J Nash's 'Real Algebraic Manifolds' (part 2)

0:00 In this database, this database offers the possibility to say, for example, looking at Whitney, he has a bibliography, he quoted certain papers. It is possible in this database to say that Wichme refers to Nash, refers to Lefshitz, and so on and so forth. okay so at this in this circle we propose to make a complete to put completely the reference quoted by the reference which are there it is It's important to find properly the set of references for which we have done this work, because it defines the quality of the information you have in the database. Because after, if we don't put the same thing, for example, for the reference of left-shed 1924, we have to know that from this point of view, okay and as it cannot be complete we have to know in what in which respect we have a kind of completion and this is very important because the interest is that it's very easy when you read Whitney to see that he's quoting Nash and Lefschetz but when you read Nash you cannot know that he's quoted by Nash By Whitney. By Whitney. So, thanks to that, looking at the reference at Whitney, you know that it is quoted by a paper in Category 3. Maybe the other papers quoting it, we will not consider that. You are sure, the point is that you are sure that if it is not quoted by a paper of this category it is not quote if you if you don't find this quotation in the database you are sure that it's not quoted which won't be the case for other references okay that's why so and the

2:30 Mark, you should just say that you have made such a database that exists, the entries are not typed, but the database and the program are made by Alain by the name of Thomas. Yes, the database contains a lot of data, and with real algebraic geometry, it contains, for example, the complete bibliography of the two books of Marie-Françoise and her co-author, and not only that, which is a starting point. Okay, so these days will be rectified, I hope. By authors of other books. Well, so if you want to make correction to this page, you can enter all your references and the reference you quote. And the fourth circle is just all the references we want to mention to have in the database. algebraic geometry so the difference between for the fourth and the third it's just that here we have done the complete work to enter the quoted references that's the only difference between i have forgotten something you can also add notes to say what you have to describe the content of the references okay and also there are these famous keywords that so any any one in these say fourth circle and of course properties of the four circles are of course positive for the smaller ones any paper in or any book in the fourth circle as is a real algebraic geometry keywords attached to it and and also the real algebraic geometry, meaning mathematics and historical. And to make the thing work a bit, for example, if you look at a reference there,

5:00 and it is in fact a reference which pertains to the circle, you have direct access to the text, because it's on the web, so when you are there, you have the text immediately. You just have to click one time and you have the text in front of you. The backwards arrows only are from the third circle or from the fourth circle? No, from all. Every property of the fourth circle is a property of the one, second and third. every property of the third circle is also property of one and first it's an inclusion so anyone who is in the first circle as for example his own or his old bibliography including the database the world papers which are on the database where all the bibliographies included and nevertheless they have not been read carefully with comments about them and so they are not in the second Another way to say that is that the work done on the reference increases when you go to the center of the circle. I guess the question I'm not positive about, the diagram you have just to the right, with these citations, it seems to me with this, what are the paper's texts that will have such arrows? Those which are in the third circle. Just in the third circle? One, two, three. Because three means one, two, three. But those in the fourth circle... There will just be references, there will just be papers that are part of real algebraic geometry, but nobody took the time to include their whole list of references in the database, because it is work. It's more work to put all the links than just to type the title of the paper. Because you see, necessarily, one reference quotes references which does not belong to our domain. You see? We can quote references in which we are not interested in. Tarski can quote many things in logic, which is not part of religion. and we decide for example to put Tarski 1948 paper we have to put the complete

7:30 reference and most of them we are not interested in but we have to put them in the database because we want to know all the references quoted by Tarski for for example, in this paper of Tarzan. Yeah, but I don't know whether there is some difference between the historian and the working mathematician, especially working mathematician like .. Working historian, please. Especially working mathematician like Guy, myself, that lived about 65 million years ago, tend to use indices, for whom the indices are very useful. an index of terms. No, no, no, no, no. There is an index. Index. Index. Index is different. The way I proceed is, I take a book or something and then look at the index. Sure, sure, sure. You know, and look for a specific term and then go. That's it. Exactly. So it's included I'm not in one and two. One and two contains one. Rather than the references, you see, eventually the references will come later. You know, the sequence, the web of references will come later. But initially, the process is what I'm looking for, whatever, a specific concept. Here I have a question. Alain describes only a small part of what TAMUS can be. TAMUS is a . You enter the references of the paper, you enter the date, the title, the author, the journal, etc. And then you have the possibility to indicate links. So one type of links, and this type is what we have to do, is quoting some other authors. So you have a link in this database, which is Whitley, Coles, and you have another little

