Opening lecture / Away from set-theoretic geometry - schemes vs spaces

0:00 Thank you very much. We will start with the executive order of 10 o'clock. I would like to thank the various institutions of the University of Florence for their support in the project of this meeting. And to the department of mathematics and the department of philosophy. This is Professor Vincenzo Antoniou, who is the director of Professor Florence, and maybe you can speak to one of them. We are very impressed by the high level of the teacher. My personal experience goes back to the work of Alexandre Dominique in algebraic geometry, derived perfectly by William Stanley. And much every time I run into Helleberg clinicians, I wonder how is it possible

2:30 And in the end I came to the conclusion that the so strong achievements of the theory are the starting point. So what I think for a mathematician is a really great opportunity to listen to the talks of the workshop and take notes of more than a million notes of candidate theory in many different domains. So I wish success to the conference. Today we have been addressed by the director of the Department of Philosophy, Andrea Cantini, who Dear invited speakers, dear participants, let me first apologize for not being able to attend the opening ceremony around the recent Catholic Reveal, contrary to the announcement of the official program. The reason is that this full term of lecturing from Lawrence today is a week of my enthusiasm. Please now stand and let me first welcome all of you and wish you a pleasant stay in Lawrence. In the name of the Department of Philosophy of the University of Florence, I would like to express special thanks to all the speakers for accepting and contributing to this workshop. Due to the outstanding level of the scientific program, I am sure that the atmosphere will be highly sedated and that there will be a chance for a fruitful interaction between mathematics and philosophy. So, very nice to receive this important thing that he is not present for this seminar. Then there is another message from the General Department of Professional Sciences, Professor Favorelli. First of all, I wish to express my warm welcome to all the participants in this workshop at Symposium on the Mathematical-Catholic Theory. I am glad that this event is hosted by the University of Florence and I hope that you will have a good graphic waiting for us. Since its birth in the 40s, Catholic-catholic theories are now recognized as the source of the new and fundamental way of human education.

5:00 The effects of this on education are of primary importance for us, especially in times such as this in which the role of computer science is perhaps superfluous, but also a risk to increasing the dog with less and less aspects of mathematical education. As far as I can see, the use of a category of theoretical notions and methods affects the understanding of basic linear aspects of space, quantum quality, and ultimately the grounds for logical reasoning. This approach has not only a general philosophical relevance, but also specifically a pedagogical one. For example, its contribution could more deeply imply a picture of different domains of mathematical activity, and favor the growth of completely direct linking between spatial and logical fields in high school and university students, and even enhance a different methodological approach in the teaching of mathematics at a more elementary level. And hopefully the organization of high school and university people will have a benefit from the use of the category theory and the adoption of a kind of seminar in Walsall College meant by young people in the area. Due to the previous attending offices, I can't be present in the opening ceremony of this workshop and this symposium. I apologize for my absence. So, that's all. Thank you for your attention. Thank you for your attention.

7:30 You can see the Y-coordinator hiding in there, because you have these E's of A's of A's. Well, yeah, that's the kind of diagonalization here. But in your case, all it gives you is that sigma has the fixed-point property. Why is that? That's the beginning of that. That's good, because then you know that sigma has the fixed-point property. So one of the things you want to show very early on is that all maps from sigma to sigma You can't invert the top and bottom, and that's exactly because sigma has fixed point properties, so if you could invert the top and bottom you would get the map without the fixed point, so you know you can't do that. So that's other proofs of, that I'm aware of, I see. But the kind of diagonalization that one does on natural numbers, or on personal functions, on the numbers where you number the functions. Diagonalize. Usually those are thought to be the fixed-point property, but that would be the fixed-point property at n. So the axiom is that there is an ultimate that gets... Exactly. So, well, but... Done again. Now sigma to the n makes 1. Oh, there you have it. Nice. It's just amazing. It's too easy, this is the first recursion theorem. Every map on sigma2p has a fixed point. Well, the first recursion theorem is a little more specific, it has the least fixed point. But, you get a fixed point when you just do a map. Well, then you can get the least fixed point as well by doing this sort of thing in an order Thank you for your attention. This is a peanut roll that drew my attention today.

