Houzel / Houzel, finitude, quantification, deformation

0:00 The French University does not generally have a passion for those who focus on the borders. The question is unquestionable, and I leave you with the exercise of determining what exactly is unquestionable when it becomes unquestionable. The superior number from 56 to 60 does not disappear immediately, since it is the same number from 60 to 60. It is known, on the one hand, because it is exposed to the third feminine constant in local geometry, which we will talk about later. But also by the writing of a course on the theory of potential, by an exhibition at the Voltaire Museum on the function of the dictator, by his participation in the exhibition Remarque, published in the National Journal of France. We can already see that the landscape is described, it is someone who works on many subjects in many directions in France. He has three years of CMRS, he is named Chargé de Bourg. All of these have disappeared today at the University of Nice, where they became a conference center in 1970. I was a second-place professor at the University of Paris XIII, and I stayed in Paris XIII in 1991.

2:30 And it is from these years in Paris XIII that the first works published in the history of mathematics have been published. In particular, the first one in Paris is Le Lair and the formation of formalism in a collective framework published by Maspero in 1908. There are a number of works on various subjects such as elliptic functions at the birth of the theory of equations. In addition, Rachel will talk about the works of Fugel in the field of history and science. A very large activity is also the exhibition of mathematics, the notices for the universalist, the participation in the orientation committee of the Science Museum of the Guilherme and so on. He sees the importance of the history of mathematics for teaching and participates in the creation of the Inter-Irem Commission on the History of Mathematics. He is still taught in mathematics by several publications in the 1980s. In 1991, he was appointed professor at the UFM in Paris and director of the UFM in France. He was also appointed director of research at the CNRS between 1999 and 2004 as director of the archives of the creation of mathematics. He has taken on many collective responsibilities and has been president of the SMF from 1982 to 1984, founding member of the EPEIS in 1983, president of the group of reflection on the teaching of mathematics for the management of high schools and colleges from 1987 to 1990, member of the history of mathematics of the International Mathematics Union since 2007. And donors, some of whom are correspondents of the International Academy of History and Science and of the Academy of Sciences in 1996 and were created in 2020. Perhaps as a sign of the diversity of these activities, the diversity of the organizations, of the laboratories that have participated or patronized the Tournée de la Méditerranée,

5:00 And the History, Science and Philosophy Center in Marne-et-Légal, the REI, the foundation of the Human Sciences Museum, the Mathematical Laboratory of the University of Paris, the Mathematical Department of the University of Paris, the History and Science Department of the University of Paris, the Mathematical Department of the University of Paris, the Mathematical Department of the University of Paris, the Mathematical Department of the University of Paris. I've been studying science for the last 40 years, more than 40 years. I didn't know very well how to finish this introduction, but the solution often comes from a book. I don't know the name of the book today, so I'll start with the book of Dr. Postnikov and Dr. Denidov, respectively directors of the Banidov Institute. History of Science and Technology at the Russian Academy of Sciences and the Head of the Department of Mathematics History of Science and Technology at the Russian Academy of Sciences. Professor, congratulations! Your colleagues and friends congratulate you for your research and your work on mathematics and the history of mathematics. You have gained international recognition at the National Academy of Mathematics. Congratulations! These are indispensable studies for researchers. Your passion for research, your talent for languages, your dedication to art, make you a versatile personality who understands himself as an extremely talented master and an exemplary leader of science. We wish you good health, prosperity, and a lot of success in your research. Cécile Niaz, Director of the Department of History and Geosciences of the University of Paris. Dear students, it was time for us to pay tribute to this man. I fully associate myself with the name of the Department of History, Astronomy and Sciences, which has largely contributed to the creation, with the advice of Rossi, Michel, and in another way, Friedel. I would have liked to be present, but I am happy to take on such a role. At the risk of being attacked by the notice that is proportional to the statement of my lawyer, I cannot help but say that I will bring back certain facts.

7:30 You are undoubtedly one of the expositors of knowledge of the history of mathematics and the most important mathematics at their level. From the beginning, I have heard about the history of the theorem of Fermat, on the works of Lincoln, on the mathematics of Zadar, in any case, as we have witnessed with the students of the laboratory, you are an appropriate example. I have heard recently in Bologna, at a conference in Italian on the work of Poincaré, who participated in an innumerable number of lectures on the history of mathematics and mathematics. I have always been very impressed by the subject, which has always been one of our infinite talents. I have always been impressed by the extraordinary heart with which he lends his expertise in the various fields in which he has participated. I hope to find a long time that this role is really absolute for us. I think that the colleagues present will talk about mathematical experiments in complex geometry and algebraic geometry, about his role at the UNR and in the development of mathematics. I would like to start by talking about the type of history of mathematical physics that I am working on. First of all, and this should be a change in the logic of mathematical comprehension of the texts I am working on, is the constitution, the reconstruction of the mathematical directions of the texts. All the resources of history are in the work and the composition, but it is always to promote the mathematical substance. This substance is what I have taught to mathematicians and non-mathematicians. The only critic of formalism was Witten, and from this point of view, he was rectified by the formalist of mathematics, a very good friend of our fellow philosophers, and who paid particular attention to the importance of studying the fight against the monarchy of mathematical historians, a position that he defends, dare to say, of epistemology and the best of them all. Finally, the last point in this letter is not too clear, but I would like to express my feelings about Witten. I have never met anyone who is like him. In any of these subjects, a complex theoretical theory of geometry is used. The construction of a subsexual or notation of mathematical methods can be done at any time in any discipline. I only wish that, even if it takes a long time, I would like to give to your mathematical expression the red flags of the large-spread Michel Taki.

10:00 I would very much like to present to you, on this day of March, the contract between students and mathematics colleagues, mathematicians and historians of mathematics and historians of sciences in a more general way. I do not deny it. There are also major forces, such as the coincidence of this subject and the modest colloquiality of the papers, which take place in the same time in Tunisia, and which we could not, of course, derogate from these forces, but from this. The organizers of the two events were not synchronized, and each of them ignored the colloquiality organized by this principle. The aspects and the relationships of these pieces are not known to their other forces. This ignorance of these forces is often the most real aspect, which is much lower than the representation of each of them. We know both of them well, but also for the fragile and contagious energy of relationships and dreams. While I had suggested to the organizers of the colloquium that I would do a lot of research so that you yourself could also help us with what you buy for 15 years, although it will unfortunately be limited to 10 years. I learned that you yourself had invited the organizers of the colloquium to do the work, so it would matter a lot to you what you buy and what you use frequently. Knowing this made me very happy. It was a cycle among others that we lived in different worlds. We are all aware of the importance of a considerable amount of work that we have had to do together, Roski, Kouak, Christy, me and Michel, which has led to realizations that each of us can appreciate today. We are all overworked with work and education, and the youngest are probably even more so than we are today. And we will not intimidate anyone by saying that no one dreamed of celebrating the anniversary of the 20th anniversary of the foundation of the Béthelis team of the DEA. There is also the subject of doctoral training in the disciplines of history, science, and mathematics, depending on the first years at the Paris University of Physics. The years 2003 and 2005 were quite long, but of course we did not have to work for such a celebration. The important thing is to include, not in the non-directed courses, but in what has resulted in the realization of the movements to be induced, which is why it is necessary to explain to each of those who are involved, the importance and the current of ideas and works that emerge from them.