10:00 fish with Nash, and so you have a link between two, the two of them, Whitley, Coles, Nash. But then you have a list of other types of things. For instance, you can have a link, use this concept. And so in this case, I mean, it will act, I mean, and I think very important to pick up that or Krivine doesn't quote Stone, for example. It could be a link saying that it's not quoted, for example. It cannot be complete on those links. No, but I mean. It's not only text, but also mathematical tools or mathematical objects. And also, to make some more advertising for TAMUS, then the idea of TAMUS, so that the idea that there will be some restricted project, which is real algebraic geometry, where the key words of real algebraic geometry are present. But TAMUS has also a lot of other projects on other parts of mathematics and other parts of history non-mathematical and and once you enter a reference which is so it means the reference exists in some way you enter it for yourself because you are interested in this project in real algebraic geometry but then it is also entered for anyone who's in the future might want to look at it from another point of view for example I don't know maybe someone in model theory will want to and then the references some references in the data theory will be already present and they can have their own keywords of model theory and So there is various levels of publicity also in Saithemus, right? Okay, so is it more or less clear, I mean, the picture of what we... So there is a list of journals. So, of course, we would have liked to present Tammuz and to even practice Tammuz here in Bel-Isle, but there were problems with the internet connection. But people will be lucky enough to be in IHP during the trimester.

12:30 We'll try to have a very active Tammuz activity during the trimester so that people will be invited to enter their own references. So, for example, let me tell you that the last time I worked on TAMUS, so I was trying to enter the new references for the new version of our book, and I noticed that Journal of Symbolic Computation was not on the list of journals. So, in fact, I had no time to do it. Even Alan told me I should do it. But okay, but I mean, I had no time because I was busy for some other reason, but I can enter, when I'm working on TAMUS, I can enter the title Journal of Symbolic Computation, and then next time that someone will want to use it, he will not have to retype it again, he will just be part of the list of journals, for example. So it's very convenient. very convenient. So it's really... and you can suggest improvements to Alain also and And sometimes, you just... But for example, the... How is this accessible? Just ask and ask. And since you're in Germany, in Germany, you have assistance, you know? In a way, sign it. So I can tell you... you know that you would realize that you have to speak again. Yeah. Sorry. And you want to include also translations? Yeah. There is a link from the . Yes, of course. Yes, but typically, of course also the existence of the circle too compared to the book means that these are translations in various languages, they can be included in this crown and maybe not in this one because we are not going to publish several. Because a given reader can prefer to read, I don't know, French to German, for example, or the opposite, so it's... Fabricio can tell you how easy it is to have an access.

15:00 Or both, or both. Tammuz, pourquoi ça marche pas? He has an access. Tu as un accès? Et ça marche? Oui, mais j'espère... I hope to put some things inside the back of the time. So it is convenient of the only real constraint that you have to be connected to the web. So for example, here, it was really bad. It was everywhere except Belil. So Belil had a very good, very good characteristics, as you noticed. So, you can explain why TAMUS is not an acronym, TAMUS is not an acronym, TAMUS is not an acronym. There is another website, so I have to find a certain name for Tammuz, and the third would be Latour. But Tammuz is not. It's an old Egyptian king. that you have potential controlling access to the third slide, or I want to open it to the general public. You mean that I accept your contribution too easily? You have the choice to make some part in this book for a storybook. You have the choice to make some part in the book in the general public. You can also enter notes and decide that the notes, The point is that if you enter a return, you are the only one who can change it. Anybody can complete it, but you will be informed that someone completed it. So then you can correct it.

17:30 if you are not the group. So the person we enter, the public part of the referendum, the title, the name of the author, what you can imagine, is the owner of the referendum. And then everybody can add some remarks. But as Ketcher just explained, you won't have access to the remarks made on the referendum. So the idea is to have a group which would be a source book in real algebraic geometry which will be a collective so that things we type inside and then so if you type a reference and you decide that it is inside this project then it becomes open to the possibility of adding keywords for example from the real algebraic geometry list and this will be visible only to our for the moment. And if a way that takes the return to keywords So, for example, I decide that the return has to do with real and in fact nonsense that there will be no keyword on it, so it won't appear, it won't appear, it won't show me a CS6, because it can't be returned, so it's not because someone entered a return to a lot of territory, we won't see it. The idea is that the proof is not at the right place alone. So if you don't have a cure to accept your own, you won't see it. But so what we'll do, I guess that many people will spend some time in Paris during the next trimester and we try to to make really as a very active and it's good if people can they enter their own list of papers ready of course okay okay so now let's

20:00 so let's now look at what was originally proposed as a as a list of papers to in part say one union two in part two of this diagram so this list comes with colors so as far as I can see blue color means that this text has been read in the working group we have in Rennes which is which started a little more than one year ago and then there are more colors And then, so the red text are those that we've been discussing here, and of course here it's not complete because Hermit we've been seeing other texts. And yes, and then there are... And then the black ones are texts which were included in the list initially, and we've not read them yet, neither in Rennes nor here. Okay. So we can... So it was on the website and maybe it's complete because I have my laptop, it's an old version dating back from January, I think. No, but it looks... Maybe not to date. Looks more or less, okay. Okay. Okay, so they are in... It's around in chronological order. Descartes, that's for the rule of Descartes' rule, the memoir by Sturm, the text of Sylvester on comprising an application to the theory of this function. Text of Hermit, which stands for the text that we have seen here. The text by Armack. So there's a text by Commissati, it's a representative of the Italian School of Tobacco Germanic and the Italian Commissati

