Discussions, incl. (pre- transit) FW Lawvere, C McLarty, M Wright — Algebraic geometry & other topics

0:00 We have the history of the quantum and the mathematics of the 80s and 90s. And the huge... So he was, how long was he going to be a scientist? Exactly, so he was old, he was almost 80. He was born in 1985. The things he said about had a... His internment basis was Audemars. Audemars was the center of the mathematical. You don't want to do my theory. ...organization of mathematics by a general... ...but he must convey it in some way because he's almost my only source for that time. And I do have that information that the Adamar seminar was in for. So I must have sort of gotten... Well, there's a big book on Adamar here, which I also have. Again, sitting in those bloody boxes. I'm trying to get my library out of trouble. And you don't want to swallow all of it, but you certainly want to read the book.

2:30 Yeah, the informative, informative. You get that? I mean, many of these are wrong, and the whole plot is wrong, but part of the information is correct. Well, it's conceptual reconstruction of history, as Burbaki would like it to have been. I think that's the point. It's hard to read. If you don't know a little about it, you cannot learn it from there. You know, Dieu Donne probably didn't have, you know, democratic discussions and fights over texts and books of that sort. No, no, I think that was left to the secretary after the party. It was written mainly by Dieu Donne. Right, that's what he tried to get. Without any discussion. Exactly, right. Whatever he thought it would be. Stalin did, but in this case it really did happen. It was the secretariat was directing everything without consulting the members of the party. And also the exercises I'll say. I mean, the exercises I'll say, don't say by do's and don'ts. They just have to be everyone's say. And they are good. Perhaps this is the best thing we've made of it in the Burbank table. They are high, it's a complete hollow earth. A lot of people think of him as a serious historical, well, yes, I've forgotten that.

5:00 The one I'm thinking of is in the Kitcher and Asprey volumes. Well, that's the one I'm thinking of, too, the Kitcher and Asprey volumes. I would say philosophers of mathematics do regard him as a major historian. A smart, careful guy doesn't... He's still pretty active, isn't he? He must be quite, he must be getting on a bit now, on a, well, of course, yes, because he must be in his 70s. I've been into philosophy because of that now. Bill Tate keeps up with me. Bill Tate gets me out to lunch every once in a while. There's a lot I don't like about Bill, but, you know, he stays close to this guy. John Bell and John Mabry both, I've read some. When you say there's a lot you don't like about him, you talk about his work, his blinding. It depends on what time of day the train is. I think it can be between 29 and 30 minutes.

20:00 I think it's in some sense, it's a model of that. Well, it's okay, I'll get the model. I'm just going to get rid of all my loose change. Well, actually, having said that, actually, we want to keep some of these, because when we get to the Gare Montparnasse, with all this luggage, it would help if we could use the chariots, and for those, you need a one. Well, I can give you a change. Well, no, what I'm saying is keep hold of these, keep hold of the ones, because with these, we can get the chariots, the cars, when we get to the Gare Montparnasse, which will save a lot of time. There's quite a long walk at that station from where the main TGV train comes in to the subway station, so I think it should be helpful to have that. I've done that many times and it's a pain in the butt, actually, walking out there. Where do you want me to get back? You've got enough there. Yeah, more than enough. I think it may need to be...

22:30 The third story is in any way... I mean, the fact that it's a proper class is easy to understand also because, well, I mean, in various examples, like graphs, where you have these chains of monomorphisms that are arbitrarily long and there's no way you can collapse them. You just reminded me of one topic that I never get around to discussing, which Alberto submitted on the list of his suggestions, but I'll... We'll get round to it, we'll get round to it. What topic was that? Ah, it's this whole theory of the origin, a symbolic one, quite a very interesting idea. These people are saying, they'll rather kill me than not let me be in a funeral. No, okay, we understand, we understand, you can't offend them. You are in Paris and you will not come to... No, no, no, you can't, I understand. But do you know for certain which plane you're catching yet? No, I... Because if you're staying till tomorrow, we, you and I, and John and me, if you like, could go and have dinner there. Okay you will be at this hotel? Yeah. You know the phone number? Yes actually I think I do because I had to ring them yesterday to confirm the booking so I think I have that here.

