Local & global in mathematics, Riemann sheaf

0:00 The book had left me with the first impression that I understood what it was saying, but not really what it was talking about. After I stopped studying mathematics, and after I chose to return to the history of mathematics, these are the same theories. I did not want to be left in debate, so I chose to take a lot of research. I naturally found a lot of readers in mathematics. I also engaged in the field of mathematics, which is an education in quantum mechanics. I did a major in mathematics at the Centre for Contemporary Mathematics. I also did a major in abstract mathematics at the University of Paris. I also did a major in the more recent book of Thomas Conyers on the history of quantum mechanics. The theme was different, but I was also very interested and influenced, I think, by the problems of quantum mechanics. The emergence of structures in the modern world, and therefore the taste of the reader, led me towards these galactic geographies, either on a notion or a theory, and choosing a fairly long temporal scale, from 50 to 100 degrees. It's a bit in this kind of thing that I wanted to start. My more specific project is an idea of theories on the subject of geometry. You know the kind of things I like to read or maybe I would like to write. My more specific project is perhaps Hawking's lecture, which is very much related to the theory of reading. Generally, we read more and more on the subject of the history of geometric theory. We encounter the question of the local and the global systematically and a little bit every day. A modern reader cannot talk about these theories without using these terms, so he meets them systematically, but he always meets them indirectly,

2:30 since it is at the occasion of reading Riemann that we translate the term Riemann in terms of the global case, it is at the occasion of the work of Dick, Thomas Hawkins, all the two sentences, the methodology, in parentheses, of course, but here it is the global case, where, beware, it is not global and so on. So, as they are systematically encountered, there are always gaps between them, so it creates a motor that leads strictly to a notion. I therefore conceived the project to address the study of this element of the global model, not with two gaps, but with three different ones. Then, after some transversality, different theories, and to address different authors. This is the global model that showed the result. And I was happy to see that Christian Hossein was willing to comment on this theme, which interested him. This choice of studying the emergence of the global local in geometric theories also provided me with an angle to approach not only the theory that I was interested in, but also mathematics. But also a subject that allows us to encounter two other visual and important questions, to many of our colleagues, that we will probably encounter in historiography, namely the general goal of the ensemblist transition, but not from the angle of the ensemblist theory of the non-foundational side of the ensemblist language. And this subject, as we know it, seems to me to be a good occasion to encounter the theme of the structure of mathematics, but under a rule that is not, for example, supported by the ecology. This is a classic example of algebraic structures in which topological structures have already been used. So, the model is a formal language that combines them and the model is the question of mathematics and structures, but with mixed structures with analytical topological structures. This project is in formal language. There may even be two subjects of study, depending on the choice of each one. I consider that there would be a double subject, and neither could be completely separated.

5:00 A double subject with, on the one hand, a documentary file in France, and on the other hand, a work on the explanation of the global locale, which would entail the reflexive and mathematical nature of mathematics. This is how, on the one hand, Parts of a documentary file, a three-part documentary file, to be constructed and presented with the necessary decisions. So, a work that would be similar to a thematic monograph on a range of global theories, on the passage to the global with important theories, but initially conceived or taken locally. A thematic monograph on two points of general tables, depending on what you want. Several disciplines are also involved in the course of the course of the course of the course of the course of the course of the course of the course of the course of A discourse that places statements, denounces systematic errors, analyzes a priori all the objectives of a theory or another, by declaring an aspect of the reflexive nature of mathematics. This is a double object, which I found as an entry point to reorganize the big documentary base to present. I found the entry point on the side of the reflexive aspect. By identifying two important periods of explicitation of the global block of the period of 1900-1913, the first period of explicitation is fair, and the period of 1945-1950, the second period, where this sphere appears in front of the scene, and this appearance, of course, has a very complete renewal of the languages ​​of geometry, geometry and geometry of physics. So these are the texts that I read during these two periods of explicitations, mainly at the beginning of the 19th century, in 1913, for the first phase, and the texts of the 14th century.

7:30 It is from these texts that I identified the corpus, the thematic, and the temporal units of my research. The temporal unit, ultimately, is just a rival, since the period of explicitations is 1913. There is a lot of re-formulation and feedback on the theories that have been published in the journals. So, I would definitely like to take your help. I am 152 years old, you are 153 years old. Thematic life too, since I am a theorist and I have not yet addressed the aspects of the journals. It would be necessary to integrate, to review all together, a more global explanation, but that's it. So, once the temporal and thematic limits have been set, some remarks on the second part, the complexity aspect of mathematics. So, not on this part, on the history of global theories, but on the history of local-global themes, local-global passages, to approach the complexity aspect of mathematical knowledge. This return to mathematics is in mathematics, often without mathematical means, not directly a return to mathematics, but often without mathematical means. So, other approaches to this question of the complexity of mathematics are unavailable, but not borrowed. In particular, not borrowed from the Kerelesian gap of the duality of mathematics that we will discuss. To find out if they were adapted to understand each other. There seemed to be a lot of elements present in this long story of the global realm, but I did not have the means to pursue the director's field together. Besides, I did not take up the subject of Leo Cori. I found myself more adapted to the description of the evolution of a discipline.

