and then jesus who has created even the very thing that's coming after him so if the devil will go after eve with the word twisting the word and the devil will go after jesus and try to attempt to twist the word what do you think he's going to do with you the same thing and oh by the way you fall faster you fall just like i do we fall a lot faster than eve or jesus jesus didn't fall you would before the devil would have gotten a half a syllable out of his mouth you and i would have eaten the fruit the fruit would have been bitten before the devil would have said when the devil says did god before he gets the word out we would whatever the fruit was we we would have eaten it he wants to twist the word and so our only recourse is to well one fight him and go to the word even more so go to the word even more so but we think that because yeah we shouldn't we we shouldn't do we shouldn't think that much too much thinking is a bad thing or let me just let me just put it the way that that voting puts it because everybody knows if you really want to love god you got to be ignorant yeah we we we should be ignorant and the way that we can be at our at our ignorantist i just made that word up our our ignorantist is just simply not read the word now paul did say i don't want you to be ignorant i would have you not ignorant i don't want you to be ignorant i don't want you to be annoying and we think that the more you study the the more listen the bible study scriptures is not a holy spirit repellent no reading the bible does not repel the holy spirit reading the bible invites the holy spirit it brings about a closeness to you yeah but you're already saved you're already saved you got the holy spirit in you so you don't really need to read as much but can i go in and take you to the words of a very famous modern-day prophet i like what this modern-day prophet says matter of fact i think we haven't we haven't used his sayings in quite a while but this very famous modern-day prophet he gives us some words of wisdom you can be saved and stupid tweet that you can be saved and stupid i agree amen you know what you want to speak in tongues go ahead and speak in tongues on that go ahead and speak in tongues on that because what he said was correct you can be saved and just because you say does not mean you're not but and paul says that we should grow in the grace and knowledge of the lord so as we're growing we're growing in his grace as well as the knowledge of him that's how we know what he wants you don't know what god what god wants if you don't know god the only you know god is know his word that's the only way all scripture has been given it's been breath out and is profitable for our instructions for reproof for correction and so that every person can be complete not lacking anything now i want to play this this other clip this other clip i played this before but this is the height of just it's just silly for it for it to come out of his mouth but again is it dangerous to study too much theology in my opinion yes my opinion is this if you become so head knowledge and knowing all about god like the Pharisees did they knew all the laws they knew the um all the scriptures by the way if you actually did study you would know that the Pharisees did not know all the word of god as a matter of fact the Pharisees were horrible when it came you know what Jesus never said to the Pharisees that is if you actually read it if you studied it you would actually find out that Jesus never once went to them and said you know what you guys you guys really know this word you guys are smart you guys you guys are really something else when it comes no Jesus Jesus never committed them for their knowledge of the word what he did attribute to them was their disobedience their hypocrisy and their love for self he knew they knew some words some of the scriptures of course but he would say he would always rebuke them if this is not the way it was think about when they were talking about marriage Jesus rebuke them saying it's not so but out of the heart of your heart this is why they didn't know Jesus never said that the Pharisees you guys know the scriptures very well he find that he doesn't yet they didn't know the messiah when he came this is what we're going through in romans they thought by learning the scripture by doing all the rituals that they were saved and they were spiritual and they were christian and it did the adverse effect they knew so much that they missed them again what Pharisees were we talking about in romans none none sire when he came so should we know theology yes theology is the study of god ology is the study of theo is god so theology is the study of god i think it's great to study theology i think it's good to know theology but when you know so much and you're not a practitioner it creates pride at puffs up okay well let's just handle that with the scriptures what is the bible says matter of fact let me go ahead and go to the king james version i like the way the king james version reads on this party he says study huh isiah study why study to show yourself or i'm sorry if i want to read the king james version i must read it the way the king james version reads study to show thyself approved unto god a workman that needeth not be ashamed rightly dividing the word of truth yes so should we study yes how often as often as you can as often as you can now obviously you have to live and you have to live it out the model here when it comes on the screen whenever you see that little red outro or intro have you ever had let me ask you guys a question have you guys looked at the words when it pops up do you all know what it says do y'all how many y'all know know what it says do y'all remember what it says when that when the red when the red intro comes on i'm just wondering i'm just and i'm hoping i'm hoping you guys remember it i'm hoping you guys have seen it hurts my heart it hurts my it will hurt my heart if you guys don't know if you haven't thank you jeff jeff knows it learn it love it live it y'all don't believe him here it is yeah you should love the word so much that you learn it in order to live it how well you live it by in your in your in your practice in your dealing with others but also how you convey it it's of the it's what he left us with it's the one thing that we can literally put our hands on and touch the word of god love it learn it and live it so that's why you ought to study as often as you can and then go and put it into practice why is this important why because the word of god according to the bible is living the word of god that we're studying we're studying what is living and active and sharper than any two-edged sword and piercing as far as the division of soul and spirit both joints and marrow and able to judge the thoughts and intentions of the heart now let me say this whether you believe in a dichotomy of man or trichotomy man it didn't really matter in this point the only thing that can cut through the soul and spirit if it's possible is the word of god so should we study it absolutely now you can study it in a in a way that that is incorrect because bad and i like how jama carter puts a bad study is almost like not studying at all the importance of careful biblical interpretation couldn't be overstated it's impossible to overstate it misinterpreting the bible listen is essentially no better than not believing yeah but when you have someone that brings about a misinterpretation of it and you haven't read yourself how will you know it would be impossible for you yourself to know it would be in it would be difficult for you to know what the word says because you're taking someone else's understanding because you haven't read it yourself so what does what does paul say what does paul say about the matter in 2nd Timothy 4 he says i solemnly charge you in the presence of god and of