 everybody for this very happy meeting, mathematical meeting. So we are very happy and lucky to have Professor Carolina Rao from IMPA, lucky also despite the effort of Lufthansa that was trying to steal her, to keep her immunic for all the yesterday and today, but in the middle of the night she met then. So we are particularly thankful for her to for this complicated trip that brought her here. So Carolina is a good friend of ICTP, I mean, but before that, I mean, she's, as I already said, she's a professor at IMPA. She's one of the very prominent figure in algebraic geometry all around the world. And she's an affiliate member of the Brazilian Academy of Science. In terms of our ICTP friendship, I mean Carolina is one of our most distinguished associates of this part of the Simons Associates Schema since 2015. And she has been, she has been actually giving a number of seminars. Some of them are actually on our YouTube channels, our basic notion seminar, for example, and but number also research seminar. And she has been lecturing in a number of schools, I would say actually all around the world in Chile and Mexico and in Trieste a few times in our activities here. So we are very, very happy of this connection. And as just a highlight of, of course, of her standing in the mathematical community, I just remind the fact she's been an invited speaker in the International Congress of Mathematicians, which in our communities, of course, is a very high recognition. And she's also the vice president of the Committee for Women in Mathematics of the International Mathematical Union. I think for in all these capacities, for all these reasons, we are really very happy to have her here. And she's giving this, the talk of today on Algebraic Geometry, History, Application and Current Trends. So thanks a lot. Thank you, Claudia, for the introduction and for the invitation. It's always a great pleasure for me to be here at ICTP. So I, so today I'm going to tell you a little bit about Algebraic Geometry, which is my area of research. It is one of the oldest fields in mathematics and still very active, because it has connections with many other areas within mathematics and other sciences. So when I send the title and the abstract, I was planning to talk about Algebraic Geometry, History, Applications and Current Trends. And then when I started preparing my talk, I realized that in fact, I probably don't have time to tell you about all this. I will tell you something a little about history and applications, but I will mainly focus on current trends. So let me start by telling you what is Algebraic Geometry. So Algebraic Geometry is the field of mathematics that studies solutions of polynomial equations in several variables. So this is where the algebra comes from, because polynomials, they have a very rich algebraic structure. And so to manipulate them, there we have many tools from Algebra. But this set of solutions, they actually have a geometric manifestation. And this is where the geometry comes from. And so what we will see today is that Algebra and Geometry, they actually play a dance together to one complimenting the other. And from this point of view, I like to quote this quote by Michael Atia. So he says, should you just be an Algebraist or a Geometer? It's like saying, would you rather be deaf or blind? So we really need all our senses to perceive the world. And this is how Algebra and Geometry play here. Okay, so what are the objects of studies of Algebraic Geometry? So as I said, these are, we are interested in solutions of... Number six, Algebraic. Okay. Okay, so we are interested in studying solutions of algebraic systems of algebraic equations. So here I get some very simple examples. So this is the equation of a line. So this is a line. So maybe I should just for completeness, so probably we like it. So this is a linear equation. The first one, the other one is just, you think of the equation of the circle. This one here has this equation. So these are polynomial equations. So here the degree is one, the degree here of this polynomial equation is three and so on, and they give us this picture. So the point here is that what we want to see, so this is the algebraic part. So these are the polynomials that define those set of points. But when you look at the picture, you really have some other filling and some other grasp of these sets. So they do have, so for instance, this point here in this third example is what we call the singularity. So here the curve is not smooth at that point. But also these are simple examples because these are solutions of one polynomial equation in one variable, but of course you can put more variables, more equations, and you have much more interesting varieties. So now polynomials, they actually, they are used to model many real-world phenomena. So they really appear in many parts of mathematics. So let me just mention one application here that could be interesting, for instance, in terms of computer vision and so on or computer graphics. So imagine that you want to plot a certain algebraic equation. So how would the computer, so computers only deal with polynomials. And so how would you say you would plot one surface like this? So one idea is that if you have a surface which has a certain equation, and let's suppose that you can parameterize this equation. So you find polynomials, say X, Y and Z polynomials in terms of other variables. Here I put U and V. And so this will help you to parameterize this variety. So a computer could plot this surface and you could actually visualize it in 3D or in 2D, in a projection through 2D. So for applications, it's actually very important to be able to