10:00 and less transdisciplinary work. So, as an example of a flexible approach, we will start in the second or third part, to show the variety of flexible approaches, perhaps the fact that we do not have a framework to theorize this flexible approach in a unique way, to show the variety of these flexible approaches by using the explanation in mathematics, but not in different moments, in different contexts of the academic field. I will quickly repeat the words of William Togleswood, who is part of this history, the first who gave birth to the systematic literature and the French language, a didactic theory, which is exposed in the lectures, a didactic theory of Goleba, a second discourse, which is part of it, to place the theories, to give the titles to the chapters, to summarize the different stages of the demonstrations. And a discourse. The discourse is always the same theory. But at Euclid, we find a unique thing, in fact, a definitive definition, which we hear, for example, in Grosseuil. In this way, we can identify the syntactic form of certain fields, not the nature of certain fields. A field based on neighborhood can be a global field, it is not the nature of the field that makes a field a global field, but it is the type of dependence that the different fields present in a complex field maintain, which makes the field a global field. So, this type of explicitation is the result of a certain definition of a global paradigm. I will try to find a historical reason for the use of this phrase. There are different explicitations, for example, by Abel Amar, who wrote the book here. There are also different explicitations of the structural interrelationship of the item A and B, or other structural explicitations, for example, in his work on Erasmus. We can see the work of the students as a difficult structural work, a structural transformation of the tracks launched by the case of reinterpretation, a work of the second level, which is not the one at the end, but the one at the end, which is similar in the sense of the one at the end, in contexts and with imaginative practices that come from the case.

12:30 In other words, the construction of the documentary dossier on the history of global theories, the emergence of the first global theories, the global passage of theories that were previously conceived or adopted. Once again, I would like to summarize and present two points that have helped us organize and problematize this mass of documentaries. So, two elements of our problematization may not be the general idea of the ASB, which allows us to organize the parts and the coherent models over quite long periods of time. So, we will not talk about the question of the place and then about the competitors, the global local. So, this question of the place, it is in it that I found a way to organize it in a coherent way and find myself baptized. The part that contains several steps, which in fact are parts 1 and 2, which are very different from each other. I found this key that for me was useful to organize the corpus. So this question of the reference to a place, namely the question of the pertinence of the reference to a place, in a certain way, and how to make this reference pertinent when there is no pertinence between theory and mathematical theory, the question of the force of the reference to a place, which is written in a non-anthropistic way, the question of the obligatory factor of the reference to a place, which disturbs the modern reader who is used to the systematic reference. The link between these two fields is also maintained with a particular expression, i.e. the field, which appears from time to time, but at a certain point in time, it changes. The question of the different fields on one side, the question of the practices of the field on the other, is, as I said, an intra-administration ego,

15:00 In the meantime, there are the aspects of circumference, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion, perclusion. So I choose the best to be the best, even if I find myself in the same planet by multiplicity, by gravity, by gravity, without being 100% consistent in both conditions. And so to realize these aspects of reference to the best, I built the world of grandeur, which allowed me to at least This has allowed me to better read mathematics or mathematical texts in which the elements that articulate our mathematical language are based on zero language. There is a distinction between the set of elements, the possibility or the possibility of division of intersections, there is a distinction between the set and the application, between domain and image, there is a distinction between, on the one hand, the domain of X and the domain of L, and, on the other hand, the continuous applications themselves distinguishing the applicable application and so on. All these elements are not elements that live, but that live in artificial spaces. So, if we were to attack these elements, it would be a good idea to understand them more authentically, what we call the world of the future, the world of the future, in which we find, for example, the availability, a priori, of numerical spaces in the world of the future. The first role of dimensional structuring I think that the intersection with Piedmont is not affected at all in the same way as the intersection from Port-à-Bord. Another element that I liked a lot is the barrenness, which I think is very important. For example, the premium of the notion of multiplicity, the intersection between two things, and the instability between narratives.

17:30 I have read, for example, from these tools, the work of O'Neill. Not in the first part, only in the second part, which is about the work of Riemann, which I wrote a year later in fact, and I re-read with this title the work of Weierstrass as an explicit work on these frames of the world of language, as well as, for example, to show that the work of Weierstrass is largely driven by considerations of dimensions, for example, the neighborhood, it is a definition of continuum. We spontaneously have the idea of reading as a double-edged sword, and I think that it at least raises the definition of what the double-edged swords are, because they are full parts of the first year, and the double-edged swords, at least in some of the texts, will reveal that. So this part of the work on the place, the relevance, the reference to the place, the practices of the place, and the construction of the creative domain. The world of grandeur may have led us to cross the border that separates history and psychology, which we think is practical, of course, and the border that separates psychology from history at that time. However, it seems to me that the construction of the model is a manifestation of a moment to return to the scientific education, which is to say, to integrate into the history of the world of mathematics and physics. Another element, another red thread that will serve to organize the vast mass of these key terms is There are a few red lines of competitors in the global locale. Today, there are two. There are more competitors who are stuck in the middle of the road and some more locally. There are two red lines that follow almost all the work. The infinitesimally finite competitor and the competitor of the intrinsic value of the intrinsic value. As for the infinitesimal and infinitesimal as a couple, the concurrence with the global local as a couple is permanent. Thus, the differentiation between the infinitesimal and the local is something that is gradually built in the second half of the 20th century,

