Building a Source Web-Book on Real Algebraic Geometry — Discussion

0:00 So, I'm a little surprised is that the next approximation, for instance, of differentiable methods in two steps. First, by going to a 90-1, then approximation of 90-1s by . So the little surprise is, why not choose the first approximation theorem? And in fact, the proof of Westras approximation theorem, if we choose the kernel to be, I mean, You could just approximate the given function f by some kind of gap. And if you choose this kernel to be something like, so it will directly give you a polynomial and a polynomial. No way So yeah, this is also a question. I don't know. I was thinking but this kind of approximation would also approximate the derivatives So he does it using some kind of Fourier.

2:30 So another thing that struck me was that in the modern exposition, this aspect of this paper of the manifold given abstractly and you define as some sort of abstract algebraic structure on it is not there because everything is after all embedded, so one could just start with an embedded smooth manifold and try to approximate it by an algebraic variety. So this extra level of abstraction is probably not necessary, but probably it comes from the fact that at that time people were really defining these abstract manifolds and sheaves of functions. It's actually the definition of the products of the ring here, it comes almost by definition of what is called now Nash Manifor. Also that Nash function and Nash Manifor came after The first part of this paper is the presentation of the notion of Nash function which relates it to the main current, say, of algebraic geometry. by making link with, say, presenting a Nash function, which are algebraic analytic function, as a terminology, as a real algebraic analytic function, is the function that you need to If you want to have a new class function there in algebraic Germany, if you want to have

5:00 a inversion term, it would disappear for instance here that this function which sends the point to the nearest point of the algebraic clarity, this you get by first function, but if you start with, this function is not polynomial or rational, it's a Nash function, analytic algebraic function. So that's what you need with the inverse function, to have inverse function term in the algebraic geometry. And this is related also to the, I think they found it in terms of the pan-lapse in Italy. So this is related to the type of biology which was then developed for biogenetics and so on. Potentic and other people. So it was a junction between the approach of NASH, which was the insulated plane, and the main current for a bunch of lunges. And this is this beta of Hachmini laser, which gave the name of the NASH function, which was used as well. Well, so clearly here some sort of algebraic geometry in mind when he defines essentially the spec in terms of his r and identifies the points of the manifold to the maximal idios. It appears that some, yes, it is clear from this paper that the Nash function is a Nash

7:30 This kind of real algebraic manifold may be seen as a link between the differentiable domain and the algebraic. So it proves that every compact differentiable metaphor that has an algebraic model exists. And also in the paper of Armin and Masler, they use this algebraization of differentiable objects. The fact that every diffeomorphism, differentiable diffeomorphism between algebraic manifolds may be approximated by Nash diffeomorphism, algebraic manifolds. The use is to have results on the dynamics of the differentiable mapping from functional dynamics to counting the period points, showing some exponential behavior, number of period points for generating your points by using the fact that you can approximate them by Nashmatics, using the Ajabai Cart of Nashmatics to reduce some count on the number of ballet So this, I remember also there is some notes by Ballet, which were never published, stressing also the idea that Nash-backing, Nash-manic form are a good tool between differentiable and I have done it. But this enthusiasm for Nash methods was one of the pushy by the

10:00 which I already mentioned, that with the Nash functions, Nash has no good grammatical properties so that all the tools that one is used to have complex algebraic, or even the real analytic case So I'm no longer available in the form of hash functions. So I think that becomes a consideration of hash functions. and only a few people used to be interested in theology by geometry. There were many works by Framson, and also by Sucsida, we had a pushback, and also some works of Bissler, and we discussed about properties of national actions. So it may restrain the development of the ultraviolet generators, not much used by the people in the United States. The only other remark I can make is that I once saw in one of these centennial volumes published by Amherst, some list of problems in an article by S.D. Yao, and he posed the problem of trying to prove some kind of quantitative version of this Nash approximation theorem in the sense that you are given a Raimonian manifold and Nash approximation theorem tells you that you can approximate it by this sheet of some variety, so you would like to know a bond on the degree of this variety in terms of, well, this epsilon of the approximation