12:30 which contribute today for the most important to the vitality of the discipline, especially in foreign countries, by the international regulation of the IAEA and in particular by the education of the students of the University of Paris. We have said at the center of our objectives to take into account the relationship between the ethics of the history of biology and the history of science and real science as they are today and in the proximity with the scientific and also as they are done in history, Without omitting the political and social contexts of physics and the diversity of its dimensions, we could not avoid the first interpretation, which in response to all the rest would be of the urban psychos and derived from the modern civilization. I kept in mind the rich conversations that we could have during our coherent and long-term work, and the huge culture of mathematics, which made all of us generally, generously benefit from it. When you listen to those who approach you, and also to your openness and your understanding of mathematics, which is particularly important to me, and which concerns too little of my historical and mathematical science, the relations between mathematics and physics, so rich, yet, in the world of science and mathematics. But in particular, to meet with you, an attention and understanding that is much greater than that of men and women who know the aspects of the question concerning the problems of the interpretation of quantum physics as you want to formulate them today. And this proximity of thinking is for me a precious simulation. I would like to mention other significant moments that make us rich, such as meetings and meetings. For example, in Brazil, a few years ago, it was the mathematicians of Rio de Janeiro and of Sao Paulo who appreciated you very much and knew your works, but also in the old libraries of the Crescent, where you interact much more with the manual work of the French mathematicians of the 19th and 20th centuries, He testified for a more and more powerful force to a few major parts of linguistics and, in particular, to the languages of the Middle Ages to compete with the most well-known and most influential in this field, so from the bottom of the heart, to the mind, to the homage of the Romanticists, to the historian of mathematics, to the researcher, to the teacher, to the man of basic culture and the need for moral gratitude, so we are proud to see that this is a work that we will perhaps pass on to the program of the Inter-Iraq Al-Zoumou in five days.

15:00 The first erotic orator, Machel. Christian Witten, historian and writer. Hitler. Christian Witten, historian and writer. I think we can hear you well. It's okay. Can you hear me? No, I can't hear you. Then move on. I can hear you well. It's the violin that has to cry. The violin has to cry. So, inviting you to learn the word today, it's with pleasure. But I knew that the task would not be easy. Because, at NAMI, we are not the only ones who can bring facts and analyze them. It is true, however, that the philosophers have said before me that sympathy is often the best way to understand. To understand the course of the questions of the young, we must return to the young normalians who, certainly, are well destined for mathematics. I have already attended the lectures of archaeologists and have enjoyed the reading of linguistics.

17:30 This frequentation is the only one in which we will feel the early course of the character had a certain impact on its destiny. At the Normal School, he directed the preparers at this school and wrote alone, in a rough blur, the elements of the classical theory of the potential of the Velo. And with others, the Banach Seminar, which took place at the school in 1962-1963. It participates in the Boudement Seminar on the Autonomous Function by the study on the function of the integral line and the Cartan Seminar by the study on local analytical geometry. We are then at the very beginning of the 1960s. Christian Hauser had already chosen the field of his research and his discoveries, analytical geometry. His research will soon focus for a decade on the theory of analytical spaces. I leave to your colleagues the opportunity to analyze Christian Hauser's contribution to these fields, as we will do together in the future. I will not go into the context of the time. I will return to this work in the history of mathematics. I started with a bouquet of courses on the field of geology and linguistics, language. The path was already paved for the history and of course for the history of mathematics. The historian indeed is already well known among mathematicians. He is clear without doubt. and certain singularities of the path of the Christians to Earth. It is probably a creativity in mathematics that turned, as it often does, towards history. On the contrary, it is this historical inclination, which has already led productive young mathematicians on the field of current affairs, which naturally includes the history of mathematics.

20:00 Organized by scientists who were then scientists, the young mathematician developed his most essential research on the history of algebraic geometry. Without exaggeration, it is the different disciplines that have prepared his recent equipment. Algebraic equations, algebraic curves, collective geometry, integral calculus, and ancient analysis. I have just pointed out that new forces have been drawn up in order to penetrate this region where we have to visit the ancients such as Descartes, Euler, Bausch, etc. As well as his first work in mathematics history, i.e. in his article entitled Euler and the Apparition of Formalism. In his master's studies, his electrical and integrative function at IELTS, he left directly from the original memories, without stopping, for so many years, at the time of IELTS, secondary literatures, including the famous Encyclopedia de la IELTS. A good part of the historical work of Christian Cossette therefore bears the legacy of the Indian community, its long-term origins, as well as its contemporary current. If I had to characterize these researches, I would consider them to be inferior. And in fact, Tancuzel is a rare historian sufficiently armed to traverse the times and dominate the different periods of the history of our second, very distinct, a scholar of the original world. He has terribly refuted, on the one hand, the simplification of the incredible works of the... On the other hand, the ideas received are well-inherited and follow the spirit of the time of Sturgeon and Menard. A learned reader will be able to discover, in front of the diversity of historical works by Christian Rosel, a certain feeling of expression, a feeling that quickly disappears.

22:30 When a reader knows more, he can devise authentic strategies that enrich his research in history. This strategy has the main objective of avoiding those who threaten those who write a history of mathematics on a long-term basis. I look at the problem of science. Consciousness, first of all, and then imagination, if only. For a moment, the mathematical knowledge that must be dominated, the languages that must be mastered, the knowledge of the indispensable, To read Apollonius, we have Kawarizmi, Kaya, Bombelny, Lachey, Fermat, Euler, Lagrange, Gauss, Riemann, etc., before arriving at Pei, at Gruby, among many others. Yet, what are the lectures that historians offer if they refer to a serious history of adult geometry? Even Andre Beuil, with all the science that was his own, in his famous book on the history of the natural world, after a brief chapter on the biopharm, has really only started performing to serve the people. But this difficulty that an individual, isolated, cannot overcome, is not the only one. The history of a non-linear human being also exists. All of these topics are well-known in the world of mathematics and algebra. With Apollonius, as I said, he supported the data. A few years later, what kind of realizations were adopted? When can we say that a period is over and we move on to another? To which logic was each of them submitted? What are the beneficial elements? There is no answer to this question. On the contrary, it would be necessary to place at the top of the research that we should recognize both the textual tradition and the conceptual tradition that underlie it.