20:00 and probably in the first half of the 20th century. Similarly, the distinction between the past and the present and the past and the present and the past and the present and the present and the present and the past and the present and the present and the past and the present and the present and the past and the present and the present and the past and the present and the present and the past and the present and the present and the past and the present and the present and the past and the present and the present and the past and the present and the present and the past and the present and the present and the past and the present and the present and the past and the present and the present and the present and the past and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present and the present There is a great deal of complexity in the field of quantum physics and what is called the field of local research, which does not mean that quantum physics is only used for local work, since it is not systematically a local point of view, but... We observe that indeed the work that is being done today by Lebeau, apart from these works, does not take place only in this period of time when Lebeau appears. Lebeau competes with that of the late 7th century. I quote at one point a quote from Czern, who writes in another article that, of course, since he verified the local formulas that Lebeau had in his book, So they are global. But this articulation is so fast that it turns us off from an article of our importance on the general model. Of course, historically, there is no evidence, and as much for a perfectly clear articulation, as much, for a long time, the two themes have been competing. Of course, in our notion of variety, we articulate both the multiplicity of local maps and the global organization of the system of maps. The two levels are integrated in the definition of variety itself. The level of the multiplicity of a local map and its possibility. The subjects are legitimately free on a variety and the studies relate to the variety and not only to the artifacts on the map.

22:30 For me, it is almost the same gesture. I consider the articulation of the map to be the same. However, for a long time, the emphasis has been on the concept and on building the question of the global. My hypothesis would be that there is a real construction of a random point in the sense that, in order to make people understand what is happening in France, we have been forced to hide the type of nature in these global languages, for example what I have chosen, the different genetic structures, we must not distinguish the white from the blue. So these are the two pedagogies of what we can understand as the concept of a commercial geometry. Besides, we still hide certain things. We are not totally illogical in the field of geometry, but it is a bit difficult. So that's what I'm going to talk about. I hope you can understand what I'm going to talk about in this topic. I hope you can understand what I'm going to talk about in this topic. I hope you can understand what I'm going to talk about in this topic. I hope you can understand what I'm going to talk about in this topic. I hope you can understand what I'm going to talk about in this topic. I hope you can understand what I'm going to talk about in this topic. I hope you can understand what I'm going to talk about in this topic. I hope you can understand what I'm going to talk about in this topic. The project and the scope of the work, which is a fact that has increased over the last few years, is not an archival work. I have only worked on sources that are not applicable to the project as a whole. In fact, it is a series of different disciplines that were in the same room. And on this point, we would have liked to... We will have more details and we will be able to vary the observation chains in the language, including 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 62, 62, 62, 63, 62, 62, 63, 62, 62, 62, 63, 62, 62, 62, 63, 62, 62, 63, 62, 62, 62, 62, 63, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62 In fact, I chose not to focus on the topic of a lot of aggregation, a lot of metaphysics, which have been associated with no text, and which did not necessarily find material for a long exposition or no echo in other parts of the field, and so I played the game a little bit of leaving this part of the architecture visible.

25:00 By letting go of everything that could let go of them, they could one day make a track for a future work, even if it seemed like a problem just to touch it by passing, and not deduce more than that, so it would still be a problem. One last problem of this approach. I know that the documentary aspect led me to present quite a lot of texts, especially very important ones, in the global economic agency, but on which I did not have much of a special to bring. Sometimes the presentations were flat, which seemed a little dramatic to the interviewees who, on Riemann, or even on Kramer-Weil in 1913, commented on them. Well, on Riemann, I think it brings a little something else. When I come back to it, when I talk about the fact that, a year later, on Riemann-Weil, perhaps it is the comparison between the idea of quantum physics and Azadeh, which brings something else, but there were, by the project itself, some obligatory passages. I don't know if this is tense, but it is pleasant, that is to say, a work of study, the end of the study, I hope, we see it, a work that is, well, to learn the job, I must say. So I had the impression that I was learning a lot of things. I was reading a lot of texts that you didn't know. Before starting this work, I wanted to tell you that I was analyzing a lot of square points, the general dissertation of Sir Pascot or De Gaulle, the work of Berlingo, but I'm not playing the role right now. I think I made progress in German, which was welcome to confirm during our last interview. It may be a delicate way to make you understand German, but I learned a lot of mathematics. And it is particularly special that I know you in connection. I will not deny the representation of the groups, it is simple. I'm not going to go into too much detail, but above all, I would like to introduce you to the theory of linear and complex theory.

27:30 It's a work of study and therefore a work intended to launch, to accumulate a first material to be able to continue the research. I would like to take as an example, as soon as possible, what I say in chapter 14, which is dedicated to Fibret, It's a bit of a critique. It's a bit of a mouth-to-mouth discussion on a campus that was not built in a very original way, but it appears to be a bit of a mandatory point of passage. I have not problematized them, so I have problematized them in terms of periodization, since I see, in a sense, the end of the era of remodeling, from Riemann to... Up to the work in the 1920s included, and the global passage at Du Cartan, in theory and geometry, and in differential geometry, we are still in the moment where it is the universe of coatings, plants, the global passage, the fiber passage, change the law, and so on. This area of ​​renewal with its familiar elements, the problem of local conservatism, the duality between its technologies and its groups, the implicitity of the idea of ​​common sense, the idea of ​​partiality, and so on, or even, we move on to another period, in the case of such fields. This being said, the points of globalization that are present in this field, for example, the evolution of action practices. With, in particular, the passage of the human sense in the form of mathematical practices such as Arnaud Van Gogh which is very different from the one he did in 1913. Van Gogh comes back to the task of axonatization. He gives them another task of axonatization which is already very hard. Already, some of the main problems were forced. All of these have the same function, namely a change in genetic practices with the passing of the arm around it.