christ jesus who is to judge the living and the dead and by his appearing in his kingdom preach the word crooks on preach it call it out yell it preach the word when he says be ready in season and out of season well even if i don't know when the season is and when i'll know if it's either in season or out of season it's both times all the time be ready to do so reprove that means when someone says something incorrectly what prove them right well how do you how do you how can you possibly prove someone who's giving bad information excuse me how could you prove them right or wrong well you you must know the scriptures yourself reprove rebuke we shouldn't rebuke we should just pray for people we should we should never correct anyone we should never call them that we should know that's not what the word says rebuke exhort with great and exhort with great patience you do all three of these things you encourage people you rebuke people and you prove them you teach them how with all long suffering with great patience keep doing it so how often will we call out false teachings according to the scripture with great patience and instruction how often would we teach a particular passage a doctor and so forth with all long suffering over and over ever and ever ever and ever how how long is ever and ever so long we took the four of ever and ever should we ever rebuke someone hey you're doing when should you how often should you call someone out whether it's rebuke with great exhortation with great patience and all long suffer all the time but Cory there's always going to be somebody who is there there's always going to be somebody that doesn't agree with this doctrine that doctrine so we're in the bible do we do we hear give it to them once and then hope they get it throw whatever you can to the wall and see what sticks that's not that it's not our mission how long did it take for you after hearing the word of god for you to come you i promise you i don't care what you say if you tell me differently i know you're lying and you won't lie to me how often how long did it take for you once hearing the word of god to decide you know what place your faith in christ you did it the very first time no you didn't you did not place your faith in christ the first time you heard no no so while you have time what we do we keep trying we keep calling out we keep doing as it's going to be this way until the end of time why why because somebody hadn't heard this i can promise you you can say the same thing a thousand times and someone's going to say i've heard enough times yeah but of course he just heard it and i can promise you there'll be a testimony from somebody who's like man i i thank you for what you just said for what you heard i've never heard that you keep doing it you and at some point in time it might sink in because you don't know what position you're in you don't know if you are in the position of the one that's planting or the one that's watering that god is using ultimately he's going to give the increase so therefore you keep doing it what did paul say in season and out of season in season and out of season keep let's continue for the time will come which is now when they will not endure sound doctrine wanting to have their ears tickled wait a second court no we should just listen stop with all that theology all that doctrine all that studying why it's literally bread it's literally my life it's your life it's our life it's all we got it's all we got yeah we have a holy spirit that's i'm i'm not minimizing holy spirit because if you don't do this in concert with the holy spirit then you have nothing an unsaved person can read this but that unsaved person that reads this they are damning themselves they are heaping judgment upon themselves because they're literally looking at what they need to know but those of us that have the holy spirit what do we do next turn to turn to it for our instructions so he says uh but one have their ears tickled they will accumulate for themselves teachers in accordance to their own desires what happened with eve her own desires what does james say we are led away the lust for our heart we are led away by our own desires but be so be sober in all things endure hardship do the work of advances fulfill your ministry what does he say now something interesting that paul says verse seven he says i have fought the good fight well there are some people that that would not want us to fight the good fight there are some people that would rather as you know what just say what you got to say and leave it alone say what you got say it once hit it once and bam that's it no this fight is an lasting fight i have been fighting the good fight i have i have fought i have fought i have i've fought and fought and fought and fought and fought i fought the good fight the good fight in a quick fight now as much as we would like it to be we would like to be in the fight where it takes us just one hit and we can we can knock our opponent down that's it but that's not really a fight is we have fought the good fight and look what paul says as he is as he says i fought the good fight i have finished the course i have kept the faith this is the part i want i want to focus on i have kept the faith me highlight this let me highlight this how did i do that okay let me wait a minute i don't know how i just did this whole thing like that i did not mean to highlight the whole part okay now let's make it better i have there it is why is it doing okay whatever the whole thing's highlighted i'm not gonna even fool with it i have fought the good fight look he says i have kept the you know what i know what i'll do i know what i'll do this thing is not going to get the best of me what i need to do is study this this software to figure it out but what he says here i have kept the faith this you see highlighted in the blue i have kept the faith he has kept the faith the tenets of the faith you only keep the tenets of the faith if you know the tenets of the faith you only know the tenets of the faith if you read the tenets of the faith in the book that's the only way you cannot get this understanding by osmosis by putting the bible on your head you put the book on your head and that's that you know that that's that's not how we're saved that's not how we grow okay now i understand i understand in the early days of the church um they didn't have bibles like we had they couldn't read like we wanted to read now think about this though and this is to our shame he says cori has a little grandpa chuckle i never thought i probably do now now now you've got i'm gonna be so self conscious about it i'll be thinking about it now i don't even want i don't even want to chuckle now i can't do it dog on rock you don't you don't mess you don't mess the whole you don't mess the whole video up but don't you think about something though in the day at the founding of the church they didn't have what we have they didn't have the research they didn't have the they didn't have the bible like we had they they just didn't what would they have killed for could you imagine could you could did anyone graduate from Marcus Ryder's seminary yet no i'm didn't want to apply could you imagine though the envy of them could you imagine because do you know what paul didn't have do you know what peter even peter didn't have do you know what john didn't have they didn't have the word like they didn't have the print word where they could just open it up no paul did not have the print word you know how we know well by reading the bible look what he says verse 11 he says only lucas with me pick up mark and bring him with you for his useful uh useful service to me of usefulness i can't even speak it for he is useful to me for service now let's drop down he says when you come bring the cloak which i left at troas with carpas and the books or what books especially