parameterize a variety. So now let me just give you this question and we are going to come back to that later. So a very natural question which is important to application is to know which algebraic varieties admit algebraic parametrizations. When I say algebraic parametrizations, I mean that, so let me just give you, so I was working out this right before the talk, so I hope I got this right. So for instance, for this curve here, this curve can be parametrized. So let's say with a parameter T to T square minus one, T cubed minus T. So hopefully I got this one right, but this is what I mean by parametrization. In other, in more dimensions, you would have more variables. And here I also allow not only polynomials, but ratios of polynomials. So this is what I mean by an algebraic parametrization. So this is, sorry? So you mean what you usually call rational? Right, yes. Right, so yes, I mean rational parametrization set. This is what I mean, the quotients of polynomials. Okay, so now, so we are going to come back to this problem. So you can think of this problem, this is a very, I think of this as a very algebraic problem. You want to determine whether you can find a parametrization. And the point that I want to make now is that now we are going to move to geometry. And we are going to see that a simple algebraic question like this has a counter part, a counter part, a geometric counter part that is very interesting. So now let's move to geometry. So let me just start by talking about curvature. So maybe I think you all know what a curvature is, but let me just remind you very quickly. So a curvature, by a curvature, I mean it's a measure of how much a curve, so this is a curvature for curves, deviates from being a straight line. So how do you associate a number to a point on a curve that makes such measure? So what we do is, so suppose you want to measure the curvature of this curve at the point P. So I take the circle that best approximates the curve at the point P, it will have a certain radius, and the curvature is one over this radius. So you see, if you have a straight line, you can think that it is approximated by a circle of radius infinity, and this gives you the curvature is zero at that point. And so at different points, I can have a different curvature. So this is how we define curvature. Now, let me just mention that the curvature is not something intrinsic to the curve. It depends on how this curve is embedded in the space. So for instance, let me just give you an example. I have this, I can make this into a straight line or I can make it curve if I change the way that it's immersed. And so this is going to be very important. So this, the curvature for curves depends on the embed. So now let's move to higher dimension, and now I want to explain what it is the curvature, one way of measuring the curvature of a surface. So let's start with a surface, and I want to measure, I want to associate to it some number that, sorry, that measures how this surface is curved at this point. So this is, I will do the following construction. I have, at this point, I have a tangent plane and I have a normal vector. Now I take any plane that contains the norm, passing through the point containing the normal vector. So that is going to cut my surface along a curve. And then I can compute the curvature of this curve. And now I can vary my plane and compute the curvature for all the curves that I obtained in this way. So if I do this, there is going to be a maximum value of curvature and a minimum value of curvature. Here it is important, one point is important, is that when I do this, I have to assign also a sign to the curvature. So we can say that if something curves upwards, then I will tell, I will say that it has a positive curvature while if it curves to the other direction, then it will have a not negative curvature. So for instance, for this surface here, I have here the two planes that cut the curve, cut the surface in the curves with minimal and maximal curvature. And one of them is going to be positive and the other one negative. So if I call this two extremal curvatures K1 and K2, I can define the, let's say, the Gaussian curvature to be the product of this, to the minimum and the maximum. So this point that you see here, what we call a saddle point, is a point of negative, negative curvature. While if I had a sphere, this would be a point of zero curvature. And one important, sorry, positive curvature, and one, positive curvature. And one, if I have a piece of paper like this, this has zero curvature. Now one important feature of the Gaussian curvature is that it is intrinsic. So it does not depend on the way I embed my surface in the space. So for instance, if I curve this sheet of paper a little bit, you see that most directions it will curve downwards, but I still always have one direction here where it's flat. So in the end, my, in this case, the Gaussian curvature is always zero no matter how I move this, I move this piece of paper. So I will never be able to make it shape like a saddle unless I destroy the, or fold the paper. So this is the definition of curvature that we will need for a surface. And from here, we have this great trichotomy. So these are the three different fundamental different behaviors of curvatures in surface. So we have negative curvature. So this is a saddle point, a zero curvature and positive curvature. So this trichotomy is going to carry on when we start to talk about classifications of algebraic varieties. Okay, so now let's move a little bit to algebraic. So this is, so this, these are concepts from differential geometry. And now I want to apply them to algebraic geometry. So to do