Conversations with Alex Afriat post-ENS Seminar Riemman

0:00 Thank you for your attention. I think I've got enough space. I think I've got it right now. Unfortunately, I've got a dash, because there's a thing in the support I'm supposed to be at, which is supposed to finish. Which is finishing, in fact. Well, they'll be running late as well. Would you like, would it be the way you can download it from the website? Yeah, but how long does it take to burn the first piece? Yeah, it should take no time. Just let me make sure. Thank you for your attention. Oh, it's not going to fit. Won't it fit into your... No, no, it should fit. It should fit. It should fit. Oh, maybe you can switch it on. No, no, it's okay. As I said, I'm not sure there will be enough space in there or not. Otherwise, I'll just give you a few webpages.

2:30 I wouldn't say very late. Thank you for watching. I'll see you at about 4 o'clock in the morning, so make sure you're there. I want to get your order ahead, excuse me, so there was a lot I wanted to ask you about. Please, please. Okay, but I must go. Yes, yes. I'll be there at about 4. Cheers, bye. History of attempts to prove the parallel postulate are very...

7:30 I didn't think they were still doing it in the 1830s, the trisecting the angle. To be honest he was being very irritating today, Jean-Jacques. Not only did he... And also he really does piss me off when he gets up... I mean, why when somebody brings him on his bloody mobile and you or anybody else is actually talking, is delivering, you know, is actually delivering a conférence, why he can't go out in the couloir and talk instead of having to walk up and down in the room? Yes! Yes, on the top of his bloody voice. Well, you were busy fielding questions from... Oh, at the very end. Yeah, at the very end.

10:00 But they, you know, they shouldn't be allowed to degenerate that far. Maybe I'm being a bit prissy, haven't I? I'm used to Italy. Yeah, much, much worse than Italy, I agree. Is that what we saw? I mean, Odile, during his... ...all exams, so he's examining the student, while he's examining the student, and while he's ostensibly listening to the student's answer... He's busy texting messages. He must be very willing to be able to text, most, most, most, most deans are too... Too much old farts to be able to text messages. They say nobody over the age of 35 can really text messages. My ex-boyfriend taught me to text messages. For about a year I was quite good at it, but I've totally forgotten how to do it now. Mobile phones are quite useful. Yes, I don't have one at the moment. I refuse because it's such a bloody nuisance. They're such a distraction. If you haven't switched on, then they're constantly disturbing you. And if you have them switched on all the time, what's the point of it? You might just as well, you know, check your e-mails. I mean, for coordinating. Well, things like, yes, coordinating meetings, I suppose, it is quite useful. I should, I actually do have one at the moment, but I need a new SIM card for it. I need a new SIM card for it. It's still got my old UK SIM card in it. So if I use it over here, it costs an arm and a leg in roaming charges. Can you just get a SIM card for the... So tell me exactly what is happening right now about your, you know, your French top applications. And I've applied. Well, anyway, then February 28th, a bunch of jobs appeared. None, however, in the section 72, which is the one that I have my fingers on.

12:30 But my qualification in California was to put in the jobs in 17th philosophy. So there were four jobs in philosophy that were more or less relevant. There was com, musique et finition. Oh, that's presumably Ivan's old job, yeah. When I mention two of them, but why would you want to go to a place like Caen? Well, that's not the point. Caen is a very pleasant city, actually. I don't think it's as, you know... All French intellectuals have this attitude, you know, this... Well, I'm perfectly happy, as I say, not living in Paris, because we're just commuting here and spending as much time as I want to here, apart from anything else, you have half the living costs. I mean, it's just me, but I've never understood this, this absolute addiction to having to live in Paris. I mean, you can spend, as I say, as much time as you want to with Ferris. You don't mind that I've been buried in one of the most beautiful places on earth? Yeah, sure, in one of the most beautiful places on earth, yeah. That's stunning. No, both of them are, I mean, so amazing. I want to come down and see you in Avina before you leave. Never having seen Urbino, of course. I almost did it in the last month, because I was originally planning to go down to Greece, but unfortunately I had to blow that up because, well, I couldn't afford it, basically, so I ended up having to get the cheaper flight into Athens, which was a nightmare! I would so much rather have come down and spent three or four days, especially since I have long, long overdue for me to visit Alberta. There's loads of things I wanted to talk to him about, including about synthetic differential geometry and stuff. We were talking about it in Greece. The plans for this big meeting next year with Bill O'Hare and Anders Koch and Steve Shandell and various other people. And then of course I wanted to come and spend a bit of time in Albino but it just didn't work out. So I ended up having to get a cheaper flight into Athens and they were in the middle of a general strike, you probably saw. There was general striking groups for about three weeks. Because the government just introduced this... Why the hell didn't I know about that? How annoying. I wish I'd heard about it. Damn! Why didn't I know about that?