25:00 Oh this reminds me you did all take your pads didn't you? No I didn't. But I mean that some people have written quite a lot. Oh you took the stuff you'd written good because if not I can mail Here we are. Here we are. Here we are. Yes, I do have their number. It's this one. It's 0140378850. That's the phone number for the Hotel Milan. It's the same train, right? Same train. We're all going on the same train. What's the name of the hotel? Hotel Milan, like the Italian city. And where is it? It's in a street very close to the Garda Nord called the Rue Saint-Quentin. Well, we can... U-U-E-N-T-I-N. It doesn't matter. But it's where you'll probably be leaving your luggage anyway. No, I'll take it now to the hotel. Oh, you'll get straight to your hotel, okay, because Bill and Colin... It doesn't make much more sense if they left their luggage in our room at this hotel and then took it on later to the airport. Okay. It's okay. I have the number. I will leave you a message if they know exactly the number. That's the number of your hotel at the airport of your booking. That's the reservation number of Bill's booking. Yes, yes, I rang them yesterday to confirm it. Good thing I did, because in fact we thought we'd done it over the net the night before, but we hadn't. I was right, my suspicion was right. They had been registered the booking, because they didn't take a credit card, but I gave them the credit and they do now have the booking. Just a minute, if I am in Paris, I dial 01 or no need for the 01? You do dial the 01 even in Paris. What you don't dial is the 01. Oh no, no, no, stop. Here it was the 02.

27:30 Yes, here is 0-2, Paris is 0-1. You don't do 0-1 even in Paris. I also have a location map of your hotel. It's okay, we'll sort all that out on the road. No, it's not the first one. There exists a stochastic section. Already exists one. I don't know, I mean, I'm just starting to play with this, but right now it looks real scheme theoretic. Stochastic sections never fail to exist for attention reasons. If an extension would give it, it already exists. Uh-huh, that's my question. Yeah, in the cases I've looked at, yeah, I mean, you look at, say, the square root over the root. It finds the imaginary roots over here, even though you didn't ask it. You did this all in real coefficients, real polynomials. You know, it looks at the scheme points. Yes, I see, yeah, yeah, yeah, which are... So you've got x squared plus one. Yeah. So, yeah, when I first drew it, I drew the suit over here, and I drew a sort of a... The stochastic sections react as scheme points, so they already orbit the action of the orbits, I mean, in connection with this definition of the stochastic section.

30:00 When you look at a variety defined over the reals, you look at x squared plus y squared equals one. We're going to think of this in terms of real polynomial. And so you think, well, that's the circle. Well, but it also, it has roots. It doesn't have, but it does have these degree two points. These are the root and its conjugate. The scheme already has this, because the scheme registers the whole arithmetic of the polynomials, and x squared plus one is a perfectly good polynomial, even though it has no real roots. It's a good real polynomial. And the scheme registers that as a kind of point. Now, if you pull back over the complex numbers, this becomes reducible. The factor is x plus i times x minus i, and it appears now as two separate points. Which is a Galois orbit. Right, I understand now the connection. I mean, this analysis, in terms of points, in some senses, you have this, in some sense, more variables than the other one. A few lesser and more variables. So what the stochastic section does is, given a function that depends on the more variables, it averages over the fibers and gives you a function. So if the fibers are the roots of the polynomial, whether the roots exist or not, you can still average any function of two variables with respect to that. So this is why you spoke of it as a kind of coarsening. Well, no, the coarsening I was thinking of was... What is the time difference between arriving with the bus and the train?

32:30 Oh, huge. Huge? Okay, because it's a... We'll have over half an hour when we click. Oh no, we'll have all over half an hour. The train is at five minutes past two. The bus only takes an hour and then we'll have, we'll have half an hour. The train station is? It's right next to it. I should have connected the distance from here to that building. You were saying that there's this different aspect in connection with which you introduced the notion of this kind of coursing of the, I didn't understand that, I'm sorry. That you might consider the existence of A is less important than B. It might not exist where A and B were originally given, but it might exist on a covering. You have a covering of the situation where they are, and you pull them back, and then you might find a less than or equal relation there. But that defines a new relation on the original things, which is a coarsening of any original relation. It's really just one of the hybrid categories, a functorial situation where you have this distance of a map. That's all we're really talking about. So you have a partially ordered set which is a functor, a contravariant functor of some base objects, but each one has its own ordering. But now there's a new functor. I say the elements that exist here are said to be in the order of the objects that have to be covered. Absolutely. So again, the point is that distance may just be existing.