so, let me talk a little bit about Riemann surface. So if I have a, I start the simplest Riemann surface I want to talk about compact Riemann surface. So if I start with a plane, but it's not just an, I don't view it as an R2, I view it as a complex plane. So I have the complex plane. So the Riemann constructed what is called the Riemann sphere which is a complex, it's a complex variety that is obtained by attaching a point at infinity. So one important thing here is that, so if we change coordinates at this point of infinity and write w for one over z, so we see that once we add the point of infinity we can make this change of coordinates and see that it behaves as any other point of the complex plane. So we can, we can, we can multiply and we can divide points in the, in the plane. And then we can, we make this change of coordinates to look at the point at infinity as a point in the complex plane. So this is a Riemann's, this is a Riemann surface, the Riemann sphere, and of course this has positive curvature as we can see. And now let's look at more interesting Riemann surfaces. So now what I will do is, I will do the following. I will take, I will take a polynomial f of an x and y and here I just have to be a little bit careful so that the associated curve does not have a singularity at any point. So that means that the two partial derivatives of f cannot vanish simultaneously. So if I choose such f, but now, differently than I did in the beginning, I'm not interested in real solutions but I will be interested in complex solutions. So now x and y can be complex numbers. So if we do this, we actually see that before we looked at this as a curve but now we can view it as a surface because each variable can take any complex value. And now, okay I start with this and now just like the previous slide I had the complex plane and I could attach a point at infinity and make it compact. So I can do the same thing here. So let me just explain quickly the construction of the complex projective plane. I will draw a picture for a real projective plane just so that we can see. So the idea is that we want to, we want this to be our plane, c2, c cross c. Union, I want to add one point for each tangent direction. So one point tangent directions of the plane. So if you, you can think of this in order to visualize, let's think of this as R2. So we have the plane and we want to attach one point for each direction. So the way that I can do is the following. I can look at, so this is actually defined. I can look at this, all the space of all the lines. So if I take all, so if I look now at c3, so we coordinates x, y and z. I can look at all the lines that pass through the origin. If I look at the space of all the lines that pass through the origin, notice that if I take a general line, it is going to cut, so this is, let's say that this is the plane z equals to one. So if I take a general line through the origin, it is going to cut this plane at exactly one point. And the lines that do not cut this plane are precisely the lines that are contained in the x, y plane. And there is one of those for each direction. So the way that we can define this then is to define this as a c3 minus the origin when I, so I remove the origin and then I just quotient by the equivalence relation that identifies two points in the line through the origin. So it identifies a vector v with lambda v for any lambda different than zero. So this is the projective, this is what we call the complex projective plane. So our, and it contains as an open subset here, our plane c2, viewed for instance, are the lines that cut this plane. So now if I do this, my s lives inside c2 and I can take the closure. So this is a compact, complex, a compact variety, compact complex variety, and I can take the closure of this s inside here. So usually what I will do is I will write the equation, what we call the affine equation, but I think of it, I will always be thinking of this as the compactification obtained by taking the closure in the natural projective space. So if I do this, I will then my, what is before it was a curve, we saw it as a curve, it is actually a surface and it will be compact, meaning that you can think of this as being bounded, but without any boundary. So here's for instance an example and all Riemann surfaces, so all the surfaces that we can construct like this, they will have this shape, but I may have another different number of holes. So this is what we call in topology the genus of the surface. So this again, this is an intrinsic invariant of the surface, you may try to deform, if you look at this surface that I have there, you can try to deform it and you see that no matter how much you deform it, you will not get rid of these two holes and you're not gonna make another role. So this is what we call topological invariant. But now let's go back to the question of curvature. So suppose that I want to know what is the average of the curvature of this Riemann surface. And there is the famous Gauss-Bonnet formula that tells us that the average of the curvature is well, two pi times two minus two g. So it depends on this topological invariant. So you can change the Riemann surface, try to deform it, but you're gonna make it more curved at some points and less curved at others, but in the end you are not going to be able to change the average of the curvature. It only depends on this topological invariant. The g zero, right, correct. So for the Riemann sphere, the g zero, and see if you put g equals to zero, you're gonna get four pi, which is the average of the curvature in the sphere. And so see, for genus, so we have genus zero, the curvature is going to be