10:00 Well, you should talk about this on Friday. It would be the right place for it. Galileo would appreciate it. That is Galileo. Isn't it? I suppose so. Sure, it's gotta be. Then I should take a picture of Galileo. That's what I already said, that was my joke. Oh, you put him in your... I didn't realize. You put Galileo into your catalog. Yeah, of course I will. Really. It must be Galileo, isn't it? Put him into your gallery of... How much time have we got? Your gallery of mathematicians. You have 10, 15 minutes. Okay, I can do this the right way. Without the flash, it will look great. Thank you. Thank you for your attention. Thank you. Thank you very much for your attention.

12:30 Subtitles by the Amara.org community Thank you for your attention. Thank you for your attention. There may be a category of equations, or a common one, there may be a category of equations, or a common one, there may be a category of equations, or a common one, there may be a category of equations, or a common one, there may be a category of equations, or a common one, there may be a category of equations, or a common one, there may be a category of equations, or a common one, there may be a category of equations, or a common one, there may be a category of equations, or a common one, there may be a category of equations, or a common one, there may be a category of equations, or a common one, there may be a category of equations, or a common one, there may be a category of equations, or a common one, there may be a category of equations, or a common one, there may be a category of equations, or a common one, For the definition, we can say topos, but let's talk about another category. What is the object of classification? The object of classification... It's a general classification, right? No, you have an exact category with added sums and with double types. Ah, so this is the category. The definition is as follows. Here's another opportunity to get Bill with the statue, you know. Do you have Bill already for your gallery?

15:00 There are many classifiers, let's say, but the activity of predicativity is born when you crush the roots of the classifier in the same level, and you make a hierarchy, basically. From what I have seen, you said that you discussed the small sense, and then you said that the classifier is not of the small sense. No, it is small. It is small. The definition of a tiktok teacher is not a tiktok teacher, it is a model of a tiktok teacher, if used in a different category, it is equal to the type, in this case it is what I call a model of a tiktok teacher, but a tiktok teacher is not a tiktok teacher, it is a model of a tiktok teacher, but a tiktok teacher is not a tiktok teacher, it is a model of a tiktok teacher. It is difficult to say that Topos has all three parts, while quantum mathematics does not have them. So, yes, Topos certainly has them, but it is not like in the theory of the modification of types, that is, there is a geometry that you construct from the bottom. This is what we have seen with Palgram in the other lecture. And then, in particular, There is also the object of the W-type, which is a particular type, but it is predicated, there is nothing predicated, and it is the objects, the trees, the trees, and there is nothing to do with the topos in itself, there is nothing to do with the topos, no, no, in fact we do not work with topos, we work with less. It's remarkable, yeah. It's a strange axiom. Thank you for your attention. It's Cantor and Dedekind and Ironberg and MacLean.

17:30 I took a picture of Galileo for my website. Did you recognize the other ones? You got all the other ones? Yeah. He can't be more than 25. There's a little bit of a... What? I don't understand that. You know how he works? Yeah.

1:30:00 General properties that he studied in a move back and forth from the restricted form. Yeah, yeah, that's certainly good. Oh, if you see, oh, it should, it should, with one hand, as fully as possible, the general...

1:32:30 It's pretty sketchy. How's this? Oh, no, well, I just wanted to check, actually. Andre, what do you notice? I captured the idea. Yeah, that seems to be fine. Which is not maybe as well understood as causal chapters, which is the influence of the theory of algebraic groups on the development of quantum mechanical work, more than algebraic geometry, I mean, schemes. And the other way, by making some research, I discovered that schemes are not invented by more than one group. Skills is dutiful than three other occasions and would discover, let's say, distributions of generalized functions.