22:30 So the text is the text of Martin Schreier and Artin, that we have seen here. Karski, Nash, oops, the text which was mentioned by Vakage, triangulation, the text of Mid-Marius, which was distributed, but not right here. My main or the basic number of real values So that's some mounds of the space in numbers. Tom, which is a companion to the place by Milner. Also, the results of Tom and Milner and the neural algebraic studies. Sullivan on combinatorial topological properties of analytic and neuro-analytic and neuro-atubaric spaces. This paper by Arnold, which is representative of the, which marks the beginning of modern some work of Russian school on neuro-rubial geometry and all the work of Arnold and people in Leningrad at this time, the school of Berkeley and his students.

25:00 Collins about the algorithm of microfiber elimination by hydrology by decommunation. So, paper which is part of the decommunation. This paper with Baker and Broker, representative of the relation between quality forms and real algebraic geometry. I mean it's not the best choice in this domain that this description of the reduced filtering and the track detection of signatures plays an important role afterwards with the study of signature bikes and so on. This art, semi-algebraic, you can calculate in semi-algebraic maybe, this is one, there are several papers by art of semi-algebraic sets, 25 sets and so on. But this one is one of the very important results in the semi-acryphometry. And Romansky sets, say, some expansion of finite S-resour to some transcendental functions. And also it's important in the way of, for instance, development of, you know, now as the hominimal, hominimal geometry. and so on, so this is from the perspective of this one, and Viro we have mentioned before doing methods of construction of curves or more generally of algebraic, real algebraic

27:30 Krikoyev-Korobyev is also related to hydroids for quantifier elimination or decision problems. Real solutions of polythonyal system of equations in the world. Well, of course there are some more questions, I think, concerning the result papers, which include or not, it's not very easy to say, it's not, it's probably this, which was, so of text, for possible text for this meeting, also the website. The meeting, the original part of Julia, can help in the future to try to define the identity of the text we want. Which I didn't know the problem. But I do not understand the so-called logic that goes from to the dead, so it's a well-fined interval and coherence. So we have to think of the kind of coherence we want of our people to reach, So, there were also suggestions by members, and I remember in particular a long list by chip, but unfortunately we don't have, we are not connected here, so I don't think

30:00 it's printed, I don't have it with me, for example. Maybe I would like to remark on phototic forms. Michelle mentioned that maybe Beckerberg is not the best choice. At least it's not the oldest choice. Sure. I think it's the paper of Vista where he presents his local princes, at least as basic as this one, or even Witt's paper, which is not particularly on the real people, but where his local princes about real curves. But it depends on how many you want. And also, I mean, since we have also this second circle, that for the moment, we are not really excluding anything, maybe. And then, yes, so certainly, I mean, who? But this is the reason why I asked earlier. What is the definition of real algebraic geometry? What real algebraic geometers do? working as opposed to the non-working I didn't find that I remember some of them could you, yes, is there something that I don't know is there, I mean, can you remember, yes, well, because the list was long, but maybe the ones you'll remember are kind of significant. There were a couple concerning the Fourier-Houdan theorem. Yes. So that's in the link with Descartes. Yes. And Sturm's theorem. It was related to Sturm. Concerning Descartes, I mentioned a few papers where they actually proved Descartes' rule because because the cards himself did not publish proof. You mentioned down the graphs. Oh, yeah, several of the Newark papers. Newark, 1888, on the non-representability of certain things, the sums of squares.

32:30 And maybe that's enough in the book, 1888. Because that uses some real algebraic geometry, algebraic curves in its proofs. We already mentioned Klein. Oh, and well, I guess Arden's proof of the uniqueness of the real closure was, he said, based on Gauss's proof of the fundamental theorem of algebra They also had the same ideas, and there was a fundamental there about the book, basically. Or if you want the next guy who had a little paper or similar to the book? No, no, but it's not the whole literature. It's this proof by Laplace and Gauss, which is very algebraic, and which plays a key role in the equivalence of various notions of Reuel-Busfield. So it cannot be. Yes, but it's not the whole topic of the theorem of algebra. That's certainly not possible. I think the sort of catching the specifications made by Artin, the way to prove the fundamental algebraic lab lab, was it not that clear? Yes, I would say. But from this point, I think Catherine discussed. Par rapport à la preuve algébrique? He called Gauss, so he can keep Gauss, but then there was a suggestion that he called not because of the reference, but because he had also the same proof. No, no, it's the same proof. But they could be... Yes, but he... Yes, no, but We are referring to a precise proof. There's that one proof where they reduce the power of 2, and then