30:00 Another possible problem is a red line in a slightly tinnitus-like way on this table. It is the question of the value of the values. In fact, the real subject would now be the value of the values of the triplets, the value of the values of the values. And so, it is important to focus on the fibers and the other elements of the thesis, the elements to continue in this direction. If, for example, the construction of the regular fibers is in the first place relatively indifferent, the problem is quite visible. In the case of Steyr-Roll or Steyr-Roll-Méplore, we will realize that the new framework leaves a lot of problems, in particular in the first place, in terms of direction. It will not be that bad. There is the question of the biological nature of mathematics, but how do we go from the level of mathematics to the level of quantum mathematics? How are they constructed? How do they differ? How do they be combined? The construction of the place is not difficult to associate with a vessel. So, this passage from the optimal scale to the optimal goal, having its own place, can itself be linked to the passage. There is also a great deal of knowledge and expertise in the field of mathematics, physics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, It's still a very specific knowledge of the practice of mathematical physics that we found unchanged in Darboux and Cartan, for example, but again in the early days of Ramanujan and the popular era of his time. It is therefore a very specific knowledge of mathematical physics that very largely disappears due to reformulations in the field of invariability.

32:30 Even though Penrose is still borne by some transitions, it is still a student of Cartan. I would like to end this presentation by thanking the people who participated in this lecture. I would like to thank the people who participated in this lecture. I would like to thank the people who participated in this lecture. I would also like to thank Pierre Cartier and tell him that if he is not here today, it is entirely due to the incomplete knowledge of the organization and if I may, to the pleasure that I have of being here, I would like to be very attentive to the concept of the day, in terms of the question of what seems to be the line and the depression of physics and the line and the depression of physics. Thank you, when you came to see me a few years ago to prepare the thesis, I was first frightened by the ambition of your project, but finally you have accomplished this project by doing this monumental work that we have under our eyes. So, you have a solid math culture, and you have completed it to the extent of your needs, in the preparation work of the thesis, and it has allowed you to use with pertinence a whole series of texts, some of which were really difficult, and some of them, very large founding texts, that you have read with...

35:00 There is an extreme focus on the texts of Griemann and de Poincaré. You have read these texts in a scrupulous, attentive way, attentive to the way they are written, by carefully distinguishing, as you have said, the different types of texts, that is to say research articles, Exposed research programs such as the Riemann inaugural dissertation, page 76, didactic treatises such as the Neumann treatise or the Rostov-on-Don report. You address in this way This is a study of what I would call mathematical rhetoric, which is a necessary component to make the history of mathematics, which has unfortunately been brought up quite little, and it is through this rhetoric that you highlight the emergence of the global local group, with the attention you have to what you have called the syntaxic force of the statements, which seems extremely interesting. So, you distinguish the narrative style, for example, in chapter 4 of your memoir, page 248. You talk in a very interesting way about the abandonment of the world of grandeur and the language of grandeur. I know, it's on page 250, somewhere there. The reference to the place, the point of view of the local universe, all these things, you have highlighted them very well, in a more specific way, for example, from page 20, you point out a non-assembly reference mode to the place. Or, a little further, page 48, about the Riemann surfaces, you talk about the Riemann surfaces as a place of a generic point, so all these remarks seem to me to be really very interesting and really bring the arguments that allow us to show how the emergence of...

37:30 In a more general way, I appreciated the comparison between some authors such as page 448 and page 450, the comparison between Oswald and Hadamard. So, it is always this attention to the text that you have practiced throughout your research that has allowed you to notice a number of things about the texts we are going to talk about, which are quite rarely noticed, but which seem important to me. You may have minimized your merit in your presentation because there are remarks. Very interesting, as a proposal of Riemann's surfaces, to point out the distinction that Riemann makes between the discrete or continuous parameters that we can depend on. The nature of Riemann's space depends on whether we consider only topology or the analytical structure. This is from page 80 and you report it again in page 646. I also appreciated that you interpreted the study that Riemann makes of the hypergeometric portion, page 88.

40:00 In Dizon, he gives an almost axiomatic definition of this function, which is quite fair. You have also noticed, which is not often noticed, but which is very fair, that Poincaré, in his work on functions of two complex variables, Note that we can cover the arcane with balls so that the non-empty intersections are at most 5 balls. This is quite interesting because it is the definition that Lebesgue will give later for the dimension of topological sense. In my career I have already seen that. I have never seen in historical literature this kind of remark and it is quite fair. Page 324 is about homology, so there is something quite interesting about homology at Poincaré because it is rather page 335 after, but in my opinion you could have said it right away, the very definition of homology at Poincaré, which is page 324, It may seem a bit obscure, but as you just said, it obviously depends on the fact that Poincaré considers these varieties as domains of integration and that, consequently, this arithmetic calculation on the varieties comes from the calculation on the integrals. This is what explains the definition of topology. You should perhaps be a little more, instead of saying it later, say it right away when you comment on this point-square passage. So I have a few questions to ask you, to allow me to add a little to the comments you have made on our work. On page 51, you say that, commenting on Riemann's memoirs, you say that we will not address his section 2, which is devoted to the functions of state, the memoirs on the functions of state.

