 I feel like an actor. So it's a pleasure to meet you. It's a pleasure to present the first last Ignatian seminar of this year. So our speaker is the founder in the mic section. He's Alberto Velcozzi from Mexico. And he will talk about the Poncaredis and non-Ecrian genitalia. Well, thank you very much. I feel very honored to be back to this talk on basic notions. I thank the mathematics department section. So I will speak about basic notions. And of course, one of the most basic notions starts with Euclid, 300 years before Christ, with Euclid and Geometry. The founder of Euclid and Geometry. And in fact, the notion of basic notions is in fact born with the Greeks, the notion of postulates, axioms, the Earth, et cetera. So I think it's very appropriate to choose the right topic. So Euclid created a series of postulates, axioms, with which you create Euclid and Geometry. And in fact, you don't need to know what Euclid and Geometry is. You can play with it. So let me give you a metaphor. The metaphor is the game of chess. You can play chess without having a precise definition of chess. Because no matter what definition you give me of chess, I can give you another way. Say, well, it's a square 8 by 8 with black and white. I say, no, no, they are red and green. So it doesn't matter the things you can play. And that's the way it goes with Euclidean Geometry. But in fact, there's a model. It's a universe in which you can play the game. That universe is the Euclidean plane, R2, provided with a usual coordinates, x, y, provided with a standard distance. And this beautiful object has the property that is homogeneous with respect to a group. It's the group of Euclidean motions that present the orientation, which are of the form of rotation. Let me write it also as, can be written as column vectors. I like to write them as that. So it goes to sine theta, sine theta. So this transformation, as you know, is a rotation of theta. And also you have translations. And so you have the group of composition of translations and rotation. And that's called the Euclidean motions, the rigid motions of the plane. And it's SO2. Well, it's written in this fancy way, so the direct product of the orthogonal group R2. And according to a famous German mathematician, Felix Klein, he gives a birth to a child in the hospital. He goes to FATO. Nento. Nento. Nento. Nento. Nento. Ah. OK, so Felix Klein, in the elderly program, she described geometry as the story of the set of invariants of a group in space. That is, for instance, this group acts on the plane. And it acts in a beautiful way. You can go from any point to any point by a rigid motion. And you can go from any direction to any of the other directions. There's a rigid motion that takes this vector to now. So you translate, and then rotate, and it acts like that. So without further delay, let me explain what the title of my talk, which is the Poincaré disc. Well, the first place, the Poincaré disc was not created by Poincaré. It was actually created by an Italian, Eugenio Ventrami. And what is important is that every Poincaré is one of my heroes. He was born in Nancy, and died in Paris. The use he made of the Poincaré disc is what makes fantastic. Because as you will see, it's a combination of complex geometry, hyperbolic geometry, number theory, group theory. The whole world enters into one object. And it's an amazing object. So without further delay, then tell you what the Poincaré is. So in the first place, we can always identify the plane, Euclidean plane, with complex numbers by the usual rule. So we can think respectfully, either the plane or the plane provided with a complex structure, namely C. And in this C, you know the things that you have taken in your school, complex conjugation, the absolute value of a complex vector, how to multiply is a field, how to add, et cetera. And you take the unit disc, consisting of all, called a delta, consisting of all complex numbers. Such that it's known, it says that one. So it's an open disc. So what's a great invention is because this disc is not only a disc, it's provided with a beautiful metric, which I will describe. It's described by the following. The S, a nice plane, all the notation. Can you write a little bit bigger for us? Excuse me? Can you write a little bit bigger for us? Yeah, OK, bigger? Right, bigger? A little bit, yes. OK. Thank you. Well, this in complex notation is simply written as Z square. And what this means is a line element. And one is a university, you don't understand exactly this notation. And this notation is actually due to another great mathematician. Can you see? 1, 7, 7, 7, OK. And when we see this notation, this notation is a particular case of a general notation invented by Gauss, or described by Gauss. Here's the following. He describes S2. So you take a region on the plane. So I take an open, let's say open connected set of the plane, understand the connected, take an open set of the plane, and consider the following expressions. Of course, I'm going to define a demand and metric of the plane, but let me explain a little. So what is that? I say, what's that? There are many young people that, of course, most people know what it is, but sometimes it is supposed to be an element. So this is simply for each point x, y on the region. You take a set of four vectors anchor at the point that's called the tangent space at the point