15:00 I might just subscribe to this talk. No, no, but I do. You do subscribe. I do, I do, I do, but I don't have time to look at everything all the time. There's so many other things to do. I mean, you know, unless you're spending, you have, you know, unless you're looking at everybody day. I actually have a list of all the observatory meetings. How are they? You hold there all the functions, all the good thoughts. And I use it, yeah. It's very tempting. As a matter of fact, there's a thing about, an issue about... I've got something I wanted to ask you about Covary and functoriality as a matter of fact. It's an interesting question which came up in discussion with John Mabry when I was talking to him a little while ago about the example of a natural equivalence which Eilenberg was playing, given their first paper. It's a very interesting question which he put to me. I'm going to stay until Wednesday. Why did I send you a message now? While you're doing that, why don't I get you a drink? No, no, I'll send you a message so you've got my email. Okay, okay. My message is... Move your back, sir, and I'll get you. I wanted to ask you about these very interesting conversations with Mark Lashiers-Ray that you alluded to briefly in your talk. You mentioned that you've been having some interesting conversations with him. Well, I had lunch with him today and I heard his talk yesterday. Why the hell didn't I know that? What are you talking about? There was an audience of three, namely me, Jean-Jacques, and another guy that, what's he called? I'm so pissed off with Jean-Jacques because he never ever tells people about these people. Oh yes, you have to know him. Yeah, it was the seminar where Alain... What was he talking about? I would love to have gone to that. He was supposed to talk about cosmological topology, topological cosmology, I don't know what.

17:30 Well, no, that's his great area of interest. He didn't, he didn't. This Hoffman guy, who's a nice guy, I talked to him the other day, he's a nice guy, very friendly. There's nothing at all about relativity. There are a lot of mathematicians like that. He's a pretty extreme geek. Cartier says that Grothendieck understood nothing whatever of relativity. But anyway, so now I have to explain to him Quinn's paradox, proper time, all kinds of things. Basics. I mean, I didn't know too much about it, but even I understood those things. Oh, come on, you would understand a great deal about relativity. Because the other time he invited the salams who were there, who were speaking about topazes and stuff. You've been there once, and then the salams, did they remind you of that? Oh, the guy who talked about type theory and topazes. Yeah, yeah, I couldn't get into it at all. They were strange. Those were strange. But I do wish I'd known that Mark was talking, because I could easily have come. I was here yesterday. It was frustrating. And also, I didn't realize Jean-Jacques had been talking as well. I met with them together, Jean-Jacques and Martin. Did you know about this? No, we had the drinks after. It's so annoying. I could easily have got to that. Okay, well, never mind. I'll just have to try to pick their brains afterwards. I thought you were coming today. Well, I was originally, but then I had to come yesterday because of this problem with Pavlov and his visas. So I could easily have got there if I'd known. But never mind. There we are. There we are. There we are. Never mind. Can't be helped. No, the question about, John Mabry's question about the example which Eidelberg and McLean give in their very first paper, or I think actually the second paper, the 1945 paper, their first example of a natural equivalence. Actually, I'll tell you what I'm going to do. I'm going to go back and look at the notes that I made of this conversation I had with John, but it's the example I think they give is the dual of a vector space. I think it's specifically the dual of a topological vector space. Yes, it is the easiest example, except that John points out that, from the point of view of a Poinsettian topologist, that there is a problem with this example.