42:30 So here I asked myself a question about the functions of state, because it is still an extremely interesting subject. For the theme of local-global, since if I atrophy, and without giving the name, if Arel atrophies from the start elliptic state functions, it is good for having a global representation of elliptic functions. They were well aware that, as it is a function of the remotes, the Taylor series is only available in a certain disc. And if we want to have the remotes in a certain global, we need another way. And that is the origin of the Teyvat forces. And that may have escaped you, but it is one of the paths that you might be able to approach in further research. Another small historical question. Page 241, you very precisely say that, at the bottom of the page, we see that discontinuity is to be heard in the general sense of singularity, it is also possible. Even if continued, although defined by Cauchy, it plays a meta role here, since it is a reminder of the convention allowing to eliminate, in the course of the work, the five points. So that's quite right, but maybe you could have insisted a little on the fact that, and it comes back to page 287, The concept of definite functions as an expression of relativity, and not at all arbitrary functions, as this notion of something completely arbitrary rhymes later on. All of these functions are constructed from functions that are made up of different algebraic processes or infinitesimal transitions, etc.

45:00 This does not really touch on the essence of your subject, but it is still a point. It is important to note the distinction between Cauchy and Riemann. It could have been repeated in page 313, where you say that it was essentially on this first phase that we were working on. It forms the context from which we could better grasp the work of Riemann, Klein, Neumann, and Strauss-Pancari. That is to say, in fact, there is a break between Cauchy and Riemann. By the fact that the human body speaks of continuity, it can enter into these continuous functions only if it verifies the definition. In Cauchy, this is not at all the case. He has a set of functions to which he thinks, and when he talks about continuity, as you say, post-241, it is simply to eliminate discontinuities in the interval to be generated. So there is a whole passage where it would perhaps be desirable to clarify a little your thoughts on this, always about the same problem of the definition of functions. and what it is that functions are possible, etc. So you say, with just reason, that in Cauchy, page 277, functional equalities must be considered not as equalities after the transformable expression of one into the other. This is the point of view of the 18th century, the point of view of Hegel or Lagrange. But numeric equalities, numerical equalities in the field of mathematics must be specified. This is absolutely right. I would like to point out, perhaps you do not know this author, but he is a former author of Cauchy who had already had this point of view, and who had been appreciated by Gauss because he had clearly highlighted this point of view. A Portuguese mathematician who taught me in a military school in Portugal, whose name was José Restácio da Cunha, in which he defined the fundamental functions by their series, the exponential by the series, the fundamental functions by their series.

47:30 In order to define something, one must first give a definition of the convergence of these terms. And one does so by the trick of Gauss. This book by Capugna was published just after the death of the author in 1745. That is to say, again in the 18th century. So that's the little point. Just to fuel your thoughts, page 35, you are very rightly right about the arrival surfaces, what we see as a description of the surface above the plane, etc. And then, such a relief is described without ambiguity by the pre-mutation it induces between the surface parts itself, described as immediately deriving from a path on which it passes through the flow of ramifications, etc. So that's quite clear. There is perhaps a source, it is quite likely that Riemann had read... The memory of Puiseux on algebraic functions was published just before, in 1850, and Riemann published it in 1871, but it is quite probable that Riemann published it, and precisely Puiseux studied algebraic functions by considering the permutation of the branches when we do a turn-return of an algebraic function. So from here, there may be more sources of answers. I also point out that page 58 tells us that, I remind you that this is the memory on the abelian functions. The number of sections to be practiced is P, without P being called the genre here. And why is it not called the genre? Because this term genre was introduced by Clepsch.

50:00 We will talk about this later, in 1868. So, always questions of terminology, because, as I told you, you are just waiting for a slightly rhetorical study of mathematics. It's good to be attentive to the terminology because I feel like you're talking about the term used in French to translate German as mathematical, variety or multiplicity. I did not find the reference, but I would like to point out that the case of variety has been introduced in French on the advice of an Italian author, I do not know if it is Cremona or someone like that. It's in a letter, but I read it once and I did not find the reference. Unfortunately, he advises his French interlocutor to use a variety rather than a complexity. So there are some small warnings that I point out for the correction, it's not very serious. For example, page 404, there is a place where you say that Picard uses Dirichlet, which is obviously related to Dedecky, about the elliptic modular function that Picard uses to demonstrate his theory. A little further, page 473, you make a reference to Weyl when it comes to Weber. So I don't know why, but you talk about Weber for quite a long time while using his algebraic language. But in fact, the whole theory you're talking about is that of Weber's deductions, which has been used quite a bit before. I also point out that the algebraic function of the body of a variable is written as C of t, with a parenthesis, as if it were the body, quotient by... no, not quotient by the body, it is obviously the ring of the polynome.

52:30 There have been a few small errors of date, but I would like to point them out in a very particular way. There is also disavowal, such as, for example, Pasquale Sant'Antonio, who said that you were talking about the evocation of pilater surfaces, that is, non-orientable surfaces, which are obviously unilaterals. Page 551, you have a reference to the elliptic geometry, the Euclidean and elliptic geometries, but in fact you say that it is called Rn, but it is obviously the hyperbolic geometry. The way in which Schreyer constructed the representation of the universe in a continuous group, in a topological group, was defined by the generator of the relations, and the relations, you write them at the level of... Well, it's me who is bad at reading, by the way, it's probably not clear enough, but your lines are a bit... You have to read U1 bar multiplied by U2 bar, and not U1 U2 with a bar above. There is a small interruption. It's so small that I can't get it out of my head and I haven't seen it before. But if you put a weight between the two, we'll see what happens. Yes, you know what I'm talking about.

