 So, I begin by thanking the organisers for inviting me and for organising this colloc. You realise I have also to apologise that I have a title in French, but I will be speaking English. I'm sorry about that. I also have to apologise for the technology. It appears that my presentation did not wish to come to France. So I brought it on this machine which is not exactly compatible with the machine. Here you will see immediately why we have this delay and also in per grossier, no? So, the story begins with Gauss, who will do two things. He will unite the theory of this differential equation, the hyper geometric equation, with the power series you see below, which is the hyper geometric series, and he will show that the series is one of the solutions of the equation. To do this, he plays games. He changes the function in this fashion. He increases or decreases one of the parameters by one. In this way, he gets several different functions. Any three are related essentially by the hyper geometric equation, but what he published is only the study of the series and the functions it represents. Unpublished and in the same sequence of numbered paragraphs is the study of the differential equation itself. Here he finds an independent solution from the one before. He has a basis of solutions. He transforms the variable in certain ways, five of the elements of a particular group of transformations. He experiments with changing the solution and the equation we're looking for to simplify things. And he finds that by formal manipulation, you reach an hyper geometric equation which is quite certainly false. This intrigues him. And he explains how this could be. There is a clash between the solution function and the series that represents it in a certain domain. The many-valued function is defined everywhere except at zero, one and infinity. The infinite series only on a certain circular domain is variable as complex. And he explains that this impossible equation from before is akin to this mistake. You have many valued functions and you would not infer from the behavior of the arc sign function that 30 degrees was equal to 150 degrees. So he is studying analytic continuation of complex valued functions of a complex variable defined on a certain domain. And he compares this with the representation you get by means of power series. The analytical continuation of a complex function outside of its domain of convergence will lead us to a regouse and us to monodromy. That's why the story starts really here. And it does not start with Kuma. Kuma's study which later mathematicians use is almost entirely about a real variable, not a complex variable. And monodromy is a complex variable phenomenon in this period. So I'm sorry again, Mr. Riemann has had rather more good dinners than he should. We move immediately to Riemann and his dissertation of this year 1851 in which he defines, it's still our definition, what it is for a function to be a holomorphic function. It's to be complex differentiable. He explains that these functions are conformal except where their derivative vanishes and it has a branch point. These ideas he would have taken from Gauss. Gauss had made a study of conformal mappings in 1822 in which these ideas, without being so tightly connected to complex function theory, are there. And Riemann of course adds that the real and imaginary parts of a complex function are separately harmonic functions. But he makes a great innovation. It's an astonishing innovation for its time and we take it completely for granted now. That the domain of the complex function should be really something you get with only this property that at each point you can move in two directions and you can then have the Augumi-Koshi-Riemann equations. Such a domain need not be part of the plane. It certainly need not be all of the plane. It need not be topologically equivalent to the plane. This is a complete new departure for function theory and he gives in 1851 these examples, the third one is not a planar domain. It's a domain here with arms that come out and connect in this fashion. So you're looking at something which is not even a planar domain and a few years later he turns to the hypergeometric equation and he remarks that the unpublished study of this series is in Gauss's nachlas and amongst other people who worked on this problem since Gauss there is of course Kummer and he actually has a very big general setting for this topic which is linear differential equations with algebraic coefficients but I shan't take us in that direction at all. It's because Griemann has read the Gauss nachlas that he discusses functions which he called P functions and the notation P, the letter P is taken from Gauss. So this is how he defines them. He says they have three branch points. There's a linear relation between the branches. Any three branches satisfy a linear relation with constant coefficients and the function can be written in terms of what you see locally in the fashion indicated here. So they look very like the solutions of the hypergeometric equation. That's really where he's coming from. He's formalized the, if you like, geometric behavior of the solution of the hypergeometric equation and he's starting from this more abstract point of view. He makes certain preliminary assumptions to avoid troubling cases. The exponent differences are integers and their sum is one. The exponents are coming from the branch points. The exponents are what will give