Afternoon & evening talks

0:00 I was at a meeting with him, actually at a summer school in Roskilde near Københagen, where this was the topic of this series of talks. He'd mentioned that he'd been at that summer school. Yeah, but suddenly that was what he was interested in. Well, you know, he was the life and soul of the workshop in Oberwaldbach. He was great. He said they'd had a huge amount of admin to catch up on, so he wasn't going to be able to come in this afternoon. I think he plans to be there tomorrow. You can imagine all those idiot things. Oh, God, the Luigi's been around since the mid-1560s. It's clear that if art perceives their place, you know they must be worthy of it. Of course, he says that, but one of his own pals has the C.D.O. in 1991, call it the Medici bag.

2:30 Yeah, he's brilliant. He says it in a way. You can take it as a kind of a large, because many people made the same mistake about trusting the original Medici, of course. But they think they were moving in the right circle. Actually, let's just assume that all objects are moving in the right circle. So that's why I have this mission question that... Silvana? I never knew she'd been in the UK. I'd love to see her again. Well, of course she's now, though. Happy mother of your... Is it two children she's got? She has one so far, yes.

5:00 Is she still in Calgary? No, no, she moved back to Oaxaca. Ah, right. Where is Oaxaca? Oh, right. A lot warmer than Calgary. Thank you very much for your time, and I hope to see you again soon. That was actually the first paper I made digitally, because I was forced to, just to have it printed. Charlotte, one of the biggest bankers. But that was on a very primitive level. Well, they were. So, of course, they redid it, but it was before tech anyway. So at some time it has existed. It's very electronic form, but I'm sure it does not anymore. But maybe it does on MathSciNet. I'll check it out. Do you know that, Michael? Does the archive of rational mechanics and analysis exist on MathSciNet or JSTOR or something like that? It certainly should exist on JSTOR, but has it actually ceased publication now? Or is it still...? But if it's still being published, then it should be on JSTOR. But even if it's not still being published, you'll probably find it on JSTOR. MathSciNet, I'm not so sure, but certainly it should be on JSTOR. You should be able to check. Actually, we can check when we go back tonight for you, if you like. We only take a couple of minutes to check. I suspect it's probably on both.

7:30 JSTOR is not something that you have free access to. No, but you can tell, but you can access the thing which tells you which journals they have, you know, whether a journal is on JSTOR. You simply do Google on the name of the journal and then you'll see if it's in JSTOR or in MassSciNet. Even if you can't access it, you won't be able to access it directly online without your password and being signed up to JSTOR. But you can find out if they take it. I've downloaded it and found that it's illegal, so I put it on my homepage, I don't know if it's your own work, they usually, with maths, I don't know, there's protocols for them, if it's more than a certain number of years old, then I think you are entitled to do that, but it'll tell you on there that they have a page with all the rules for authors. I think that certainly with... MathSciNet, there is some, after something like 10 years, they allow people to do that, but it would certainly be able to find out very quickly, well, I'd love it, because of course, unfortunately, not being affiliated to any institution. Why wasn't the analytic topos there? It was. But it sounded very much interesting. It's a big interest. I even cited him. That's true, I do remember that, yeah. And I cited Gaga. Which I don't know whether has anything to do with the analytic topos, but I think it's sort of a comparison between the local ring classifier or the italotopos or something, the cross-italotopos on the one hand and the topos, the analytic topos. I don't know. I have never looked into it.