20:00 In fact, there is a proof that it's unique, that there can only be one functor that satisfies, but it's a very difficult, it's a very demanding technical proof in point-set topology. You can use the axiom of choice. No, I'll explain, I'll explain, I'll explain the point to you when I've gone back and refreshed my memory by looking at John's notes, but John was talking to a very good... Classical topologists, no, try and say topologists at Bristol, who pointed out that the problem with this example is that you can use, as long as you can use the axiom of choice, which you certainly normally do in the case of topological vector space, you can cook up examples which are actually contravariant funders to the identity, and therefore not examples of natural equivalences at all, and what is there to support the assumption, which Maclean certainly has made both in the... In the original paper, and every time he cited this example afterwards, that this is necessarily a unique transformation. I think there is a way to get rid of that and just consider a linear function. Well, no, I'm not saying that you can't. Well, yes, that's just to say that you can give other examples which definitely are examples of natural equivalences. I feel a very good... It was not a historically important example, it was absolutely kind of a dummy example. It was a dummy example. The real example is that Zuzane... Nobody, nobody... Nobody is disputing that there are not very good examples of a natural equivalent, but it's just that that first example they gave may well have been not at all a good example, because it may not even... It was not really the first, I think historically it's wrong, because it's... It is the first, it is the first example in their 1945 paper, actually. Ah, in paper, right, but it's an absolute dummy example. It's exactly... And then of course you just say your functions are linear, which is reasonably... Yeah, yeah. You make this... Yeah, yeah, but then you're talking about... You can show that the dual of a topological vector space is uniquely defined, I mean, as a natural equivalent, as a natural transformation. Yeah, double dual, as I wanted to say, double dual. But you do actually have to use a fair bit of point set topological machinery to prove it. But John had another point, which was not directly related to this, which is very, very interesting, which I'd love to understand better,

22:30 which Richard may well be able to talk about, even when he comes to Paris, because he's been talking a lot to John about it. It's all business about what was going on, what was the motivation that Doudonnet and Cartier had in mind when they were working on the old, the original Brabacchi theory of structures, you know, in the late 40s, early 50s. And John thinks that there's some very interesting ideas from type theory that are involved there. That in fact, the burbaki theory does give the resources for... Now, getting around this problem, which he thinks he's identified, about this, what he calls set-theoretic mobility, there's a kind of ambiguity as to when things occur as elements and when they occur as subsets, which he claims actually buggers up the category-theoretic way of treating general structures, which is why you're led into all this problem about great and deep universes, you know, the size of the, well, the problem is the problem of large categories. It's the problem of the absolute in mathematics, the problem of absolute infinity. It's the problem about how you can actually justify the closure conditions on large categories, and one knows that Grotendieck justified them as it were internally because he just wanted nice environments for the right kind of functors to land in so that the properties of the spaces that he was interested as carries of homology would be exactly what he wanted them to be in order for the kind of constructions that he wanted. You could be right on the nose, which is great, and it's the way that a very, very powerful algebraist thinks, but there does actually have to be, at some point, there does have to be a kind of ultimate rock-bottom guarantee that you are talking about something that, well, you start, you, John, no, no, well, John is very, John, I think, has convinced me that there is a problem, there is a problem, the problem of large categories has not been solved, and the problem of growth in the universe, The device of growth in the universes doesn't solve it, and it's the problem of large categories that has not been solved, and the solution to the problem, he claims, actually lies in the old, original, Deutonic-Cartier theory of structures, which Babocki worked on and then abandoned, as it were, in favour of category theory.

25:00 No, he's writing it. He and Richard are actually writing something now. It's very interesting. He and Richard are actually writing something. I think it would be very, very interesting to get them to come to Paris to talk about that. No, no, no, it's his turn. You got the last one, I got this one, and he's getting the next one. But I don't think he's going to get a drink. He's just going to have a piss. Oh, I don't know. I never know with Andre. He's a lord himself. He says prospects in front of a book. Well, yes, but mind you, I don't know why he's saying that when he's just managed, finally, after hanging on by his fingernails for, well, since, you know, since Moses was a child, to get a job. No, no, no, but he says for these, specifically for these method of confluence positions, he says. Ah, yeah, yeah. He says it's easier, if you've got the habilitation, it's easier to get a job as a professor than as a method of confluence. Could be. I mean, he's got his ear much closer to the ground than I have, obviously. I mean, I was hoping you might be able to land one of these Victor Clebson things, but I think they're all gone now. I mean, you've done very interesting work which bears on philosophy and mathematics. All that interesting stuff we were talking about, about the Legendre transformation. Besides, they need some good philosophers of physics in France. They're very short, thin on the ground here. They're very thin on the ground. And they could do with some good philosophers of physics. In fact, frankly, straight off the top of my head, I can't think of anybody apart from Michel Bitmore, who even, well, I said, who even holds himself out to be a philosopher of physics. Well, I think of him more as a kind of philosopher of geometry, but I'm thinking more as a historian of mathematics, not a philosopher of geometry. Yes, that's true, that's true. Yes, I agree, he's a much brighter guy. No, no, no, I like Jean-Jacques Fichini, actually, enormously. It's just he's so totally, heroically, firmly disorganized. I mean, even for an academic, and even, even, even, even for an academic, and even for a… A math philo-academic, he is so disorganized, you can just, like, you know, but he's a nice guy, he's a very, very nice guy. I would love to come and hear him talk on Tuesday, but I've made a deal with Andre, I'm going to record the second day of this resize thing about functions,