25:00 It is only at this point that we will be able to isolate and write the structures of effective rationality that coexist and exist. All of these are the bodies of the discipline. She made this choice, and for at least two reasons. First, the concept of research satisfies her demand for rules, because it responds to an interest already expressed in her first work on History, devoted to the role of analogies in discovery. In this first team, entitled Euler and the Apparition of Formalism, Christian Auzel sees, in the writings of Euler and especially in his famous introvert version of Grabo-Truth, the final point of a famous debate on the metaphysics of infinity. Euler specifies all his work on the calculation of an infinite execution and the calculation of infinity. This approach of Christian Auzel, who describes it as marvellous, All these terms represent a new orientation that has transformed the metaphysical problem into a purely mathematical problem, where, as described, the formalist position adopted by Heidegger thus allows him to evacuate the infinite of the infinitesimal calculation. Concreted with the help of metaphysics of the infinite. However, these debates on the metaphysics of the infinite of the people are in fact only analogies in mathematics, even if the words are not pronounced. Analogies between the calculations of finite differences and the calculations of infinity. It is these analogies that will later find their place in the theorem on the limit. It would then present itself as a first step in the transformation into a purely mathematical universe of vague analogies grouped together under the metaphysical and infinite titles.

27:30 A deposition of history, Oselle himself called it a new strategy. Christian Oselle therefore thought of a reasonable approach that would allow him to write... This is the long-term history of algebraic geometry. It consists, on the one hand, of describing the history of algebraic geometry, of Agarwalism, and the history of elementary algebraic geometry as we see it today. We are preparing to study the history of electrical functions and electrical integrals from Broglie to Poincaré, as well as the work of Epicard on the surface and those of Humbert on the super-electrical surface, the theory of the stream and the prehistory of architecture, not to mention the history of quantum mechanics. Here is how Christian Auzel declares the universe in his image. I quote, The algebraic geometry, as a mathematical discipline, is of relatively recent origin. It could be situated in the third or thirteenth century, at the time when algebraic theory was introduced. But its essence has been preceded by a long maturation with research in different mathematical sectors, primarily without a link between them. Solid complex, algebraic equation, algebraic curve, positive geometry, integral calculation, and quantum analytics. We have been conversing for 10 years and we cannot afford to study the history of modernity if we want to understand its history properly. However, Christian Rosen had to choose several points of attack. His study seems to reveal one of these points. The same, by the way, that he had aimed in his lecture on Euler, to show the unity of the same thought of Christian Rosen.

30:00 The real object of this article is still the analogies. These three are still the name of algebraic functions. The underlying question in the study of Christian Rosen is how mathematicians can even think of these analogies. And we are talking about the algebraic books of Cravalier. Where we encounter for the first time the analogy between numbers and quantum quantities. To then examine the analogies between the numbers of all the numbers among mathematicians of the 10th and 11th centuries. Then the series ends with Newton before stopping at the algebraic function theory among the successors of Riemann. He has always placed algebraic function theory at a higher level, in order to avoid analogies with numbers that have disappeared in a diagram. It is Didikine who takes the time to refine these analogies before interpreting the arithmetic function of algebraic numbers. For this reason, Didikine makes the algebraic population of the work of Riemann appear in this brief and regular article. Christian Moselle retraces the stages of the history of these analogies. We start from the analogy where the world is a continuous quantity, the founder of algebra, until the theory of the world, algebraic, and the theory of function, algebraic, both appear as particular states of the theory of the diagram. Following these states, we assist in the transformation of the variable notion of the analogy And these are implicit terms in theorem and mathematical theory. As long as it is a core, the analogy is evaluated once it is itself founded. We have followed there one of these heuristic ways that dominates mathematical research. At this point, the historian has discreetly put an hour of his time down. All of this could have been classified as serendipitous philosophies, or even as epistemological and mathematical history. Surely, this is not the so-called history, and perhaps it is this construction of history that has led Christian Rosen to address this kind of theme.

32:30 In this article, she reflects on mathematical theory, invented between 1957 and 1936, which led André Pau, in 1949, to announce this conjecture. This time again, it is a very common theory that holds Christian Rosel's attention. Their analogic theory. Here is what she writes. A certain metaphoric character is generated, analogous, and it is first inspired by the scene of the algebraic number, Artin, to go to the scene of the algebraic function of the barrier, Schmitt, to go to the scene of the algebraic function, Witt, with André Veil, excuse me, at the beginning of the algebraic courses. The construct itself results from the existence, for algebraic varieties, of a cohomology theory that is similar to that of the cohomology of algebraic varieties on the complex core. The starting points of the story are the analogy used by Dick Green in 1957 between the arithmetic of polygons... All of which are proficient in an entire corollary of the first number and the arithmetic used of the entire number. But they are not the same as the blue of the world. For those who have developed the numbers, the analogy that would lead them would be the number of the first in an arithmetic combination. And so many arithmetic studies have been applied to the quadratic extensions of the corollary of the rational function In an analog way, mathematical physics is a quadratic function of the relational body, already developed by David Christian Rosen, to pursue his study by examining the generalization by the pursuit of mathematical physics and the works of Hassel, and finally of André Bell.

35:00 We cannot here resume the article of Christian Rosen, already very dense. We only want to highlight the recurrent presence of this problem of the transformation of analogies by and without precision into mathematical propositions and of this illogical way traced by research to found the analogies. If we do not belong to this kind of history, we also belong to the thematic monography or to the monography dedicated to the French mathematicians. We have, for example, a monograph of a thematic, its lecture entitled is the algebraic resolution of equations, completed by the history of the theory of algebraic equations, Lagrange, Galois, and by Thésardier, Le Founy and the resolution of the theory of algebraic equations, general of the fifth degree. If to this we add his lecture on Troussy and Newton's polynomial, In fact, we have a thematic monograph of the history of the algebraic equation of Havarism and Galois, unified by the continuity of interrogations and the constancy of preoccupations. Another thematic monograph is the classic one today, on the function of the algebraic integral T. Christian Rosel, on the other hand, dedicated a few monographs to a single mathematician. In these two genres of monography, we successfully analyze the mathematical writings to obtain the history of a conceptual tradition that is formed and modified both by rectification and by ramification. It is by learning the causes that Christian Auzel opted for this difficult subsequent story, insensitive to the modes and laws of biology. There is now a story of differential analysis, to which he devoted all his life. One is entitled An introduction to the history of quantum mechanics, the second one is called The theorem of Fermat through the history of quantum mechanics, and the third one, less general and deeper, is about Poincaré and quantum analysis.