10:00 Is this what we call a challenge? No. Anyway, there's a fundamental result of complex analysis and theory of the analytic functions of several areas in a compact domain space, and you have to think it really has a great experience to say you are a movement of science. I agree with that. And so on and on and on and on and on and on and on and on and on and on and on and on and I'll be very appreciative of it. I'm really appreciative of it. Thank you very much. I'm excited for you. Thank you very much. Thank you very much. There are a lot of great questions that we have to make about it. I'm really engrossed in everything I'm going to talk about. Not exactly out of there, but I'm interested in it. In fact, it's even called Riemann's Existence. It sounds like a sort of functional thing, but it's generalized into something called the Existence. I thought that would be very interesting. So, I'll see you in a couple of weeks. Take care. I think that, at least, I don't know the size of the paper, but those papers back in the fair, I don't know, it's been three years. But you've got to talk about the names. The correspondence. The correspondence. I don't know the correspondence. I see it historically in my future. I'm going to take a movie, a birthday film, a film that I'm going to make. I'm going to make a drama piece, because it's a drama piece. I'm going to take it back to analysis.

12:30 Or you could have used these forces of the universe, with the suitability of the equation, or logic, or quantum mechanics. And to compute the linear dual, which is the function of our dual force, which is the general concept, but looking at its significance, we found out that if you consider a function in an open region, it seems... Then the dual space is also consistent. In other words, all intents are on a different domain. We move the complementaries around. We look at the complements, and on that we look at... Not to go general on it, but there is an exponential growth in the physics of the story, and that's the duality of it, and how do you make the duality, push and pull, to take the integral of f of w and g of z divided by z minus w. The difference makes sense because Z is in this region, W is in the top of the region, and W is in the bottom of the region. If the F varies, we have a linear function of F. So we move from... So what about the domains of these? What kind of structures do they have? They moved over from science and analysis in general to complex, complex, complex geometry, analytical. Then you come across Gagarin and you say, well, a really interesting case is what is facing you. Geometry algebraic and geometry analytic.

15:00 Yeah, it was a collaboration, yeah. Yeah, yeah. Yeah, yes. And Grotendieck. He talks about it in Recolte Semae and Sewings and Harvestings too. I think he gives it as the first example of what he calls a yoga, a notion of which he became rather fond. That's where I grew up, algebraic topology, using only simplicial sets. Did you sort out your own satisfaction in the discussion we had on, I think it was on the

17:30 ...versus cubical sets for the... that's the correct environment for the... I can't remember how that particular line of discussion ended, whether we reached a definite conclusion. ...the extra napkins, not me. Yeah, right. I think. I'm not sure. I haven't really looked at my shirt yet. I thought maybe you were an origami expert. Well, I would say there's no... One correct notion, I mean, that I have. Oh! Oh, wonderful! Oh! She can sense that something very bad has happened. I suppose the little girl can sense that something very bad has happened. I mean, because of Manuela being so upset by... Do I use that example? A fact of mind. Ah, yes, I know, I know, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the.

20:00 No, really, it's like that. So if you are a mathematician, especially a logician, playing is, you know, being an interpreter is quite satisfactory because it's a mathematical exercise, but also it's a kind of easy mathematical exercise because... Well, depending on the score, of course. Yeah, but you know, I mean, the structural aspects of music are relatively... I'm not sure I'd agree with that, actually. I'm not sure I'd agree. I'm not a musician, but I would have said that. I would have said, you try composing symphonies like Mozart and Beethoven, I think you'll find it's at least as great a challenge as me. Proving the deepest mathematical theorems, I would say. I would suspect it's an even rarer gift to be able to compose great music. It is to do, well it's certainly as rare a gift as it is to produce great mathematics. Yeah, of course, it's very demonic, it's very demonic indeed. The topos of music, well there is actually a book about that I thought. There's a whole seminar in Paris called Mammouth B. I've been to a couple of their... I must admit I was very unconvinced by the stuff that they came out with. Our friend Benabu was in fact speaking just this weekend, the same weekend that the PISL was taking place in Cambridge. He was giving a lecture at Mamoufi on the subject of the applications of topos theory to music. In Mamoufi it's this seminar that they have, monthly seminar which they have in Paris at the stands for mathematics, music and philosophy. And it's a seminar which has been going for a long time now, for about 20 years. No, no, because I was in Cambridge. But I have been to previous ones. And it was actually started by Pierre Doulez.

