 For the last lesson of this course, I decided to give you an overview on Riemann's surfaces. I decided to choose this as the last topic, because I believe it is a very nice topic where you can see the tools we have started applied in several ways. And in some sense, it's a good field for those who are interested also in other subjects. So first of all, well, Riemann is a family name. But what about surfaces? Are you familiar with surfaces? Do you know anything from real geometry which reminds you the word surfaces? Well, it's natural to have at least a naive idea of what a surface is, right? Surface is, well, something, some deformation of a plane in the space. This is what, say, if you ask anybody from the street, it's like, well, surface is, well, something wrapping, right? To be more precise, well, let me tell you what is a Riemann surface. And I start with this very strict, but in some sense, also very condensed definition. And then I will go into the detail. So one-dimensional, connected, complex manifold. So I want to just give some ideas of what a manifold is. A manifold is a generation of a surface. So if you don't know what a surface is, it's difficult to say, well, a manifold is a generation of a surface. But in any case, what we can do is that if we give a definition quite general, you can see that it can be generalized as much as you want. And you obtain the general definition of a manifold. So one-dimensional will be explained very soon. Complex will be explained very soon. Manifold will be explained soon. But at least connected is something you can understand now. It's a connected set. But it is not necessarily assuming a general definition, since you can talk of several connected components of a Riemann surface. So in any case, connected here means exactly what you expect to have, this set which cannot be split into two parts, or more than two parts. So let us go to the basic definition. A one-dimensional complex manifold is connected. We are assuming that it is connected. So connected house of topological space. House of means, you know, you should know this, T2. So you can separate points with neighbors, neighborhoods, right? And then we have also a complex atlas, a maximum complex atlas on this topological space. That is a family of local coordinates. There's pairs of sets and functions like this. And the description is here. So uj for each j is an open set in x. x is a topological space, so this means something, right? Furthermore, if you consider the union of all uj's, then you obtain x. So it is what is called covering, but not in the sense of covering space, right? A covering of x. There can be overlapping parts, but no. So it is an open cover of x. hfj maps uj into c. This is the only reason why we call complex here, OK? If you put r something, it is real, right? c and then c1, I omitted one, but it gives you the dimension of the manifold. Any case, each uj is homomorphically mapped onto the open set fj uj and c. Sorry, I forgot to put a j, right? fj uj, I'm sorry for this. So this means that we have a topological space. And luckily, since any point is at least one of these uj's, which is called everything, and luckily, you have a function which maps neighborhood of this point into an open set of c in this case. And these are called local coordinates, because the idea is that while the surface is like Earth, and luckily, you have the coordinate, geographical coordinate, or charts, right? And you transform something which is not flat into something which is flat, because c is like r2. When you have something which locally is like an open set homomorphic to an open set of r2, then you are talking about surface, and c is like r2, right? So in general, here, you can have a very, a more, well, you can even have a Banach space. But you say, in general, you have Rn or cm. And in the first case, you are talking about real manifold. And the exponent represents the real dimension. And the case you have ck, k is the complex dimension of the complex manifold. So one dimensional complex manifold in this case is what we are starting, can be also regarded as a two-dimensional real manifold, because the underlying structure of r2 is here, right? OK? So when I say dimension, I have always to be more precise and to indicate with respect to what, to c or to r. Up to now, we are not using any regularity. I want to remark this now. Fij is just in the class of function which gives information about topology. So homomorphism, right? We are not assuming that Fij is something else. But we cannot, because Uj is just an open set in a topological space. We cannot say, well, I want Fij to be, I don't know, complexity, harmonic. I can't, because I cannot define, well, up to now, I cannot define homomorphicity, analyticity, harmonicity in a topological space. We just have a topological structure on x. However, we transfer charts locally this vague topological space into something we know. So the ground is, in our case, c, right? In general, it can be cn or rk, or what I said before. Then the last property we require is what makes you understand why we are transferring information from one set topological to the other. Because whenever we have an intersection of two neighborhoods of the family, Uj, so you assume that Uj and Uk intersect. Of course, they intersect in an open set, right? And these two open set are the set of definition of Fij and Fk respectively, right? Because for each Uj, you have also Fij, which is a homomorphism. Then you can consider this function here, Fkj, which is Fj minus 1, and then Fk. But Fj minus 1, this, remember, is an open setting c. So this function here is a function from c into c. It's an open set of c, the domain of this function. And the value is an open set, the range is an open set in c, from c into c. You see this? So Fij maps homomorphically x, so a small neighborhood of a point in x, onto an open set in c. And then I have Fk, similarly, mapping an open set Uk into an open set in c. But then I consider this diagram. So starting from here, from the ground, I go to x, and then map into the open set Fk Uj intersect into k. This is a function from c into c, from an open subset of c, this, into an open subset of c. And then I can use my notion of holomorphicity. This is what I require in this case. But in general, what I have for granted