37:30 In this part, Christian Hauser analyzes the memory of Poincaré in 1901 on the arithmetic complexity of algebraic curves. Poincaré finds in this memory and independently the results obtained a year earlier by Hilbert and Horwitz concerning the research of rational points on the algebraic curve and the variance of these problems by irrational transformations to rational conclusions. They show that these irrational transformations provide a procedure that forgets all the differential equations of people at zero. Poincaré not only found these results, but more than a year later, he studied the rational point of the curve of genre 1, and he decided to make the curve of genre 1 a rational coefficient. Christian Rosette spoke in the term, the way of life in Poincaré and the resulting results. He continued the course of this research until Montpellier, in 1922, is our guitar. Even if they carry the essential of algebraic geometry and the adjacent disciplines, the historical research of Christian Rosen did not stop there. He wrote many stories and parallel theories on the mathematicians of Baghdad, Zahra, Kouroua, Mohr in 1901, and on one of his students, on Descartes, etc., etc., etc. He also wrote on Roubaix. On the occasion of this last year, Bourbaki brings me to talk about another dimension of the activity of Christian Rosel on the history of mathematics and science. I recall here two examples that I know directly and well. The one of the foundation of the research team and of the doctoral school, which Claire did earlier, which is now in the scientific universities. Secondly, the foundation of the archives of mathematical creation, whose first project was precisely the establishment of the archives of Robatt.

40:00 For the last hundred years in France, the history of mathematics was not endowed with an independent status, neither in the field of teaching nor in the field of research. At the Sorbonne, it was associated with the field of philosophy. and disposing, as I know, of a single position at the CNRS, at the possible commission. Also a professor of science and mathematics, it was an advertisement for the clear hours to which we would go to the retreat. However, there were very good scientists, such as Alexandre Poiret, director of the Religious Science Section of the Pratik-Rozet School. Jean Etat, who taught mathematics in the laboratory, René Taton, CNRS, and a few other people of value here and there. Things remained as they were until Georges Pannier gave life to the field, both on the plane of thought, which was not as useful as on the plane of institutions. The history of science has emerged from the marginality that has been relegated and tolerated. It is indeed under the influence of George Kandilien that researchers have begun to think of intellectual and institutional conditions in the development of science history. On the other hand, three people, I say three people, including Christian Auzel, who participated in the development of science history. These are the basic concepts in the context of new objects in the field of cosmology, history of science, and scientific institutions, that is to say, these reservist groups, which should bring together mathematicians, physicists, biologists, and philosophers. The ambition was to make them experience this tradition of history of science that is technically peculiar and theoretically clear. The AIS had also established a teaching program for future science historians without any precedents in France. This project was also prepared by a seminar on mathematics, first in Paris XIII, where Christian Moselle was a lecturer,

42:30 then at the Hormage School, before being simply created in Paris, which moved away from the village to be well received in Paris VII. In the last ten years, Christian Oselle has spent neither his time nor his efforts to ensure the success of this book and this doctoral school where he was the founding member. The visualization that I would like to evoke here is that of the archives of mathematical creation. Christian Oselle was the founder and director for four years. This project is a testimony to the construction of Christian Oselle. It is always made of mathematical history. His multiple training, through his experience as an active mathematician, through the various responsibilities he has assumed within the Mathematical Community of France, and through his contribution to the project of La Villette, Christian Oselle has managed to develop a form of conceptualization of the history of mathematics of the present. In mathematics, each essay distinguishes itself from the others in this sense that history always occupies the present. The position of Montreux-Jadis is a definite achievement. Contemporary mathematics are also, in this sense, objects of history, a history that allows us to better understand the near past and also to prepare grounds for future historians. The one who undertakes these tasks must therefore grasp, in this way, the sense of mathematical understanding accomplished under these eyes of logic. The future of reasoning archives is animated by the number of projects that Christian Rosel has written about the studies of Burbaki, the rest, etc. For example, in an article published by the Burbaki family, it shows the development of Burbaki. A Phoenician scholar from the 50s to the 70s has explained the origin of Burbaki by a dominant return of geometry. In another chapter, no less than 30, Christian Ozen, a mathematician and historian of the present, draws the sketch of the historical table of the mathematics of the 21st century, the pastiche of Lander de Condorcet, and draws the constitution of the archives.

45:00 In the course of the centuries, Christian Ozen had conceived the projects of these archives, which were transformed by interviews with mathematicians in Paris, in Paris, in Paris, in Paris, in Paris, in Paris, in Paris. The study of mathematics techniques began with Bourbaki. Once this project was developed, it was expected that the institution would provide the means for its realization. It was the CNRS that made it possible, thanks to the intelligence of Jean-Michel Demers, who was the director of the department of mathematics at the time. It was then that Christian Rosel founded the archives of mathematical creation and that he engaged research in his field before leaving for the U.S. four years later. To this founding role, it is necessary to add that Christian Rosel, a teacher and researcher, has been brought to the community. These students know and value the responsibility, the happiness, and the dignity of the teacher he was. These colleagues can bear witness to the quality of their writings, but also to their exposures and interventions at research seminars. All of them will also bear witness to their frank and disinterested collaboration with scientists. Before I give the floor, I would like to mention a few personal remarks of these scientists, friends and experts. These almost endless discussions enrich my thoughts and stimulate my own failures. These always precise criticisms, which I have never accomplished, have allowed me to avoid any setbacks and have always helped me to move forward. These encouragements, without malice, have accompanied me in these moments of encouragement that win all researchers. As for men, women and consultants, they know how to be very strong. It is necessary to defend a friend instead of a calumny, moderate, simple. The scholar has at least one passion, that of ideas. It is one of the people I know whose faces are clear of choice, before he is alone, at the announcement of a discovery made by another.

47:30 Thank you. From the history of mathematics to mathematics, it is assumed that there is a real border between the two, and therefore I give the floor to Pierre Javier. Well, thank you. First of all, I am very honored to be here in honor of my friend Christian. I hope that you trust me. So, I would like to... well, I have prepared an exposition which is quite easy. First of all, I would like to say that we have spent about 20 years together in Wittanus. We have shared a lot of points of view, both good and bad. We have participated in the creation of the Specialist Commission, where there were a lot of people in Paris who said that Wittanus was good. All of these have been written together in a book that was sent to me in the history class that I would not have written here, and I have used it a lot. I want to try to speak with you. We have also shared conversations about biology, because they always have to try to move the teaching a little bit. So I think we can talk about that.