35:00 Yeah, this is meaningful. At each step, and then you have a new, so they're different. One function and then the coarsening of the other, induced by that. Indeed, if you pass the sheaves with respect to that notion of covering, it would be limiting your world so that the foreshortening looked like the original one. Looked like the original one, so it was no longer a cause, it just was the idea, I see. Yeah, I think I, I guess I do. It's very subtle. I was trying to think of an example. There seems a good age, like you say, it's quite old. Oh, a good age. Okay, that's very, very helpful. I mean, because obviously if you took a general map, it wasn't necessarily a covering. Pulled back two elements of the codomain back to the domain. Of course, they might even become equal, and they weren't before. So in particular, they might turn out to be an orderly division. But if... You know, so it's much more restrictive to ask that for a map, which is sort of an interesting... I mean, Grotendieck's topologies always seem to be connected with vibrations. Objects, a category of objects that depends on a parameter space pull back punctors between these categories induced by maps in the base space. And then you want coarsening properties in general, stating that some relation... There once, on one, over one base space, is actually true attributing that back to the original one. There could be some sort of sheafification. Of course, this was how I first understood, you know, how to think about sets as fitting into those pictures.

37:30 The sheaf's on the one point space, so there, of course, is the sheaf. Somehow, the descent for a particular vibration, the notion of a geobranding topology of fluid... They all apply to lots of vibrations. So typically the Grotnik topologies, which are interesting, are chosen in order that the descent problem will have a nice solution for a given, for a certain vibration that you like or are interested in. What does descent mean? It means that you have a thing here. Well, what if I had it after covering? What if I give it after covering and then moreover I give a notion of equivalence of such thing by going to a further covering? Is that enough to say that this all came from something actually at this stage? Yeah. Descending back. If you just take, you know, so the base, for which that's always true, the reasonable notion of Grotendieck topology on the base, because then you can, because then these maps will become epimorphisms. Yeah. Even if they weren't basic results, epimorphisms are the descent morphisms, just for... Well... Well, why did it say on the timetable? Well, I'm fortunate I left the timetable back at least. This is crazy. Are you going to Rennes? And where's the next departure? Yes, it's in 15 minutes. 15 minutes? Yes, it's 45 minutes. It's not a departure at 12.30? Oh, okay, thank you. And that timetable was a current timetable. Yes, yes, yes, there is a, apparently it's not till 12.45. I don't know why. My timetable, well you saw it, said there was a 12.20.

40:00 But it also said M.E. on it, you know, the one from Winston. Well, no, I looked very carefully and it didn't say, it was the 12.05 that it said it was Mercury only. This one was supposed to 12... I'm only going on what was printed on the timetable that I had, which I checked was a current timetable, it said there was a 12.05 departure on Mercredi only, which didn't go to Rennes anyway, and then there was a 12.20 departure, Lundi-Avandredi, which went to Rennes. It turns out it's 12.45. Okay, they will appear to have changed the timetable again. Guys, I don't run the bus company. Even the 12.45 is still going to get us to Rennes in plenty of time to catch the 5 past 2 train. And in the event we miss that, there's a 5 past 3. It'll still only take two hours. You'll still be in Paris by just after 5. But I agree, it's a pain in the arse having to sit around here waiting. Hmm. Oh well. No, there definitely is a 12.45. Ask those people and it's printed on the timetable. What did they say? There's definitely a 12.45. I don't know what's happened to the 12.20. It was printed on my back, which is what we want. 20 minutes delay. Maybe it is the 1220, it's just been delayed. Ah, maybe it is the 1220. Well, okay, well, the thing is, it's our bus and it's going to rain. Let's not look at the controls in the mouth. Mr. Sir, we see this back. Oh, okay.