x. And you decrease an inner product. An inner product that will depend on x, y. So you take two vectors, v1 and v2. And the multiplication of v1 by v2 is given by this expression. So I think for matrix, I convert this expression in a matrix. And I simply wear the coordinates of v1, i1, a2, and the coordinates of v2 or v1, v2. Of course, to be an inner product, this has to be possibly definite. It has to be symmetric. It's symmetric automatically. And positive definite by Sylvester's rule that we all know in the first year of high school, in Russia, not in Mexico, maybe four years. Then in this case, well, to be positive definite, it implies that you need e-positive. All the minus has to be positive. e-positive and the determinant positive. So e-g minus f squared has to be positive for every x, y. So this is a Riemannian metric. It's the data of an inner product point by point. And what's without saying that I require that all parameters, everything, varies smoothly or even reanalytic, not to have it. That could even continue. And that's what is called a Riemannian metric. It's a beautiful invention by Carl Friedrich Gauss. That's very useful because it allows us and one across the boundary that I don't need that is. And that's extremely useful because in that way, one can actually construct the length of curves. So you have a smooth curve or even piecewise smooth gamma from an internal AB to omega. Omega is this kind of vision. Then you'll define the length of gamma as usual as the integration of the speed. So the length of gamma will be equal to the integral from A to B from the norm. And I explain what it is, all the derivatives. But this is the norm with respect to the inner product. So this is the norm. So what you see, Euclidean, might be different with this. So therefore, this is a special case of that in which G is 0. The middle coefficient, S is 0, sorry. Only you have dx, dy. You have a mixed term. And furthermore, the two E and G are equal. It's a special case. So these are very special cases of metrics for all the center of the plane, which are of the type, rho of z, a positive function, times metrics of this type. In other words, are a multiple, positive multiple, of the standard Euclidean metric. So this is the standard Euclidean metric with the inner product, the standard inner product. And this square is called a conformal metric. Of course, z is the explosive y. So let me write like that. And let me write it as a square. Give the size that is positive. Well, this is called a conformal metric. And this metric is very beautiful because the angles of this metric, or the omega, are exactly the angles of the Euclidean metric. So you see two unit vectors. And I put expressions like that. So the metric then, you measure the Euclidean angle and coincide with the Euclidean angle. So this metric decrees the same angle. But of course, it decrees different lengths. That's what is called conformal. And therefore, this is the date. Therefore, the disk with the Poincareto-Washevsky metric is the object of my talk. It's the disk provided with this metric. Now, there's a beautiful theorem. You see, it takes a long time to explain what curvature is. But if the metric is given in a beautiful way, then there's a formula for the curvature. The curvature of z at a point on the plane, at a point in omega, is given by the beautiful formula, rho to the matrix 2, with a fashion of the lower end of rho. With a fashion is a standard fashion. The operator, you need to cooperate with that one. With a fashion, you know. And so it's a beautiful formula to compute. It's called the conformal curvature. And now there's a beautiful theorem that actually is also due to Gauss, says that if you change the coordinates, you can always choose an appropriate chase coordinates. So as a metric involving E, F and G becomes conformal. These are the so famous isothermal coordinates. And the existing was proven by Gauss himself in the reanalytic case. And then in the Hilder case, by Colin Liegekestein. And it goes all the way to the beautiful theorems. Therefore, there's no loss of generality if I give this definition of conformal curvature. It's like that. And in particular, for this case of choice, except there's a four, so this is four times the real metric, weighted with this function. For this metric, the curvature of the disk at every point is constant equal to minus 1. So this metric and Poincaré has a beautiful way of describing this. It is beautiful books. You know, like Science and Hypothesis. He imagines the disk with a distribution of temperature that decreases as you go to infinity, as you arrive to the boundary. The circle is the boundary. But in our case, we think of it intuitively like your ruler becomes smaller and smaller as you tend to the boundary. So you keep walking, and you don't cross the boundary and fall into the precipice. But you keep, your steps are more smaller, smaller, smaller, and never reach the boundary. In other words, as I always say, the metric induced by that is complete. Every Cauchy sequence converges to a point there. So now, this introduction, I described for the famous Poincaré disk, and you will see that it's incredibly rich in structure and incredibly beautiful. This is one of my favorite objects in all mathematics. Because now, consider the following. So it's the disk, the complex plane with that metric. I don't write