the parchment paul did not have all of the writings you know you and i have even on our own little phone even with we have all we have a wealth of information at our disposal we can we can go and look up stuff even if you've never bought a book if you don't have a cordon's bible software somebody's gonna ask what this is this is a cordon's bible software if you don't have a lot guys if you don't have some sort of bible software that's fine even google as wicked as google can be youtube is what you can go there and you can just type in something and get information you can look at maps you can find out what the customs were if you don't know greek or heber you can act like you do and look up someone else can help help you out and tell you you've got a wealth of resources at our fingertips and we don't even use them they would kill to well first of all they would probably kill to have a cell phone but to have what we have in the palm of our hands we've got more at our disposal now what they did have though since they didn't have the bible they did have the apostles well that's pretty cool that's pretty cool what bible software do you use i use a lot this is this bible software right here is a cordon's but i also use lagos i use them both i just put a cordon's up because a cordon's just looks better it's more aesthetically pleasing on screen than than lagos lagos it kind of jumps around a little bit it just i don't like it i can do this a lot quicker i put a 30 passages and move fluid not fluidly what's the word seamlessly through it that i could with a cordon's but the point is i mean than the lagos but we've got all this information we've got all this information and then someone will come and say you don't need that much all all you need is the holy spirit all you need is the holy spirit well sure all you need is the holy spirit for salvation you're absolutely right but what about growing so don't diminish what the holy spirit has given us which is his word do not give the whole do not diminish what he's given us which is his word this is what we live by guys this is what we have at our disposal this is what this is what keeps us fat this is what keeps us moving this is what keeps us growing his word why do we need his word because you've heard some of the ridiculous statements that were made earlier you've heard some of these things and it's just it's an affront to god than what he's given us amen that chosen reborn guys thank you for soup chat he says reading the word makes my day so much better same here i find myself reading every day without fail thank you know it's funny sometimes i'll read and i don't i don't have enough time to go through what i need to go through i just i just don't tend to thank you for the super chat uh as well as tie your guy i mean your guy tie thank you as well but just being saved if if a person truly is saved if a person truly is saved with an amen praise god i listen i'll say it again you are not saved by what you know say it again you're not saved by what you know but what you know helps you get closer to who you know you're not saved by what you know you're saved by who you know knowing christ is what makes you save but what you know about him draws you closer you get to know how he moves the mind that that's been given to us in in print form he's revealed himself in a few ways in nature even the even the heavens declare his handiwork he's done that he's revealed himself in the person jesus christ and he's revealed himself through his work now we don't know about jesus christ as we weren't there except through his work we get to see his character and his nature so the fact of the matter is if you are listening to someone that tells you that too much study is a bad thing and they want to pit the holy spirit against studying reject that person reject that person the most dangerous thing as a matter of fact from the very beginning we are in the position that we are in because someone some demon some devil some serpent came and made it as though that the word of god is not as important and so i will leave you with this because it's true like what he said you can be saved and at the same time not know much but you should there's no excuse for not knowing the word of god there's none whatsoever so again you can be saved and stupid you can be saved and stupid treat that you can be saved


Well, good afternoon, everybody. I'm very happy and honored to introduce you for this special lecture today. So Professor Shigefu Mimori has been very kind to visit us, and we are very happy about that. So let me say a few words about to introduce you, even though he's, of course, extremely well-known in the mathematical community, probably beyond. So now he's a professor at Kyoto University Institute for Advanced Studies. I mean, he's visiting us in mainly two capacities, as a great mathematician and president of the International Mathematical Union, which is an institution with whom we are collaborating and we are actually planning to collaborate more in the future. But a few words about Professor Mimori as a mathematician. He got his PhD in 1978 at Kyoto University. And actually, he has held professorships in many, I mean, Harvard, the Institute of Advanced Studies, Columbia, Utah, and so on. But he has been, I would say, very faithful to his origin in Kyoto. He has been professor at the Research Institute for Mathematical Sciences at Kyoto University, which is one of the most distinguished research centers. And actually an institute which he has been also directing for four years between 2010 and 2014. So on top of this, of course, he has been recognized with many prices and many distinctions. I mean, actually, I printed the list. The list is a bit too long. He is, of course, a honorary member of many Academy of Sciences around the world, including the Russian Academy of Science, the American Academy of Science, and, of course, the Japanese, and many others. He has been awarded a number of honorary PhDs degrees around the world. And of course, probably the most famous recognition is the 1990 Fields Medal he got for his work in Algebraic Geometry. So that's on the side of the Algebraic Geometer. As president of IMU, he has been president of the International Mathematical Union for the last four years, which IMU is by far the largest and most important and most influential organizations of mathematicians in the world. For those who are not mathematicians, probably just to give you an idea of what it is, the IMU, it is the organization which selects and gives the Fields Medal. So he's not just a recipient, but he's also the one who. So this is an institution, of course. We have been collaborating with ICTP for a number of years, in particular, it is the institution with whom we collaborate to award the Ramanujan Prize, together with the Indian government minister of research. And actually starting next year, we are supposed to organize together the UNESCO International Day of Mathematics. It has been recognized March 14 for the Anglo-Saxon. You understand why? Only for the Anglo-Saxons. I mean, you write it in the way, otherwise you celebrate 143, which makes no sense. So that's something we are actually supposed to do together. But it's going over and over our interaction. For example, I just remember this morning, we are actually organizing a school and activity together in 2019 here at ICTP and not actually not just mathematical related. We are having a big meeting on the global approach to the gender gap in mathematics, computing, and natural sciences, co-organized with the IMU. But I mean, the plan and, of course, part of the scope of the visit is to extend this interaction to a broader schemes. That's it. So besides repeating how happy and honored we are of having him here, we just found out also in these discussions these days he is probably a mathematical hero of