35:00 by the 3. But if you call it category 4, or category 3, I don't know. No, I don't think so. It's a source of real exploration in the universe, and that I don't know how to prove it. No, I don't. It's that the proof of, say, we need, in order to prove the equivalence of notions of real close fields we are using a specific proof which is very simple and that's to this proof that we are referring we are not referring to other algebraic proofs like there is another one by lagrange and others it's for us we consider this as a source it's the only anything we can tell you, I mean, so... Laplace? Oui, this proof. Laplace? Yes, Laplace text. Laplace text. Yes, Laplace and Gauss. In the outer range. That's exactly, yes. I wonder how this has been compiled, just as you said, So what we did, well, so it's really certainly not at all systematic, and that's the reason why we are now having this discussion. So what we did is we started with a list of keywords, in fact, which was compiled by Savas And then we try to illustrate these keywords by what seemed to be a significant source for the keywords. Also, national, national, national, national, national. Yes, well, so, of course. Yes, for the moment we have just Krivine, for example, yes, we just have Krivine, so that's typically...

37:30 The difference of the Nollsteinsatz was frequently quoted in all the texts of Yes, the deficiency of the numbers of dead-blooded females were to affect the population. It was explicitly said because of this deficiency, it doesn't make much sense to them. So, then... The question is for real geography, this data or... Krivine. We've seen it yesterday. We've seen it the other day. We've seen the pro by Krivine. What we have seen is that Krivine took it, you haven't seen that it was the first. No, no, but that's the reason why Krivine is in this list, basically. And the version was stronger than the one of the . But, of course, that doesn't mean of... And I believe that in this second circle, there may be several references for each topic, but what we tried is to make it as small as possible, with a variety of, as diverse as possible, and as short as possible. So, for example, so you squirt Stengler, for example? This of course would be something. But you're... Karine on the roof. Okay, so cut out Stengler. Is there any reason that... Well, excuse me, but... Yeah, it could be online... Sure. ...comparison to Antwerp Frost, so... If we keep Stengel, we must keep... We must... We must... The press approved it, too, in... ...1975, or... ...1974... So it was, that was before. Stengler is 76, right? No, Stengler was 74.

40:00 It was 75, something like this. It was 74, 75. So what would Well, it was to have some Americans from time to time It's because there were a lot of Russians already, so again, but, no, but of course. No, but because maybe, no, I think again here we have to think in terms of this bigger circle. Like we said several versions of Tarski, then we can have Colleen Petroski and Tomei Miller. Now, the day where we'll think more about the book, maybe we won't have everything. of Abraham Robinson in his 1955 paper. Oh, that's true. So Robinson is interesting because we've been building it in Rennes, but it was not included in the list. Because he started this business of Marlott Derey. Rennes also had the Lounge. And Lounge. Oh, yeah. Here we have three of them. Thank you. So, no, I'm wondering, I think it makes sense to how it was forgotten, some parts of it were improved in a few years, maybe three years, so to create more context.

42:30 But that you have also to, there is also the databases which is including all its own references. So not all the text can be, because to include it you have to, we have to see that to include it in this set as a cost. the cost is that we have to read it and to write comments and uh and to present it and so forth and otherwise we have just you know we can put it as something really very interesting and then we're adding and we know who could sweat but we don't we don't read it in details writing comments so i wonder it seems not very costly to have backward references completely considerably in terms of how do you, where are things led to? That's exactly the point of the third circle. Once you enter the references, you'll know who, for example, you'll know where Tarski has been used. Well, then I guess I'll miss it. If you know that all these people here, if you know, so you have several hundreds of in this third list and for all of them we have the papers they are quoting then we can know no but that's the whole point that then what's the information in the database you can say who has used storm in his life and you'll see all the all these people because all the references have been included doesn't mean that we'll have taken the time to read and write and take comments so we have this so we have to this and again there is a cost I mean it's more costly to be here the cost is a human work

45:00 there is this Yes, except that how many men a year, how many men a year do we have to do it? No, the idea is exactly... No, no, I agree completely, but that's what we said already, but the question is... So, would it be possible if we say, right now we just include a large number of papers, probably to say, post-dicture and the text-nour-stern, and certainly, if somebody takes the time to read three of them, they can be moved... Of course, they will be moved, exactly, that's exactly the whole idea. Let's already be complete in terms of cross-references and include everything, at least in the third circle. And then we can always have someone who reads and includes it. But for the moment, the idea is to discuss what was potentially... ...into. ...don't find that... ...they stay in the firm level of the relationship... ...everybody who says, I don't... Thank you.