positive. Genus one, the average curvature is going to be zero. And positive genus will have average negative curvature. So this is the trichotomy that we have in mind. But now let's remember that this surface was defined by an algebraic equation, a polynomial f in two variables. And the beautiful thing is that both the genus, which is this topological invariant, and the curvature, which can be computed from the genus, they are actually only depend on this, on very simple algebraic properties of the polynomial. So what is the simplest algebraic invariant of a polynomial one can think of? So this could be, so this could be the degree of the polynomial, right? Even if you change variables, the degree is not going to, if you make a linear change of coordinates, the degree is not going to change. And so as it turns out, for Riemann surface, which from now on I will call a complex curve, because it is really, the complex dimension is zero. Sorry, it's one. So it has real dimension two, but complex dimension zero. And so if, oh, sorry. It has real dimension two and complex dimension one. So I call it a complex curve. And we want to compute the average of the curvature, depends on the genus, but as it turns out, the genus only depends on the degree of the polynomial. So it can be computed with, so this is what we call the degree genus formula. So this is one of the first things we learned in the first course of algebraic geometry. We obtained this. So we see that, so what we see is that, that trichotomy for complex curves, say plain complex curves defined by a polynomial like this. So now I put the degree D here. So D here is the degree of the polynomial F. So the curvature of the Riemann surface that we obtained this way, we'll have positive curvature if the degree is one or two, zero curvature if the degree is three, and negative curvature if the degree is greater equal than four. So we see that in a sense the simplest curves, so the lines and conics, they have positive curvature. So positive curvature, maybe what I want to think of in the algebraic geometry, varieties having positive curvature, they tend to be very simple. And now let me go back to a question that I posed before. So it seems that this has nothing to do with the question whether which curves admit algebraic parametrization. So this seems not to have anything to do with curvature, but in terms it does. So the answer is the curves that admit algebraic parametrizations are precisely the ones having positive curvature or genus zero. So from one side you have the geometry that is measuring the curvature, from the other side you have this question of parametrization and we get, but they are actually, this is a niff and only if. Now, maybe if you will remember, I think I gave you the parametrization of a cubic curve, this one right in the beginning, I showed you that this curve has a parametrization, it has degree three. But this is what I said, I have to pick up my polynomial carefully so that it does not yield singularities. So when you have a singularity, then there is an adjustment to make. So this is still a curve of genus zero if you define genus appropriately for a singular curve. So I don't want to get into that, but what I want to say is that, so here in this discussion I'm considering a smooth variety, so I'm ignoring those that contain singularities. Of course these are very important and they are very much studied, but they will bring new ingredients to the discussion. Okay, so now what I want to do is to move to higher dimensions. So this, so in complex algebraic geometry, one of the central teams and the most important directions of research is towards a classification of algebraic variety. So what do we mean by a classification? So we have our algebraic variety, so this is a zoo of algebraic varieties, and we would like to classify them. So we would like to divide them into certain boxes according to their shapes. And what we want to do, so for curves, this is a accelerated trichotomy that I told you before, that depends only on the genus, and we have these three classes of varieties depending whether they have positive, zero, or negative curvature. We want to do a similar thing in higher dimensions. Now of course this is much more complicated, so I'm, but this is possible to do, and this has been some of the most important developments in the last decades in algebraic geometry. So let me just briefly tell you how it goes. So we start with a complex projective variety. So again, I always consider something that is closed in the projectives, in the projective space. And I will define the canonical class of X. So the canonical class of X is the first-turned class of the tangent bundle of X. So for those that know, so for those that know what I'm talking about, let me just tell you that our canonical class lives in the second cohomology group of X. So if you don't know what this means, let me just tell you, let me just give you an idea. So an element here in the second cohomology group, what does it does? It associates to a compact surface inside my variety, an integer. So here I take, now when I say a real surface, so a real surface, as I said, this is a complex curve. So it associates to a complex curve, always a number, an integer. And this number, if I take minus K, this is exactly the multiple of the average of the richer curvature of X along C. So what I wanted to know is that we have this gadget here and this is intrinsic, this is canonical, that's why it's called canonical class. So it's the first-turned class or minus the first-turned class of the tangent bundle. And in a sense, it measures the curvature of X. Now, and so what we would like to do