27:30 which does look interesting because Jamie Tappenden and Jim Ritter and this other guy are all speaking. Jose Ferreros, Jamie Tappenden, and David Caulfield too. And they always, well certainly Jamie Tappenden always has very interesting things to say. He's an extremely good, very scrupulous scholar. You know, same old as Colin McLarty. Oh, why don't you go to one or the other? I mean... Oh, but you feel you've got to go back to Urbina? See you on... But the only thing is, I need to speak to you on... I need to see you on Sunday if possible because I've got to give you a recorder to record on Monday because I won't be able to get there. Well, I will, but... Well, no, I'll tell you what I'm going to do. I'll come and then I'll have to leave to go to sort out Pavlov's fucking visa. It starts at 10 o'clock, doesn't it? The resize thing on Monday. I think it starts at ten. Nine? That's extremely early for resize. Okay, I'll check that. Well that's good actually because then I can come and leave and get back to the Rue de Washington. But I'll be there on Monday. Oh, I gave David LeBlanc the tape, but I need to give you the tape to give to Frank... I can't pronounce his name, the Polish guy who's... was in December, who's been pestering me ever since to let me have a copy of it. Jeff Halefsky, do you know the guy I'm talking about? Well, where about? I wouldn't say no. Yeah, but you'll have to tell me where it is. Well, can't you just send me an email? I'll check my emails. I'll check my emails tomorrow. Yeah, I'll call you tomorrow. So it is, in fact, very, yes, but it's not just the natural, it is actually, yes, yes, it is the mapping, yes, of course it is, it's the mapping, as you say, from a vector space to its double dual.

30:00 Velocity to momentum. Yes, yes. Or momentum is dual to velocity. But it's not its double dual in the case of the genre of transformers, it is just its... If the condition on the Hessian determinant is satisfied, it's invertible, and therefore it would be double dual in that case. You're quite right, yes. No, no, no, I'm not saying, I mean, but that condition essentially says... No, no, no, but right as you say, the determinant is right, so it would be a double dual. So certainly an actual transformation, but it's also, according to MacLean, actually an example of an adjoint functor. I'd like to understand that better. No, that I don't know, but... No, he's quite specific on that, I remember him... In general... I remember him saying specifically in Bordeaux that it was an example. For an arbitrary Lagrangian, it is not convertible, and in fact I can tell you that... I don't think precisely why it is an adjoint construction, but it's not in general invertible, because adjoints are not in general, you know, inverses. I mean, Galois correspondences are the first, you know, really good, clear example of adjoint functions, and they're certainly not generally invertible. It's typically invertible, but not generally invertible. ...invertible, because in mechanics you tend to have a quadratic Lagrangian, and a quadratic Lagrangian with respect to, well, the geometry of it is this. You imagine a kind of background Euclidean metric on your configuration space. You see the boson there, a kind of dynamical metric. Sorry, what do you mean by dynamical metric exactly? Well, I'll explain, thanks. ...kind of dynamic metric whose metric tensor has... well, suppose you have two particles of mass, m1 and m2, and no constraints, so you have a six-dimensional configuration space, and you would have a metric tensor whose eigenvalues are m1, m1, m1... M2, M2, M2. So you sort of recalibrate by maths.