it three times. Now, consider the following expression. Delta z1 c1 c2 equal to z1 minus z2 1 minus z1 c2 bar. Now, consider the following transformation from the disk to the disk. t of z equal to t to the i theta c minus a 1 minus a bar c to the second percent of the square t. That's what I needed by it. OK. You see that expression? It's a fractional linear transformation. It's the quotient of two linear functions. And here, I required theta as a real number, mod 2 pi, a real number. The only thing I required is that h is a third one. So this is a holomorphic function in the disk. Of course, it has poles outside the disk. It's a holomorphic function on the disk. It's an easy exercise that it preserves the disk and it's a projection. And it seems this is also this type. So this is exactly the set of all holomorphic maps from the disk to itself, completely characterized, depending on three real parameters, a point of this, two dimension, and a real number. So this group, very easy to see, that it acts transitively on the disk table. You can go from many points to any other point by a transformation like this, an easy exercise. But also it acts transitively on directions because you see, if you are in the order, this contains a particular case, rotations around the origin. Transformation of these types, you see a rotation around the origin. And therefore, since you can take any point to any other, you take a point, rotate whenever you want, and translate it. So this group acts transitively, it's called, on unit vectors. You can go from any vector, unit vector, to any other vector. So it's what is called a homogenous manifold. And this one is called two transitive, two transitive, namely vectors go to vectors, as transitivity of vectors, you can take it. And therefore, this goes, fits very well with the philosophy of Fennish Klein, the Erlangen program. This will be the universe of hyperbolic geometry. We will, in a moment, describe geodesics and describe geometry, which will be beautiful because it's exactly a model of non-Euclidean geometry. And non-Euclidean geometry has a beautiful history. It's essentially always Gauss. Gauss discovered it and never published it. At that time, it was very secretive, mathematics. In fact, you know, if you, I recommend you, go in a little outside of the road, recommend you highly, a book by Fennish Klein, Developing on Mathematics of the 19th Century. And then you can see the characters of the, which, of course, for him, the 19th century, starts in Germany and with Gauss. And, of course, you keep your secrets. And we go to Jacobi, the existence of the ICTP. Of course, that's a job, but what I mean by that is Jacobi was an open personality. He started, liked to talk with everybody, organized conferences, and the notion of that, actually, we own it to Jacobi. And it's a little bit of a joke, saying that we own it to Jacobi, but there's a little bit of truth. The idea of sharing knowledge was not always universal. And diversity was not always universal. And so this way, so we have, okay, we have this group. And this group has a beautiful following property. If you apply this operation to that expression, this is preserved by the transformation. So these two variable functions is invariant on the transformation of that type. And it's almost like a distance, but it's not quite a distance. Can you see, can you see? Okay, so now we reconfiscise, and we let Z1 tend to Z2 because I'm infinitesimal. And Z1 tends to Z2 means this becomes DZ squared, and this becomes one minus, no, this becomes DZ, and this becomes that. And of course, all that, all that you can make rigorous by taking curves. And that implies this preformer exercise, take curves and make that 10 to zero. And this implies that this theme is exactly the group of orientation-preserving isometries of the disk with the Poincare metric, Poincare-Lowacevsky metric. So now we have a space with a metric. I have to find the metric. So the metric, I define the length of curves, and to define the metric, you do the usual way, you take two points, take the set of all curves, piecewise, differential curves that join the two points, and take the infimum of the length. In this case, it won't be necessary because we calculate this moment precisely the distance between the two points. Okay, so, okay, so now we take the disk and remember, geodesics are differential curves that minimize locally distances. Okay, so I take the disk and I take, I want to compute a geodesic starting from zero and going along the real axis. So I parameterize it in the obvious way. Half of t equal t, t belongs to zero, open one. So I take a curve like that. Now, it's clear by the nature of that, that this curve, that if I go like that, this curve would be half length lower than this one because this function is radically symmetric. It's invariant on the rotations. And so if I take any other path, I arrive from here to here, we say from zero to r, r is less than one. So I take a, if I go any other way, I'm bad. So I go like that. So this will be geodesic and I only have to be, to compute the length, I know my formula is the parameter is t, varies from zero to r. And my formula says this, is this formula. 