is that this function here cannot be an homomorphism. The remark is not correct, I have to say. Fij is just an homomorphism. So it is invertible, and the inverse function is still continuous. So what I can definitely say, if I don't assume anything about, even if I don't know anything about this set, but of course I have to put some topology on this set, well, this composition of two functions which are homomorphism, is necessarily a homomorphism. So these two sets here are homomorphic. What I cannot say is that this is an invertible homomorphic function. I have to require this function as an homomorphism. I cannot even have for granted this is homomorphic if this is c. I can say, well, this is a complex value function, but not this is an homomorphism. In general, this function here tells you the regularity of the manifold you are requesting. So if you don't require anything here, you have what we say topological manifold. And in fact, these are the important functions you have to consider. And they are called change of local coordinate. That's obvious. Move from one chart into the other, or transition function. And in general, this function here, according to the fact that pj is homomorphism into something else and c, like cn or k or whatever, can have different regularity. For instance, if this c be replaced by r2 or rn in general, and fkj is supposed to be a real differentiable, you are talking about a differentiable real manifold. Assume that the transition function is real analytic. You are talking about real analytic manifold. So the information of regularity is not on the local coordinates. It cannot be regular, but on the transition. This is general. OK. So now the reason why this one-dimensional complex manifold are called surfaces is because, well, surface is a two real-dimensional manifold. And one complex dimensional means two real dimensional. That's the reason why surface was used. But in general, you can talk about manifolds. Manifolds. So now, with the definition of local coordinate, we can give also the notion, the definition of homomorphic function between two real-man surfaces, but in general, more in general, between two complex manifolds. And the reason is the following. You start from a function f between two, say, one-dimensional complex manifolds. This is the function f between x and y. And it is to be homomorphic at a point. Remember that we define homomorphicity locally at a point. And then we had the neighborhood, where you were. OK. So locally at a point, we can use the change of coordinates and transfer everything into C. And use the coordinates of p and of f of p. F of p is also a point in a real-man surface. So locally, it can be red on the ground floor. So you put a composition like this. And you consider a function from C into C. So you have a function from two topological spaces, which locally are homomorphic to C. And this local coordinate system allows you to define homomorphicity, for instance. In general, you can define what? Harmonicity, real analyticity, and so on and so forth, according to the properties of the change of coordinates. So I summarize this idea here in this remark. If you just assume that transition functions are real differentiable, then the manifolds are real differentiable. Or Ck, or leapsheds, or analytic, and so on and so forth. And then also, the function between this class of manifolds can be defined to be whatever you want. Whatever you want. Whatever you can say. So in general, you have a topological space, locally homomorphic to something else, to some mobile space. This is the idea. And what you can always do is to use topological characteristics. You can say the f is continuous. This doesn't require the structure of the manifold. But if you want something else like analytic regularity, you have to work on the model instead of working in these topological spaces. This is the main idea. And this applies not only for one complex manifold, but for real, more than one dimensional complex manifold and so forth. This is a general idea in geometry. You can have, well, we have models. Of course, we have this. The standard surfaces in R3 are models like this. If the surface is not odd in the sense that there are some singular points or singular curves, so if it is smooth enough, so you can locally transform everything into an open set of R2 with functions which are sufficiently regular in a sense that they can be also not only continuous, but also, well, maybe more. And then you can define frames. You can work on the tangent. You can define indicators of the orthogonal directions and so on and so forth. So you can talk about orientation and all this stuff, which is quite natural to do. But in general, well, this is called embedded manifold because you have the universe which tells you what to do in some sense. If you want to measure a vector in the tangent space, you can obviously apply the standard Euclidean scholar product you have from R3, which is natural to consider. So it is just a vector, tangent vector. Imagine you have a curve in a surface. You can always consider the tangent vector to be the velocity. It is not a point of the surface. It is on the tangent plane, which moves smoothly on the surface if the surface is sufficiently good without odd points. It's not like a cone or like a roof like this. But if it is smooth, you can go around and live in this tangent plane moving around. Then you can measure distances using scholar product. But the scholar product is given from the structure, the Euclidean structure of R3. So the fact that what we have in mind is that R2, so the surface is embedded in R3 gives us many other information. But in general, you have a not embedded manifold and you have to define a way to measure tangent vectors and so on and so forth. And since you have the choice of changing the way of measuring this, you can define different ways of defining a scholar product on each tangent space. But this is beyond the problem you will have, of course, in differential geometry in the second term. So in case, keep this in mind when you will see it. I will not go into these aspects of Riemannian geometry. So this is the first very important result, which is not obvious to prove. We have proved one part of this. Essentially, that we know any simply connected domain, plane domain different