12:30 and some global property of the manifold, and the one that he mentioned was some kind kind of average of the mean curvature, so I don't know whether people have worked on it a lot, but at least some people still are interested. I guess that was much of another point. So Nash-Wizard says that, from the point where MEO-algebraic writings have, well, say, national military, which competed, which competed in Nash-Wizard, say that the non-singular neuro-algebite varieties have nothing special, in the sense that every compact which actually were made for this ever-algebite model will be realized as non-singular neuro-algebite varieties. And so this, in some sense, was a starting point of the program to characterize the real algebraic varieties, possibilities, possibilities, singularities, among, say, the stratified So the main idea of this program was that in the non-singular case there is nothing special. Every differentiable manifold is a different model. So I started with some stratified set. We try to build a tower of the singularization in order to arrive to a non-singular set, then take a model and then go down some low-ups on the right side, go down to the other side in order to arrive to an algebraic model of the singular set. And this program was developed by a huge series of papers, and it was only, say,

15:00 martialistic sense that they arrived with a big characterization of two dimensions, really, But there are some problems with this process of desangularization going down. So that's not... Most of the... It's not the computer system. Now the computer idealization of the computer. So we want to go into that. In any case, it would be interesting to know if the algebraic model is in the same space. That's a lot of questions. That's a lot of questions. Agro-Gothè King calculates the abstraction, which is in an e-cromology group, but in this group it's different from zero or not, it's a long system. So, in Nermaut, Nash considers only the local question, but the fact that the near point is unique is also coming from global questions, because maybe there is another part of the variety, coming back to the variety, not only depends from, maybe the, if the variety is not compact, maybe there are several branches of the variety, many from passing a, a compact, but there is some piece depending also from the position that you think of, the neighborhood, the neighborhood, and depending from the curvature. It's a little bit more... No. If you have variety as this, this point, what is the near point? This or this? So the tubular navel can be a sweet tubular navel. In the Nash paper the question is that it's

17:30 Locally in the projection, but if the body comes back, there are two points, so one can prove that there exists a suitable neighborhood where the point is unique. But for compact manifolds? Compact manifolds? Yes, but where is the proof? where does he use that it's compact yes take this point yes where is the point the near point no it's in let me see it's just sufficiently small but sufficiently smaller since the last point of view is a point in the variety. Really, the point is outside the variety. So maybe the near point is on another path. Well, in the beginning of page 409, it says that we shall show that if x is near enough to x0. Now, but you take a point in the variety. Then there is a neighborhood. Well, all the points in the neighborhood are in the variety. Okay. That's because of the embedding condition. and then you have to restrict maybe to half of this neighborhood so every point in this new neighborhood and the way another part of the variety don't pass here it is too small enough for each point then as it's compact Sorry, but... So, in a sense, tell me that it should be... Maybe we should interrupt the long discussion... Maybe it's not a general discussion by your mind. No, no, only one read this, that maybe... I should be evidence that it's not completely... Je propose qu'on arrête la discussion générale. I think we should stop the general discussion and go on in the final discussion. I'd like to say something very smart. For instance, these kind of things are not completely explained

20:00 but there are some other things which are explained in very much detail maybe it's like the movies the part of the movie that we don't have today in our paper but for instance in the beginning of page 413 it explains that when he puts matrix l into a polynomial he will get the same eigenvectors with different eigenvalues which are the the evaluation of the polynomial in the values. He makes a very long explanation. What is this? What was it? 4, 1, 3. 4, 1? Here. So when I read it, I was a little bit surprised, because, I mean, in these papers, you don't get this explanation. He says that his device of putting a matrix into a polynomial, it's summed up. It's sort of a little inconsistent, because he explains that in such great details, but then he uses in page 415 something like the paragraph just before theorem 2, he uses the fact that by classical algebraic geometrical method of generic linear projection is possible to project without introducing new singularities and I think that this probably should have had a reference or something. It's certainly more sophisticated than the fact about eigenvalues. Yes, and also when he explained the conventions in page 410, then it starts with convention, and you have like one page long explanation of the sequence. It's not just one approximation, but a sequence of approximation, but it is said in a very

22:30 long way. I don't know, it's surprising. So also the modern exposition of the proof as it is in the book in BCR is in some sense much more sophisticated than this because more or less the same thing is done to this projection differences, but the map that one approximates is the, so you have a map from M and sort of the Gaussian map that takes point of M to the tautological bundle of the Grasmanian normal place and so and that's an atline variety and then there is a proof for that if you have such a map on which lands in an atline variety then you can you have some kind of general approximation theorem for this situation national proof is kind of it involves the it avoids this terminology of grass When you said such a projection introduces no new singularities, this was the general projection. This was, I think, quite classical because it was just a consideration of the dimension of the variety, say of the lines which were tangent to the variety. This can give some point where there are singularities. And another case is when there is a line that intersects twice. And then it is just a calculation of the dimension of all such lines, which allow you to project up to 2L plus 1, which is 1 plus the double of . This is completely standard. Then, if you want to still project once more, you have to be careful just to avoid bad lines.