2d t, one minus t square. And we know how to integrate that. That's the logarithm of one plus r. Use partial fractions, you try this, like one minus t, one minus t over one minus r. So this is explicit formula from the origin to a point in the real axis and a euclidean distance r. The euclidean distances r, because that's, but the hyperbolic distance is that formula. And now this group properly adds a transitive on vectors. And now I take any point, take a unit vector and take the unique transformation of the type t that takes that to that and I computed explicitly all the geodesics with any given initial conditions of unit vector at any given point. Just using group theory in the spirit of felskline. And now something beautiful because this graph, if you see this, it's actually a web use transformation. The general web use transformation is of the type. Also called homography in French, homography in Italian. And these are transformations of a Riemann sphere of this type. This is a, so this is a special case of that, you know. And there's a beautiful property of transformations of this type in the Riemann sphere. So I add a point at infinity. It has the beautiful property of preserving cross ratio. And the cross ratio of four points in the Riemann sphere is defined the following way. So we have C1, C2, C3, and C4. Four points in the extended plane. I do define the cross ratio of these four points with some exceptions, we see one different group. You know, it's defined for almost every quadruple. And this is by definition C1 minus C3, C1 minus C4, product C2 minus C4, C2 minus C3. It's the cross ratio of four points. And web use transformations is the first thing of the loop by alpha, the first beautiful thing. Web use transformations preserve cross ratio. And also the loop by alphas, I recommend the complex value that you will take. Four points are in a circle, if and only if the cross ratio is real, beautiful fact. And web use transformations preserve cross ratios. And therefore, same points with real cross ratio to positive cross ratio. And therefore, circles go to circles. If web use transformations are homomorphic, therefore preserve angles, and send circles to circles, are co-circular. And therefore, because this is a circle, I mean it's a line, but it's a circle of range infinity. So the inverse, and preserve angles. So the image of that geodesic, this is a complete geodesic, will be a circle. And because this angle is straight line, the image of that, image of that complete geodesic, negative or positive, is our circles. Because this is a circle of radius infinity, this image is a circle, and the angle is 90 degrees. And therefore, all it's possible image are circles which beat orthogonally the boundary. Okay, so now I raise the blogger, and look at this beautiful model. As you wanna give one test, I plan to give it a little faster. So therefore, I have now it's points, and it's geodesics, complete geodesic lines. Geodesic line, non-parameterized geodesics. And the geodesic lines are diameters, or circles, actually are two circles, which are orthogonal to the boundary. And now, I decreed that point is a point in the disk, a line is one of these geodesics. And with this, all the axioms of Euclidean geometry are valid, except for one, the axiom radius. Cause given a line, remember that's mine, a line and a point, et cetera, to that line, I can consider infinitely many lines. If I go through the point, I do not touch so it's unlike Euclidean geometry. And that is what makes the difference. And therefore, this is a beautiful property of that. And therefore, you can do geometry, you can do everything, you can actually construct Euclidean polygons, you can do triangles, you can compute many things. And actually, there are other models of hyperbolic geometry. And the great British mathematician, Kayleigh, you have this mathematician that it would be impossible to read their papers, because it's like Poincaré, you go to the library of Poincaré. If you understand a little bit, that would be good. So Kayleigh, consider the following transformations. C of z from the extremity sphere to the riemann sphere. C minus i, and this has a beautiful property of sending the upper half plane, the point of the type, positive imaginary part to the circle, to the disc. So the upper half plane, both the disc. It's really easy to see because, you see, if it points in the real line, this is the conjugate of that, because z is real. And we know by our high school complex variables that a complex number divided by its conjugate is a unit, has unit one, has number one, and then your i goes to zero, is injective, so that the upper half plane to that. And therefore, you say, well, why don't you use that by injection, and actually it's a homomorphic map, to transport whatever I know from the disc to the upper half plane. Okay, if you do that, you can also pull back the metric. And the metric becomes the Lovachevsky metric, which is the formula. It's again conformal, right? It's a, the factor is one over y. And that's the Lovachevsky state for the upper half plane. Now, it's provided with a metric of constant negative coefficient minus one. And now, because this is wave-use transformation, circles go to circles, and therefore, these things here become circles, half circles that are orthogonal to the boundary. Okay, now this is not, I'm glad I have a little spot because otherwise I will ride with my circle. So, therefore, you see that the geodesics here, vertical rays, or circles, half circles, are orthogonal to the boundary. And that's