from C is by how morphic to the unit is. This was already something. But there is a more general and more, I'd say, and a deeper result, which tells us that, well, simply connected Riemann surfaces are essentially up to biomorphism, only three. The disk, the plane, and the Riemann sphere, right? So in the plane case, we already said that the plane and all the other simply connected domain are into two classes. But this is something else. To start from a Riemann surface that is a one complex manifold, we assume that it is simply connected. Then it is necessarily biomorphic to one of the three models. And this is the starting, very important result, starting point of our consideration. Now, if you have a Riemann surface, then you can always cover it with a simply connected Riemann surface extilda. So now we use something which comes from covering, topological covering. And this proposition can be rephrased in this way. To start from a Riemann surface, not necessarily if it is simply connected, uniformization theorem tells you that it is biomorphic to one of the three. But if it is not, then it can be always covered by another Riemann surface, which is simply connected. In other words, you can say that we can take as a covering a universal covering. Remember that universal was the precise terminology we use, we introduce for covering with the covering space which is simply connected. And this covering spaces, which in principle is just one topological space, turns out to be a Riemann surface. So it has a structure of complex manifold, one dimensional complex manifold. So you can cover a Riemann surface with another Riemann surface, which is simply connected. This is very important. So it means up to a Riemann surface, you cover every Riemann surface by the disk, the plane, or the Riemann sphere. Furthermore, the covering function P turns out to be even more the continuous as a objective. It turns out to be holomorphic. So it's much richer than the topological definition we gave in general. So if you start from Riemann surface, the class of covering you can consider is very precise. Universal and with P, the projection, also holomorphic. And while X tilde is uniquely determined, well, this is obviously the uniformization theorem up to biomorphism. This is holomorphic universal covering of X. And now we can classify. So a Riemann surface is called elliptic or parabolic, or hyperbolic. This is just the terminology. This terminology went and used many, many years ago. According to the fact that universal covering space is, respectively, the Riemann sphere, the complex plane, and the unit disk. So we are talking about elliptic Riemann surfaces, parabolic and hyperbolic F, respectively, the universal covering spaces, Riemann sphere, plane, and the disk. In fact, the three models represent three geometry models. And one is elliptic, the other is parabolic, and the third is hyperbolic. Well, I just want to recall you that we are now going to apply the notions of automorphism of the covering in this setting. So an automorphism is nothing but transformation. Well, there is an extra dot here. I'm sorry. There's transformation which sends fibers into fibers, right, in general. And as I said last time, this is a group, right, with respect to composition, the automorphism of the covering, that transformation for my group. And this group turns out to be, since the covering is a universal covering, turns out to be isomorphic to the first fundamental group of x, right? Remember, in general, you have the quotient. If you have a regular covering, you have the out P over P star of blah, blah, blah. But in this case, the P star of P1 of x tilde is trivial, because P1 of x tilde is trivial. And the group, well, this is another important property, it acts transitively on the fiber. So it acts on the fibers in general. So you know that one fiber is mapped into a sub-automorphism, because P of P is P, right? This means exactly this. P can map point on a fiber into the same fiber. But here I can say more. You take two pairs, two points, and the fiber for any pair y1, y2, and the fiber of x, then you can also find an automorphism of P such that y1 is mapping to y2 by P on the same fiber, right? So it means that the action of out P on the fibers, on the fibers, is transitive, right? We already introduced this terminology. Transitive means that two points can be mapped one into the other by the group. And finally, well, this is natural, because you can take the orbit space as the quotient of x tilde over the action. So all the fiber is mapping to one point according to this relation. And so x is biomorphism. You can see this as a quotient. Of course, I'm not going into the details of the proof, because this is an overview of the subject. However, what I'm saying is that you can read the proofs. I think that you are smart enough, of course, and you have all the backgrounds to understand also the proof. Of course, I'm not going to enter the details of the proofs of these parts, at least. Now, I can consider a subgroup of the group of automorphism, right? Remember, this was done already, right? And we say that gamma acts freely on x if no element of gamma, but the identity, of course, has a fixed point. That's why I stressed several times about the description of fixed point set, not only because I like it for my research, but because I think this is, well, a geometric property, which is important also in dynamics, right? When you study orbits, well, of course, the first points you want to study are the fixed points, because the dynamics represent some special points, right? Or periodic points, in some sense, OK? But this is just the definition about the group of the subgroup. The action is set to be free, or the group acts freely on a set if, beside the identity, no other element of the group or the subgroup has a fixed point. Of course, the identity has all points fixed, but this is kind of unavoidable condition, because you need to have the identity, right? So do we have examples of subgroups of the automorphism groups we study in the three models which act freely? Well, we do. If you remember, in the case of the plane automorphism, the plane is one of the models, right? In the plane models, we add linear transformation like az plus b with a different from 0 as possible transformation. The