1:35:00 It seems that in the three cases, there is one person who installs the notion in the most general and practical way. It's certainly Einstein and Minkowski who form a special relativity. But the low brochures, then in the Stalin way, and again those give a number of people country collections in the 50s, I mean the mathematics in Paris was centered on the character and the seminar. They called normal, later on it developed and now there are 70 Paris universities plus another higher education institution, but at the time there was one university and the colonel was part of this university. So, it's clear that the idea of skills came to many people. First of all, certainly, Serres complicated to do that, Schrodinger complicated to do that, and so other people as well. So, the time was right, and many people tried to fend the necessity to define such a concept. Not only to think about the new motion, but to have a working group of researchers to work on it for 15 years with a large group of researchers and to develop it in a more systematic way. Maybe a little too long to expect, but never too late. And so, I want to take this to the beginning and also I want to... There is some misunderstanding about the definition, the distinction between schemes and spaces. Schemes and spaces are not exactly the same. And if we think about the meaning of the word, the deep meaning of the word, you can see a scheme as a skeleton, a schematic.

1:37:30 So the notion of a scheme is a skeleton. That's my thesis. Manifold, as we know, is difficult to ascertain for one person, but it's certainly a creation of the 1930s, and certainly Hermann Weyl has a quite clear definition in his mind of what a manifold is. He doesn't spell it all the way explicitly, but he has it. In his mind, Édith Cator also, who has a few competitors in differential geometry in the 1930s, claimed in some cases the notion of the manifold is too difficult to define, but nevertheless I will use it. So, certainly Édith Cator was a more pragmatic and intuitive mathematician than Hermann Weyl, and no reservation used in the notion of the manifold. So, we can say that at the end of the 1930s, the idea of the manifold was quite clear in the mind of many mathematicians, and as usual in differential geometry, you have to credit Ehlers-Mann for his definitive definitions.

1:40:00 He developed a category that was a very unpleasant thing. I mean, we had two competing schools in France in the 50s about categories. Erosman and Mouhamad. And they never spoke to each other. Never, never connected. They didn't know completely. Erosman formulated the notion as well as a number of erosion-based notions in differential geometry. So, he put a point in a mind quote, in a theoretical mind quote. Well, this idea comes from certainly Gauss and from the geography, the needs of geography. We can't go as far as a number of charts, of maths and charts, and we have to ascertain how to relate different charts and maths in overlapping heat. For instance, in the 18th century, a great concern was mapping the various countries, and France was mapped by the French Academy of Sciences, and Germany was also more or less mapped in the first guidance of Gauss from this part of Germany, and there was a practical problem to relate the maps obtained from different countries, obtained by different teams in different matters. So, we have charts and maps with different regions and we are overlapping. This is the basic notion in Manifold Theory and the idea is that you need coordinate systems, but that one coordinate system cannot cover the whole of your galaxy. Even on elementary needs, you have a sphere, you see dimensions of space, there is no coordinate system which is valid all over the sphere at the same time. At the least you have to know one point to see if you have the coordinates in which to step or not to step. So this wasn't a good idea. But implicitly in that you have the notion of topology because the domain of the chart should be used as an open set.

1:42:30 And so implicitly you have the notion of topology, open sets. Amstorff was the one to formulate this. But in Hilbert, the idea of a human surface is already at least in the two-dimensional case. I mean, this is a prologue, because the space is well understood. A space has points. Whether the space is a collection of points or not, it might be the only one. Certainly, the collection of points is not all of the space, but at least it is. You necessarily have a collection of points of base in your space. That's the way we look at geometry in space. After a contour, but it's not certainly... It's not made of... We have corners, we have circles, and when two lines intersect in a point, and the point is something which has no extension, no dimension, but it's not certainly considered as the film store of the space. It's just a rather tensionary... Let's say, in a more global way, they do the foundation of projective geometry, and there are various axiomatics of the projective space, and the best is using lattices.