beautiful, you can do geometry like that. And each model, I will show you three models, has its own virtues. This is fantastic to compute areas. Because you see, you see the area form for the Riemannian metric of Gauss. Remember E, F, and G, the area form is this form. This is the determinant of the matrix I wrote, remember the matrix E, F, square root, E, X, D, Y. So, this is the area form of this Gauss metric, this Riemannian metric. And therefore, for this particular case, the area form is the X, D, Y, and the Y square. You compute areas in the upper half plane with this beautiful curve, with this beautiful formula, using your calculus, you know, courses. And then, it's beautiful, because I'm gonna show you the following beautiful fact. Now you can speak of a regular polygon. A polygon is a convex sense. You make sense to speak of convexity. A regular polygon means, or a polygon, not necessarily convex, is a sense, which is a general curve, which is composed by a final number of jodesic arcs. That's a point. And here, of course, jodesics are the arcs. So, if you prolong this, this is supposed to be autogonal. So, here you have a hyperbolic quadrilateral. And very easy computation, because this, you can parametrize in terms of cosines and sines. You know how to parametrize circles. You can easily compute the area. And the area of a polygon, they say of a convex polygon, p, will be equal to n minus two, beautiful formula. n is the number of sides times pi minus the sum from i equals to one to n. So, we have n internal angles. So, you have one phi one, phi two, phi three, phi four. So, the four minus two times the formula. In particular, for a triangle, n is equal to three, three sides, three angles, and the formula is exactly the area of a triangle. Well, this is a bad choice. Let's call it out. It's equal to pi minus alpha minus theta minus gamma. Where alpha, beta, and gamma are the internal angles. Beautiful, because the opposite of Euclidean geometry. The sum of the angles is never pi. And in fact, it becomes pi only if the triangle is ideal. So, it doesn't happen in Euclidean geometry. You can send the vertices to infinity. And the area of that will be pi. But as soon as you have vertices inside, the area is strictly less than pi. And you have that, and it's a beautiful construction. And now, hyperbolic geometry has some other virtues. Like for instance, it has, let me use a key, height, I say I have personal notation. Hyperbolic equals complex, because this will join two subjects of mathematics, hyperbolic geometry and complex. But I want to not forget. So, has right angle, hexagons. Something that doesn't exist in Euclidean geometry. You have a complex polygon with six sides, and the angles are right angles. Beautiful, right? Because I might say, there are many formulas of hyperbolic geometry. They love science, they love psychoscience. They probably won't have two sides. Many formulas, which they're worth learning. And these formulas imply the existence of an hexagon. In fact, many hexagons, one, two, three, four, five, six. You see, these are geodesics, don't suppose to be circles, arcs of circles from talking to the boundary. And these, all of these are right angles. Now, by the formula there, the area of that is pi. And now, I become, with this, comes the idea and become a hyperbolic tailor. I learned the word Schneider recently. So, I didn't know Schneider, it's a hyperbolic tailor. So, you couldn't piece, now all the material will be hyperbolic. So you take a hexagon. By the way, hyperbolic geometry tells you the following beautiful things. Opposite sides have the same length. So, the set of hexagons are in one to one correspondence with triplets of positive real numbers. So, that's one. So, some of the, when you go to the couturier, you know, you say, these are my, well, you have to do exercise first. These are my, this is my size, so, ABC. And then you take your hyperbolic, so you coat your material with sort of a circular, see, so, and coat one piece like a hexagon. Now, take two hexagons of the same sizes and really correspond to C with C, A with A, and what you get is beautiful. It's a pair of pants. The boundary is geodesics, because you see, if you move that with that, where are the hexagons? Well, one side, second side, one, two, three. So, you see, it's the front part of the pants, though, and the other one, like that. So, this part, of course, they have to be on the same side, otherwise you're gonna look like, you know. Okay, same A, A equals A, so you glue two, boom. And of course, you glue that with that isometric, you don't wrinkle them. So, you take two hexagons, you glue them with an isometric of the size, you pair with pairs, and you get pair of pants. Beautiful, though. And with the property that these are orthogonal. Once I have a pair of pants, and I have as many as the positive ordnance, A, B, C. Now, give it, now, take two pair of pants of the same sizes, the cuffs, the same cuffs. I don't remember this, since you already, it makes sense, but it's not universal. So, you take two copies of that, use the hyperbolic tailor, this side is equal to that side, the legs of that curve is equal to the legs of that, legs are not equal to the legs of that, that equal to that, and you glue them, boom, boom, with isometries. And now you have a pretzel, I saw it in the series two, but with a beautiful hyperbolic