you the particular solutions of this equation. The exponent differences will control the quotient of the solutions. So we'll need them later on. His P functions can be continued analytically in a loop around a branch point and he explains that in this way you get a matrix of constants which are determined by taking a particular branch of the P function around, in this case, he's going to say the branch point at z equals a. You have another one for b and another one for c. If you think of those three points on the sphere, going around two of them is like going around the other one but in the opposite direction. So if you go around or three of them in this order, you could essentially perform the identity transformation. So the matrices, the monodromy matrices that he has satisfy this identity. And he claims that the coefficients of these matrices completely determine the geometrical behavior of the P function. So before we proceed a little digression on the words, the word monodromy is due to Kushi in these years 1851-52 when he's engaged in a race to write up his ideas of complex function theory for the audience they never got 20 or 30 years before. And this is what he means by monodromy because it continues it's single valued as a function. Hermit 1851 introduced the idea of using matrices to describe how algebraic functions are branched. Historians hate first the first occurrence of, we had a very nice joke from Colin earlier on. I don't suppose anyone else wants them to be a last use of the idea of monodromy. But it may be that Riemann is the first to consider products of monodromy matrices. The term is used extensively by Camille Jordan in the Trité that Caroline Erhard mentioned earlier on. And the term became popular because it was used by Jordan and by Felix Klein. So this is where the term monodromy or monodromy comes into mathematics. A little hint of what Riemann was to do with this, we have the branch behavior described at each of the singular points little a, little b, little c by a matrix. So he now wants to take his basis of solutions around one of these branch points and carry it around another one of these branch points and see what he gets. And so he starts writing down what you get, you get more matrices obviously. And he satisfies himself that the coefficients of the hypergeometric equation in this case determine all of the entries in all of the matrices. And so the equation, the differential equation determines precisely the monodromy of the solutions. And then he reverses the argument in unpublished material and shows that the equation determines the matrices completely, not just the ratio of the coefficients. And then if you allow yourself certain simplifications that allow you to make a path to a problem in which some of the exponents are zero, two of the exponents are zero. The exponent differences remain the same as they were before the transformation you've made. But the exponents themselves become zero on two occasions. Then actually the monodromy determines the hypergeometric equation completely. There's only one equation with that monodromy. It will not be true if you have more singular points. So this is quite a profound study of branching behavior, monodromy and differential equations. And now we pass to, I suppose, the local hero. And a very familiar story I should say, of course, as I should have done before, that everything I'm saying is, I'm sure, very familiar to many, or perhaps most of you. So Henri Poincheret, I'm hoping you see four pictures of a torus, at least the one on the left and the one on the right with a homology basis drawn in. And probably at the age of 10 you were told that the pictures below are also pictures of toruses and mathematicians glue the edges together and all of this. And I'm hoping, first of all, it appears. Yes, here we are. I'm not at this stage wanting you to see this picture of a torus. This is also a picture of a torus. As you know, we're going to explain why in a minute, but I don't want you to see that one. I want you to see the other pictures, first of all. Back to Riemann for a moment. Riemann had discussed branched covers of the sphere and how branched coverings of the sphere are related to algebraic curves. So that's the story that mathematicians then tried to understand after Riemann's work how you could have an algebraic curve, that's a polynomial in Z and W, and you could realize it as a branch covering of the Z-plane or the Riemann sphere. And you could then cut it up in the fashion we cut up the torus before and obtain some kind of understanding of the cut-up surface, which would be a four-piece side of polygon if the genus of your curve was P. These ideas are coming in after Riemann. They're coming in perhaps more in the German world, thanks perhaps to Klebsch, than they were immediately in France. And they are, of course, rather difficult to understand because the happy little picture we have of the torus doesn't really show you how it is mapped down as a branch covering of a sphere. So there are all kinds of conceptual or technical difficulties in the way of understanding all of this. But that's the background or part of the background of this story. Another part of the background is, of course, the Franco-Prussian War. So Franco-Prussian War, I need hardly say, was a catastrophe for France. And it provoked an enormous debate about the decline of French science, what might be done to revive French science, and how in particular