50:00 I want to remind you of Viltaneux, we talked a lot about him. There is one, I'm not at all talking about mathematics, but he reminds me of a seminar, the Benabour seminar. Witten explained his categorical view of Tchaikovsky's works, which was extremely interesting, in fact, I did not have time, I had never written them, I was very impressed. Well, then, I still want to come to another subject. So, I'm going to talk about the field of finitude, a subject to which Christian contributed, but it is not him who has ... There are some very interesting ones that will be published in a few years' time. We have published a lot of history that I have written down. So, finitude. These are functional years. Personally, I hate functional years. I was too saturated with them in the 70s, when we thought that we were going to have all the games in the functional years. But... There is one domain in which functional analysis is indispensable, and that is to demonstrate finitude theorems in analytic algebra. For example, if you take a variable function, a differential operator, and a differential operator, an ordinary differential equation, The finitude of the nucleus and the co-nucleus that do not go through functional life is interesting. The finitude, I think the first theorem, well, yes, it's the following, I think it's Marcel Ries. Pay attention, there are two Ries, and as I said, I'm not a historian. I think it's Marcel, I don't know the exact date, but it should be 1930.

52:30 He says the following thing. If you have a space of Banach, well, if you have the unit, the Fermi of course, is compact, in the other direction it's easier, the dimension of the space is finite. I almost stuck a license exam with Hicks-Mead because I thought that the Fermi-Bornet was going to be compact. Hicks-Mead wasn't happy with it. So that's Fermi's theory. He has variants. I'm going to give you a variant right now. Take a complex Bornet. A complex of space of the type of, let's say, a nuclear flasher, or even a black flasher, that's enough, and suppose you have two complexes, so I'm going to use the notation E. There are several versions, suppose two things that each morphism of EI in ETI, so suppose for example that The two complexes are very complex complexes because the morphisms are compact and also because U is an isomorphic case, that is, the cohomology of the two complexes is isomorphic. So the cohomology of the two complexes, which is sufficient to understand that they are both isomorphic, this cohomology is of finite dimension.

55:00 Maybe you can read L'Horange-Farge, but not on this one. Maybe L'Horange-Farge is a fresh chain, a space that is both fresh and smart, it's a dual of fresh and smart, but with the same dimension. So, this kind of theorem is very useful. I'll try to explain why. So, this is the functional list. But the interest, as I said, is to apply this to geometric problems. So, a theorem. The definition of finitude is the following, you take X, an analytic variety, and you take S, a coherent plus 1 or X-module, a sub-coherent, introduced by Lancashire. This means that, if you will, locally, S is defined as the nucleus of a matrix of functions at the base. And finally, I don't know if it's the interest of, let's say, of cartons-serres, because there is the duality of serres and the finitude of cartons-serres, well, it's a bit difficult to distinguish this kind of collaboration. The interest of cartons-serres is the finitude of cohomology of coherent beams on a compact variety. I don't want to bother you with mathematics, with too much technique, I don't want to stop before that. How do we go from the abstract theorem on the right to the geometric theorem on the left?

57:30 For simplicity, I think L is equal to 8. Well, you take, I'm going to give you several examples. You take a unit curve, and you take two curves, So, you are making finite recurrences, and you can make recurrences with relatively compact Ui in Bi and the Ui in Bi in Schwarz. So, W1 is written in derivative-derivative language on the Ui. This thing here is represented by the Chech complexes. We are not giving you a lesson on the theory of the scissors, so you are making Chech recurrences. The Chech complex is associated with a recurrence, and then you take the same associated with the other recurrence, then you have more than to apply... So, we can use the acyclic recurrence of the ray C-complex to calculate the topology, and you have more than to apply this ray above it, because you have two complexes. There are many types of Schwarz-Schweitzer, Schwarz-Schweitzer-Witter, with compact morphisms, called Montaigne's theorem. We understand very well.

1:00:00 In terms of finitude, in fact, it is an example. We can give another demonstration, a generalization, in the following way. Always take X, a compact variety, and below that, take the real analytic. A de-module is an elliptic system, i.e. a system of partial and elliptic equations. I'm not going to give you the definition, but I'm going to give you what it implies. It implies that the solutions C, etc. of Petrovsky, the solutions C-infinite and the solutions of distribution are isomorphic. So, I'm simplifying, but I'm going to calculate them. The solutions... So I write the infinite solutions here, the same thing with the distribution solutions, and this arrow gives an amortization on cohomology. So, if we write this, by simplifying a little, we find that the infinities on them have certain powers, a matrix of differential operations in the first complex, and the second complex with its distributions, it's the same thing. With distributions, i.e. n does not make the presentation that I am talking about and that such resolutions exist locally, you have to paste them, there is a whole technology that is quite unique and that we are not going to do here. So what can we say? The field of regularity tells us that this vertical arrow is an isomorphism, i.e. an isomorphism on the cohomology.

1:02:30 The bottom complex is a complex of the type of nuclear cooling and the bottom complex is a complex of the type of nuclear cooling valve. So a variant of the theorem that is above, instead of having two cooling complexes, you can have a nuclear cooling complex and a nuclear cooling valve complex. If they are almost identical, you have finitude. These are always variants, except the theorem of Marcel Ries. There are many solutions of elliptic systems. X is compact when it is marked. Oh, I forgot that it is compact, sorry. I didn't say that the three harmonic functions are called dimensions. So, I want to give you a third example of classical finitude and then we will move on to the variants. So, all this is terms of finitude on C. So, the cohomology is of finite dimensions. The last theory of finitude on C, by Kachubara, says that if it is a variety, then it is a complex case, M, 1, x-module, or the other, then the complex of the solutions of N to Y.

1:05:00 So, what does holonome mean? I'm not going to say it now, but let's say that in the case of dimension 1, it means that you have a differential equation, other than zero, a real differential equation. In the upper dimension, it means that the characteristic variety is the Lagrangian. And now, we will move locally, let's say, locally, OX plus M0, OX plus M1, still with the matrices of operators of different scales, because here we have several variables, we are on CM locally. And the fact that I am at the bottom tells me, so let's say X, second X, and let's say B. Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Connes, Hawking, Epsilon, Witten, Con All of these are quite small, that is to say that the topology is the constant term on the balls. So, if you apply the vector section on an epsilon ray ball, there you will have a nuclear fracture complex. If you change the epsilon, you will have another nuclear fracture complex with compact morphisms in the area of my head, hence the finitude.