1:45:00 So, the projective space is not just a collection of, it's a collection of and the notion of . Let's say, a projective plane is a lattice, I think there are points, there are lines, and there are two extremes. You have an empty set of counts as nothing which is a certain flat of dimension zero and you have the whole space of dimension two and it's stratified as dimension zero, dimension one, dimension two. So, but if you want to have a very symmetrical description of the project, you take all this collection of objects, not only the point, but the line and the plane. You can say that, and there is a notion that the ball lies on the line. Now, out of the axiom of the geometry, you can ascertain that if two lines are exactly the same incident point, then they are the same line. But it should not be taken as a definition. It's just the goal of the definition. And you can imagine situations where this is not true. So, here in the plural of differential geometrical equations, In the literature of geometry, we assume, let's say, after Cantor and after the birth of Mars, we implicitly do not know what is at all. Okay, but this is not enough. Amos Mann himself was very much concerned about what is a feeling, what breathes. The idea of touch, in the way it is explained in Bohm's book, in the last... In the first chapter of the Settler of Obagi, they claim that they have a field of sculpture which is, let's say, categories for the poor man, and they did great effort not to mention explicitly the word category, but of course, when you realize that at the time it was written and published in Cuba.

1:47:30 There was a very active member of Bobecki at the time, and McLean was not only a member of Bobecki, but he was closely associated as a friend of Bobecki or a brother of Bobecki. In order, what Bruegel had in mind is the notion of isomorphism. Why isomorphism? Because it's just the image from the Klein program, the Erlangen program of Klein, that the geometry is characterized by the group of automorphism and isomorphism and justice writing. And at the time, a great emphasis was put on the so-called transporter structure. Which, in modern terms, means that in order to define the structure, you need a fine tool which goes from the set to the set, but not with a full math, I mean the set with bijection, invertible math, into the same category. So, in order to define the structure, you need an end of math tool of the category of sets with bijection. You may choose of that definition in combinatorics, specializing to mathematics. But what we have in the Obakir definition of the fraction is a rather absurd class. It's interesting that when Obakir published his definition, he was very absurd about it.

1:50:00 First of all, there were people who were not interested in foundations. It's just general nonsense. On the other hand, there were people who said, well, but of course we know that the truth is with categories, and I will not have to tell you the inner battle of Borbach-Yaroff, whether it should be good category or not. This story has been told in the House of Committee formed by Armand Borrell and so on. But now, people who believe in geometry went one way or the other. André Ré mentioned that one of the purposes of Baubach was to be strong, and Baubach was certainly one of them. Erlenbein was not so well understood, but he was very much influenced by the German mathematics. But Erlenbein was very little sort of light and not dogmatic enough, certainly not a dogmatist, and Baubach was rather dogmatic. And finally, the obituary of Hermann Weylitten by Chauvelet and André Lecq in Enseignement Mathématique.

1:52:30 And you will see it. So, but for us, what is important is that Chauvelet is a symptom. First of all, long concern about founding. Second, he was, well, he was a very romantic mind, which means in the positive sense. Which means that if one had a certain idea, he would go straight to Google and download it. It's interesting that a curator is wise, and it's interesting to look at his notebooks, so he could write something for 100 pages. And then, all of a sudden, he stops. He crosses the last page really carefully, line by line, line by line, and then he starts again from the very beginning. So, Chevalier has always been very much concerned about what is an invariant definition of a knowledge. Now, before the development of 20-year-many-lives, I really just sketched some of them. Certainly not all of them. The German algebra. And ending in Chevalier.

1:55:00 Chevalier was a good actor. He got his education. He was always a very faithful follower. For instance, he published a book about the algebraic function of one variable, which is an expression of the algebraic function of the positive and the positive. And there was a review of this book by his supercontra, a very common reading, but a review by his friend, where he says that it's completely wrong. So straight, I mean, once he had defined his line, he followed it without any compromise. So, this is the general algebra, and it was neatly expressed by the paper of Dedek and Fabel, where they define an algebraic curve or a linear surface. How do they define it? They start with a field of rational algebraic functions. So, in modern terms, a finite algebraic extension of the field of rational functions. Out of that, you went to geometry. You went to geometry and the point, the non-singular model, complete non-singular model, the number of times it vanishes or even more. So instead of defining the point, you take the collection of functions as they enter and you define the point as a process which assigns to any function the order to which the function vanishes or the order of the point. So, a collection of such orderings or, if you want, the question in law-breaking space was defined as a collection of laws.