55:00 Well, it's not... but it helps. So, another thing that I would like to point out, which bothers me a little bit, is the suppression of nouns and the use of adjectives as nouns. The page 735 of Adolphe VI You talk about a circled domain, etc. You say that all the coordinates are multiplied by a complex of volume 1. I like it when we say that we have many complex of volume 1, especially since in these theories, the majority, we always use complexes in the sense of algebra and cohomology. And there, there is a possible construction of the concept. It's a little bit embarrassing. So, another small possible confusion when you are going to point out the risk is about duality, which you saw in the sense that there are two things, but it is a little embarrassing in this mathematics to talk about duality, while there is duality in the sense of the term. So, you have banned it in the amount of places on the board, but there are still 4 or 5. There are also retellings. It's a bit annoying because there are moments when you have made a quote, it's long, and then you take it as it is, with the translation. In general, these are quotes in German, with the translation and the text in German, as it is, you take it as it is, or you take it out, or you take it out. The course is a little embarrassing, you just have to refer to the page. It is not worth doing this debate. So, for the writing, I have to say, it is really very enjoyable to read. I think you apparently have a pleasure to write. It's written in a way that is easy to understand and pleasant to listen to. Maybe you get a little carried away by your pen, which means that there may be some length. And I think it's something that needs to be published on the Internet. It's an extremely interesting moment, but to publish it, you have to tighten your head a bit. That is to say, for example, chapter 4, an elementary chapter, as you said, which needs to be narrowed down a little bit.

57:30 I also think, perhaps chapter 11 on Raoult-Zeig-Materiel, where you may need to narrow it down a little bit, because it is all the more so that your conclusion is that in this homage there is not at all the global global term. So you could come to this conclusion in a way with a little less development. Otherwise, there are still a lot of things to work on. For example, you had problems with a second S when you were studying. There is no S, I suppose you understand that. You also wrote a lot of third person of the future blog with an S at the end as if it were the first instead of a T. There are several places where... You do not take into account H. Aspiré, Henry Cartan, or Hadamard, while it is two Hadamards, H. Aspiré, etc. On the contrary, there are places where you do not do the edition, so you have to do it. You have two Osgoode, so you have to write Osgoode. I will point out in detail all that. There is a twist that I don't like too much. It's as if I'm living from a past participle, for example, but without verbs. It's a twist that's a bit vicious, I don't like it too much, but I also like it. There is a twist all over the place. It's really... it's aghorel on the subject matter. So, well, you talked about German, you have mastered it quite well, your translations in general are good, there are a few false phrases in your German citations, there is one translation that you need to understand, it's page 520, where your translation is really wide, because... The German text at the bottom of the page, Haber, Entschluss, Erkenen, Siegdorfer, Stuttgart, etc., you translate but if they were able to forge such a key, it is because the lock has been studied in all cultures after we have managed to forge it.

1:00:00 The text can say but if they can build the key, it is because after a successful drilling, They can study things from the castle, from the castle, from the castle, from the castle, from the castle, from the castle, from the castle, from the castle, from the castle, from the castle, from the castle, from the castle, from the castle, from the castle, I am very happy with your work, especially since you are the last student in this department, and it is very pleasant to end on such a high quality work, so I congratulate you for what you have done. The thesis is in the form of a vicious one, I can't talk anymore. Thank you very much. In the case of Sheckman, there is a big difference between the function of mathematics and the function of quantum mechanics. Sheckman uses numerical mathematics. It is the optimization of mathematics.

1:02:30 But he did not use abstract functions. This is a new generation. He uses mathematics in opposition to classical mathematics. He does not deal with the subject matter. Thank you for your attention. The four periods are a few more than the time you have. You step here, and you're kind of here. You go down, and you step here. All of your motions are here. They're more than you have. Yeah, cool, right? Because I want to start with a short expression of what I see in many of your pieces, and I will come to remarks and questions from you all. So, first, of course, the evidence for everybody who has seen the species is that it is a huge amount of well-documented material, over about 100 years of mathematics. And, of course, we have heard the presentation of the author. It was collected and discussed in an article of what the author called this couple of global, sometimes, edified by the third concept of infinity. So, of course, in reading this... Let's say I want to express the subject. I, as a reader, and a reader coming from the history of mathematics rather than from the epistemology of mathematics,

1:05:00 I wonder what this couple of the script will refer to. And apparently it's a way of looking at certain types of mathematical concepts, symbols and objects, etc. And as you expressed in the introduction, it does not refer to mathematical concepts in the narrow sense or to objects in themselves. We don't write a history of mathematical concepts, we don't write a history of mathematical theories, and we even say in some places, we don't want to write a kind of history of mathematical ideas in the sense of, let's say, I call it the old history of ideas style. So, what is it? Well, there's 30 of them! And then I come to the conclusion that it seems to be a kind of mathematical epistemology which uses a historical approach, but maybe you would like to turn it around. For me, it's a very epistemological look into the history of mathematics rather than a primarily systematic one from the point of view of epistemology or, let's say, a more traditional historical. So you are establishing a nice passage in between these two levels of the history of mathematics, or the traditional history of mathematics and the systematic history of mathematics. And, of course, I found you are only reflecting about it in, say, one part of page 1772. It is in this level of thought when the problem is posed, more interested in the why with the why, less in the motivation for the technique and more in the form of the reason for the reason for. Of course, that's now a very, let's say, that's a very reflective answer of what we are looking for, and this results, of course, in a, in the end, in a, what I would call an epic style.