1:07:30 So you see, there is a mixture of geometries. For example, the Lezour character, Lagrangian, which leads to the existence of these balls, which leads to the fact that the balls are non-characteristic, and then there is the functional analysis. So as a case in particular, you have the case of an ordinary differential equation with one variable, where, as surprising as it may seem, I think there is no demonstration of finitude before 1970. So now we're going to move on to other definitions of finitude. So, the first situation of finitude where we are no longer on C is the Bauer's theorem of 1960. So, this theorem says the following thing. You have a morphism of complete variety. You have a coherent vessel on X, where the object, the derivative vector, is important. And you assume that the morphism is true on the support. And the conclusion is that the direct images of the coherent space are Y-coherent. So in case of a solid, that is to say a finite dimension, we can easily return to the case of a product. This is the case of the previous one, but with a variable base, with parameters. The core of Mars is no longer a core, it is an anode, but it is not even an anode, it is an accessorium of anodes. So there were various demonstrations.

1:10:00 I doubt that no one ever understood the demonstration. Well, except maybe those who wrote the real articles. There was Forster, Leutine, and Guilbertier. And there was Husserl. In any case, I think it's 71, 72, 73, things like that. But not all of that, there is a lot to be said about each other. The advantage of Mouset's demonstration is that there is a general abstract terrain behind which we can apply in any situation. I will perhaps say a few words about the abstract terrain of Mouset, because the variants do not give a more general terrain. There is a lot of technology, but they have a special capacity of the terrain of Mouset, which will help me with what I want to do next. I do not take the term physics alone, I take it on a fixed basis. We take an exact, a complete CAC, maybe even in the case of an interest, I take the term of the DFS type. And above all, the important thing is multiplicatively convex. Multiplicatively convex, that is to say, any border can be absorbed in a convex circled border. So, as you can see, these things, whether they are B-bordered, B-B, bordered, convex, circled, circled, which means invariant by the action of the circled, as E-B is made up of a multiple of non-nulls, that is, a multiple of multiples, and B-P multiplied by B-P is made up of non-nulls.

1:12:30 I'm going to go through the technical names, but let's say that the core, the spirit, the generalization of the collisual to the cardinal is to replace C by an algebra of this type or even better by an algebra cluster. So an example of A, it could be the function of a gene in a point. We will take other examples later. But that's not enough, we need to take a bucket of objects, a bucket of things, and I don't give them here. As I said, the field of Cartan-Seine, in fact, is generally without difficulty, with elliptical modules, and even the field of Grauer, we can put modules in history, and it's an article we did together, Now I would like to come to the type of your exposé where I think there is a word, quantification by deformation, because it is another, it is not very different from the modules, but it is another example, a new chapter, we will talk finitude at the end, first I would like to explain what it is, why it is natural, especially deformation by quantification.

1:15:00 So, let's take the word for Witten, which is an open atiyah, i.e. an open vector space on C of infinite dimensions, which is complete. So, if you have a differential operator, you can associate the symbol of that, it's a collinear, it's a collinear and it lives on the cotangent space. It's a function on the cotangent space. It's the power of minus x here and the power of x here. Of course, it's not intrinsic, but at this level, it doesn't matter. And then the product, the denominator of the product is given by the formula, sum of the value of the denominator of the denominator, which is the fact that it is given by the sum of the denominator of x, which is given by the factor. This is the formula for mathematics. Excuse me, we can't say absolutely anything else. So, this kind of calculation leads us easily, well, not easily at all by the way, not easily but naturally, to differential micro-operators on tangent spaces. So, first, the formal differential micro-operators on an open U These are series of infinite worlds with a certain M in Z, two pj of x and y, the pj are holomorphic but homogeneous of two pj. It is quite natural because if you want to invert an operator, you will come across such series.

1:17:30 So, in this case, you have a nano with the product given by, I'll write it down, given by the formula of the nanite. So, there is a sub-nano that is more interesting, I'm going to manipulate it, but I'll come back to it later, in historical terms. I haven't changed my text regarding the fact that Witten is in the sand, it was planned. So, historically, we have to mention Sato, Karouane, Kajiwara, who have done all the theory in a complex way, etc. But before, in a situation a little more particular, there was an article written by Witten and Monbelle, which defined the analytic products of the century. So the problem is that, well it's not a problem, but here we are working on what? These rings, O or P , they live on P , but they have a homogenous symplectic structure. If you remove the null section, you can see on P , which is a variety of contacts. So here we have quantification by deformation. And finally, it's about symplectic varieties, not homogenous. So how do we go from homogenous symplectic to symplectic? Well, as usual, by adding a formal variable.

1:20:00 So we're going to ask why I'm doing this between the variable and its inverse. We're going to ask that if there is no h bar, then our symbol will be a series of pj. H-bar is the time of j, and the time of j, from n to the negative n, to plus infinity, or H-bar, it's like the inverse of the Fourier transform of time, if we add a parameter to time, after all, and then, said like that, we have a very analogous calculation, and we can... In general, you have a theory on contact varieties, a theory on synthetic varieties, and on synthetic varieties, you have this H-bar parameter. Why H-bar? It's to have the artisticians, but it's also because there is a founding article of three authors whose initials I know. They have been the founders of the Euclid analysis and the H-bar parameter. Well, not the H-bar parameter, but the H-bar parameter. So, now if we write the unit formula that I have erased, we are immediately there. So now I'm going to work, to simplify, only with the operators in order. So, as I said, W is the sum of all these terms. So, first of all, what do we work on? We don't work on C, that's the subject of this lecture. We work either on what I call the 4-0, or on the square of the uniformity in H20, The interesting theory is not the formal theory, it is another theory that brings us closer to our subject, or K0, it is the formal theories, such as the existence of positive constant C, such as the series A, J, the factor J, and so on, and so on, and so on, and so on...

1:22:30 In other words, it's not the germs of analytical functions, it's when we divide by factorial g, it's analytical. It may seem strange, but it's not at all. This is what happens naturally after the factorial constructions of Sato, Kawa and Kachimura. This is also what was in the paper you just read. It's these hyper-differentials there. So, on a syntactic variety, we have, well, in the case, we have these algebras. So, now we can generalize. Historically, deformation by quantification did not appear, at least not explicitly, the authors did not say, as a generalization of the theory of the differential microbes, although it happened afterwards. But we can, we will forget that we are out of time, so we forget all that. Now we have a complex variety. And, look, OX, the series of coordinates in H-bar with a star-produced. So a star-produced, that is to say that the product... I'll take a little bit of your B-H-bar, then I'll spread it to you by linearity in H-bar. The product has its sum for J plus or minus zero. Pj G H-bar plus H-j. Where Pj is a differential beam tracer. In some conditions, P0 of FG is produced by R, and Pj1 of F is equal to PjF1 is equal to 0, for J less than 0. So, X is a complex variety, and it is a variety of fish, because there is a fish hook, which is given by P1FG minus P1.