only automorphism of the plane are like this, with two parameters, a and b. And when I started this automorphism, I said, well, if a is equal to 1 and b different from 0, they are necessarily the transformation. z is mapping to z plus b as no fixed point. So translations are examples of a subgroup which acts freely on c. So this definition can be applied. Similarly, in the unit disk, we have, you remember, the possibility to have one fixed point inside or one outside the disk, or the two fixed points are on the bottom, distinct or coincident. So for instance, if you restrict your attention to rotation around the origin, for instance, well, the origin sticks for any of such rotation. So this subgroup represent a group which does not act freely on the unit disk. So this makes a difference if you ask something wrong. And this second definition is a bit more technical. But however, it is unavoidable. So I have to introduce this way. So the action is proper discontinuous, or gamma is properly discontinuous. The action is properly discontinuously. So sorry, the action is proper discontinuous like this here. At a point, if there exists a neighborhood of x such that, you see, the range gamma view and you do not intersect, for just a finite number of elements in the group gamma. This is a bit technical, but this is what we need. Properties discontinuously. The number of elements in the group having this property is finite. I make a remark. I talk about a possible subgroup and the automorphism of C acting freely. Translations. And I also said something about the rotation in the unit disk. I couldn't say very much about free actions of subgroups and the case of automorphism of the Riemann sphere. Because if you remember, in that case, in that class of function, there are always at least one fixed point, right? Do you remember this? We show that if you start from a general linear fraction transformation and you impose a z plus v over cz plus t equal to z, you always find a solution in c hat, at least one, which means that any automorphism has at least one fixed point. So none of them can act freely except, of course, the identity, which is allowed to have infinitely many fixed points and represent a trivial subgroup. So in that sense, we can say, well, no trivial subgroup of, no known trivial subgroup of, what did I write it this way? Well, I should have put here c hat, right? Can be fun. OK. This proposition explains why we are assuming that the action of the automorphism of the Riemann surface is supposed to be properly discontinuous. Because then the subgroup of the group is discrete. It cannot be differently. Because assume that the group is not discrete. So it eventually has a limit point, right? And the limit point is an automorphism. I'm saying this because, well, you might say, well, we have seen that in case the function is just one to one, it's injective. Not necessarily the limit function is injective. It can be constant. Remember this? This was an application of Hurwitz theorem, right? You start from a sequence of injective function, holomorphic injective function. And the limit function might be also not injective. It might be constant. Remember, this was one of the problems also. And the proof of Riemann theorem where to restrict our family. Yes, please. Discrete means it doesn't contain an accumulation point. It's not, well, it has to have, it can be considered continuous. It is not continuous. Discrete in general means no limit points, right? Well, let us go to this point. Assume that you can find a sequence of automorphism in a subgroup acting properly discontinuously on X and having a limit. I'm saying that the limit is still an automorphism. And this is because you can always take the inverse also because you are dealing with automorphism. You start with a sequence gamma n, for instance. You can also consider gamma n minus 1. So if gamma n tends to gamma and gamma n minus 1 tends to delta, I can show you that gamma and delta are one, the inverse of the other, right? Because the limit is the identity of the composition of the two. This is a uniform limit. And this guarantees that the limit function turns out to be an automorphism. But if it had this property that starting from a sequence, you have a limit point, so if you are assuming that gamma is not discrete, then you can take gamma minus 1 gamma m as a new element in the group. And of course, it is in the group. And this tends to identity, right? But then as a seller, the identity makes this action to be not properly discontinued for infinitely many, right? Elements of the subgroup, I had that intersection of view and gamma view is different from the empty set. So this tells us something so that we have to deal with, if we are interested in something, reasonable, we have to deal with discrete subgroups. And discrete means, well, in this sense, there is no limit point. So if I want to consider, for instance, a subgroup of, say, for instance, in the case of translations in C, well, I cannot say, well, take B and C as you want. Because I have to take a discrete subset. So at the end of the story, I will have a lattice instead of an open set of parameters to describe this. This is the idea. And this is, in fact, the fundamental theorem which put together all this information. You start from a Riemann surface. And you consider the automorphism of the universal covering, which exists, x tilde p of x. While this automorphism group is properly discontinuous and acts freely on x tilde, you can prove it. So if you start from a Riemann surface, cover it with a universal covering. The automorphism of the transformations of the universal covering acts freely on x. And vice versa, if gamma is a properly discontinuous subgroup of the group, sorry, this g is not precise, of the transformation acting freely on x, then you can always consider this quotient space and put on it a structure of Riemann surface in such a way that projection which comes from this quotient is the canonical quotient map. And this function here is the universal holomorphic covering of this Riemann surface. So in some sense, you can describe all possible Riemann surfaces as soon as you know the uniformization theorem and the fact that you can describe them in terms of quotient of x tilde, this plane and Riemann sphere by the action of a subgroup