many members of the ICTP mathematical group. So I just leave the stage for Professor Mori thanking him again. Thank you very much for a very flattering introduction. So I'm here just to explain about the classification of algebraic varieties. And possibly by going through how my interest grew, I can give you a more friendly and better introduction. And that's what I intend to do. So I will give you my personal view, not an official view. So in 1975, I was 24, just finishing a master course. And I understand that there are quite a few diploma students, which means more or less my age that time. So it will give you the kind of feeling how it was that time. So algebraic varieties. So I do not exactly know the audience. So some physicists and some mathematics oriented, but not algebraic geometry. So I have to be a little bit careful. But nevertheless, I don't make any definitions. And I even don't give you any equations. So I just say that algebraic variety is a kind of a figure defined by algebraic methods. So of course, that feels like a differential geometry, where they can define figures by differential equations, or differential methods. So algebraic geometry handles smaller class of varieties. And because of this, there are pros and cons. And I'll explain that later. So given an algebraic variety or algebraic figure defined by the complex number field, it may have bad points, like holes or some kind of singular points. And by the film of Hirunaka, you can always blow up and make it into a nicer variety. And that will be x over the original z. And so instead of starting this figure with singularity, we work on the smoother one. And on this smooth x or a manifold, there is something called a homomorphic volume form. And it gives a canonical divisor or canonical divisor class. Just something you can play with. And the first thing you get out of it is something called the canonical ring. So for any non-negative integer i, i times kx gives you some divisor and the global section. They form a finite dimensional vector space. And taking the direct sum of all non-negative integers, you get a vector space. But not just a vector space, but a graded algebra. So if you take a piece from the i-th factor and the j-th factor, if you multiply them, it will be a member of i plus j vector space. So it is a graded ring. And it turns out that it's a birational invariant of x. And so it is even a birational invariant of z. So it doesn't, you may change x depending on the resolution. But still, rx does not change. So it is invariant of z. Out of this, one can define the codal dimension, which is this graded ring. If it has a non-constant element, then take a quotient ring, a quotient field, and take the transcendence degree over the field of complex numbers, minus 1. And that will be the codal dimension, kappa x. And if there is no non-constant section, then it's just minus infinity. So it can take values between minus infinity, 0, 1, and all the way through dimension x. And if it's dimension x, it's called the general type. x or z are called the general type. And it is, again, a birational invariant. And the idea is to classify algebraic varieties using up to a birational equivalence. And I'll explain a little bit. To classify a variety up to a birational equivalence is more or less classifying function field. But sometimes it's more complicated than it looks. So let me start with an easy example. So if you take a projective space with dimension n plus 1 and the quadratic hypersurface. And it is a hypersurface of degree 2. But it is actually birational to Pn, n-dimensional projective space, or hypersurface of degree 1. So hypersurface of degree 2 is birational to hypersurface of degree 1. And you can realize this by projecting Pn plus 1 to Pn from a point q on this hypersurface. And the easiest case is the one-dimensional version. So quadratic hypersurface, in this case, is just a circle. And it's birational to A1. And so this is the picture. So circle and pick up a point here and take projection, meaning that take x1, x2 in the circle and draw the line joining minus 1, 0, and this point. And then it intersects the circle, sorry, the line at one point. Whose value is t? And or pick a point t and draw a line. It intersects with the original point. So using this relation, one can express x1, x2 in terms of t or t in terms of x1, x2. So in this case, a point, a circle, is rational parameterized by one variable t. That's the definition of a rational curve. So it's a curve with a rational parameterization in one variable. And that's basically the tool I use in my research on algebraic varieties. So let's start with dimension one case. In dimension one case, that's not that much important. In dimension one case, you just consider the smooth model, smooth compact model. And they have, so they are Riemann surfaces. And depending on the counting, how many holes the handles they have, you can model express for them. So it's called the genus, G, genus of C. That's a number of handles. So if G is 0, it's a Riemann sphere. And if G is 1, it's like a surface of a donut. And if G is bigger, it's some kind of donut, but it's more complicated. And if you compute the coder dimension, for G equals 0, kappa is minus infinity. And for G equals 1, kappa is 0. And for G bigger than 1, kappa is 1. So for later reference, I also want to consider how the curve C is curved at each point. So in case of an algebraic curve, one can introduce a very nice metric. And in terms of that, for G equals 0, it's a Riemann sphere. So it looks like this. And given a point, you take two transverse geodesic. And in this case, they are curved in the same direction. So it's positively curved. And in this case, it's flat. And in this case, it's like subtle. And in this case, it's negatively curved. So that's what we mean, how the variety or how the curve is curved at this point. But anyway, this is something done in the second half of 19th century. And dimension 2. So in this case, a bi-rational geometry is already non-trivial. So starting with a surface downstairs, a pick-up point P, then you can do an operation called the blowing up of a point P. So it is to replace P by a rational curve, let's see, rational curve C, keeping the rest the same. And this rational curve C is a self-intersection minus 1. If you compute the topological intersection number, you can see that it has minus 1. And that's why it's called the minus 1 curve.  And that's how you look at it from downstairs. But if you look at it from upstairs, you can say that it's also a blow up of P, but it's a contraction or a blow down of minus 1 curve C. OK, so this is the process you do in studying bi-rational classification. So if you compare upstairs and downstairs, upstairs you have a curve and downstairs you have just a point. So intuitively, the surface S is smaller than S prime. That's the basic attitude. And in the case of coder classification, its intention is to generalize the classification of curves to dimension 2, up to this kind of a blowing up or down. This is done based on the minimal model program in dimension 2. So it was done around 1900 by the Italian algebraic geometry school. And let me explain a little bit that minimal model program abbreviated as MMP. I put it traditional. I mean, it's not an insult, but it's just a different thing. I will change the notion a little bit to give a different name. I just put traditional. So as I said earlier, if a projective surface has a minus 1 curve, then you can safely contract it to another smooth projective surface. And since X prime, in this case, is smaller, it's natural to work on X prime than X. They are bi-rational to each other and X prime is smaller. So it's natural to work on the one downstairs. And we repeat this until there is no more minus 1 curve. Because each time you contract the minus 1 curve, the number goes down by 