32:30 Yes. That's the only free parameter. I mean, that's what you say, but you don't have any constraints. Well, no, but if you have constraints, you may have fewer dimensions. I mean, they may kill one, two, three, or however many dimensions. We double it because we have two particles, but they're not exact copies because one of the faces you think of is having a particle of mass or money, and the other face... Well, since your metric is based on kinetic energy, there's an intimate relationship between mass and length. Ah, I see, I see, okay, so they may have different... And so, in physics, the metric tensor is the mass tensor. Ah, I see, right, so... So if you had a different mass for every direction, you would write m1x, m2x... But if you don't have an isotropic mass, mass is isotropic, I mean it's the same for every direction. So there's a degeneracy. Take M1, M1, M1, and then M2, M2, M2. Right. Um... Now, then you have a... So then you take... That's your configuration space. On every point... At every point of the configuration... You assign, you, you, you find tangentials. Yes. That would be a six-dimensional linear. And so the velocities live in that. The velocities at that point. Yes, yes, yes. It's a very kind of synthetic viewpoint, really. It's the way one should think of the problems. Then you write down a Lagrangian, which would be a function of position and velocity. So it would be a function not on the configuration space, but on the... Then you consider the restriction of the Lagrangian to your generic tangent space, so you fixed a position, now you consider this tangent space, which is a function of velocity, and you consider the values of the Lagrangian in that tangent space.

35:00 All we want to get to is a mapping that will take vectors, namely velocities, in that tangent space, to co-vectors, namely momenta, in the cotangent space, along the cotangent bundle, on which the Hamiltonian will define. Hamilton is defined on the cotangent bundle, but LeBron is defined on the tangent bundle, which is exactly where this construction that Maclean suggests as an example of an adjoint functor construction is coming in. Thank you for your attention. Now, so you imagine this background metric with eigenvalues 1, 1, 1, 1, 1, 1, 1, 1, and then on top of that you've got this mass metric with eigenvalues m1, m1, m1, m2, m2, m2. Now, this is my way of viewing things. As I say, you use this real-valued function on the tangent space, Lagrangian, you consider its restriction to the tangent space at a single point, Q, and in mechanics it will be a quadratic function. There'll be quadratic in the velocities. Yeah. It's like the formula 1 half mv squared or whatever. Maybe he didn't like the 1 half, maybe he just wrote mv squared, I can't remember. But hey, so...

37:30 So then you consider the level surfaces of that structure, and in fact, you know about 19th century mathematics, in fact then they call for that structure, and I'm about to develop it, this is pole and polar. Polar. Pole and polar. Pole and polar, sorry. I think that's what it's called. No, I don't, I'm not familiar with that, but I'm not, well, I'm not familiar with the history. I know about polar coordinates, but that's not what we're talking about here at all, no, it's completely unrelated. Construction is a kind of duality. Yeah, I can see, it's a kind of dual, it's a kind of dualization, it's a duality, it's a duality principle going on. It's a very nice one. Yeah. So the way to express this duality is, so you're wondering, how do you turn a vector into a plane? That's the pole, and that's the 19th century, it is Giergon, and you should see it with a couple of French mathematicians, one called Giergon. Giergon, yeah, I never knew that was how you pronounced it, but yeah, I know who you're talking about. I wish I knew more about 19th century geometry. Anyway, so the duality in question is, given a vector and a vector space, how do you associate... A plane with that vector. Well, by plane I mean if it were a three-dimensional vector space it would literally be a two-dimensional, a literal plane. Exactly. You can picture it in your mind's eye precisely that way. As you say, it's the plane in which the vectors are pointing in whatever direction. No, no, no, it's not. It's not. No, no, no, no. Oh, sorry. So this is the construction. So the tip of the vector would be on a given level surface of this function, which in the case of mechanics... Will it be the level surfaces of the restriction of Lagrangian to that tangent space? So you imagine that Lagrangian is having each of these level surfaces. Yes, yes. Sort of foliation. Yes, yes, exactly. So the vector expression, which in mechanics would be a velocity. The tip of the vector will lie in one of those level surfaces. That level surface will have a tangent plane.