is to divide of us, classify our algebraic varieties into three classes, according to positive, zero, or negative curvature. Well, this is unfortunately not well, maybe fortunately, otherwise things would be too easy, but this is not possible because in general, so this is, you can think of this as a function that associates to every curve a number. And in general, the sign of this function is not well-defined. You could have some curves for which this number is positive and other curves for which it is negative and other curves for which it is zero. So in general, this is not well-defined. When the sign of the canonical class is well-defined, so these are very special varieties, then we have these three cases that for simplicity, I will call them, so those for which the canonical class is positive are called of general type, and they're called of general type because most varieties with well-definite sign of the canonical class will lie in this class here. We have those K equals zero, which I'm calling calabiol. So these are the, for instance, the calabiol varieties that appear in connections with physics. And the ones with negative K, which are positive curvature, are called funnel varieties. So these are very special varieties, exactly because the sign is well-defined, and the tools to investigate, to use to understand these varieties are very different. They have very different behaviors and very different, many different properties. So let me just illustrate for the simplest case. Now again, suppose that my variety is given by a single algebraic equation. So I'm looking at complex vectors that satisfy a certain polynomial of degree D. And again, I choose this polynomial carefully so that I don't get any singular points. And if I do this, I can easily compute what is the canonical class in those cases, and I get exactly this trichotomy. So canonical class negative are those with small degree. So you can think of these as the simplest, simplest algebraic varieties, for instance, planes and connet, quadrics, and low-degree hypersurfaces all lie in this case. Here, again, I still have something finite. And then here it becomes wild. So as soon as the degree is large enough, everything that I, every hypersurface will have positive canonical class. So I have been studying funnel varieties for many years, so I'm not going to tell you, I'm not going to get into details, but let me just tell you that there is, we can define what is called the index of a funnel variety, which measures how positive the anti-canonical class is. So see, this is something that can be made precise. For instance, there are many ways to define this index, but you can see, for instance, what is the biggest integer dividing the minus k inside this cohomology group. So the biggest, this integer, the more curved variety. So this is measuring positivity of the anti-canonical. And so for instance, funnel varieties with high index are extremely special and classified, so high-rational homogeneous spaces all appear in this class. And I think one of the biggest achievements in complex algebraic geometry in the last decades is what is called the minimal model program. So what does the minimal model program say? It says that these very special varieties, the ones with a definite sign of the canonical class, they are the building blocks to all algebraic varieties. So this, I assume that I'm saying this in a very vague way, but so the jargon in the area is that all complex projective varieties can be built up from these three classes. So how they can be built up from this, then maybe this is a bit mysterious and it would take more time to explain. But I heard that Mori was here last year, right? So Mori was the one who set up this program and who completed in dimension three and he probably spent more time explaining the details here. But here, let's just assume that this is what we get. Okay, so now what I want to do now is to, okay, maybe go back to a question that we had before. Oh, okay, so maybe a little bit about the minimal model program, some history. So in dimension two, it was so for, so first in dimension one, Riemann surfaces, this was just done by Riemann, so this is 19th century. And then dimension two, this was done by this Italian school in the early 20th century. And from dimension two to dimension three, some very important new ideas had to come up. And so this was proved by Mori in 1988, you see here a huge cap and that won him the Fields Medal in 1990. So what Mori realized is that, well, among other things is that if you look at dimension three and higher, there is no way that you can avoid singularities. So for curves and surface, you can do the classification scheme without dividing into these three classes and making the constructions to get there by avoiding singularities, starting in dimension three, this is impossible. And this has led to an important theory which is a theory of singularities, of pairs of the minimum model program, which is very important, also in algebraic geometry. And dimension four, this was proved more recently by Birkart, Cacini, Hayk and McKernan. And again, this was a major result. There are still some things missing in this theory here, but for purpose of the classification, their result is enough. And let me just go back to the question that I posed before about which algebraic varieties admit algebraic or maybe rational parametrization. So these are called rational varieties exactly because they admit this rational parametrization or uni-rational depends on your definition of parametrization. And so for instance, assume that we are in this in one, we consider varieties in these three classes. So which of