1:25:00 So, in general, most people have the opposite problem. They talk about a variety of fish and they are looking for an ancient star, such as the associated hook, or this variety, or this structure of fish. This is the famous Ptomsenic. He says that in the real case, this is a complex case, in the real case, it exists. Since there is a variety of fish, we can find a star product that gives us this hook. We start from the hook and we look for the star product in the real case. But then in the complex case, it's more interesting. But then for X, complex, there was already in 96 Kashiwara, but in the case of contact, complex. And in 2000, so it's X again, in the case of fish. They have shown that there are indeed such algebras in the condition that, without algebras, they are replaced by algebroids. So what is it? There are two words, algebroids. I don't know how to pronounce it. It is necessary to pass, so this is important, like every insect, we have passed functions to the beam. There are a number of categories, some of which I am optimistic about, some of which I am not so optimistic about.

1:27:30 The reason is quite clear. On P1c, the non-morphic functions are generally defined on P1c. This is not the case. There is the same problem. If x is a quantity of fish, Well, there is no water stream, well, there seems to be no water stream, I think there are ontological instructions, there is no water stream at the IC3, at the start point. However, if we go together, category, there you have function, stream, algebra, algebra. Once you reach 30+, you move on to the two categories, to the stack, and then to the stack. And here, what we call a stack in Algiers, is an atymoid. So I had totally... I didn't even have time to explain what it is, but I'll tell you now. Let's say it's 30+, in this mountain path. So an atymoid, similarly, like an algebraic system. But it doesn't hold up well. And so all this, all these two cases live, in the case of Poisson, not on C, but live on either the form of a sub-anode, K0, which is of the type of a multiplicatively correct agent, of the type of DFN, that is to say... There is an algebra on which the Eusebius theorem applies perfectly, but we must not try to have the finitude theorem on C because they are wrong.

1:30:00 And in the case of contact varieties, we do not have the finitude theorem, which is interesting to note, the finitude theorem of Cartan-Seine. He says on a compact variety, we have finitude. So we could say now we see the microdifferentials and on a variety, in the case of projections, a variety of contacts. We have finitude, but finitude is the direct image of the object. And in the case of contact, we don't have finitude, and that's obviously because a point is not a variety of contact. On the other hand, in the case of a spectacle, we can go on a point, but on a point, we fall on the stage. So we really need this field of finitude. So if my time is up, I can... apparently there was... There is a lot to say, too much to say. Let's say that in the U.S.A. we have demonstrated finitude theorems in this case. In fact, we have done it in the former case, and there is also Schneider's theorem that is not published. Schneider's theorem is in this case. We have finitude theorems, which are generalizations of what they are, We can talk about the holonoid system for the fish algebras, and we still have the When I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say we are interested, I mean, when I say

1:32:30 There are also the terms of finitude or duality, even the terms of Riemann-Roch. In this case, your remarks or comments. To make the transition with history, we can note that mathematics, in fact, have a historical dimension, that is, it is subjective, and that if we do not capture it, we make the archives of mathematical explanation. If we do not capture it, when it is done, it may be lost. So without further ado, let's go to the next speaker. So there is Erwan Pachev, I will not introduce him in detail because there may be a few points on the subject, but of course we will include Pachev, Erwan Pachev, and perhaps it is the last one on the list, in any case one of the last on the list is the exhibition Debt, which he published his Debt in 2006. Speakers include ATER at the University of Paris-Seville, the Department of Science and Technology, and the University of Paris-Seville, the Department of Science and Technology, and the University of Paris-Seville, the Department of Science and Technology, and the University of Paris-Seville, the Department of Science and Technology, and the University of Paris-Seville, the Department of Science and Technology, and the University of Paris-Seville, the Department of Science and Technology, and the University of Paris-Seville, the Department of Science and Technology, This is a text that I studied in my first year at the University of New York, so what I wanted to tell you is the development of the effects of mathematics. We can't hear you. I can't hear you. Yes, it's very difficult. It's very difficult. Yes, it's very difficult. Yes, it's very difficult. Yes, it's very difficult.

1:35:00 Yes, it's very difficult. Yes, it's very difficult. Yes, it's very difficult. Yes, it's very difficult. Yes, it's very difficult. Yes, it's very difficult. Yes, it's very difficult. Yes, it's very difficult. Yes, it's very difficult. Yes, it's very difficult. Yes, it's very difficult. Yes, it's very difficult. But in fact, it's a text that contains two aspects, a geometric-algebraic aspect and a theoretical-algebraic aspect. And I'm going to try to explain how the two aspects combine. So, in fact, the concept that I'm trying to study first is what I call the same rationality of the rules. Which corresponds to the notion of an extension of the whole of the national numbers. Precisely, an algebraic extension of a transcendent extension. This is how it is constructed. This is the form of an extension that contains a certain number of transcendent quantities and then an algebraic quantity. These n are relative to each other. So, there are two... in his work, there are two... On the one hand, it is the study of algebraic extensions of the set of rational elements. So there, there is only one entity, which is not a political entity, which is good in this case. And on the other hand, there is the study of purely transcendent extensions, And then there is the third case, which is the end of the lecture, which is the case where algebraic quantities are added compared to the number of people who have never been on the continent, compared to the number of people who are algebraic on this planet. And to add a little bit to all this, the first case had a finite point of view. Some people said they refused the existence of infinite ensembles.

1:37:30 Well, it is possible to conceive a dwarf if there is no infinite form, but in fact the question does not arise so much, that is to say that it does not resonate in terms of the whole, but it still has a notation that defines what we call a dwarf. So it may pose some philosophical problems, but we can translate the mathematics of Croy-Eclair into the language of the dwarfs. So that's what I'm going to do. Well then, the question that arises is clear. In this case, if we note this form, it is clear that, well, we are going to note the major cases, rather, we are going to note the small cases of the National Assembly. In this case, we have the same density, of course. We can ask ourselves about the possibility of depending on this anti-anomaly in the same anti-anomaly in Algeria, where the probability is high or low, and where it is no longer present in Europe. We can ask ourselves the same question in this context. In this case, the anti-anomaly has several possible choices. For example, z, x1, x2, etc. We could also take, for example, 1xn-1. In fact, it is enough to take a factorial anode, and that gives us a dental anode that allows us to conceive a theory in this framework that looks a little like the FDU dental theory. So now we can play the question of how to expand this dental anode into a dental anode contained in the algebraic extension. And this is the question asked by Konekar at the end of this text. So I will try to deal with all these cases.