of out p, so the automorphism of respectively Riemann sphere, plane and disk, unit disk acting properly is continuous and freely on x tilde. So two Riemann surfaces are biomorphic if they have the same universal covering and if the fundamental groups which are related to the subgroups of out p are conjugated in the group of that transformation. So this is quite easy to see, putting together the previous results. This is a proposition that we put. So this is what I wanted to show you. We have some classification. So there is only one elliptical Riemann surface up to by the Riemann sphere. Why? Well, you start from Riemann surface and you say, OK, we have three classes. It can be covered by C, disk, and Riemann sphere. Let us start from the Riemann sphere. What are the possible choices of a subgroup acting freely and property is continuous? There is only the trivial subgroup. As I said, all other groups do not act freely. So only the Riemann sphere can be considered as a quotient of itself. So for the others, what can we see? Well, the property discontinuous subgroups of C. Remember, the out C, the automorphism of the plane are described in this way. The general automorphism Z map is mapped into A, Z plus B. We want to have an action which is free. A has to be equal to 1. So we have the freedom of choosing just B. But B cannot be chosen randomly. It has to be chosen in such a way that the action is property discontinuous. It is to say it has to be discrete as a subgroup. So either we have a translation along the reals or along the reals and along the lattice here in the plane. And tau is what is called the module of this lattice. And there's an element in the upper half plane. So it has an imaginary part positive. So we have a translation along the reals and a translation with a component along the vertical line. Another vertical line, definitely. These are the only possibilities up to conjugation. Consequently, these are the parabolic Riemann surfaces. So covered by C. Either C itself, obvious, or C minus 1 point, punctured plane, which comes from here, imagine that you are wrapping everything. And the final and interesting case is this one. So you have a lattice of points. And imagine to have this as a fundamental set. And everything is identified. So the sides of this rectangle, not the same rectangle, but rhomboid, whatever, are identified. So typically, these two sides are identified side by side, opposite side identified. And this gives you a structure of torus. That's why they are called complex dory. But beside names and beyond names, these are the only possibilities in terms of discrete subgroups of outside available, acting freely. So I want to repeat this. A is equal to 1 because we want the action to be free. So no fixed point beside the identity. And then here, we cannot put a continuous set of parameters. B, because otherwise the subgroup described is not discrete. It would have a fixed limit point. And these are the only possibilities, either, while in this case, t, this tau is 0, so translations along the reals, or translations along the reals, and independently also along the vertical axis. OK, let us make some observation about this. The Riemann sphere is connected, but also it's like S2, right? It is simply connected, right? It is simply connected. So the fundamental group can be easily found. Fundamental group is, remember, the class of loops. You will calculate it. In any case, it is simply connected. So each loop can be contracted to one point. However, this also turns out to be true for C, but not for C minus one point. So 0 is one of the points, right, as puncture plane. Because if you take a loop going around the origin, you cannot deform it to a point, right? This is an obstruction, right? And well, this is not simply connected. This is not simply connected either. But you can calculate and show that, well, there is just one generator. If you remember, like in an analysis, right? You start from one important loop going around the point. And all other loops like this are combination of this generator. The other loops which do not go around this point, this origin, this case, can be contracted to a point easily. But all the others can be deformed to a chosen one. So there is one generator. And that is enough to say that the fundamental group of the puncture plane is Z. It's isomorphic to Z. Whereas for the torus, the torus can be seen as, well, this is complex torus. But the torus as S1 times S1 is generated by one generator here and one generator here, right? We have this structure, the donut, right? One generator is here. We tell you the number of times you are moving this way. And the number of times you move this way. So you have two generators. And with some, you can imagine at least that the fundamental group is Z direct sum Z, right? In all these cases, what we have is either it is simply connected or the fundamental group is Z or Z direct sum Z. In all these cases, these fundamental groups turn out to be a billion. They are a billion. But as I said last time, in general, the fundamental group is not necessarily a billion. So we're debating on tau. Sure. Yes, but tau is fixed. No, no, tau is fixed. This varies. And tau is fixed. It's a point where we have to learn how to do it. Exactly. Exactly. So in fact, this is gamma tau. Yeah, no. Tau is not changed. Well, I probably have to, yeah, we're right. So mm varies, and tau is fixed. Right. Yes, notation is probably misleading. You are correct. Tau is given, right? I could use this correct. What I wanted to say is there is a real part which varies on the discrete set like Z, and here I have a part which is real and imaginary, right? So in this part, in this set, the imaginary part is 0, right? So what you change here is just the real part. Here you change real and imaginary part, but correct. Tau is fixed is given. OK, I will adjust to the, thank you. Also from here, you can have an idea that while you see n here, it tells you that every time you identify everything as a cylinder, it is like s1 times an infinite line. So you have just one generator, and the infinite line can be squeezed to a disk, and then you consider just contour, one generator. Here you have two generates, m and n, which tells you that you have, in fact, one s1 and one s1 working independently as generators of all