do we French catch up with the Germans? They seem to have understood and grasped something that the French had understood at the time of Napoleon, but it seemed to have lost control of. And so there are many prizes offered by the French in various subjects, which have as a subtext catch up with some German scholar and explain it to us so that we can keep up with the Germans. And this is one in particular. The judges, Amit is the important one here, Bertrand, Bonnet, Puiser, Bouquet, are involved in this. The German they want to catch up with in this case is Lazarus Fuchs, who's a friend of Amit and has been doing important work in the Viestrassian spirit on differential equations for some 10 or 15 years by this time. And the subtext here is perhaps you might like to join in on Fuchs's theory of differential equations. And that's what Poincaré proceeds to do. It's one of the three things he starts doing in this period. There's number theory, there's real differential equations and flows on surfaces, and there's the material we're going to have here. So he's become a lecturer at the University of Caen. This has rescued him, or at least it's his choice, I suppose, from being a mining inspector, which was on the agenda at the time. He makes one contribution to this prize competition and then withdraws it. He makes another one in 1880, and this one is eventually published only posthumously in these places after Mathematica, after the First World War, and the first volume of the Irv, which is the second one published. What he does is he goes back to the hypergeometric equation as an example of Fuchs's work, and he considers not the two independent solutions, which you might say are formed basis, he doesn't pick a basis for solutions and study the basis, so much as the quotient of a basis. Under allelitic continuation, this quotient will reproduce in the fashion I've shown here, okay? Each member of the basis comes back as a sum of the basis vectors, the quotient comes back looking like this, and you can stop two thirds of the way down the slide with a many-valued function, fine, that's not a problem, or you might think it was more helpful to study the inverse function, the set-theoretic inverse function, which is, in some sense, periodic, it's automorphic, we would prefer to say today. And Poincare thought that it was sensible to track this problem geometrically, that the, if you start with a hypergeometric equation and you suppose that the equations has real coefficients, then what you are really looking at is the image of the upper half plane, which you must think of as a triangle. It has a vertex at zero, a vertex at one, and a vertex at infinity, it's a rather strange looking triangle. The angles of the image are determined by the exponent differences, okay? So the image will look like a triangle, the image of the lower half plane will look like a triangle, the image of the whole plane minus three singular points will look like a quadrilateral, and now as you do your analytic continuation the quadrilateral has picked up and moved around and joined on to what you've already had until perhaps it overlaps itself. So Poincare, but not Fuchs, spent a lot of time thinking well what about the shape of this net that we're building up as we put the quadrilaterals down next to each other, what am I going to get? In this particular case you will see that the angles are given by pi over two, pi over three, pi over six which adds up to pi. We have a triangle whose angle sum makes it a Euclidean looking triangle and Poincare investigated what happens and he found that he would be able to obtain a parallelogram which was mapped to itself, it's made up of several copies of the upper and lower half plane put together and in this way because it was a parallelogram that was moving around you were seeing really an elliptic function for the inverse of the quotient of a basis of solutions and so he drew it. Now this is a prize competition, I have to like Poincare for a number of reasons, this is a prize competition. So he is writing on individual pieces of paper for his work and he comes to make a drawing of this and he gets it wrong. So he hands it in, he doesn't scrap that piece of paper, throw it away and do a neat drawing, he just sends it in. So I thought I would exert myself and show you what he couldn't do, it's the only thing I can do that Poincare can't, I can draw you the net of the triangles forming quadrilaterals, I can put them together and make a hexagon, I can add the extra ones and make a parallelogram. This is what Poincare failed to draw. So in this case you have angles of pi by two, pi by three, pi by six, you can see the angles, they fit together in this fashion and eventually you will wind up somehow or other with a periodic, doubly periodic function and elliptic function. Then he took other angles, again we have here one over n, so the angles are going to be pi over n, this is to ensure that they fit together nicely at the vertices, you should find four fitting together, five or four, eight fitting together if the angles are pi by four, six if the angles are pi by three, twelve if the angles are pi by six, he finds, and this is one of the drawings from the published version, and now he finds that the triangles do not go very far, they are all trapped inside a certain disc and all the edges of the triangles he's drawing are perpendicular to this, the boundary of this disc. So everything is happening inside