40:00 That Tangent Plane is basically the mental... Oh, I see, yes, okay, I was thinking, yeah, yeah, yeah, yeah. But, you see, if the Level Surfaces... I was thinking the Level Surfaces was the Tangent Plane, but it's not, of course, it's just... But the way I'm doing it, I mean my construction is slightly artificial, I'm calling them ellipsoids with flaxseed. Such that the ratios of the principal axes are the ratios of the masses. So there would be an ellipsoid in a six-dimensional space. Yeah. With principal axes whose ratios are like m1 to m1 to m1 to m2 to m2 to m2. Those are the ratios. So it's a degenerate ellipsoid. Yeah, yeah, yeah. I remember you talking to me about this when we were down in Croatia, or more accurately actually when we were in the car on the way going back from Croatia to Venice, I seem to remember. I remember you talking to me about this ellipsoidal construction. Because of course if you take it as a metric, you only consider the mass metric. Then, in fact, it will look like a sphere. Yeah. It'll be... But if you... if you wanted to... You're taking into account the velocities as well, then it's going to be, you know, a little science. If... if you... if you... if you represent those level surfaces with respect to what I'm calling the background Euclidean method, Yeah. the eigenvalues, the identity, one, one, one, that's it, then it'll look like soil. Yeah. Yeah. The principal axes of ratios as well. But of course, if it's an ellipsoid or a sphere, it will be invertible because for every vector there will be... For every vector, there'll be exactly one tangent plane. You'll never have coincidences, you'll never have, for instance, if you have, imagine you have a surface that sort of, that isn't convex, so it goes in and so you have, well then you would have the same momentum, so they map the same momentum, how do you come back?

42:30 What? That can't happen in this situation. What? That can't happen in this situation. Well, you have no simple rule to invert. Yeah. I mean, you can go one way. Yeah. Because both the velocities map to a single level. Fair enough. You're surrounded with porpoise. Yeah. But then your transform is not invertible. Because you have two velocities corresponding to a single momentum. So which velocity do you map back to? Yeah. So it's got to be okay, so it's got to be invertible in order to be And so, the non-vanishing, or whatever it's called, the Hessian... The Germanic, yeah. So, geometrically, you wonder, why the hell does it work? Geometrically, what does it mean? It means that you've made the thing invertible. You mean, but geometrically, that's because you're looking at the level surfaces of the Leblancian. Yeah. A given level surface will never have the same tangent plane at two different points. That's a geometrical explication of that highly algebraic evolution. It's a beautiful connection between, as you say, the algebraic and the geometrical contents of the condition. But usually in mechanics books you just see the algebraic essence. Then it's trivially satisfied. Yes, exactly. There's a very interesting connection here with Hamiltonian, well, actually, with the way one ought to think about Hamiltonian conditions, Hamiltonian constraints, I think. There's some interesting connections here, I suspect, with what's going on in geometric quantization, where the quadratic, well, the restriction to the quadratic case.

45:00 It makes things very easy, very simple. There's a thing called the symplectic capacity of a cell in phase space, which precisely involves having a kind of ellipsoidal cross-section for things which preserve a kind of ellipsoidal form. For elements, for volumes, for volumes in the phase space. And that condition holds if the Hamiltonian, if that holds in the case where you have quadratic Hamiltonians, but it doesn't hold in general. And when it doesn't hold in general, then you have to go to very... Really very powerful algebraic topological machinery for modern algebraic topology, mainly Gromov's squeezing theorem. In order to recapture the same kind of algebraic conditions on the phase space that you need in order to carry out this kind of generalization of the geometric quantization program. It's very, very interesting because there's this guy de Gausson, I think I've mentioned him. There's this extremely interesting guy de Gausson, Maurice de Gausson, who I think is one of the most brilliant mathematical physicists working in the present time, who has written this. I think tragically neglected, but that's because there are very, very few physicists, I think, frankly, who are up to the math, which does involve a lot of very modern algebraic topological machinery, particularly to do with the squeezing theorems and the Gromov's work, who has argued that, well, he actually claims that he can derive the Schrodinger equation. He can derive the Schrodinger equation, so a priori. From this theory of symplectic capacities in phase space, in the way that you treat Hamiltonian systems in general, it's basically a very modern, in terms of the geometrical machinery, way of doing old-fashioned rational mechanics. But the same structure shows up in... A number of areas of classical physics, for instance, it shows up in classical optics, it shows up in the the iconal equation, which is to do with the focusing of lenses.