possible loops and the torus. Well, this is probably something beyond your knowledge and algebraic topology, but what I want to stress is here at this stage is that, however, what you can imagine is that these fundamental groups are all billion. Therefore, whenever you have something, a Riemann surface, whose fundamental group is not a billion, you can say, well, it is not elliptic, neither. Parabolic has to be hyperbolic, all right? So that's the first point. So every domain in the Riemann sphere, whose complement contains at least three points must be a hyperbolic Riemann surface. Because if it has more than two points, at least three points, then the fundamental group turns out to be generated by two generators, but the fundamental group is the free group zz, which is not a billion. So if you take two points out from the plane, which is the minimum request, if you remove one point, the complement is infinite in one point, right? And we have z as generated. But if you take two points, well, it is like an eight figure in the plane, right? In the eight figures, it's not like s1 times s1, not at all. We have one generator and another generator, and you have a combination of these two independently. So it's not a billion group. So this is interesting because it leads to many consequences. So every bounded domain in the plane must be a hyperbolic Riemann surface. And this is another important fact. Remember that we have the definition of holomorphic function between Riemann surfaces. And remember the last time we were talking about coverings. I was also saying that it is, in general, possible to consider the cover transformation. In general, cover transformation is the function which plays the role of commuting the diagram at a level of universal covering. And this is always possible to be found, right? So assume that f is holomorphic and x is non-hyperbolic, whereas y is hyperbolic. Go to the level of covering. You have something which is either the plane or the Riemann sphere. On the other hand, you have that y is covered by the unit disk. So you would have a diagram closed by a function between the plane or the Riemann sphere into the unit disk. So it would be a holomorphic function, entire, and bounded. So it would be constant. So necessarily, f is constant. And putting together all this stuff, since we can say that every domain in the Riemann sphere whose complement contains at least three points is a Riemann surface, which turns out to be hyperbolic. Then we have a little picard theorem now proved. Every entire function missing two values is constant. Of course, the plane is covered by the plane. The plane minus two points in the complement of the Riemann sphere has three points and is covered by the disk. And this shows you that a heligant, now the little picard theorem, becomes. It's a consequence of all this machinery. So let me quickly go to this. So here is the description of all Riemann surfaces up to biomorphism with a billion group, fundamental group. So it is trivial. If it is trivial, so it is, well, necessarily, it means that x is simply connected. So it is by unionization theorem, either the unit disk or the complex plane or the Riemann sphere. And if it is z, it can be either the punctured plane or the disk minus one point or the annulus, up to biomorphism. And we know that they can be deforming. So it is a line of possible choices of, remember, we have shown that two annulus, two annuli are biomorphic, if and only if the ratios of the radii are the same. And finally, well, the direct sum of z, so z, direct sum z is the last possibility. And then, necessarily, x is a complex torus. This gives you the complete descriptions of all of them. And two complex tori are very much studied. And I explained you in the last 15 minutes why. Are biomorphic, if and only if. So two complex tori are identified using this tau. So gamma tau and tau is fixed. Take tau and the tau, tau 1 and tau 2. And this gives you two lattices, two groups, discrete subgroups, and then two tori in the quotient. And these are biomorphic, if and only if, there exists two by two invertible matrix with A, B. Oh, sorry. This is A here, right? No, it's 2, right? Of course. Sorry for this. Such that one tau 1 is mapping to the other using this very simple. This is a linear fractional transformation, you see? So one torus is mapping to the other. And this guarantees that two tori are biomorphic. So the class of non-biomorphic tori have been heavily studied. Because it is easy to say, well, I can pass from one to the other with this very simple condition. And this is the difficult part. So given a complex torus, you can choose the tau 1, so the module, to be with, of course, positive imaginary part and the real part in between minus 1 half and 1 half. And the tau itself is in modules greater than 1, all right? In case it is 1, then necessarily R2 1 is greater or equal to 0. And you contain just one of this module in this strip. It is an infinite strip, as I show you in the next picture. In fact, it is this strip here. The condition tells you the following. You take any torus in the plane, sorry, any torus, any complex torus, so any parabolic Riemann surface. Then up to biomorphic, it is just represented as, sorry, as the quotient C over gamma tau. And tau is taken in this infinite strip here. So the real part is in between minus 1 half and 1 half. Modules of tau is greater than 1. And so this means that we have complete description. This set here, this gray set, subset of the plane, describes you in each point you have a different complex torus, not biomorphic to the others. So there are very many, not biomorphic, right? This is called the fundamental domain, the fundamental region for the torus. And assume that you move like I'm moving the mouse here. And imagine you have a complex torus associated to each point in the center of the cross of the torus, right? You have different tori each time I move, and two of them are never biomorphic. Just this precise description. Whereas if I go here and here, of course, I obtain tori, which can be biomorphic, right? Now, just a few slides tell you the important relation with projective algebraic, projective plane curves. So a plane curve is given by, as a zero set of a polynomial of degree d. And then we obtain in CP2, for instance, OK? It's a plane