1:57:30 Now, of course, what we have, which is so special, essentially, by the proof, is that when we give the field of algebraic functions, which is any function, there is exactly one geometric function. About the same time, in parallel, there was an Italian faculty chair. So here you can see dimension one, algebraic theory. Now we have a parallel development, which is a pure algebraic surface. It's now developed by Henriques, Rémi, as they work on the Italian side, and then Van der Waalen and the German side, to state many things. But then we have dimension two. And then, of course, people try to imitate, that means, while they went through competing developments, a purely geometric world, you have poles, that means you take a projective space with three dimensions, and they're constantly into complex numbers, coordinates are complex numbers, and you have an algebraic surface, and an algebraic surface is a collection of poles that is like a certain equation, a familiar equation. So, an algebraic surface is a collection of forms embedded in movement. Now, of course, we can divide the algebraic function on such a space. So, we have a field of a rational function, which is now a finite algebraic extension of a field of the algebraic function. So, the function is two variables. And again, but it develops slowly, this problem develops slowly, and it's basically statistically understood in his study of binational transformations that to any point of the surface, or to any curve of the surface, there is a certain probability embedded in the field. So, but then, the point is that you have binational transformations, that means if you take two algebraic surfaces, each one is embedded in the project.

2:00:00 And sometimes to remove the singularity, to go from two-dimensional space of three dimensions to a high-dimensional space, but this was quite understood in the Italian school, so you have a clear notion of what is a projective algebraic surface without a singular form. But, unlike in the case of curves, you may have two such surfaces, which are bilaterally equivalent, meaning they lead to an isomorphic field of function on them, which are not isomorphic. The transformation from one to them has necessarily something wrong, where the transformation is not impeccable, it's not complicated. It's generally meant to remove some exceptional from both sides, but it's not fully impeccable. And therefore, we can say that, as it was in the Cervantes surface, there is no unique model of a algebraic surface. It took many, many years to understand this phenomenal environment in which I can see. So, but then there was a shock. The field of rational function was not enough to reconstruct the algebraic circuit. Now, in between we have a definition of the natural gravity height. In the 1940s, we knew that taking the complex number as a concept was not enough.

2:02:30 If we wanted to read a computer, we should have number of arithmetic applications, and the merging of algebraic geometry and number of arithmetic was one of the main motivations of underlying mathematics. Therefore, we had to deal with other problems. The German School of Archaeology is up to date. But, for a long time, it was assumed that the constant can be replaced by a complex number by any algebraic algorithm of Cartesian theory. And there is a joke that Zaisky told Paul Sartre and he said, let's take the algebra of Cartesian theory and then he developed his geometry. And then the student erased whatever letter was used at the time. But, for the purpose of number theory, we need to find out the guidelines of speed and amenity that people were taken in the 1930s. It was a... it's Mathematician Sci-Shit. It's a volume of Mathematician Sci-Shit, where there are people by fashion, by doing, and so on. It's a check. And so, they understood the necessity and asked them... So we had to invent a method. So we had to invent a method. Our model was differential geometry.

2:05:00 Difficult. All areas have been embedded into a particular dimension. I have to give a lecture. I have to give a lecture about this. So, for Van der Waalen was the first emotional numbers, that means, if you want a field that belongs to the field of rational numbers, so you say that you have defined orders of the field of rational numbers, now you are seeking real forms of solutions. So, it's basic to have two fields, Q and C points, and in one field where the equation of the equation, one field belongs to the other solution. In fact, you can trace back this idea to Gunn-Weiners, but it was systematically developed in the 30s by von der Waalen for the purpose of... But von der Waalen, this is more or less what is usually told, but when I came here, a lot of it was a equation of it. That came out of the concern of von der Waalen that the construction of the algebraic code was not a concern, and he was very much concerned.

2:07:30 There is a kind of connection. So he says, oh, I like to enlarge my view. On the other hand, the opposite was maybe out of his philosophy. He considered again and said that, well, you know, not to enlarge unnecessarily my view. I think he has a very large view. Let's say the view of Kant's sentence, I suppose, is now. It's the overarching view. It's a universal field. And so you have a collection of fields, a collection of sub-fields in a given large field, and it's very ambiguous on the very definition, on the very definition, it's very ambiguous whether we define a variety as a field. It has a pleasant description, a variety, an algebraic variety, and you assume that it exists, that it is a smaller field, that it takes the coefficient of field, it's not intense. I could not divide one book by a paper because it was full of data.