the unit circle if you like and everything angle, every edge he draws is an arc of a circle perpendicular to this boundary or perhaps a diameter of the circle, so this is what he discovers, and I now have some pages of quotations in this very famous essay he writes much, much later, 1907, 1908 when he talks to the Société de Psychologie in Paris, and he explains how he came to these discoveries, but if you don't want to read all of the French, and you surely don't want to hear my French accent, then look to see right at the bottom where we have a mention of Fuchsian Functions and the Hypergeometric series, so this is what he has been working on, he's been blocked for some time, he has a cup of coffee contrary to his usual habit, and ideas surge up in his mind at the cost of a night of no sleep, which generally speaking for pancreas he considered too high a price to pay, he was not willing to crank out more mathematics by giving himself coffee and bad night's sleep, he was very worried about his sleep, but it's productive on this case, and he comes to this conclusion that he can construct these particular things, but he's actually blocked at going beyond triangles, so differential equations with three singular points he can do, four or more singular points he can't, and of course he feels acutely that this is a problem, and also he jumped ahead, he also at this stage wants to explain something else, he has been looking at triangles which are images of the upper and lower half plane, he's been doing analytic continuation, at some stage he also thinks of these transformations in a different way, this is the famous story of him boarding a bus on a trip, it's a mining engineering expedition, and he's talking to somebody and as he leaves the bus, he realizes that the transformations of his triangles have been transformations from non-nuclidean geometry, well he's in conversation with somebody, he can't check this now, he waits till he goes home at the hotel in the evening, and he checks it in the hotel in the evening, we should all have such bus trips it seems to me, what has happened here is that the version of non-nuclidean geometry that was known in Paris at the time and regarded as the best mathematical account is due to Beltrami, and in Beltrami's account, you draw inside the unit disk, geodesics with respect to the non-nuclidean metric, as straight lines. Now here, Panqueray has been straightening out his triangles to try and understand the analytic continuation of them, so in his head as he boards the bus, he sees a picture of a disk with triangles drawn with straight lines in, and he goes, oh a disk, straight lines that's Beltrami's non-nuclidean geometry. And then he goes, but when I began this story, my triangles had particular angles and my map was conformal. So can I get that model of non-nuclidean geometry and as I'm sure you all know, you can get that model, you can convert the straight line picture of Beltrami to the picture we had before, where geodesics are arcs of circles perpendicular to the boundary. Or they are perhaps diameters of the circle. So this is what he sorts out his satisfaction. And now he goes back to the prize competition and he says, okay let's take a general case, but still only triangular and see what we get. And he pushes this for two more supplements, one right at the end of the period. These were lost in, I have to say, the Académie des Sciences for quite some considerable time and were eventually discovered and published by Scott Walter and myself a few years ago. The Suplement a marvelous addition to the original essay, he still didn't win the prize. But what he has in the Suplement is a way of thinking about how these quadrilaterals are fitting together. It's not analytic continuation now. It's congruence. Non-nuclidean isometric equivalence. So the quadrilaterals or whatever they might be are now congruent copies. So he has a geometrical way of thinking about the moving around and he can begin to study what happens as you rotate about this. So rotate about that. He calls it crab-wise. I sort of imagine it creeping across the disc in this kind of fashion. And he says, well this is geometry and what is after all geometry? The blue here is his underlining and the red is just for us to follow our story. To study geometry is to study a group, he says. In this case it is the group of non-nuclidean geometry, the geometry, the pseudo-geometry the Loboshevsky and that's all we're doing. And he has this view I should say throughout his life that to study geometry is to study the action of a group. And it gives us a convenient language, he says for explaining what we're doing. Now I can catch up with the story I started before. He still can only do three singular points and there are differential equations with four or five and any number of singular points. He's upset, he's stuck, he goes for a walk and he realizes that what he's been studying when he was studying number theory, a la amit is just the kind of transformations he needs. This is the if you want to connect it with our PIVA story, this is the geometry model you get as the disk as one half of the hyperboloid of two sheets you can think of the hyperboloid of two sheets here's one half, here's the other. Each of these carries non-nuclidean geometry in a natural way but the group now is SO21 I would say. So he spots this and this enables him to study differential equations with any number of singular points and to go beyond