1:07:30 And rather than, let's say, a more dramatic one, which one usually finds in a historical description, in a historical magnation, concentrating or emphasizing active persons, development stories with underlying causal or quasi-causal dynamics inside the knowledge development, social work or cognitive networks, etc. And of course, in this sense, it is uncommon, but nevertheless very, very rich historical presentation. Now, to resume, my strongest criticism is for this style of approach that in the 80s has become mainstream. And I guess a cut-off would have been helpful for the reader and probably for yourself also, because then you would have forced yourself to concentrate on, let's say, certain phases inside this very broad, epic narration. Okay, so that's more or less my introductory part. A lot could be said about the concept of this ethical panorama. But the 19th century material is covered by the historical literature, much denser in the 12th century part of your ethical narration. So I would argue it comes in the 12th century part. And here I remark that your thesis... Connect things which, if they have been covered at all by historical research up to now, your presentation establishes interconnections which have not been present up to now in the historical literature. And just to concentrate on certain points... To explain what I was thinking about, when you talk about wise, you don't need to talk about the term, between 1909 and 1809. I think we are doing very helpful work of finding things together which are interconnected, so that's the part which is really new for the, for, inside the historical context.

1:10:00 That holds even more so for the development or the presentation of the development of part of structures, part of spaces, and later resulting here in one part of that which does not yet seem extraordinary or the same in the historic, in the historian's history of mathematics. Of course mathematics is looking back into the history of that period of written power. In my view, these chapters contain the most important new contributions to the history of mathematics that could come from an area of professional development. Now I'll come to some questions, but in order to hold the question, I will start first with more localizing math. The question relates to, let's say, the relationship between five, between physical geometry and quantum mathematics, and just to put it that way, because purely physical geometry was not so purely infinitesimal in all respects. Why? He wanted to construct the spaces from this new chronological protocol of the infinitesimal generation, but the groups operating in these infinitesimal structures were the finite groups, so that was different from Carton, in the sense that Carton was more radically infinitesimal than Baye at that time, let's say. Although Cantor's structure is quite general, he talked about the roots, but apparently, obviously, these roots were still there. So the step towards localization lies here.

1:12:30 All of these are justified later in specific eras. So my question follows. I would like to have your remarks and your comments on this question, the comparison of Bile in the early 20s and Cartan in the early 20s, and my question is... It's really complicated to me how aware were Weil and Carl Kahn about this different view or usage of the algebra versus the groups in their respective programs. And there is an interesting... And of course, he presents that he discusses both the groups and the infinitesimal, so he has, let's say, There is a question which leads directly to your thesis, when you talk about the gestalt switch inside Cartan's work on the grooves in the rock. Now the question is, is it rather obvious? Did Cartan change in this respect, also in this way of perceiving this as generalization? So would there a kind of gestalt switch also become Cartan's differential? About the middle of the 20s or maybe in the 30s. So that's one complex of questions. May I ask the other two questions after the comments of the, or maybe in a more sense, not the continual accumulated questions. I'll ask you a second part of the question. Certainly, the thoughts of the same constructs which, in some sense, look at this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and this and

1:15:00 The first effects of which we show is that there are lots of these local equations that come to be certain of the life spaces of the world's global pressures, and there are links between the global topology and the various connections in the space and the global threat with respect to the energy. And yet, habits of property that the... There are a lot of good terms, particular thoughts, and good parts, if you can think of it that way. So, this is the actual concept. But, more generally, I think it's not relevant to really distinguish what Dr. Long, you know, what Dr. Long does. In legal theory and in differential geometry, both fields blend in the early 1920s. That's why they want to space theory. And the key constantly switches from one to the other. And it is quite clear to me that it has a legal... There is a legal understanding of space, a legal understanding of potential geometry to do, so it's no wonder that when you do it, the rules change. The view of potential geometry changes all the time, at various times. As to the first part and the fact that maybe Gauss-Kahn was more strictly intelligent and strict in generalizing space than Bayer was... I don't think, I don't think it was really relaxed or closed in that community.

1:17:30 In that community. In that community. In that community. In that community. And I think that, so far, we've got no hard time. So, I may not take up the answer and come to my second question, because I don't know whether you, I mean, you give a clue that there was a, of course there is a, maybe I come back to the core of my question and now entering here is the introduction of homogeneity. In his understanding, from what I read from you and from what I read in other places, is that he had the self-perception that he just made explicit cartons out of wood. And for me, it is not so clear, because... Just to put it that way, if you really write Hermann Weyl's early infinitism, or John Pure's infinitism, you can do it rather well with the language of the five elements of physics. So why? Because in these infinitesimal structures, from the point of view of Hammond-Weill, it was really the finite group, the intermediate group, that was the problem, and that's it. And this is not the case in the early countries, when it comes up. So your answer, that there were the kinds of structures which we can count on in the middle of the 20s, it referred more to this question, that you of course have this question of . Let's say locally flat spaces, but it does not yet answer whether he entered the gestalt which also saw for, let's say, the fiber structure, if you would translate the early cut down, the fibers of the early cut down, could be the, somehow be the real component. And there is a rather recent work by Char, which reinterprets Carlton in these terminology, without you knowing it. So, just to make it more explicit. So my question, coming back to Iris, is, so first, is it true that he made, let's say, a later Carlton?