2:10:00 But other interviews were very important, which were obtained by gluing, in a way. So, it imported that notion of charge, the domain of charge, and the exercise of algebraic geometry. It did not introduce experience. If you look at his book, Foundation of Algebraic Geometry, you will see that the Zariski proposal The first one is that in unpublished notes, Maximiliano Giacomelli, in Chicago 19, would introduce explicitly the science-fiction with the purpose of imitating the method of different principal bundles in algebraic theory. So, Giacomelli has always been very, very familiar, and one of his major ideas is abstract, abstract means purely algebraic, algebraic theory. We don't really need the problem of bundles, but in order to define the algebraic varieties and set them to us, and here comes Chevalier. Chevalier was very unhappy about this definition of André Weil, because André Weil said, in order to define a variety, I take patches, wooden patches of its domain of local coordinates, and we patch them together. But each patch comes from one chart to the other when we write it. And it was not clear, so there were two defects in this exposition. First of all, there was no explicit notion of the field of definition. It was written down. And on the other hand, I mean, the variety came in with specifics we all know from the country.

2:12:30 A sphere comes in with geographical coordinates and geographical coordinates. But you have a freedom in the Monterey definition. Now, there is another line of development in algebraic geometry which I have already mentioned, which is Zanisky was interested in the so-called bi-rational transformation and he realized in great importance there is a local thing associated not only on an algebraic surface, but each software on a positive extent, the collection of objects, and now if you start with, let's say, a projective model of a Then, you have the notion of the field of algebraic function. Then, you have the notion of the field of algebraic function. Then, you have the notion of the field of algebraic function. Then, you have the notion of the field of algebraic function. Then, you have the notion of the field of algebraic function. Then, you have the notion of the field of algebraic function. Then, you have the notion of the field of algebraic function. Then, you have the notion of the field of algebraic function. Then, you have the notion of the field of algebraic function. Then, you have the notion of the field of algebraic function. Then, you have the notion of the field of algebraic function. Then, you have the notion of the field of algebraic function. Then, you have the notion of the field of algebraic function. Then, you have the notion of the field of algebraic function. And moreover, in spite of his work on science, he insisted that you should have not only as a local institution, but also as an intermediary. This paper was published in Japan. I mean, I'm a curator of the files of Chevrolet, and I'm slowly progressing to publish quite a bit of papers.

2:15:00 Such a work, wow, tedious it is, and unbelievable it is. There is some limitation that these algebraic varieties are irreducible, which means that you can speak of the field of rational refraction, that is, to just take a simple example, if you have, say, a curve which is, of course, you cannot define the rational refraction from it, because, I mean, it's... If you can have a function, a natural function, something, a rational function of a coordinate on one piece, if you square, I mean, if you see a function which is one of these pieces and zero of these pieces, this can be expressed. If you square this function in your field, you cannot say it's E, so it's not a field. And that was, okay, so, just now, so, we can claim that the first thought came from this paper as a model of It is very much similar to the plan B of the project.

2:17:30 That means, in the project, the plane has axiomatized the plan C theory, only not only the pole has been, but the line, the planes, the flat of all linings, and you put them on the project. Why? Because you have the duality principle. That every state, let's say the protective plane, in every geometrical statement, you can replace in a given statement. If you believe in the geometry principle, there is no reason why the pole should be there, because you have another geometry where the line is the frame of the line, and that cannot be understood if you see that everything, the basis, the strike, the basis, the construct, is the pole and everything is the point of the step. I mean, it was well understood in the geometry principle that the pole can appear as a collection of lines, meaning... You can replace the pole by the pencil in the game. All straight lines go through the pole, and it's a very fruitful idea in projective geometry. So, if a line is a collection of poles, a pole is a collection of lines. Okay, but this was just an outgrowth of science, and so this key variety, according to Chevallet, is a collection of historical data. That, if you start, diagrammatic geometry is looking for solutions of such equations. All the fields which are smaller than the field of quantum mechanics, by its numbers, and so the point is that if, of course, for the purpose of dual quantity geometry, you certainly can accept that the field where you're calling is the field where your points for your equations are questions, certainly not algebraic. And therefore you can ask a natural question.