the hypergeometric equation in this fashion. So another enlarged picture I'm afraid this is Felix Klein, this is the point at which Felix Klein finds out what Poincare has been doing and he's very angry. Felix Klein is very angry because Poincare has called these new functions and these new transformations fuxin. After all he learned about it from reading the work of fux, he invented it I should say after reading the work of fux and he's called them fuxin in public now so he can't withdraw and Klein thinks this is quite shocking. He thinks it's too much credit to someone who doesn't deserve it and dan dan and backwards and forwards, backwards and forwards letter after letter complaining about this and this is Poincare's reply that really okay I would have if I'd known of Schwartz's work I would tell you, well that was in a minute I'd have called them Schwartz's but I didn't and I can't now and Klein won't stop. So this is Schwartz, this is the century of the beard I have to tell you this. And this is his famous discovery of the tessellation of the disc you see here lots of triangles which to Poincare are non-nuclidean but not to Schwartz and Mittag Leffler who has been promoting Poincare's career for a couple of years by now tells him that Schwartz is really angry, fux is delighted everything is fuxin that's good Schwartz of course is really really angry absolutely almost suffocating with anger he says and Poincare writes back to say well he had the chance and he didn't take it I'm sorry to give this in English I didn't have time to find back in my French sources but Schwartz is really angry with himself for having had an important result in his hands and not profiting from it and I can do nothing about that that's perfectly true the difference between the Poincare view and the Klein view is summed up by this picture this is where they start from of course they catch up with each other the Poincare view on the left is that you have the disc you have some group action you have some quotient space and you get in this case a double torus obtained from the octagon you can see in the middle Klein is coming from a Romanian point of view where what you start with is a branch covering of the Riemann sphere and you can immediately see that the torus has a covering and Poincare is not very interested in the fact that the torus is a branch covering of the Riemann sphere and as a starting point Poincare's starting point is a better starting point but perhaps I should go back the uniformization theorem which they are working towards now is the claim that you can put any Riemann surface in place of the double torus except the sphere two or three punctures any Riemann surface of genus one or more can be covered somehow by the disc the genus one, torus one, is covered by the plane so they want to put any Riemann surface in there and lift it up somehow that's going to be the claim it's first made by Poincare in 1881 that somehow you can express any algebraic curve apart from the elliptic ones in terms of fuxian functions and perhaps the simplest thing to say is that this is what we've all done numerous times we've done anything with the circle you replace, you parameterize the circle by these nice rational functions the claim is you can parameterize a more complicated curve by some fuxian functions in the same spirit the next year Klein has a go pardon me and Klein has one advantage over Poincare at this point which is that he can essentially enumerate the possible algebraic curves of a given genus he can say that there is a certain number of moduli which count the different complex structures on these algebraic curves he gets it wrong first time round so this is his second time at doing the enumeration and the enumeration goes like this you can roughly count you want to have a polygon that is moved around ok we're going to glue the sides together so it's a 4p side of polygon but there can only be 2p different lengths but there's an awful lot of angles and then once you've counted all those parameters he didn't really care where in the disk that polygon was so there's a three-parameter group moving it around so that's where you subtract three complex parameters so you wind up with this number it's a rough and ready parameter count but the point is it says that the number of polygons in the disk is of 4p sides is roughly speaking 3p-3 it's a complex parameter count and that's the number of moduli your algebraic curve of genus p has a complex structure the moduli space of those complex structures has the same dimension 3p-3 it's too good to be true it really must be the case that each polygon somehow gives you one complex structure and all the different polygons you can have of the right number of sides modulo of certain equivalents are going to give you all the different complex structures this is the uniformization theorem it's just as many fuchsian groups as there are complex structures and this is the theorem they want to prove Klein has a sleepless night after the asthma won't let him rest he's awake in the middle of the morning and guess what? he realizes there's a way of proving this theorem and he states it and he just decides he's had it with the holiday he's going to go back home to Düsseldorf write this up and tell people about it and that's what he says he does he writes to Pankareg he writes to Schwartz, he writes to Hurwitz they tell him about the uniformization theorem and he says this is the one that everybody