1:40:00 The second problem that arises is that Konekar used the idea notion. So, in contrast to what we do today when we talk or when we make a theory that is not algebraic, the theory of this entire universe is not in an algebraic extension at all. These are the major features of this channel, as well as the main factors of the three nodes. But this was not the ideal notion. So the way in which it was done was to introduce the forms to which it was determined. So in the case of this mass, I'm going to introduce a few notations, whether in the framework of algebraic theory or in the framework of universal transformations. I'm going to write C as the basic number, In one case, it's Q of x, in the other, it's Q of x1 to x1-1, so more generally, Q of x1 to x1-1. No, it's not that. So we thought of C as being the most common value, i.e. the value of the entries, so Q in the port where we were. And to talk about the relationship between the two, in fact, we tried to introduce the forms, i.e. There are a lot of elements of the same number on C that we call indeterminates. A finite number of indeterminates. There is no need to attribute it to infinity, but we all know that we have the same number. And so, in this formalism, the form S, the first character of this form, The term F corresponds to the ideal generated by the coefficient of F, which I will write C of F, and it corresponds to the variety of zeros of this ideal, which I will write Z of C of F.

1:42:30 There is a theory that allows you to understand the way of doing things the way you want to do them. If the gods believed in the gods, they are not in this article, but in another article later, in 1983, I would have to say that Grunitz-Hugues is 82 in 1881. So, the other thing is that if we have, so we know that the magnitudes in C are m0 to nn, nn to nn, nn to nn. And then M'0 to M'n, and we consider, in fact, we define the magnitude of M'0 to M'n in the following way, by saying that the sum of M'k plus k, so k to the power of 0 to n, is equal to the product of the sum of M'k to k times the sum of... So if we define the big N-pring like this compared to the big N-pring, the result says that the ideal generated by the N-pring is included in the product ideal of the ideal generated by the first N and the ideal generated by the N-pring, that is to say the N-pring.

1:45:00 And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. Well, the union of the zeroes and the trances of the numbers. It allows us to see that we can use the theory of the videos to think of an algebraic geometry that remains within the range of the zeroes and the trances of the numbers. Because there is this problem of radicals. The product is not equal to the ideal in the first part, but it is counted as an ideal and not a radical. Well, that was to justify it. So the second thing I'm going to talk about is how I do the geometry of the theory of the geometry of numbers by using the theory of the video. So that's the particular case of what I'm going to talk about. In this case, I'm just going to give an example and I'm going to announce the theories at the same time by saying what the example is. So the example I'm going to give is the term x. Here it's going to be a sine of 0.23.

1:47:30 So, in this case, it turns out that O is Z, of course, and in this case, O is not the main value, it is the main problem of the theory of physics. We are trying to restore the probability of factorization in geochromic factors in O , and to do so, we introduce shapes. So, in this case, an example that I cite is the same. So, we're going to look at the relationship between a value in relation to z and a value in relation to 1 minus the ratio of 1 to the ratio of 2, and it's not a principal value, for example, the ideal is not a principal value, so if we reason like Kronecker, we're going to represent the thing that we represent as an ideal, we're going to represent it by a form that is written as 2 plus 1 plus 1 minus the ratio of 1 to the ratio of 2, and what Kronecker does is that he works in a value in which the ideal is not the ideal value. And this allows him to work with us, in fact, to transform an anode into an anode in which all the ideals of an anode become principal because they are generated by a generator that is uniform like this one. So to make things clear, we have a multiplicative set of forms. When the cards are primitive, i.e. they correspond to the content of the quantum anode in the form of the content of the quantum anode,

1:50:00 and when they are together, we can consider the fraction anode in relation to this quantum anode, so this is a less than or equal ratio. And this anode is included in a quantum anode on O-4. There are a few cases in which there is less than one characteristic of Hawking. So it turns out that in this ring, the two ideas that are here are the same. That is to say, they are all equal. The ring is called A. And so, more generally, what we want to say is that this ring... And the main factorial is nothing but verifying all the probabilities that we would have won, whether they were verified by O'Connor or not. That is, all the probabilities of z that we would have been able to calculate by a quantum momentum. Another way of saying it is to say that in this... well, von Ecker also introduced a concept of co-dimensions, which he calls... All the ideas are generated by one element, that is to say that all the objects we consider are of dimension 1. I will come back to this later. I will be interested in the case of the transatlantic rings. By the way, there is another way in which we have been conscious, there is another way to transform the vocal ring,

1:52:30 All the ideas come together easily. I'm just giving you an example because I'm not able to do it in general. Let's say we're going to take an extension of k, which is the body of the decomposition, x3-x-1. We find that in this extension, all the ideas of k come together in real time. Poincare is aware of this, and he gives an example in his research, but it is really a distinct theory. Now I'm going to come back to this notion of co-dimension. So what is the co-dimension for Poincare? It poses several problems because there is no theory of ideals, so we cannot define the co-dimension when we present the chain of ideals first, as we can see. There is no topology either, so that's a problem. So the way to define the two dimensions is once again to consider the forms. That is to say, we will start from a form F, as before, where C this time is going to be. So there are only the terminated years, so it's going to cost 350. We are going to interpret, in fact, the other terminated years. We are going to write them here. So here I suppose that I have only one terminated finite term, N.

1:55:00 I'm going to introduce, I'm going to define what a co-dimension m is. So I'm going to introduce m as an undetermined system, m, then at the end of n, then what I'm going to do is actually consider the forms that I obtain from m. You have to think that m is a linear form of the undetermined 1 to n, and that the coefficients correspond to a certain ideal, which is the ideal of a certain variable. We are trying to express, by means of this form, the polydimension of this variability. So, what we are going to do is to translate this form of being, by replacing the undetermined places with n, by each one of these n systems of indeterminacy. So, for that, I introduce morphisms that exist by J, each time in C, and which send each one of the... Theoretically, there is an M-shape, such that the ideal generated by the coefficient of M is generated by the M-shape. Theta a of f is equal to theta f of m.

1:57:30 But we can also demonstrate, at least in some cases, what seems to me to be quite vast, and so it allows us to define the dimension of the two dimensions. So this way of defining the dimension of the two dimensions also allows us to well understand the method of elimination of Kornetker, which he exposes in this work, in this article. So we're still talking about forms, and the first-person definition theory will consist of trying to decompose the variety of...