1. So it cannot go on if you do many times. Then when there is no minus 1 curves, then X is either a P1 bundle of a curve or a P2. So these involve the G equals 0 case, like G equals 0 case. So they involve something of this nature, curving positively. Or a minimal model. Actually, I don't give the definition, but it cannot be made smaller by this kind of contraction of minus 1 curve. I guess that's the definition. And so this corresponds to a positive genus case, G equals 1 or G equals 2, or larger. And so it sounds like a nice theory. But at that time, it's a combination of arguments based on the case division. So when I tried to learn it at that time, it's quite involved. But that was state of the art at that time. And people, of course, tried to generalize it to higher dimension. But there are various difficulties. So in case of a surface, we contract the minus 1 curve to a point. So in case of a 3-fold, it's natural to contract the surface either to a point or to a curve. And if you contract it to a curve, there is an established theory of contracting divisor to a curve in dimension 3 as analytic space. So I don't define the exact conditions, but there are cases where there is an analytic 3-fold and a smooth curve. And by sort of blowing up some process similar to blowing up a point, you get a surface. And the surface S to C is a P1 bundle. Every fiber over every point is a rational curve. And altogether, X is a smooth 3-fold and Y a smooth 3-fold. So there is a situation like this in analytic setup. So it's natural to imitate this procedure if there is a situation like this. But then even if X upstairs is a projective variety, the variety Y downstairs can be a non-projective variety. It's a tricky variety. But you may say that even if it's not projective, it's an algebraic variety who cares. We may go on. And if we try with X algebraic variety, Y can be a non-algebraic variety, a non-algebraic. But that's something we algebraic geometers cannot accept. So we are stuck. And there are some other activities around that time. By rational morphism of 3-fold, which is not the composition of standard blow-ups in dimension 3, this is an example of Hirunaka. And since it's fun, I will explain it in the next slide. And since it's so difficult to find a minimal model, there are a group of people who try to do the classification even without minimal models. That's one of them is Itaka, Itaka's group. And to make sure that the process of finding minimal model fails, there are people who explicitly constructed the 3-fold, which can never have a smooth minimal model. So those are the difficulties. And let me show you the example of Hirunaka. So you start with a smooth 3-fold, then three curves meeting at a point transversely. And Hirunaka exhibited a way to construct smooth projective 3-fold above this one. And in such a way that above arbitrary point, except for this intersecting point, there is a smooth P1. It's like blowing up along the curve. So over each point, there is a P1 here, here. But not over the intersection point. So over the intersection point, there is one P2 and three P1s meeting like this. And if you look at this and try a little bit, it becomes clear that you cannot contract this into a sequence of smooth blow-ups. The only way to contract anything reasonable is to blow down this P2. Then if you contract it, you get a singular point, C3 modulo plus minus 1. So it fails. Yeah, OK, so these are the difficulties at that time. And that's the time I finished my master course. And some of you, yesterday, asked me what to do in a situation like this. So how to generalize a two-dimensional minimal model program, or to do without the minimal model? Yeah, OK, so I'm going to explain how my curiosity grew along with my research growth. So I was not an expert on the classification problem. I was a researcher in commutative algebra, or I may say algebraic geometry, but rather on the commutative algebra side. I'm not interested in bi-rational geometry. And somehow, when I kept doing what interests me, all of a sudden, I was in the middle of the classification. Anyway, so my basic question is, how a variety is carved? It's like going back to the original set-up of differential geometry. So given algebraic variety, you can put, assume you can put a scalar metric. And you have assumed that you have a metric. Then you can ask, how the variety X is carved at the point P? So to do this, you have to fix a real two-dimensional vector space in the tangent space. So you pick up a plane, R2, and in that direction, you ask how the variety is carved. So given a plane and take any direction, tangential direction, you draw a geodesic through that point, through the point P, and in that direction. And you change the direction along this R2. And then the geodesics sweep out some shape like this. And in this case, it's like the G-core-zero case in the curve classification. So two geodesics are carved in the same direction. So in this case, carved positively. And the next case, it's carved in the opposite direction, so it's negatively. Or zero or flat. And my interest was this positive case. So the question was, what if a variety X is positively carved like this everywhere, meaning all P, and all directions, meaning all R2? So the question is, if we always end up with this case, what happens? What is the original X? So this is a famous conjecture, a famous problem of a Frankel conjecture. So Frankel made a conjecture that if a scalar manifold is positively carved at every point P and R2 in the sense I explained in the previous slide, then exercise morphic to a projected space. For each dimension, there is only one such variety. And Robin Hudson is an algebraic geometry. He was very good at making conjectures. There are lots and lots of conjectures. And this is one of them. And so hinted by this Frankel conjecture, he wanted to create an algebraic geometric version. So in this case, this positivity is really differential geometric notion. And algebraic geometric analog is this positivity, ampleness of tangent bundle. So Hudson conjecture was that every project to manifold with ample tangent bundle is isomorphic to a projected space. And if you compare Frankel conjecture and Hudson conjecture, this positivity, sectional positivity is stronger than the ampleness of tangent bundle. So if Hudson conjecture is solved, then Frankel conjecture is solved. So that's the relation. And I'll explain a little bit later more about the fact is that Frankel conjecture was solved by Yao and Xu. And I could manage to solve Hudson conjecture. So I want to explain something about the proof. I go back to this picture. In differential geometry, you could talk about a curvature, how it is curved at the point P, but not in pure algebraic geometry. So what one can do in algebraic geometry is more sort of probabilistic, in a sense. So given a curve C in X, you can take the average of those curvature. So if you consider the minus k, the negative of the canonical divisor, canonical bundle I introduced earlier, and the curve C, you can take the intersection number, just a rational number. And it is an average of the curvature for all P and R2 in the tangent space of P. So if we just compute this intersection number, it's more or less the average of the curvature. And I should say that in dimension 1, a positively curved case happens only for P1. So all other cases are the other side. So I have to say that the negative or zero case is more likely to happen. That's the expectation. And so using the notions I introduced, I want to explain a little bit of a positively curvedness, a few measures of positively curvedness. So this one, sectional curvature, this is for the Frankel conjecture. And the ampleness of tangent bundle is used for Hudson conjecture. And anti-canonical divisor, this is simply a determinant bundle of a tangent bundle. So if Tx is