47:30 It shows up in the Goy phase, which is found both in optics and in a number of other areas, and it's really just part of the subject matter of old-fashioned, you know, when I say old-fashioned, I mean late 19th century rational mechanics that are very, like, very aprioristic, you know, purely mathematical approaches to the, um... Constraints and geometry of, you know, geometrical constraints on dynamics in Hamiltonians, particularly Hamiltonian systems. And he's got this, I think, beautiful argument that this is the, that really, this is the hidden topological origin of Planck's constant, of the Schrodinger equation, and of almost all the features of quantum mechanics. They're already there in the classical picture if you just dig deep enough into the... ...underlying mathematics. What do you think? Well, he's... Actually, the interesting thing is that he's shown how you can lift it. I mean, one of the assumptions that one had to make was that the Hamiltonian form had to be quadratic. But actually, he's shown how to do it using the double covering of the phase space by the metaplexic group. He's shown how to get out a generalization of the Hamiltonian, which is... Which is actually very, very beautiful and which seems to suggest that there is some very, well, mathematically deep or geometrically deep origin for quantum phenomena coming out of... Well, as I say, coming out of a purely geometric treatment of dynamics and mechanics in general, without having to assume anything about Planck's constant, it just drops out from the right way of thinking of coverings of the phase space. It also connects with these ideas about the role of the fundamental group in... Well, I'm hand waving, but there is some... I've heard him lecture about this a couple of times, and not only have I been enormously impressed by what seems to be the depth of the ideas here, but so have people who are vastly smarter than me, like Lorvier, like others, and Cartier, too. I think he's actually at the, well, he's not there at the moment, but he's going to be at the IHS sometime in the next two or three months, or about a year.

50:00 It's a very interesting character. He was a student of Le Ray. He's one of the, you know, the great, the guy who more or less developed sheath theory, one of the great modern French algebraic geometers. He was his very last student when Le Ray was quite an old man. By the way... Oh, I'm sorry, you want to put this off? No, no, no, I don't. I wouldn't mind having dinner. Yes, I'm so skimmed to be honest, I usually go these days on just going down to the deli and getting some cheese and some bread, but come to think of it, I probably left it too late to do that, so shall we go round to the Room of Tard and go to one of the places there? They're quite cheap and cheerful, and it looks very nearby. Yeah, I mean it's 9.30, isn't it? Yeah, it's true, true, true, I wasn't even looking at the time. Yes, I meant to go down earlier, as I say, to get myself some more Hedera and a bottle of wine, but no, it's okay, there's the places along the River of Tar which are not expensive. Alright, let's do that. He wrote a book called... I'm trying to remember what the full title is, it's something like... The word Newton is in the title, but I can't remember what the rest of the title is. It's something like Newton, Bohr, and the topological origins of Planck's constant. But if you just Google on topological origins of Planck's constant, you'll get the title. It's published by World Scientific. It came out about maybe 2000, 2001. It's an absolutely brilliant piece of work. Got an introduction by Chris Isham. Very, very impressive piece of work. He deserves to be better known. But the problem is that the people in philosophy and physics who know about Hamiltonian theory, they tend not to know about the results. The consequence of things like Gromov's squeezing theorem, which is a relatively new mathematical machinery, actually it was discovered 20 years ago now, but very few people realize how much difference it's made to reviving the geometric quantization program and to deepening it enormously by comparison with the machinery they were using before.

52:30 Well, you of all people, given all that you tell me, that I've learned from you, about 19th century mechanics and about the genre of transport, you must, as it were, feel the tug of the idea of a, you know, the vision of a completely geometrical physics. I mean, it may not work, of course, but, you know, there's these kind of controlling visions of... But you're smart enough to understand them if you wanted to, so... Anyway, I'll send you the references to the book. No, actually, this largely bypasses all the stuff about complex numbers. No, I don't agree. Actually, that's one of the things which de Gosson absolutely denies. He thinks that all this business about the centrality of the complex numbers is completely misconceived. That's actually one of the things that's extremely interesting. This approach to the geometrical quantization, this approach to phase space quantization, via deformation, it's connected with the so-called deformation quantization program and it's connected with the notion of the simplistic capacity of cells in the phase space, actually brings in all the complex, it brings in the complex numbers for free, but you don't have to start with the complex numbers. The complex numbers, as it were, come, you know, just... Incidentally, as a free gift along with this approach, he's one of the things that he's very, I mean, I'm just reporting on this, it's just simply a soundbite, but one of the things that Gosselin is dead set against, it'd be quite interesting because he's coming to this conference in Sweden, he was the professor at Göteborg for many years, but he left Göteborg a couple of years ago, he's now at the Max Planck Institute, but at the moment he's at the Max Planck Institute for Mathematics.