2:20:00 If I have equations with qualifications and passions in one another, why am I not looking for solutions that there may be a few of them or not? Let's see, take the x-square plus the y-square, the bonds of this. So, okay, the notion of scheme, the notion of collection of bonds, makes sense if the boundary is not algebraically closed, and even if the formulae of scheme is really what is instructed in... So, in this presentation, there is only one thing which doesn't have, that means, basically a scheme is what is intrinsic, I mean, I would say, mimic what Bill said at the end of his talk, as collection of equations is a presentation of something, is a connotation of something. And this something is a scheme, that is in Chevalier's way, a collection of tools sitting within the big three. The big three, the fact of it, the fact of it is a finite region and the universe of the big three. You can formulate a notion of a scheme of a variable even if the variable does not exist. You have no problem, you have no problem with it.

2:22:30 I made a statement of that kind that, of course, I knew that my foundation of a variable would be one. I did not want to spend hundreds and hundreds of pages, I know I would say thousands and thousands of pages, I wanted to defend my thesis and I could not just embark on an ambitious program, I had to write my thesis, I had to be compromised by various approaches. So, schemes were invented by Chevalier with some restrictions. At the speed of light, the point is that these schemes did not have points, but you can generate points. There is a certain construction which has to mean that you have a scheme, that means a certain finite engineering, so a couple of, a pair of fields more, a couple of energies finite engineering to take the world, a certain connection. And then you can generate the form of a sequence by looking. But Key is not enough. Like him, whatever. So, as I said, they went on to go back to speaking. And then, development went on paper, the idea of science, topology, algebra, much more than all of them. It's interesting that if you look at the various paper of Chevrolet inside, let's say, before the 60s, before, I mean, what they became at the end of the 20th century, Chevrolet was very relevant about choosing, if you look at this book about algebraic groups, it's very relevant. I think, interestingly enough, Act 2 is towards a geology without points.

2:25:00 The world will need time to elaborate on the development of algebraic groups. But algebraic groups force us to consider where we have very few platforms. But then, if you reach out at a seminar about algebraic groups, It's the first time where you have a really good combination of the ideas of Sayle, his sci-sci topology, which are just now both of the work in differential geometry, topology, and also his joint work with Henri Carton on many complex variables, where he imported his idea and he took seriously the idea of the sci-sci topology and my own contribution to the... The Chevalier Seminard was interested in compromising on Parisian between the point conflicting point of view of Chevalier and Montesquieu, and says about this notion of algebraic varieties, of using these isotopology in schemes. Well, by the way, I remember a short time back, I was still a student and I was visiting on some Sunday, say, a flat in Paris, and he was just typing his paper about this or that, and he ended up to get a... An algebraic variety is a space with a topology to enhance my conceptions. So, I think the next level was a combination of the idea coming from topology and schemes and the idea coming from algebra. So, I'm going to show you the definition of the general definition. Say, one took an opposite. I mean, he took his roots in a different tradition. I mean, the tradition of differential geometry has been taken by Kant and his class. So, we consider spaces without form. Let's say, considering algebraic rules. I consider, let's say, GLM.

2:27:30 Now, we know matrix and we have a matrix multiplication. But now we have the so-called Frobenius. It's the matrix resulted by raising each entry of the matrix. I mean, if you have two matrices. Because, if you work with a group of characters in B, then A plus B. Now, this transformation from GNN into GNN is a one, so assume that A is everybody that is close to me. It's a group of mobile things. If you want to invert a slide, you have to take a piece of wood and add it to the power of 1. So, you have a simple example of a map of a group into itself. This is 1 to 1 at the level of 1. Nevertheless, this is a group theory. If you have a group of...