I sort of know this I should have known all along this picture it's a little hard to see in this reproduction which comes from one of his students but it's the one I think we all go back to when we want to tell this bit of the story this has a 14 sided figure in the middle actually it gives rise to a Riemann surface of genus 3 the other pictures explain how the vertices fit together and I need this picture I should have got the story from just this picture that's what he's saying to himself okay it's usual when I give a talk or maybe when other people give a talk there is somebody in the audience who knows more about the subject this is perhaps an occasion where I am at number 17 to 1 but the book I refer to here has 17 authors perhaps some of you are in the room thank you it's a lovely book and it spares me having to say anything at all about the uniformization theorem and it's complicated history which lasts until 1907 before we get the real proofs but it's around from 1883 as an option Poincare has a go at proving it not just for algebraic curves now but for multi-valued functions of any kind and he sends it to Mitag Leffler who edits to Mathematica for publication and this is what Mitag Leffler says isn't that terrific in analysis there is no theorem which in its striking simplicity surpasses this but let me tell you what you have to do you have a multi-valued function so you must find the domain of the functions that are going to be used as the uniformizing parameters and then after you study the map from the this new domain which you've constructed back down to the curve or multi-valued function or Riemann surface that you're talking about and what Poincare thought you might be able to do is to identify is the fundamental part of this domain which is then moved around on block and fills the whole domain so think of the octagon in that picture before it gets mapped around or the 14 gone in the picture we had later it gets mapped around on block in each case it fills a disk the disk is the domain of the uniformizing parameter so you have to construct this domain and then you have to get a map and Poincare has the idea that the Riemann surface has lots of curves on it and I can get from this point to that point in lots of different non-homotopic ways and perhaps I just look at the different paths from A to B and the different paths will somehow tell me what the whole domain is so I've tried to draw a picture here oh good it's just about visible on the screen if you start from A and go to any of the points B you're winding up in different points in the plane corresponding to different curves on the torus the short curves are drawn on the left the complicated curves drawn on the right so this is Poincare's hope that going from A to B in different ways will somehow be equivalent to a picture of the plane in this case for the torus or the disk in more elaborate cases so he tried to run this process in reverse and obtain having looked at that picture and pictures like it to obtain the domain of the uniformizing parameter in this fashion I won't delay you on this particular account you have a starting point you look at the different paths if the paths aren't homotopic you say that the two paths are defining different points in the domain you're trying to create it's a terrible paper I've spent a lot of time this year reading Poincare and it's a clear candidate for the worst paper he wrote there's no mention of homotopy for example whole blocks of ideas are simply missing it was very charitable of Mitterg Leffler to include it it becomes mentioned at least by Hilbert and the Hilbert problems in 1900 because there really are problems with whether this process has to find the domain that you want even Hilbert's being charitable I don't think actually the original paper of Poincare makes sense about you supplying a lot of ideas such as homotopy it's eventually done by Poincare and by Kerber successfully in 1907 and as I say there's this lovely multi-authored book to which you may refer so I think I'm now moving towards my conclusions because I felt rather flattered to be invited to give such a paper as this but it wasn't clear to me exactly how it fitted into the subject of this splendid colloquial to like your talk yesterday thank you and the passing mentions of the same topics that I've been talking about this morning and I felt that's fine I'm here to sort of just give the background to some talks which are going to be taking place later on I'm very pleased to do so so this is how we get to Galois from this there's no mention of Galois I should say by name in these papers I've been discussing there are two ways you can look at the picture there I've joined point A in the bottom left hand corner to copies of it in the same place I believe in the parallelograms but in different parallelograms you may either think of this as a bunch of paths from the point A in the bottom left hand corner to any of the others or you may think of it as an instruction to move the entire plane so that the point A goes to the point A so each homotopy class paths is really the same as some action of some group element right so the fundamental group I'm going to call the fundamental group is acting in this case on that plane and you would have a similar picture in the disk model but much harder to draw so from this point of view what you have for the disk or for the plane seen as a universal covering space is that the fundamental group