those can admit a parametrization? And the answer is that again, the only ones that may admit a parametrization are the ones having positive curvature, the funnel varieties. Now here, this is not a niff and only if, and this is one of the most active also problems, also in algebraic geometry currently is to decide which varieties are rational. So there has a lot of great deal of results in this direction and connections to recent techniques from Hodge theory have been applied into this question. And so this is very active to decide, given a funnel variety, whether it is rational or not. And one important, maybe future directions that I heard recently is that not only, so is the question whether is there, can there be an algorithm that given a variety, there is the size whether it is rational or not. So this for instance is a problem. This is a question that is not known. So which is again worth investigating. Okay, so this is what I wanted to tell you about the classification of algebraic varieties. And what maybe what I want you to bring home from here is that algebraic geometry study objects defined algebraically but which has a geometric manifestation and geometry and algebra they interplay in this game. And that one of the important problems in algebraic geometry, the classification problem is to distinguish these three classes of varieties and understand how every variety can be built up from these three. And now in the last minutes maybe I want to discuss different problem which have been studied a lot like since that 10 years ago or so, which is the problem of foliation. So this, so let me start with something very simple which is algebraic differential equations. So these are differential equations again that only involve multiplication, quotients and polynomials and ratios of polynomials. So for instance these are two examples of algebraic differential equations. So these are very, these algebraic equations are very simple to solve and these are the solutions. So we see, so this is, so this is a, I'm sure that all of you have seen this but what I want to point out here is that we have two algebraic equations but the solutions may not be algebraic. So this is clear from the picture. So a fundamental problem, a fundamental question is if you see an algebraic differential equation, so when can you guarantee that the solutions are algebraic? And again I'm going to answer this question but so this seems to be sort of an algebraic question but we're gonna make, put some geometry into this picture. So before I do this let me just reinterpret this. So how can we reinterpret a differential equation? So this differential equation you can see it as it is given at every point of the plane, this is given a tangent direction to the plane. So this is, this ratio here gives a tangent direction at every plane, at every point and the solutions are just curves that are everywhere tangent to this tangent directions. So this is what we call a, a foliation. So let me tell you what in general, so the foliation is something that generalizes this algebraic equations. So a foliation, so I have here say an algebraic variety and a foliation is a partition of the variety into what I call leaves. So these are sub varieties here. As we saw in the example, we start with an algebraic variety but when I take these leaves they may not be algebraic. They may be transcendental but the algebraicity condition that I put. So if you go back to this, what this is, this equation is telling you is that this tangent direction to the curves, to the solutions, they vary in an algebraic way. So this is given just by this polynomial. So I do the same thing here. So I consider these foliations but only those for which the, so here I call the tangent bundle of the foliation. So this is at each point, I consider the tangent space to the leaf at that point. So this gives, and this is a subspace of the tangent space at the point. So what I get is a sub bundle of X. And so here what I mean here is this is an algebraic sub bundle, meaning that the tangent directions, they vary algebraically, even though the leaves may be transcendental. And the question is, when are the leaves algebraic? So this is the analogous to the previous questions from this point of view. And in order to answer this question, given a foliation, when are the leaves algebraic, we are going to again introduce the canonical class and measure the curvature of this foliation. So this is something that we, together with my collaborators, the Fandrouelle from Grenoble, we started to look at this some 10 years ago. And so we've been studying this and I'm just going to give you just a hint of what kind of result we can prove. So again, you start with a foliation. So the tangent shift of the foliation is going to be some sub bundle of X. While here, just to be precise, there are some points here where this foliation fails to be locally a product. And these are called the singularities of the foliation, but that's okay. I do admit some singularities. And again, I can define the anti-canonical class of the foliation as the first churn class of the tangent bundle of the foliation. So this is, again, this is sort of a measure of the curvature along the leaves of the foliation. And so this is the canonical class of X. And then again, we can study foliations for which, so in general, the sign again is not well-defined. There will be curves that will intersect it positively and curves that will intersect it negatively, but you can look at those very special foliations for which the sign is well-defined. And for those foliations, so this, and so this is, we started 10 years ago to, we defined the fun of foliation