curve. So we are dealing with something which is in the plane, so it's CP2, right? So x0, x1, x2 are defined up to a constant, not necessarily not equal to 0. And this polynomial has to be homogeneous, of course, to define something reasonable in this setting, right? Are you familiar with this notation? This means that x0, x1, x2 are defined up to a constant different from 0, OK? So these are homogeneous coordinates of point in CP2. And p of x0, x1, x2 is equal to 0. It has a meaning because p is homogeneous, OK? So this is the definition of plane curve. And this is of degree d because the polynomial which defines it is of degree d. Well, CP is a closed set, the zero level set, of a polynomial. So it's an inverse image of the zero, right? Of a continuous function in CP2, which is compact. So CP, so the curve is compact. And the projected plane curve is? Yes. Yes, that's natural to consider. Well, CP2 is definitely compact in any case because it is a ball, all right? So you can define the risky topology, say, if you want. So you define the topology using the zero set of polynomials. But in this case, it is obvious. You are considering a continuous function. You imagine to define a topology which makes polynomial continuous. That's a minimum request. Otherwise, you're working in a strange world, right? Furthermore, if you take the derivative with respect to xj of any homogeneous polynomial, you obtain a homogeneous polynomial of degree 1 minus, one less than the degree of p, right? So the set S of p where all the derivatives of p vanish is well-defined in this setting. And this is called the singular point set of the plane curve C. So typically, this is the case when there is a self intersection or there is a cusp or something like this, right? So this is the important theorem. Any non-singular projective plane curve. So I repeat, any projective plane curve whose singular point set is empty. So non-singular mean it has no self intersection or is a complex treatment surface. Can be regarded as a complex treatment. So why? Well, since you apply Dini's theorem or implicit function of theorem, in order to define local coordinates, and you show that since the derivative are not all 0 at any point, you can define on the entire set the local coordinates and the function turns out to be one complex dimensional. And well, if the function has some bad points or something, oh sorry, this is some mistake here. Well, what I wanted to write here is, of course, less cp without the singular points. Sorry for this. If x is a compact Riemann surface, then you can always find an irreducible projective plane curve cp, allomorphic function phi from x into cp, such that after removing some points from x and the singular points, I'm sorry, for the singular points of the curve, you obtain a bilomotor. So there is a one-to-one correspondence. Riemann compact Riemann surfaces and plane projective curves up to after removing the singularities. Well, there is also one explicit. And when I say explicit, I mean that in principle it is a way to find the correspondence for tori, for complex tori. And for complex tori, the tool used is the bias stress elliptic function, which is associated to a discrete module in c. Remember that the tori were obtained as a quotient of the plane under the action of the discrete subset gamma tau, the reasonable tori, not the puncture plane, because it's not very interesting. And this is in fact a lattice or a discrete zeta models in c. And bias stress introduce a function which is like this. And gamma of tau represents the lattice. So you define this function like this. And this is called bias stress elliptic function. It turns out to be periodic. Then if the complex torus is c over gamma tau, and this is the bias stress elliptic function, then you can show that p defines the projective plane curve cp as a zero set of this polynomial here. Where these coefficients are known, one, four, and these two, g2 and g3 are unknown, but can be calculated using this formula here. Pardon me? I'm sorry. I'll show you again. So cp2 has coordinate x0, x1, x2. Yes, of course I do have homogenous. I do have homogenous. I said from the very beginning, these are homogenous coordinates. In fact, this is a homogenous polynomial of degree 3 in x0, x1, x2. Otherwise, it has no meaning. I'm working in the projective complex plane. So x0, x1, x2 are, yes, homogenous coordinates in cp2. And what is interesting is that, well, it seems to be reasonable, well, g2 and g3 can be found. Well, g2 is, in fact, 60 times s4 gamma tau. And g3 is 146 gamma tau. Well, sk gamma tau is this. So it depends on your ability in calculating this, the convergence of this series. As w varies in the subgroup gamma tau, but it's not 0. And this was done by hand, by hand by Weierstrass. It's surprising that he showed, in some cases, g2 and g3 can be explicitly calculated. So in principle, but now we can use also the numerical analysis and the power of computer science to calculate, or at least to approximate, numerically g2 and g3 in general. So in principle, you start from a complex torus. You can see it as a point in the infinite strip I showed you in the half plane. Or you can see it as the zero set of a polynomial of degree 3, whose coefficients are somehow explicitly found. But this gives you different ways to see the same object. Well, this class of Riemann surfaces or projective plane curves, now we can use them as synonym, are, in fact, examples of elliptic curves. And elliptic curves are, in fact, algebraic curves of genus 1, whatever it means to you. Are very important and very much have many applications in algebraic geometry. They are a billion varieties. You can define an inner operation multiplication, say, with respect to which the elliptic curves are form a group, a commutative group. Elliptic curves are important in number theory, for several reasons, and have been applied in the proof of the famous proof of Fermat's La Théorem by Andrew Wiles. You know that the story of this proof has been, well, Fermat conjectured that you cannot find it. Do you know the story? Of course, some of you know the story. When I was your age, of course, it was a conjecture, because I'm quite old now. No. And it was one of the problems. And I was in a summer school with other colleagues. And at that time, we didn't