22:30 And I think, oh, who was the, I think, I think Giudonne actually was part of it at one time, Giudonne, it started as I said 20, 25 years ago, and it takes place at an institution called IRCAM, which is the school for the, it's the main sort of center of musicology in Paris, they're very close to the Pompidou Center. But they have a lot of mathematicians who come to give a talk there, both on expository talks. For some reason, French musicologists and musicians, both performers and theorists of music, seem to think that they ought to feel they have a duty to learn something about what's happening in contemporary mathematics. So they had a whole course of lectures on mathematics for musicologists there over about two years. Which were very, very responsible lectures. There was one on Lie theory, there was one on Topos theory, it was of course a very general introductory lecture, but it wasn't just hand-waving, it was a course of six lectures. And Christian Huzel, who Bill mentioned earlier, is a great algebraic geometer, a great historian of mathematics, who Bill has worked with, gave a lecture there about... Two months ago, I think he was the previous speaker, before Benabou, again, in his case, it was applications of topos theory to theories of translation, theories of natural language, syntax of natural language. They don't always just have exclusively music talks just to do with music, though Benabou's wants to do with music. They go online, you can actually listen to them. Overhead, so if you're at all interested, a week or so's time, I'll give you, it's a cycle, it's actually... The site that's maintained by the guy who does most of the recordings of seminars and conferences for our archive in Paris when I'm away, but he has his own website as well, a thing called consciousness.fr, but he puts all of the Mamouthi seminars on there, and there's some, I mean, occasionally, occasionally I think they're a bit silly, but more often than not they're very well worth going to, they're very well worth hearing.

25:00 We often have some really quite striking interesting ideas. Just on a slightly more humorous point, there was a great tradition in the Ecole Normale. I think we've seen great taste. But there was a tradition in the Ecole Normale. The weekly philosophy of mathematics seminar, the room where it took place, always traditionally took place, is two or three doors down the corridor from the room where the École Normale musical society have their rehearsals on Monday evenings and the philosophy of mathematics seminar is also always on a Monday evening. And for some reason, I don't know, it became an absolute tradition. They never did the obvious thing, which was to come into the room where the seminar took place before it started and take their piano, which is kept in that room, out. They would always come in about 40 minutes, 20 minutes, 30 minutes, after the seminar had started, and they would remove their piano, usually quite noisily. And then, because this seminar goes on a long time, it never lasts for less than three hours. Usually about 20 minutes before the end, they would bring it back in again, also with a great deal of noise and disruption. And this just became such an established ritual that everybody expected to just make a joke out of it. But on one occasion, we had probably the driest and the dullest lecturer I know in the whole of the Ecole des Mains. The remotest sense of you, glimmering of a sense of humor and um He was chairing the seminar that evening. I won't mention names. But in they came, they removed the piano. And of course you could hear them performing because there's only two doors up the corridor. You can even tell, you know, what composers they're playing. But on this occasion they came in, there were usually about six of them, removed the piano with a great deal of noise and class and took it up the corridor. And then absolutely no sound at all for the next two hours, twenty minutes. And then at the usual time, about twenty minutes before the end of...

27:30 The seminar they brought it back in with a great deal of noise and put it down and this person looked at them and said John Cage recital tonight wasn't it which I thought was one of the most brilliant bits of you know repartee you know joke that I'd ever heard anybody make and I would never until that moment have suspected that he even had a sense of humor. It was it was a pretty good joke They of course looked at him quite blankly because they definitely don't have it. For all I know it probably was a John Clegg recital. We never learn why they didn't in fact practice that evening. But it's such an established ritual that you can almost time your watch by it. They always come in about 20 minutes after the seminar starts and put it back 20 minutes before it ends. I think I'll have a coffee. I won't join you in a dessert, but I will have a coffee if that's... How about you, Anders? Actually, I think I'll join you in that as well. I'll have a half of bitter as well as a... I think I'll... Actually, I won't have another half of pride. I think I'll just have a half of larder. What? Oh, sorry, I'll have just a half of larder. A half of larder. Yeah, a half of larder. I think they have... Oh, they'll account... No, it's okay. Carling. A half of carling would be lovely. What? Oh, do you want me to go and get them? They have tea and coffee, I think. Yeah. Yeah. It doesn't matter at all what kind of a music it is. It can be boxed, it can go to the studio, it can also be rock and roll, but the worst part is what they have on television is commercial. You have what the Germans call an earworm, which can be a really serious problem.