55:00 He also has an appointment at the University of Vienna. It's all right, I don't think he's even talking to us. He's also got a position at the University of Vienna. He's coming to this meeting in Sweden, which Basil Hiley and these people organise every year, which Penrose is also coming to this year. No, he will come, because his wife is very keen on coming. Vanessa, his wife, is very, very keen on coming to Sweden. Well, she was, I mean, a number of years ago. She's the headmistress of a school in Oxford now. She's his second wife. I mean, it's a late... They have... He's got a son, but by his second marriage, Max, who's now 10 or 11 years old. All his children by his first marriage are, of course, long since growing up. All his children by his first marriage are, of course, long since growing up. But it's a very, very happy marriage. She's a delightful lady, very nice. But she is very keen indeed on his coming to Sweden, so I think he will come. No, no, she's a school teacher. She's not a professional mathematician, she's not a research mathematician. I believe she was a student at Oxford when they met, yes. But she's, um... She's really some educational administrator now. She's a sort of educational administrator now, but she's a delightful, really delightful person, but although they met because he was her student, she's not a professional mathematician, not at all, thank God, I mean, she's not a student, but... But anyway, he should be coming to Sweden. I think he will. He's promised me he will. He's promised George Whitman he will. But de Gosson's going to be there. It'll be quite interesting to hear their discussions. Because de Gosson has this line that all of this heavy, to be quite honest, quasi-mystical... ...privileging of the complex, you know, complex numbers by Penrose and his findings, particularly to do with the upper and lower halves of the complex, you know, the holomorphic... ...the upper and lower halves of the complex analytic plane that Penrose regards as, which is essentially essential to Penrose's development of twistor theory. That's all really completely unnecessary and that there's a way of doing twistor theory which is much simpler and which just basically relies on old-fashioned spherical hyperbolic geometry and which ties in with a much nicer way actually via Clifford algebras with what he's trying to do in geometric quantization but he's very much down on the Penrose kind of privileging of the complex numbers.

57:30 He just thinks it's the wrong point of view, but it'd be interesting to hear them fash out these things. I've never been sure which of his agendas he has so many. Agenda concerning the complete centrality of complex numbers. Really? Why? This is a very strange idea to me. I mean, why not should complex numbers be... Well, why should any particular number field be regarded as, you know, a central, rather... Well, I mean, the centrality of quantum mechanics is widely recognized. As I say, I know people who would disagree with this viewpoint, although I agree, it is a very wide, it is a very widespread view. I certainly, as a sociological observation, I can't argue with what you're saying. Well, SLQC, the thing which leads to the flag picture in twistor theory, that's of course Penrose's big argument for why the complex numbers have got to be so fundamental, because they're central to his vision of how one should, as it were, you know, geometrize. It's the thing which gives the transformations that lead, you know, it's the thing, it's the thing, the thing connected with the so-called flag picture of the, in twistor theory, it's a nice idea, I can see why Penrose thinks they're so important, but I think that they're, well, after listening to Kylie Dugosin and other very smart people arguing the alternative viewpoint, I think there may well be a quite different viewpoint on the origins even of the twistor construction, which doesn't...

1:00:00 Thank you very much for your time, and I look forward to seeing you again soon. Topological vector space, this point of Mabry's, which I will go back and look at the notes. No, no, no, no. The thing which Andrei was being so dismissive of earlier on, which I think is actually quite a subtle and interesting point, as to whether there isn't an ambiguity in Eilenberg and MacLean's original definition of a natural transformation. Okay, I will go back and study the notes I made of this expose by Mabry before I come out. Actually, I shouldn't have said that there's an ambiguity with their definition. There's an ambiguity with the first example that they chose. Which is, to be precise, of a vector space, a topological vector space, doesn't it? Well, that does sound like a problem. Well, it's just that there's a problem. I think there is a problem with their example. But I must go back and study my notes on this and see, you know... No, I mean, for instance, an example is, I mean, an example of that example is relativity. Yeah. This is not a general metric, and it's not a general theory.