25:00 So, I think this is quite an unusual effect in algebraic geometry, because it is just a consideration of dimensions. So we have a new session, collective discussion at five. Thank you. So, the idea of this last session is to have a discussion about the project of sourcebook in real algebraic geometry, what could be the next steps, the next tasks, and to discuss There's also webtext to include or not in the project, and first I would like Catherine to say a few words about sourcebooks. I saw that you like analogies, like analogies this morning. We find a good analogy. I'm sorry I don't speak about the bank ethics. I was suggesting that perhaps a good analogy would be museum. So I don't know what you know about the history of museology.

27:30 Museology. Museums. Study of museums. Yes, study of museums. So, we are museums here, and we are sort of here. And, um, so I will be killed by my colleagues working on the history of museology, but they are not here, so it's fine. Um, so, So, if you think 18th century, what you have is really a collection of curiosities, cabinet of curiosities, and that means something in terms of public, and in terms of, I don't know how to say that, type of organization or whatever. And the public is normally a small group of friends considered as experts. And so you collect interesting things and you look at them collectively with your friends which are considered to say very interesting things on this film. And I think here, it's very clear what it is, it's Belize, exactly what you have done, we have done this. So this is the situation now. So now, there is another step, which is the 19th century... We are a collection of curiosities. No, the paper. The way we have dealt with the text. I mean, you know, it's not better an analogy that you're bound cropping this morning. So, I'm just trying to explain a bit more about what we have in mind. So, this type you know very well. I mean, you have these collections of everything you can find of the same style, and you put them together like that and the more you have the best it is and the public is a small general general public and expert with the basic idea that there is a positivist view of knowledge which means that

30:00 essentially the general public would admire the same thing that the experts and we look at all And if I can think about an analogy for this, it could be a matter of use, for instance, directly. So you have exactly all the things you can find, and you put them together in good order. And it's exactly, you have some information on each item. And everything can be found that, it's not very that nice. Okay, so then you have a sort of modern style museum, and here, what you do is you separate the public and the experts. The experts are back in the caves of the museums, and they are looking at all the collections they want, but the general public is confronted to GELTS. So gems is, I don't know, something like that. So it's the inside of the... The gems. No, gems. Gems. Gems. Gems. Gems. Gems. Jewels. Okay, so it's a jewel if you want, crown. I mean, a very beautiful object, well chosen, and you put a lot of light, you put light light, you put a nice explanation about it, you put a specific environment and you look at it as an isolated representative of something, and it's very isolated and very naked, the environment, but very well put into life. And I would say it's the critical idea about what should be a source book normally. That is, we choose some gems in the subject and we put all the light we can find around them. And I don't know, there is a book recently about landmarks in the history of mathematics, so it's a bit like that. So you really landmarks So you really choose 10 or 20 gems and you put them really in the lightning.

32:30 Okay, so I will not comment the postmodern style museum, although it would be quite interesting to think about that, this analogy that I will spare you, but perhaps I will describe something like, I would call, as not quality as possible. It's a new type of museum. It's interesting because they are coming back to that now. So the new style in museums now, it's coming back to having much more objects exposed because the idea is really that you will put a certain amount of objects because only if you have sufficient objects and their little differences and their connections you can really appreciate the historical development or the little range of variations which is possible so you have really to present not only one object but somehow a group of things, so grouping according to some ideas and you have to think really different public because a different type of public could think differently about what they want and you will explain somehow the relationship between the different items and you will illustrate evolution, range of variations, that sort of stuff. So in some respect, you come back a bit to the first idea there, but the type of values and the type of connections which are made are very different. And in some respect, I have the impression that we are looking more, well, for that type of, so this equivalent of Sosbrug, but maybe not Google. Well, Sosbrug being web book. So this is a place where you can find not, of course, the macro views thing, but more text with a lot of connection between them. That will be drawn in a more convincing way and trying really to to understand more specific questions about the grouping of the text