30:00 That's what the Germans call it. It's slang, but they call it an earworm. Something that is impossible for you to get out of your head. It just stops you completely. So the worm is the particular piece? No, the worm is just a slang term for this inability to stop hearing a piece of music in your head. What is it in German? Ort, I can't remember. Yes, that would be great, thanks. There are two or three, there are two or three ads on TV. It's a horrible thing when that happens, I mean, I... One has been there for 35 years, and I'm just a slave of it, and so... Most people obviously, it's like, it's a bit like that thing, I don't know, there is a technical term for it, but there is a technical term for this and I can't think what it is, but you know that business about when, obviously, you know, 99% of the time you're not aware that you're breathing. But if you call your attention to the fact that you're breathing, then obviously for a few minutes, which in fact I'm doing it myself now, you are actually going to have to consciously draw breath. And there are people who in fact have this problem that they can't actually forget that they are drawing breath, so they have to continue, which means of course that they can't get any sleep. Well, eventually they can, because the brain has to keep telling them to breathe, whereas normally it's obviously completely subconscious, so you've got to continue. This is a little bit like people who have an uncontrollable pack of hiccups that they can't get rid of. It's just for brains. I'm sure I have to take 16 of those. Yeah. So here's the W.D.A. practice. And they're up there in a certain... There is a kind of lawyer who makes a lot of money by suing the insurance company. Justin Buffalo, man! Good lord.

32:30 We have those advertisements on British TV as well now. They never used to be allowed to advertise lawyers in the UK, but now, of course. We call it ambulance station. Yeah. It's a lawyer who looks everywhere for... Thank God you haven't gone. I forgot to pay. Yes, we know. We would be... Have you just paid? This delightful lady did make a very merry joke about your state of increasing obvious civility and decrepitude and the fact that, you know, those of you who are obviously your seamers at the table have not heard about it. Yes, the same story, of course, yes, because they take a percentage. Frequently an obscene amount of money. In fact, the actual victim never receives or receives them. I found out that the College Department has the same problem. That particular thing, if you turn it off in the first two or five minutes, it's stuck for days. In your head. Which is awful, because it stops you from doing any proper scientific work. It doesn't really stop you from doing work. No, but it must make it more difficult. It makes it more difficult. Well, it's a constant. It would be like having a... I have what you call a long-term relation with music. I love music, but I'm not, whatever that music is, I can't even define it. It would be even wrong if I played for you, it would be bad. I can't predict, it might be wrong.

35:00 Perhaps it is connected with some kind of memory, correct? Perhaps you are a person with a lot of memory, so that... And also, to the awareness partners, because probably if one performs, for example, as an interpreter, you keep everything under control, and so you, you know... You arrive at a theme you believe is something that you can't control, so perhaps that would be really hard for a professor, because it's difficult to understand them when you go by a lot of words. Have they ever done, not in your particular case, but I'm sure now they have such fantastically sensitive mechanisms for imaging the parts of the brain that are affected when we listen to music, or indeed even when we do mathematics. I wonder whether they've studied people who suffer from this particular tendency to be unable to get musical tunes out of their head. No, I'm not kidding. Oh, not very much. So... Well, he was in the CTO7 in Cabo Aro. Yeah, I've heard. I have corresponded a little with him since then, but... Yes, I understand. Not much I've been with him. We have to take control of this. And actually, that's one of the...