ample, then the anti-canonical divisor, minus k is ample. And if minus k is ample, then the intersection number of minus k with any curve C is always positive. So in this sense, k dot c, kx dot c being positive for all curve C is a weaker condition than ampleness. And in this case, we say that x is positively curved along C. So there are a few variations of positively curvedness. And so when I tried to solve Hudson conjecture, and I couldn't prove to solve the whole thing immediately, so I set up some intermediate problem. And one day, I thought I could prove it. But when I examined further, I realized that there is a gap. So I examined the gap. Then I realized that I proved the following theorem. If a projective manifold is positively curved along a curve, then x contains a rational curve. So this is a statement. Basically, this is a statement for a variety of complex numbers. It works for positive characteristics, but it's mainly intended for. But the proof uses a characteristic P. And the reason why it uses characteristic P is it uses power, piece power, the Frobenius morphism. So in a sense, it's a favorite object for algebraic geometry, well, for algebraics. But differential geometry is heavily disliked. It will be nice to have a characteristic Zer proof of this result. But anyways, yeah, so when I realized that I proved this theorem, I could prove the original Frankel conjecture in about a week. So this was really the key statement. I cannot give you the proof, but I can give you the feeling of how I did it. So it uses all many characteristic P. And I draw those algebraic varieties of various characteristics into one picture. So this is a picture I took from the book of Manford. It is a picture of a speck of P1, I mean, something called P1 over the ring of integers. So this is the speck of Z. So its point consists of prime numbers, 2, 3, 5, 7, and 0. For each prime, there lies above it geometry over the characteristic P. If I start with a variety in characteristic 0, I sort of spread it over the many characteristics. And for each prime P, we have a characteristic P algebraic geometry. And there, we have Frobenius, and we somehow create a rational curve because of the Frobenius. And since there is a rational curve for almost all prime P, you can somehow lift it to characteristic 0 using something called a bend and break. But that's a bit technical. So I want to explain about the outcome. So once the rational curve was found, the rest came out very easily. So I wanted to see what one can do with it, Father. Trying to find what one can do with such a rational curve, I stumbled over the notion of extirimal rays. And with this extirimal ray, I changed the problem slightly. So the new question is to find the key structure of a variety, which is positively curved along some curve C this way. How to find the key structure of curves along which the variety is curved positively? So in the earlier slide, I explained that in most cases, the variety is curved positively. That's what usually people expect. And if it is not, there should be some reason. And so there should be some geometric reason if X is curved positively along a curve C. So that's by my basic attitude. And here is the basic notion I use. So X is a smooth algebraic variety. And then R is the set of real numbers. Then there is H over 2 of XR, a two-dimensional homology with real coefficients. And given an algebra curve F inside X, you can consider it's a homology class F, bracket F. So then it's a vector within this vector space. So by considering all such curves or all such classes of curves, you can span a cone within this vector space. Then it's an invariant of X. It's a cone in a final dimensional real vector space. So it looks something like this. So from the origin issues align in various directions. And since each vector is represented by a curve, we can consider the intersection with K, Kx times C. So in other words, Kx is a linear function on this vector space. And you can talk about hyperplane Kx equals 0, half space, Kx positive or negative. And Kleinman originally considered this to consider projectivity of a variety, to understand the projectivity of a variety. So he theorem says that variety is projective if and only if this cone does not contain a straight line or in other words, there is a way to cut down by. There is a hyperplane section not going through the origin so that the intersection is compact. So you may wonder what the relation with such a cone by say a morphism. Well, this is one of the key facts. If there is a morphism to a projective variety, say contracting surface to a curve, then each fiber generates an element in here in the cone. But for y to be projective, the vector, the class of curves to be contracted should be on the edge. Should be on the boundary of the cone. So in a sense, I want to understand projectivity or X in terms of such a cone. So in a sense, I feel like a geometry or a painter. So algebraic variety is something even algebraic geometries cannot see. A complex surface is a real four dimension. So we cannot really see it. But nevertheless, it's somehow in my brain. And as algebraic geometry, I do some research. And to do it, I draw something. So I draw an invariant on my canvas. So in that sense, I do something similar to painters. So invariants are similar to abstract paintings. So differential geometries may draw a concrete painting. They have various detailed tools to describe local picture, but not algebraic geometry. So in that sense, a picture of algebraic geometries are more abstract. But it's quite different from a painting in the sense that we have to have objectivity and reproductivity. I mean, if it's art, each person draws a completely different painting. But in science, we should be drawing the same or equivalent paintings. So in this way, an invariant is defined for each figure. So in my mind, this corresponds to cubism. So this is why I say it's a personal view. It doesn't represent the view of algebraic geometries. And this is a painting of Paul Clair. And he says, art does not reproduce what one can see, but makes seeable what one cannot see. I like this sentence. But I also like to replace art by science. And so in this picture, you can talk about what does this represent, this circle, or triangle, or that kind of thing to appreciate the picture. Now, extremar ray is something like it. So in earlier slide, I draw a sort of circular cone. But in reality, it's slightly different. So I said kx equals 0 defines a hyperplane. And kx positive half space, kx negative half space. And the fact is that this cone is finitely generated in half space on this picture on the left side. So there are finite number of sort of edges. And the cone is spanned by them. But this is not completely right in the sense that there may be infinitely many edges which approach to the point on here. But roughly, I would say, away from this hyperplane, x is finitely generated. And at this point, or the ridge line of the cone is called an extremar ray. And so my question is, what is the geometric meaning of the extremar ray? That's my attitude. So given the surface and then the extremar ray are, I can ask, what the geometric meaning? So there are a few types. So p2 type, in this case, x is p2 itself. And every carbon has a class in R. Or p1 bundle type. So it looks like this. x is a fiber over a smooth curve. And every fiber is p1. And the class of fiber belongs to R. So in these two cases, the quadrature dimension is minus infinity. Or a bi-rational type. Well, this is just a minus one curve, a counteractivity to a point. So in this case, only the curve minus one curve c generate R. So there are three cases. And so a curve c on x is sent to a point below, if and only if its class belongs to R. And this is the contraction morphism of R. And that is the geometric meaning of R. If you pick up an