to be foliations for which the anti-canonical class is, so the canonical class is negative. And we started to study what are the special properties of these foliations. And it turns out that they have very, they have some very special properties and I'm going to just tell you one of them. But before, let me tell you another current trend and an important future direction about foliation is, so this is something that I'm also getting involved with, which is to find a minimal model program for foliations. So is it true that as in the case of varieties, all foliations on algebraic varieties can be built up from these three classes? So what we have so far is for foliations on surfaces, this is okay, this was the work of Brunella in the 90s, early 2000s. And for dimension three, this, we hope this is true. There has been a preprint in the archive in the end of last year that claims that this is, that the minimal model program for foliations on three folds is T-O-K. This is certainly a very interesting result. But in dimension greater than four, this is completely open. So this seems to be, so the techniques used for varieties don't all apply in this case and some new ideas and new techniques have to be developed. And okay, so now let me, let us go back to the true, the differential algebraic differential equations that we started with. And we have these ones with algebraic leaves or algebraic solutions. And this one had transcendental solutions. When you compute, so this is a set, this is a foliation and we can compute the canonical class. So if you compute the canonical class of this one, this is negative. So this is what we call funnel foliation. And this one has zero canonical class. And so one may wonder whether this positivity of the anti-canonical or positivity of curvature has anything to do with algebraicity. And this is what we have been studying for some years. And let me just give you a hint of the type of result that we can prove. And the answer is yes. In this case, this foliation here has algebraic solutions. This differential equation has algebraic solutions because of this positivity of the curvature. And so what we prove is the following. So this is just one example. So we start with the funnel foliation and a funnel foliation has negative canonical class or positive anti-canonical class. And again, we can define several invariants that measure how positive this foliation is. So I don't want to get into the details, but for instance, if you think of this, again the anti-canonical is something that lies in the second cohomology group. And you can ask what is the biggest integer that divides this class there. So this is a measure, this is an index that measures the positivity is a very rough one. So we have more refined ways of measuring positivity. But for sake of this talk, we can think of this as a measure of positivity. And what we prove is that the algebraic rank of the polliation, so that means this is again a measure of how algebraic the leaves are. The algebraic rank is bounded from below by the index of this foliation. So, foliations which have very positive curvature tend to be very algebraic leaps. So this is roughly, this is some of the results that we obtained, but we investigate this in more details and make some classifications of foliations with high, funnel foliations with high index. But I hope that for the sake of this talk, this is all that I want to do. So thank you for your attention. Thank you. Comments or anything? Usually on P2, we have this result that generically, foliations don't have algebraic leaves. But if we have more than a certain number, then suddenly all leaves become algebraic on P2. There exists like a similar result on any kind of surface or foliation. So what is, if you have too many algebraic curves, then they have infinitely many. So, maybe I should make it clear that if a foliation is not funneled, it still may have algebraic leaves. But if you look in the space of foliations, these are going to be very small dimensional spaces inside the space of foliation. So, I don't know exactly what kind of a similar result you were thinking of, but. We have more than a. Ah, so you mean in general, for other surfaces. So I don't know, but I believe that there are some results for surface. And I would hope that in higher dimensions, one can, maybe the invariant that one has to measure is a little bit different, but I would expect that it's possible to have the same results in higher dimensions also. I need that. Can you give us any feeling for what you mean by saying that all dryness can be built after these three times? Yes, yes, okay. Very roughly, it's like first with these eight geometries. Yes, I will, I can tell you something. So, okay, let me give you a crush course on the minimal model program. Okay, so, if you start with a smooth protective variety X and you would like to ask whether the canonical class is positive, okay, let me just put it this way and then I will explain better. So, one thing that you ask is, is the canonical class NIF, NIF means greater, it's greater equal than zero. If the answer is yes, you stop for the moment. So, you make this question and if the answer is yes, then you stop. And there is a way to produce a map that is going to separate the zero part from the positive part. So, here you can define a map from X to Y such that this has, again, greater equal, K greater equal than zero and the fibers have K equals zero. So, you separate and here, well, okay, maybe you do this, you do it many times and then in the end you get a vibration where the target has positive K and the fibers have K equals zero. Okay, so, in this case you are happy, you obtain