geometry. And you have as many, depends on three parameters, but actually depends on more parameters because you can glue that with that and rotate, right? So, you have three parameters, A, B, C, and three new parameters, theta one, theta two, theta three is the rotation you glue. And this is essentially a serial number. If you glue, you twist a thousand times, it won't be the same thing as you twist 2,000 times. And so, this way, you obtain that the surface of the genus to become the pretzel has a metric, because everything glued together, with constant negative curvature minus one. And not only that, you have a whole family of them. And in fact, if you have the surface of the genus G with G-hold, you can show that the space of hyperbolic structure depends on six U minus six parameters. And we are here, landing into beautiful subjects of mathematics, these are theories, modern spaces, ribbon surface, and Tajmyr's theories. Beautiful. It's one of my, I recommend, highly recommend all of you to read books by Thorston. And I can give you, if you stay, I can give you lots of material. I have thousands of folders that I, some of which I never read, as a opinion of what I do read. And therefore, fantastic. Every ribbon surface of genus greater or equal than 2, compact, orientable, and meets a complex structure. Sorry, hyperbolic structure. And now, the group of isometries of the disk, and what are the group of isometries of the upper half plate, and whether transformations also deform g of z equals a z plus b over z c plus b. Where a, you have the matrix a, b, c, b, a, b, c, d have to be real, and the determinant one, this group is called PSL2R. So these transformations present the upper half plate under the condition that a, b, c, d be real, and the determinant be different from 0. It can assume that is 1. That means that you are providing a ribbon surface not only with a hyperbolic structure, but actually with a complex structure. Because the changes of coordinates will be isometries, hyperbolic isometries, which are simultaneously homomorphic maps. And therefore, we show that every surface, compact surface of the gene of g, we have shown, I mean, I will need two books, but with my hands I have indicated that it's a beautiful way to prove that any ribbon surfaces, orientable smooth ribbon surface of gene of g, greater than 2, has a complex structure. The case of the torus or the sphere is particular. The case of the lithic curves. So this way you have that. So simply with this formula, you get that. And everybody here has, I think most of us, have seen the beautiful pictures by Escher everywhere, no? It's one of our, for mathematicians, it's one of our favorite painters. And now, you take the model of the Parcaret disc. And you notice that when z is 0, this is essentially Euclidean metric. And 0 looks like Euclidean metric. So now we take a regular octagon here. You don't see it, but so the angles here are very, very, very large. So I move the size of the octagon. It's like a stop sign to 3. Here's a regular octagon, no? It's supposed to be a regular, OK? And there's one moment when all the eight vertices go to the boundary. So you have all the angles become 0. 1, 2, 3. It's regular. And therefore I go from something large to 0, the angles. So at one point by the intermediate value theorem, I got that the angles are pi over 4. So I take that particular regular octagon. So it's an octagon, like the Asher figures, you see? It's hyperbolic octagon. With the property that the interior angles are pi over 4. And now I take that, and I tessellate the plane. Like a mosaic like this. How do I do it? I reflect because you can do geometry. You can reflect figures. I reflect the octagon by the other side. And there's some beautiful theorem of Poincare. So if you do that, you will cover the Poincare disk. And you'll be careful to paint the black, white, black, white, black, white. And this way you get the hyperbolic plane covered by isometric regular octagons. And there's a group that acts that moves the same way you remember the torus. You have to take the squares, the tessellation, and translate them. It's the same thing here. You have a group that acts, and the quotient gives you a surface of genus 2. Now you take a 4G cone, regular 4G cone, with a property interior angle. You can do the same thing. So these are beautiful Riemann surfaces with lots of symmetries. And it's part of the work of one of my other heroes, Hurwitz. You know, the client is one, Hurwitz, Riemann. All his name, no? Jacobi, I'd say Jacobi. Fantastic guy who got an idea of, see, Weierstrass. One of my heroes. Weierstrass was my hero because he started his famous theory of elliptic function when he was 40. And when you are that age, you start looking at the dictionary, the Wikipedia, to see who is mathematician. Still the mathematician at the 40. You're too young. So you still have 40 years to go. But when I was 40, let's see who, if I'm Weierstrass, fortunately, he was a very good life there for me. What do I do? He discovered the P5 function. Don't take that as an example, please, no? So well, in this case, you get beautiful destinations of the plane. And I recommend a book by Vilken Magnus, are the selections of the hyperbolic plane. And now, what about, well, let's see, I still have how much I have, 10 minutes? See, 10 minutes. OK, because I just started a little