1:20:00 Is the step of Cartan explicit, or is the step beyond Cartan? Not only in the sense of, let's say, of what you call the set theoretical term of making the structures explicit, but even more, would there be a conceptual step from the infinitesimal to the fibrous to the finite or the integral to the finite? So, first question, is the step beyond Cartan? By the way, this is my impression. I don't know if the second question is right or wrong. And if it's something wrong, it's not what we're left with at the end of the day. The second question is very important. The difference is quite clear, because we all find out that when it comes to mathematical theory or algebra, when it comes to algebra, when it comes to mathematical physics, it's very different. So in this case, the whole chain of perspective issues might strike between what I find valuable and what has come from it. As to what the whole point is about, you could put it really out of the way. If I close the study, I think it's so linked to the previous point, considering that... The physical treatment is not a change of scale, it's a change of place. I think the whole change of prediction affects both, both in terms of moves in this direction.

1:22:30 But this goes into historical detail. Another question now coming back, let's say, the comparison of this first phase of your thesis, a second phase, and from what I understand you somehow criticize the report of, let's say, an implicit thinking of local-global contraposition in this first phase. In fact, I do not really understand why, and even when in your presentation you took, let's say, the opposite pair of the explicit, the explicit, in your phrasing of the la explicitation du couple local, global, etc., in your introduction, so I come back to the question, what is your criticism of the usage and the description of this earlier phase as an... Let's say there's a slow maturation and exploration of the dialectics of local, global thinking inside these random mathematics in an implicit form. So, strong viewing and not using objective images. I've been ready to use local and global, so I've been lucky to be able to study a lot of different types of physics and people who study the history of motion, the history of the field, the work of a human mathematician. So it's perfectly legitimate to use the terms which relate to data to understand what it is about.

1:25:00 I tend to think it is very fancy to use this term to describe what is done before they are ever used. But maybe, even without, I couldn't be clear on the fact that I would not propose this kind of reading on the text. And when universities use these specific terms, I think we've switched levels and sometimes computers use them to go deeper into the facts and then try to understand other ways of writing and describing the facts at least. And sometimes I think we've used other parts of the facts in addition. But in the end, your questions when looking at the material of the 19th century was guided by this view of your searching for the metaphysical use of this local-global narrative. And you just didn't want to admit it somehow in your reflection on this material. I understand the caution. Yes, but I have been told by mathematicians to do that. I have been told by Biden, by Adelaide, by Hodgkin. So, I can say, I don't know what those words really mean, but these three mathematicians use these words to comment on the work of other mathematicians. So, that's what it allows me to do. The fourth question, which is rather general and may be leading to a long rush, never a less-sized trial.

1:27:30 Of course, it indicates your analysis of the narration and research on the, let's say, the development of the local global layer, D-layers. They look at the centrally driving dynamics inside the development of mathematics, explicitly expressed. And the first third or the fourth half of the 20th century. So, this was a time of materializing of modern mathematics, a strong sense of the word in the sense of the 20th century, and of course you refer in several places to modern mathematics in the sense of set theories, structures, mappings, in front of you. Because you, of course, need them or your authors need these new tools, symbolical tools, for the explicitation of calculus of algebra. And, of course, now coming back to the historical context, one can be quite sure that you are zooming in a very important point for this development of the conflict between modern mathematics. And so I would like, or just to put the question back, but can you elucidate a little bit the role of this local globe and dynamics in the life of modern mathematics, put it so, to speak the other way around. These instruments from modern mathematics into your narration, could you elucidate a little bit? The image of knowledge and body of knowledge will be used as a way of presenting us new insights into the development of science. We will try to do something that is from the perspective of your inner plan of learning, presenting very general conclusions.

1:30:00 And for today, I mention the fact that I was drawn to this topic because it would help me understand the standard form of questions in the organization of mathematics in the second year of my PhD in Mathematics and Chemistry. And I rightly say that apparently I forgot the questions and I looked it up and was confused with myself. These include to understand the variety of ways in which, and the variety of channels through which, this mobilization of mathematics takes place. In some way, one can argue that, possibly to that point, we didn't actually get here at the same time. This modernization of mathematics. Oswald is very involved in the universe of theory and politics, of course, and ecology and literary and so on. And that's what you find in this. And it's a completely other part of the organization of mathematics which you find in Analyze Data and Mathematics. We have reacted to the presentation of over 30,000 questions. The idea of importing instructions, using graphs to import and export instructions, is something that is hugely valuable to the president, and it's also not to the core of what the Geographic is.

1:32:30 There is a variety of challenges of optimization. It's a radically big, huge challenge. But maybe we can reduce it. Maybe that's what you're willing to do, right? Sometimes. Ethical narration against the grammatical narration. In some places you yourself explained that you... I don't know, maybe you didn't have the time or it didn't fit to your program of research that you did not, let's say, follow up the dense development of knowledge from one person to another, let's say, in the sense of this type of historically causal thinking. Of course, there are a lot of things, but they are much more reflected in this, leading to this, of course, coming from your, what I call the epistemological view of this. It's a very strong epistemological view and that this results in the style, in an epic style of narration because you don't have to tell it in the letter.