your variety. So, if the answer is no, then that means that there is a curve for which K dot C is negative. And now, what you want to do, you want to find a minimal curve in the sense that can be made precise but there is a minimal or many minimal curves with the property, this property and then what you try to do is you try then to contract it. So, if no, then you just try to contract. And when I mean contract C, I mean I contract all the curves that are in the same cohomology class or proportional to the, have a cohomological class proportional to C. You contract all of them. Now, if then what you would like to do is to, so you contract that means that you have a morphism from X to Y where you contract this curve where C is negative. And what you would like to do is to come back here and keep running the program. However, this contraction always exists. So, this was proved by Moriv since very early that this contraction exists. However, this variety here in general is very singular. So, you cannot go back because I mean some of this, some of these tools, so in principle we started with a smooth variety. And now you get to a singular variety, you can, some of the tools that we applied here don't apply anymore. And now there are a few cases. So, some of the singularities that appear are, we can handle and we can make all this machinery work for singular varieties. So, here there are two cases. So, singularities of Y, let's say Tame, and singularities of Y, Wild. So, one of the important developments in the area is how to actually quantify and distinguish these two. And what is the smallest class of singularities that you have to admit in order to have this? And this has been achieved, so for this Tame once again, you can go back and keep running the program and all the theorems and the existence results that we obtained in the middle, they still hold for Tame once. Now, when you have wild singularities, there is no hope to be able to run this program here. And then here what we do is what is called a flip, a flip instead of contracting a bad curve. Well, I'm calling the bad curve, but it's not a bad curve. It's like this negative curve. You contract this curve, instead of contracting this curve, you make a surgery. You make a surgery, we remove it, and then you introduce another curve in a different direction. And after this surgery, which is called a flip, so instead of doing the contraction, you do the flip. And this flip here, so you change something, a curve, which was negative, by another curve that is positive. So here, you remove this C that is negative, and you replace it by another curve on X prime that has positive intersection with this canonical. And again, this variety here, the contraction is bad, but this variety here has tame singularities, and we can make this, and we can just go back and run. So one important, so two very difficult results. So this part here is the singularities are tame, then you're okay, this is very easy to handle. The two difficult theorems are the existence of the flip, the existence of this algebraic surgery, and the termination of flips, to say that this program terminates. As what I said that in dimension three, this is all known, in dimension greater than four, termination is not known in general. But what they prove is that you can run a very special, instead of choosing here an arbitrary extremal curve, minimal curve, if you choose this curve very carefully, then this is called a minimal model program with scaling. If you choose it very carefully, you can prove termination of the flips. There are other questions? I had, oh, yes. It seems that there is a notion of irrationality between foliation, can you explain if there is? Yeah, so the notion, so you have a notion of a birationality of varieties. So two foliations are on different varieties are birational, if you have a birational map, that on the open set where you have an isomorphism, take one where there is a big open subset where it takes the foliation from one to the other. And then again, you want to do the same game here, but now you want to contract always curves that are tangent to the foliations. But in principle, this scheme for foliations we hope should be the same. But now you are just contracting curves that we want to contract curves that are tangent to the foliation. Yeah, so what this is showing is that every variety is going to be birational, too, either. And here, of course, maybe I did not say how there is a third possibility here, sorry, that when you contract this curve, yeah, so this is an important point. So it could happen that when you contract this curve, you obtain a vibration. So this would mean that the canonical is negative on the fibers. And then again, you have a, the canonical is going to be negative on the fibers and then you run the MMP for this one, and then so on. But yeah, so every, maybe one way to say is that every variety is going to be birational to, let's say, it's birational to a vibration where you have canonical positive, canonical of the fibers, zero, canonical of the fibers, negative. So you separate these three parts. Well, I have a couple of curiosities, but they can't wait for later. So I think it's also a little bit hot. So I would like to thank, to thank Carolina again. And actually I would like to invite everybody for a small refreshment on the terrace, just outside here, this one. Okay, so I hope this was, yeah, this was. Wait, you're always playing the key one. I hope you get the current trends. And then you know, if you're able to jump to the end of the slide with this one, I don't have you there. Oh, thank you. It's an amazing lecture. Okay, thank you very much.