extremar ray, there is some contraction morphism. That's how it goes. And let's revisit the two-dimensional case quickly. The bi-rational type means minus one curve. So if there is a minus one curve, we just contract. And we repeat until there is no R of bi-rational type, or no R of minus one curve. Then x is either p1 bundle over a curve, or p2, or minimal model. In this case, there is no extremar ray, no further extremar rays. And this is a very simple argument, replacing the case-by-case division of the earlier minimal model program theory. So this simplifies two-dimensional minimal model program. And in this theory, x is a minimal model. If kx.c is non-negative for all curve c, so this is our definition of minimal model. It's slightly different from the earlier one. The earlier one means x is more or less minimal. You cannot make it smaller. So you should note that it does not involve minimality in terms of dominance. So one can try to generalize these two dimensions. So with this theory of extremar rays, one can try to generalize to dimension three. But one still needs more ingredients. So if you remember Hirunaka's example, I started with a picture. And there is only one p2, which could be contracted. And if I contract it, I get a singular point. So if you apply this minimal model program, whatever it means, we may end up with some singularity. So if I start minimal model program from a smooth n fold, in this case, n is 3. One can get p2 that's mapped to a singular point by the contraction. So to continue further, I would create a theory of cones for x with such a singularity and may continue. But then I may get the worst singularity. So it's hard to continue. But fortunately, minus 3 to define the class of singularity that just works for the minimal model program called the terminal singularity.  They constructed a general framework of this cone and contraction and so on. And just to have a feeling, the contraction of extremality in three dimensional case of the following. So in case of a surface, we had p2 and p1 bundle. So in case of dimension 3, either sent to a point or 3, 4 is sent to a curve or a surface. There are three cases. They are final fibering. And final variety means minus k is ample. It's a natural generalization of projective space. So this is one natural class. And kappa is minus infinity. And the digital contraction case. So in surface case, we had a minus 1 curve. And it was contracted. And the same here. So since we have a three-fold, we have a surface which is contracted to either a point or a curve. In the case of a surface, we didn't have to consider anything more. But for a three-fold, there is an extra case called the small contraction. A curve is contracted to a point. No device is contracted. So you may think that this is nice. But the fact is that the singularity we produced is very bad. And it's something we cannot handle by intersection number method. So there is something to be done for this case. So in this case, we need a operation called the flip for small contraction. So we remove the curve, contracted curve, and put another curve in a different way. It's different in the sense that originally minus k is ample. But in the new case, new side k is ample. The sign is opposite. And the theorem, the fact is that the flip do exist. And in dimension three case, I could manage to prove it. But for arbitrary dimension, Heikon and McConan proved it for all dimension in 2007. So they followed the idea of Shokurov. For dimension three or larger, we repeat either divisory contraction or flip. So for surface case, we didn't need a flip. But for dimension three or higher, larger, we need flips. Then we get another easier three-fold variety. And if this process terminates, we get a minimal model or a final fibering, just like surface case. And actually, in dimension three, Shokurov gave a very simple, nice argument that it must terminate. But dimension four or higher, it's still not completely settled. And in 2010, four people, Birka, Cassini, Heikon, McConan, they established a special kind of minimal model program. So MMP with scaling. Which is still a minimal model program, but a special kind. And that MMP terminates in all dimensions and in most cases. Not all the cases, but still terminates in most cases. And using this, Heikon, for instance, established that canonical ring I introduced earlier is always finitely generated with no extra conditions. But to do this, to do this kind of thing, they had to consider a more general version. So they consider a relative version and a log version, meaning they add some divisor and consider a pair. And so, of course, I completely omit them. And so I just do the overview. In dimension two case, if there is a minus 1 curve, I contract it. And if there is another, I further contract it. And I keep doing it until there is no more. And then we get one of the following. Either P2, P1 bundle, or minimal model. And in dimension three, there are three new ingredients. X-ray, new operation called flip, and singularities. And now they are more or less working all dimensions in most cases. And because of this, MMP has become a fundamental tool to study algebraic varieties. Although it doesn't work all the cases. So coming back to the birational classification, depending on the Kodara dimension, kappa is minus infinity or 0, 1 all the way to dimension x. Let's not talk about this. It's just technical terms. Then, so depending on the Kodara dimension, there are other cases. And by BCHM, Birka, Cassini, Heiko, Makanan, minimal model exists. And when the Kodara dimension is intermediate values, we do not know whether flips terminate. But if it terminates, we have a minimal model. And if under this technical assumption that Kx is not pseudo-effective, it's something stronger at the moment than Kx is minus infinity. But in that case, we always get final fibering, final variety of final fibering. Abundance conjecture says that canonical device of a minimal model is semi-ample, meaning that some multiple of it has non-zero global section. And in case Kodara dimension is the highest, dimension x, then as soon as we have the minimal model, the canonical device is semi-ample by baseband-freeness theorem. So in this intermediate case, it's not settled. And recently, Birka proved that final varieties form a bounded family. It means that for each dimension, they are parameterized by a finite number of parameters. And it's hoped that this has some effect on attacking this problem, this termination. That's the current status. And let's go back to the difficulties we had in dimension three. So we had a situation where one can contract surface to a curve over three-fold to smooth analytic variety. Then in this case, in such a case, there is another contraction of an extremal ray. So this is not the right direction. So if you go to a different direction, you can continue. And I say, why can be no algebraic? Well, as long as we keep contracting extremal ray, we always get the projective variety. So we don't have to worry about it. And here now, as an example, well, every vibrational morphism is a composition of div-zero contractions and flips. Not the kind of answer we hoped, but that's the reality. And Itaqa group tried to classify algebraic varieties without minimal models, but the fact is that Itaqa's problems were studied by using minimal model program as well. Minimal model can have terminal singularities. So that's how things are. Here now, as an example, we contract to dispute to the singular point. Then there is no way to contract the divisor, but these are the small, each of these three can be contracted to a point, a small contraction. Then we do flip. It's symmetric, so I just pick up this curve and flip. Then they can be contracted. So this is contracted to this and that to this. Then this one can be contracted. Oops, sorry, I forgot. So this