late. OK, so I gave two models. I give two more models. So this thing with your link between complex variables are arithmetic because I said the mages could have arithmetic entries. And now, there's another model. So what you take is the projected model. And in that case, you take the following equation. So this equation R3 is an equation of a cone, x, yz. The Z4 triples satisfy this. This is one of the napkins, one of the branches of a hyperboloid. This equation of a hyperboloid looks exactly like so. This is a cone, and this is a hyperboloid. This is a brush of a hyperboloid in R3. And you see, this is actually homeomorphic to the disk. Here, you take the disk at height z equals to 1, and you project rapidly from the origin. You have a one-to-one correspondence between the hyperboloid and points of the disk. So this is a disk embedded in R3 as a branch of a hyperboloid of two branches. And now, you introduce a Riemannian metric. So you want to decreed the length of vectors, the inner product of two vectors. Well, you simply do it at a Likovsky. You take the product of two vectors in R3 by this formula. So this was a one that would be a standard Euclidean product. But here, I put a name. So this is called the, since we are mathematicians, it's the Likovsky metric. Your thesis is called the Lorentz metric, but it has the same thing. And you take this inner product in R3. And unfortunately, the inner product of two vectors could be 0. Exactly on this comb, everything goes back to the Likovsky. However, if you take a point here, I tell you the tangent plane, and you get vectors tangent to that plane, this branch of the hyperboloid, this inner product, because positive definite. And therefore, it induced a metric in the hyperboloid. And guess what? This metric is the hyperbolic metric. So it's another, and this has another beautiful property. Because the group of isometry is the group of Lorentz matrices. Namely, the matrices that preserve the inner product. So it's the center for matrix A, so it's that Aj. A transpose equals J. And the term A equals to 1. So take all matrices that have, this is J, that have this property. These are linear maps of R3 that preserve this. And you can see that, in fact, this group has transitively, and so this is the new geometry. I'll explain this another model. And there is a, but actually, if you are a little bit more clever, instead of looking at R3, you take the space of all lines in R3. And you compactify R3 by adding the plane, the real projected plane of infinity. So it's the space of all lines. And this is the project of Rp2. Real projected plane is the center of all lines. And now you project this figure. So there, the cone becomes a circle. The hyperboloid becomes a disk. You see, it's parametrized by points of this disk. This hyperboloid becomes a circle. This becomes a disk. What's the component? The Moebius band exercise. And therefore, we have shown, and pass on, that the projected plane is obtained from a disk by gluing a Moebius band. Anyhow, these transformations, you can projectivize these linear transformations, non-singular, and therefore, sends one parameter, one sends lines to lines. And therefore, it acts on the space of infinity. And therefore, very splendid cover that the group of projected transformations that preserves a non-singular conic is exactly hyperbolic geometry. And therefore, again, with the Erlangian program, saying geometry is nothing else than the study of invariant properties of the group action. And therefore, the Pochelet disk has many interpretations, can take you from all the way from hyperbolic geometry to number theory. I mean, Mother Earth Forge, that we have here, Professor Saghi is a case, you can create Mother Earth Forge. It's a fantastic subject, which I invite you to. It's a very, very basic notion that goes all the way to the most modern mathematics, like the proof of Hermit's theorem depends highly on the Pochelet disk. And last but not least, quite about three dimensions. So in three dimensions, you take the disk, which is in R3, with the metric, with this I finish, metric dx squared, the same one, that one more variable, minus x squared plus y squared plus x squared squared. And this is the model of hyperbolic. So if you take all, and then remember the trick of using a regular octagon and inflated it onto the angles, now you do the follow. You take us to the cathedrals with 12 pentagonal faces here. So I'm gonna, you don't believe me, but has 12 pentagonal hyperbolic faces, and then you start inflating it. But you have a regular to the cathedrals space, so that the hydral angles, the angle between two faces becomes 90 degrees. And then you play the Escher Poincaré game, you reflect, reflect, reflect, and you have an Escher in dimension three, and you can do that, and therefore that way is the way that Seifer and Weber, Seifer, Swiss, Weber, I think, I think it's called Swiss, discovered a beautiful manifold dimension three. But the game doesn't stop here. The game is that the arrival of another genius of my generation, 90 younger than me in terms of age, but William P. Thorstone, together with Poincaré, and William P. Thorstone shows that almost every dimensional manifold, in the sense of almost to be clarified, is a hyperbolic manifold, or at least is a piece of a hyperbolic manifold as was shown by Perlman, and it takes you to this Poincaré