 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. So if you blow up smooth point P over surface, above P, you get a rational curve called 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. And there are a bunch of people, Kamata, minus 3, Shokuro of Bebeniste, and so on. 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? Well, I have a curiosity, almost a psychological curiosity, but you describe yourself as the painter in front of an algebraic variety. 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. And so because of this, people try to come up with various notions, but still nothing satisfactory has been given. 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.