37:30 You're not convinced that... Those remarks about nuclear spaces were correct. Uh-huh. I wasn't good enough to refute them, but the flavor was not at all... He raised the question, and I tried to look on it, but I think there's just something not quite right about it. So I suggested, I was the editor, you see, or the referee, or whatever you call it. I was suggesting some improvements, asking some questions, and he made some responses, and he went back and forth several times, but I was still, so in the end, I accepted it because I couldn't repeat it. So I think if they're talking about convening vector spaces, it's just because they know that certainly that work of Berkeley and especially of... The idea that Friedel and Licor is truly relevant to this general circle of ideas, and therefore they want to have somebody who has read the paper and will describe some of the books. Actually, my main knowledge of it is from Marcelo Fiora, who last year went to Aarhus for a week just to discuss these issues with me. So my knowledge of this circle of ideas is Marcelo, and he's not going to Ottawa, though. He just had a baby. That's, so to speak, my contact with the discussion with Marcelo was, well, one of the things we could not agree on was whether to start with a Cartesian closed category and construct the linear category out of that, which is the way I cannot avoid thinking in that way.

40:00 But they go the other way too. They start with the linear categories and for some it is a closed category, all of that. Right. Let's see. Span, which is sin. Span? Span. Span. Which is just the other side of the Sumet-Gauss approximation sequence of the economy. I could, from the beginning, never see why the real needs. No, that's... Why it needs your law either, for that matter. No, no, no, no. Sorry. It seems to me the typical model is not something here. If you have a Cartesian closed category, you have to have a symmetric model. The next few years, I've heard, Solomon Edwards, when you just form your common commentary, it turns out that that, again, is the case of the Memorial Academy, you know, closed Memorial Academy. That's the basic answer, that's the first one. But all of a sudden, it carries a homonym because of its connection with the, uh, with the Cartesian case, and not a mononym because it's a linear case, and so, I mean, so you can see why it might be useful to have a category made out of these typical ingredients, namely a map that makes a space into a linear space.

42:30 Or it could be any of that space of distribution into some other form of science, such as the U.I.N. It seems like there really isn't a necessary diversion that's possibly productive to have a whole school just looking at all the kinds of things that are right here in the United States. I don't know of any other questions. If you start with the band, can you put sufficiently fine conditions on it? Well, my very rough idea is that you do produce out of a suitable linear... I mean, symmetric, monoidal, closed category, we are suitable, but we do use a quantizing closed category, such that the vector space objects in that category, but with all maps, not just the linear maps, is become closed, and such that the linear... This is the dialectics between the linear and the... I mean, this is a funny phenomenon. I don't know whether that's what you're referring to. But you have this precise and close category of vector spaces. That sounds like a nonsense quote, but convenient vector spaces. With smooth maps, is Cartesian closed. So, of course, it's not the full category of spaces, because there are many other smooth spaces which are not vector spaces.

45:00 But still, I mean, we know that... The proof that the vector space is, well I don't know whether it's possible, but I mean it's like the topos of the theory of smooth functions or something like that, the theory of smooth functions, the theory you put on the theory alone. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah. And in particular, the scene from the linear side, the idea of differentiation is put in the most detail of the Penrose, which is a construction on the linear side, and that's why... I recently have started re-reading my old joint table with Eduardo, the book where we analyze Taylor differentials or algebras for a theory like the theory of two functions. Or, more generally, for any Fermat theory, where you have a notion of K-references, and that's the way they conceive differentiation. I discussed a lot with Babsini, in particular, two years ago, because I was pressing the issue that From this viewpoint, differentiation should be a property not a structure. They put it in as a structure in the spirit of differential algebra, which I think differentiation should be intensive.

47:30 Marcelo's main interest in these kinds of definitions is the category of species in the sense of duality. That's really something. There you have the life of two. But we ended up with a sort of negative conclusion about this. Species don't have this uniqueness of different notions. There's too little cancellability on the addition side. I don't remember. All of these dimensions we were talking about, the differential algebra, the idea of a community of algebras.