way, this vibrational morphism is a composition of div-zero contractions and flips. And thank you very much. Thank you very much, Professor Mori, for this beautiful lecture. So now it's open to discussion, to questions, to comments. Actually, before as the tradition of the colloquium, then I just remind you, everybody will have to leave except the diploma students for private session with the speaker. But if before the other parties events, you are a diploma student. You don't want to wait your private session or? No, okay, you can. In the beginning of the talk, you talked about in the 70s, you introduced the metric on the varieties to study. So because there is a notion of average curvature and in general, in differential geometry, when there is a notion of curvature, one introduces one's notion of connections to transport tangent vectors. So in the algebraic sitting, is there any counterpart of that? And if not, why can't one introduce a notion of average connection or? The answer, so I explained things as if I knew various things and chose this direction, but it's not true. I just wanted to consider, I was just curious about this Hudson conjecture and then I just followed my curiosity. So I explained those things only later to explain it to other people, but so I do not know, sorry. And but you see, you could try it yourself and that may be a good thing to know. Any other question? Thank you very much for your presentation and overview of this. So it's more a comment really and just to hear your thoughts about it as well, I'd be interested. In the case of varieties with trivial canonical bundle, they play an important role in string theory as models of extra dimensions, as you probably know. And in that case, these singularities that you were talking about from divisorial contractions and non-divisorial contractions actually play a really important, have a privileged role because they correspond to very special points where very interesting physics happens. For instance, a simple example would be if you contract curves inside a K3 surface, one gets actually an analog, the physics of that is described by the Higgs mechanism as a simple example. So it's, I wonder if you have any thoughts about that or you've interacted with physicists that are excited about those singularities? Yes, you're talking about K equals zero. So I have been talking about K negative. And so when K is zero, you can still do something by adding extra divisor. So not just K, but K plus D. And then you can ask whether it can be positive or negative. And that has been the main object of study in the log variety, log geometry. Yeah, so there are people working on it, but... Any other question?  So my interpretation of the minimal model program at least is like a generalization tool dimension. You explain to us surfaces like sphere, torus, higher genus. In real three-dimension, our understanding is, maybe we can say it's the Thurston Geometrization and Poincaré. So we have a finite list of geometries that somehow are mixed in three manifolds in various ways. As a painter, do you really feel which are the higher dimensional geometries that really come into the classification of manifolds? I mean, one thing is to say, okay, this is an invariant, this number, this number. But I mean, can you actually describe geometrically, up to which point? Is it? So algebraic geometry is much finer. So you need, of course, discrete invariance, but you also need a continuous invariance. And still, if it comes to that question, it's totally unknown. So I'm sorry, I didn't... But do you think the method actually is actually helping us also in this? I mean, the construction of the minimal model you described. Well, so, at least depending on the... So for the case of, in the case of curves, there are three geometries, yeah? Depending on cut by zero, one or bigger than one. The situation is not that clear, but still there are different geometries. And for kappa positive, or general type varieties, I have to admit that differential geometric methods is more powerful. I have to leave it to differential geometers. And for kappa equals minus infinity, somehow, algebraic geometric method works very nicely. But at the moment, for k equals, for dimension three, it's totally unknown. So the only thing I could say about that case is whether the varieties, say, rationally connected or uni-ruled. In case of... So in case of a curve, kappa is, when kappa is minus infinity, x is rational. But for a surface, either it's rational or ruled. It's still very, very clear situation. But in dimension three, rationality behaves, rationality meaning the condition that a variety is rational, parameterized by three parameters. Doesn't behave nicely under deformation.  So the only substitute we have is rational connectedness, meaning that given two points, two general points, can be connected by either a chain of rational curves or one rational curve. That's rational connectedness, replacing rationality. And uni-ruledness is that given any point, there is always a rational curve. That's a property replacing ruledness. P1 times a curve. So, but still that's far from being satisfactory and no one still knows what the reality is. Any further question? So we really thank Professor Mori again. Thank you. We leave you discussing with our students and we will be.


Here is another demo of my file container spreadsheet, which allows you to store any extractor arbitrary files. And this time for the infiltration and exfiltration, we are going to use a website. The website that we will use is Wikipedia. Several years ago I made a blog post about how you could store information, any information that you want, into Wikipedia. And that's what I'm going to use here. Now you can do this on any page in Wikipedia, but I'm going to do this on my sandbox. I'm logged in here and here is my sandbox. So it appears to be empty. Now if you go into the history, you see that there have been some changes. And let me take this previous version. And now here you can see that there appears to be hexadecimal data. So I'm going to take this. So let me edit the page. I'm going to copy this like this. So copy this. And I'm going to drop this in my spreadsheet. So let's call this file 001. And here paste this. So now I have this file stored here in my spreadsheet. You see list files, it appears here. Now this file here, which is an executable, was not stored as is in hexadecimal form in Wikipedia. But I also encoded it with an XOR key. And file container now supports XOR key encoding. So the key that I used is dj. And now I'm going to save this executable. So I select it. Save files. And here in the demo directory, let me save this. And now it is decoded with this dj key from the data that is here in this file. And then save to disk. Like this. And you can see that it is a valid executable that you can execute. Now that was infiltration. So let me remove this. And now we are going to do exfiltration with another key. Let's call this secret. So let's come back to the sandbox. Like this. And now I want to exfiltrate a file. So let's add it here. So this is the file I want to exfiltrate. Okay, so now I have here a sheet in the spreadsheet that contains this file encoded. So now I could save this spreadsheet and then mail it out. But I don't want to do that. I want to use Wikipedia. So let's copy this. Like this. Go back to Wikipedia. Now remark also that I work with HTTPS. So the data being sent to Wikipedia and received from Wikipedia is encrypted. So let me edit the sandbox and drop this in and save it. Okay, and here you have it. It's in Wikipedia and now we are also going to hide it. So let's go to the history and say here undo. Okay, that's what I wanted to undo. Like this. Okay, and so now it no longer appears here in my sandbox. But if I go into the history, I can still find the exfiltrated data back.