have any internet access. Internet was not known. I'm sorry to say this, because it means that I'm very old. But in this summer school, just the news of a possible solution of this Fermat's La Théorem. Of course, there have been many attempts during the centuries when Fermat, you know, the story of the problem. The problem is a generalization of a solution in integers of the Pythagoras theorem, right? Some sense, OK? Instead of putting two as exponents, you put something greater than two. So you say x to the power n plus y to the power n is z to the power n. Does it have a solution, and not trigger the solution, in the integer? So for n equal to 2, it is known that this is possible, because all the triples, which are called Pythagoras triples, right? So for instance, 3, 4, 5 is OK. So 9 plus 16 is 25, right? So is there anything, is there any analog of this in case n is greater than 2? And it was conjectured that this is not true, because Fermat, who was not a mathematician, was a next. Pardon me? The offensive of an insurance, a lawyer. He was a lawyer, right? So he was kind of a lawyer at that time. He lived in the 16th century. But he was, I don't know, he was fond of mathematics. And he went to libraries and took the old Greek books, the Euphantine Equations, and so on. He solved all the left exercises. In some sense, there are some exercises. In these old books, there are something which is not proved, and it proved everything. But only one proof was missing. And this was the proof. So the statement is like, but he left a comment on the right-hand side, I have in Latin. I have a wonderful proof of this, but I have not enough space on this book to write it down. So people say, OK, it has to be affordable, because all other proofs are correct. But Gauss said, well, Gauss himself said, well, I'm sure it didn't have such a very short proof, because it's not short. All there work a lot on this. And none of them could find a solution. So eventually, well, of course, in the history of mathematics where there have been many attempts, all of them failed to be considered a proof. Then with the implementation of a computer numerical simulation, it was obvious and reasonable that something true is in the conjecture, because they proved it for millions and millions of numbers and was clearly true. But however, the modern proof was given, well, that was announced in 1997. Sorry, I'm sorry. It was announced in, say, in summer 1993, 1994. It announced. But yes, exactly, Taylor, Taylor, Taylor, Taylor. Because Andrew Wiles is an English mathematician. And there was also this fact that an English mathematician could prove a theorem which was French. So there was another struggling between. So for instance, LeMond announced, of course, that the first proof given by Andrew Wiles was not correct, very happily. This British mathematician is not able to prove it. However, on the other side, on the time you could see, well, a great result by an English mathematician, the last theorem is proof. What happened is that Andrew Wiles announced the result using these elliptic curves, techniques, and many others. Because he proved, well, he was obsessed by this problem. When he was very young and knew of the problem, and something completely unusual for the time we were living, he decided to work on this problem by himself alone, without the help of anybody. And without communicating to anybody his partial results. And he has worked for seven years. And well, there is a very nice book about the story of this proof. In fact, when he announced everything in summer, it could be in a seminar during a series of seminars he gave in summer 1993. In fact, he didn't write everything that he gave to a staff of referees after some months. So it was in 1994 when the referees started checking everything. Well, nowadays, around 100 pages write the proof, something like this, that's not very short. So we had the feeling that it was all right. And also the referees at the beginning said, well, this is a good idea, because the techniques are very elegant. But at a certain point, we discovered that with some assumption one part is true, but cannot be applied for the others, like in a game. So he disappeared from the world, because he received some mails saying, can you please explain how you can apply this lemma in this context? Because the assumption seemed to be a bit different. And then he admitted that, well, of course, he realized that the proof was not correct. So he spent six months to try to cover the gap, which is very difficult, in fact, after seven years. And with the help of Richard Taylor, and, well, this is very touching in a direct interview, it is very touching to see him that he described the precise moment where he realized that I can do it. Just one morning, I was in my office, because he put himself a deadline after, say, one year I have to announce that I failed. But in one, in a few hours, he covered the gap and said, OK, I went out from my room, my office, I closed the door just to be sure that nobody enters. And I went around, I came back, checked everything, it's fine. And it's the most exciting moment in my scientific life and so on and so forth. But in the meanwhile, he became more than 40, and so he couldn't be awarded by the Fields Medal, yes. However, this is just the story. Let me tell you just to complete the stuff. If you start from a polynomial of degree 3, and you consider this new polynomial, this is a polynomial equation. y squared is equal to px. p is a polynomial of degree 3. And, well, the degree 3 means that it can be split into three linear factors with different roots. So you have, on the right-hand side, three roots. Then you obtain a non-singular plane curve, and its genus is 1. Whatever genus means here, well, we know genus 1 is a description for, is another way to say elliptic curves. And so it can be written in general like x minus alpha, x minus beta is x minus gamma, but you can actually say that after we are ranging alpha can be 0, beta can be 1. And the only parameter left is the last one. So with this simple tool, you obtain any complex torus, if you want. As it is described by the zero set of this polynomial equation, or it is an elliptic curve of genus 1, or it is one-dimensional complex parabolic Riemann surface. That's it. Thank you for your attention.