is acting on the universal covering space which is the disk or the plane and the quotient space is the Riemann surface so instead of just seeing those paths you can think of it as a group action that's moving things along depends on what block you're moving around if you have larger figures to move around you have smaller groups to move around and a group moving a polygon around is a normal subgroup of a group moving a sub polygon around you've got more freedom of movement in a smaller thing about in each case you get a Riemann surface if you have two different groups in this picture is an attempt to draw a square you can draw it into four little squares if you think you can move from A to A you're thinking of the little square as the fundamental domain and you get a torus if you're thinking that you can only move the big square around then you're going to get a different torus and you're going to have a map from one to the other because if you're moving a little square the lines from A to A will correspond to group elements if you're moving the big square the lines from little A to little A to group elements so putting this together in the language we heard in fact this morning if you have a non-constant analytical map from a Riemann surface X to another Y then what you have is a finite branch covering of Y by X so here's X and it maps down onto Y now every Riemann surface has a field of meromorphic functions on it and the pullback map gives you a map from the meromorphic functions on Y to the meromorphic functions on X and as I've said before if you've got these two polygons these two Riemann surfaces obtain from two polygons one sitting nicely inside the other so the big polygon is made up of lots of copies of the small polygon those Riemann surfaces with that map from X to Y give you a situation where you have a map of the groups corresponding map in the other direction of the quotient spaces as it turns out that forms my conclusion actually to the talk thank you very much Thank you very much for this Can you ask me some questions in French if you can speak a little bit slowly The first mathematician who understood the Galois theory of extortion and the fact that it was fast I don't know, I wanted to wait for you in France I'm not sure maybe it's modern mathematicians The oldest mention I know of the correspondence between the Galois group is in a book by Hermann Weid from 1925 which was published because it was published after his death and there is a first chapter on the topology of the Riemann surfaces that you can show in the book and at the end he says the Galois group and he makes the link between the MotoPied group and the Galois group but I don't know what happened if it was known before But what I cannot remember now is if in the first edition of Weill's book he makes this reference to Galois It's not the book Okay So we have a lower band We need to go back up a little I'm sure Thank you I was wondering about the relation between various parts of Poincaré's work this one the work on the creation of topology because one common point would be Betty since Poincaré took the Betty numbers we saw how much Betty worked on Galois in his use and now we see both situations, both parts of Poincaré's work is very related and united by Galois theory but probably Poincaré did not No, he did not Two things to say When he does his work on three manifolds so the Poincaré conjecture in the end you have two handle bodies two solid double tori which you glue together in an interesting way these are presented by regarding them what he works with is Fuxian polygons so he thinks of the surface of these handle bodies as cut up into a Fuxian polygon so he always comes back to this language to see how some solid here and some solid here are getting identified together to make the Poincaré space Poincaré has a very curious attitude to algebra so he very much likes number theory but he thinks number theory is very difficult because you do not have continuity so you're driven to use analogies all the time and he gives a paper which I must publish because most of it is not published it was all published then conveniently the interesting bit has forgotten La Venière des Mathématiques the International Congress in 1908 which is a reply to Hilbert and if you look in Sciences Métode the mathematics is gone, it's lost but it's there in the original and here he does mention algebra, group theory things like that but his group theory is either infinite groups, Lie or finite groups Gaoua and it's almost the only time I'm prepared to say the only time but I must be wrong where he mentions the name Gaoua so it's not something that he seems to have had in mind as an animating analogy when he discusses something at any stage and then many, many other analogies we could pursue but finite group theory does not seem to have been a subject he particularly wanted to involve himself with but there's another curious thing he doesn't very often quite an individual group in this Fuchsian work, he said there are Fuchsian groups with different polygons and there's lots of them but it's Klein who for instance studies that particular one in some detail and it has interesting things to say about the corresponding quotient group of order 168 so Van Grey sort of floats over the surface of these things he seems to have worked at that level and not to want to go and give particular examples of the general story Jerry and Craig we can thank you once again