piece that started like a model of non-Euclidean geometry, essentially by Baudier, Lovachevsky, and Gauss, and Bertrami, never forget, G. Bertrami, becomes a fundamental object in all of mathematics, and I think the moral of the story is, the study, you go to every seminar, even if it's not your subject, learn everything and intermingle every subject with every subject, because you will gain with that. Thank you. Questions? Everybody's welcome to ask questions. I was sure I can answer them, but I tried to. It was so clear that there won't be any questions. So in these three models that you present here, the piece, the Poincaré disc, the projective, the projective, the Poincaré disc, the opera played, the opera played, the big... What was the first one? The what? The first one, in the history. Ah, the first one was the disc, I think. I think the disc. I think it was the plane, because it was a little isomorphic, no? It was the plane. Essentially, Bertrami gave the formula. For the disc. Like that one. Yes. And the rest were essentially pictures. So as such, as a complete Riemannian matrix, I think Bertrami. And includes all the mess, all of them. And, well, actually, Hilbert's, I recommend that Hilbert's do the same following. The metric, actually, can be given in terms of the logarithm of cross relation. Very simple formula. And Hilbert abstracted that. The beautiful theory, which is actually current research, isn't it? Because you take any complex set of the plane. Not necessarily, let me write it. Not polygon, but take any complex set of the plane. Bounded complex set of the plane. And take any, I will give you a distance. You have to show that it's a distance. That's what Hilbert shows. You take two points, x, y, a, b. Throw the line. You have x by complexity. This line intersects in two points of boundary. And then you take the logarithm of the distance of x minus a, x minus b, y minus b, y minus b, y minus b. Exercise, by Hilbert's, that this is a distance. It's a complete distance on any complex set. So any complex set has a natural, any complex set of the plane has a natural metric, which is called the Hilbert metric. It's a very rich gem. In fact, Garrett Birkhoff, the son of David Birkhoff, proves that Perot Frobenius's theory we're using this metric. There are lots of useful things. In fact, with this hyperbolic geometry by Gromov, et cetera. And this metric also is hyperbolic. Okay. This metric also is hyperbolic. No, it's a beautiful question. Question, when is it for the disk? It's hyperbolic. What about ellipse? Also for ellipse. That's because projective geometry can take it from a conic to another. But if I give you a horrible complex set, no. And it's actually an open question to characterize the Hilbert metric that correspond to hyperbolic metrics. More questions? See, yeah? So you said that I take the pants and I twist one of the... I cannot hear well. Can you speak louder? Sure. So when you draw the example of the pants. I see. The third pants? That one? Yes, so you glued two of them. Why? Excuse me? You glued two of them. Yeah, I glued two X's. The front part and the lower part and the back part. Yes. So you said that if you twist with 1000, it's different from twisting 2000s. Okay, but it can only be in. I cannot hear very well. I apologize. Come here, baby. No, it's fine. No. If you twist... I come here. No, I can't. Sorry, for just one second. Go on. So you said that if you twist 1000, it's different from twisting 2000. Yeah, yeah. So I just asked you... Why? How do you know you twist? Because I thought, I mean, if you do a choice and... Actually, if you do one turn and two turns, it would be the same thing. Yeah. A very good question. This is the first thing I say. Why 2000? Because it's a very good question. Why is it of the same thing if I turn one and then change the number of times? It's a very good question. I tell you, it's slightly nontrivial. You see, because... Okay, one invariant of this hyperbolic surface is the length of the geodesics. It's very important. Now, if I go turning one, suppose I have a curve that means transversely here. If I turn many times, the length changes. You change the length. But I have a friend, Dennis Oliver, tells him, imagine you are a plumber. Okay, you have these tubes for water. Okay, and you glue it to go down for you half something minimum. Here, it looks like you're the same thing, no? You turn. But here, if you take this curve and turn many times, you change the length. Think, you can do the same thing with a choice. Take a cylinder and do it with a piece of paper. And if you go two pi plus epsilon is not the same. If you go four times, it's not the same thing as you go two times. Because the length of the geodesics transversely changes. In fact, what's that the bundle? Como? It's like I have a, it's like I have, I have a 10. I mean, okay, so it's not different, right? But it's like I attach a bundle and then I rotate the connection I have. And then that's why they run. That's easier. You have an essential piece here. My way of explaining is the simplest way. If you do that, you change the length of geodesics. It's an operation that changes. This is called the length of the spectrum. It's an invariant of any pre-manial manifest. But this is a good exercise to see why. If you do it with a desolation, you'll see immediately why not. Okay, that's all I have to say.