35:00 and try to think about the different type of publics who could be interested in such a such a website concerned with algebra real algebraic geometry so both the specialists but also, for instance, historians, or perhaps mathematicians who are not specialists of the subject, but would like to understand the connection of their own subject with this subject, or perhaps would like to have a view about the general subject, and so on. And so for all these types of subjects, we can think of different types of grouping, which is, of course, helped by the fact that we are thinking in terms of a webbook. So I don't know if you have any comments of this very... So we could attach to that also, but I have not done that, but different types of history. So the qualitative history is based on accumulation, for instance. And you can try to develop the type of values and historical things that are put here. And, of course, the type of mapping, representation mapping that Alain drew this morning, more on that type, and also the type of thing I did with my Hermit, so my chief geist of Hermitian text, which I explained in the talk on Hermit yesterday was more also in this So in some respect, it was trying to explain what we have in mind compared to perhaps the type of source book you could have in mind. So, in here, we marked one. Right, it's everything to have his picture in mind to, during the discussion, to, as a point of view, to have some, Robert, the idea of such a source. to put a number of sources on the web, or publish a book? For the moment, we would prefer, given maybe this picture, and given also the work we've been doing,

37:30 we would prefer to talk on the source web book, and maybe there will be also a real book as part of it. But we think it's more adequate to think in terms of this web of text, in a way. Because if you start to think of the book, then the number of pages is restricted. Then you have to choose the gents, as Catherine is saying. And also we cannot vote, because it's not... We are not restricted. The difference between the book and the web book is mainly a difference of number, quantitative literature, mainly. So it's just a matter to define the restrictions. It's not maybe the first thing to do to start by defining the restrictions. Because it means to restrict, say, for example, the number of pages immediately. So you are kind of creating some constraints, if you think of a book. think of a web book, it's kind of more open model, and it's starting by the problem, so. So, do you want to present this idea of this various set of texts? Yes, if you will, then after we shall then put on the list. So... So... C'est peut-être mieux. Non, c'était bien. Je voulais pas. C'était bien, mais... C'est bien. Les grands cercles. Non, the concrete organization. The work to be done. If somebody from outside would ask you what is real artificial geometry, what would you answer? Read the source book. Nothing. Read the source book. Okay. I would answer, we are building a source book and when it's finished maybe I can answer. Okay.

40:00 Well, I'm going to start by the center, maybe the source book, and again the structure of the next circle will be the same. So the source book, and correct me, or those who are aware, correct me when I'm in the source book or the book? Source book, I'm starting by the center, the first circle, will contain some text. I don't know. You can put it in? You can put it in. You'll have other sources. Yes, of course. The text... Yes, but it's not that it will be a problem. It will be common to all things, but it won't be common. There will be a lot of confusion. Sometimes, the confusion will be a sense, and sometimes, it will not be a sense. Never mind. So, the text with the kind of commentaries we send to you by email, making distinction between the kind of commentary you are doing. You remember what I've been sent to you. So it is an analysis of the text, paying attention of the various kind of notes you can write on the text. Okay? So a complete analysis, explanation of the text from this point of view. Also, so explanation, commentaries, and so on. Also, a general presentation of the text. I guess what you expect to do, what you are expecting when you come in this business. Something that means you, what from a mathematical

42:30 point of view, you will find the text and present a historical presentation of the paper, okay? So, then we have exactly the same thing, the same thing online. What I mean by the same thing, it's the same construction, but with more text, because online it is possible to, we are, we can put whatever we want mostly. That means, for example, that for Tarski, we don't have to choose precisely one edition, if you want, and if we can put all of them, we can do it. Okay? So we can be...it's more easy to put...we have less constraints regarding choice. Okay? And the same work has to be done. Okay? So, there is no differences. That's why Marie-Françoise wanted that I reach quickly this point. There are no differences on the word done, on the second third, and the first. In fact, we have to start with the second, and the one will be the projection of the second. Okay? So, another point is to add a glossary, which is what you expect. We have, as you know, already we, that is, Michel-Marie-François et savoir-spiritéus We find a list of mathematical keywords, which appears to them to be covering the papers and the book we want to look at, and we try with Catherine to give you an idea of some historical keywords.

45:00 So, the keywords and all what we are going to say has a mathematical part and an historical part. It's always... You have always those two aspects. So, if I omit to mention one, nevertheless, it is there. So we have to write the glossary, and for some items of the glossary, which can write a longer development, both, again, mathematical or historical. Okay? Then, yes, if you want. We would like to produce also a chronology, both on the web and the version of NIT in the source book. A chronology, again,