Discussions FW Lawvere & others

0:00 I can't just miss Kate Garber or S.J. Epstein or other people I regard as really spiritually enlightened. I think William, I suppose, is an observant. Thank you very much. I don't think he's a big Christian. He's great, right? No, I haven't used him. No, no, no, I say it again. Deliberately, he's a great Christian. But he is a great Christian writer. He is a Christian writer. Is he indisputably Christian? Oh, no, no, you can't. You cannot. You cannot understand Osteovsky as anything other than effectively, in terms of, you know, How life is to be lived, as a trustee called it, he has no interest in society, he's interested in individuals and their experiences, well, I mean, only in the record society, you can't, you can't, you can't, you can't, you can't, you can't. If you look at the Brothers Karamazov, it's arguably the greatest novel ever written. It's certainly got to be on anybody's list of the five or six best novels. I would be proud to say it is the greatest novel ever written. What happens between myself and my brother? I don't have brothers, but I know... I just had to spring a leak before the next the next thing you see so we had these two two two examples well the notion of doctrine which had no definition yet

2:30 but in any case it's going to be vindicated because there's third example But it has a very strong heuristic motivation, which is this point about invertibility. So Beck has been studying Grotendieck and Chevalier and so forth, this idea of descent. This is the following situation. This puncture that you're trying to study, an inverted whatever, it's really just a piece of the vibration. It's the puncture between two fibers induced by something in the base. Well now in that case, you take something in the fiber here and you move it over This is a completely different natural ocean structure because you can take the pullback in two different ways so I can give myself a nice amorphism and even it would be clear. Yes, yes. Because the most notion of isomorphism there is keeping track precisely of the interrelationship between the categories that are involved, from which the two maps are coming, which a general notion of structure can't do. The context, the context, yeah. The context is such that we have this third idea, such a structure, right, and we can always consider a ten-minute discussion. Any child can see, ah, if two doctors have an overlap, they might agree there. The idea is that the general discussion should probably be that if you want to descend from the code of nature, back to the brain, there really is a kind of descent. So I forgot to say that. What's particularly called descent is merely one doctrine of what structure is and therefore in particular what descent can mean. But because in the case where the transitions and the fibrations now have adjoints, then of course the momenticity doctrine also applies, and one could look to see to what extent, so now we have an overlap of the domains of applicability, we could ask to extend these two kinds of structures to the same, the second is a distinct question, to what extent is the perfect descent for one to correspond to the perfect descent for the other?

5:00 I wanted to tell this story because I think this philosophy is very good, it's very important. Incidentally, I can see, I think I can see how this insight into these aspects of dissent, you know, as you say, the dissent theory, just being as it were one aspect of this general understanding of structure in this, but the whole point is that the organization of the understanding of structure around doctrines depends on this sensitivity of the notion of doctrine, relativization to the relationship between the different categorical structures involved. And particularly the point you, the stress you laid at the beginning on this point about typically one starts from a situation in which This must relate to what you and Martin were talking about in Calais, about abstraction and abstractions and presentations. The very notion of abstraction operators is really a kind of... It's a kind of reflection of this situation, but from the other side. I'm not putting this at all, I'm afraid. But the point about the monodicity and the overlap of the relationship between the algebraic structures... The overlap of the domains of applicability of two doctrines of structure. Yes, but the thing which you and Martin were talking about, which is precisely the way that this functoriality that's involved in the notion of abstraction... Which also involves a kind of inversion operation.

7:30 I think it's clearly connected. What you may postulate, it's not like the property of all these categories, but talk to concept. You're correct. If the category has the natural nuance, something like this, in one case. But as you say... In fact, algebraic theories is also a fibrous category, so it isn't just for rings and modules that you have this direct comparison, but for any algebraic theory. In other words, for that case, let's say you have an abelian group. Is it the underlying abelian group of a ring? Because there is a trivial function for rings to be a group. It's monadic. Monadic. So that kind of thing is to try to find the G and the A.

10:00 The topology itself is a doctrine. Yeah, oui, oui, oui. Non, merci, merci, madame. Merci. Yes, yes, all right. I now see the huge flexibility and ubiquity of the notion of doctrine that it's, uh, and that it's, uh... By the way, there's a lot of cohomology in that way. He actually, that was the context in which he kind of developed this, uh, doctrine. One thing that he said, uh, that's going to boost us this morning, Richard, which was the, the point about the connected components factor, the centrality of things. As far as I could follow what he was saying, that seemed to be... Remember he was talking about the connected components? Oh yeah, yeah, yeah. I don't remember any context of the... I'm trying to think what one should... Part of the... He said it would preserve all the components. Well, maybe you said it wrong and I heard it wrong. He just certainly didn't mention products. He said that it preserved, I can't remember now whether he said it preserved finite limits. He only spoke about limits and co-limits and he didn't talk about, he didn't speak about products at all. Of course, that doesn't have an expression of limits, but he thinks that... It doesn't preserve limits. I must say that I'm rather exhausted. Yeah. Well, you're obviously not exhausted from thinking about math. Or explaining it. Well, do you want to sort of head back to the hotel? I'm not sure I would, I have to be honest, I would like to listen to Isar Stubber's talk because he's quite an old friend, but I, you know, if you, if you want to head back, I mean, I, this is the last talk. This is, no, I think it's the last one. The last talk is George. It's your channel, isn't it? I'd like to. Interesting. Yeah, I think having stuck it out this long, I probably will try and stick it out to the, to the end. But I imagine you'd like to get some rest tonight.

12:30 Yeah, definitely. I wasn't even going to ask you to come out for a drink or anything tonight. No, let's leave it. What time do you want to kick off a Mechelen in the morning? I already turned down another. No, that's okay. I quite understand that. I think I'd like to get an early night myself. So I think I'll go in early. I've got to check it. Yeah, okay. We can leave for Mechelen. The only problem is I haven't checked out what the Sunday try and train times are. I could do that from the station. Subtitles by the Amara.org community And then we can meet at the station, unless you want me to come round to the hotel. Oh, I see, the station. Well, no, I can come round to your hotel, because it's actually pretty close to the central station, which is where we go from. You're nearer than I am, so I can come round. In fact, it's very easy for me to come round via your hotel. If you wouldn't mind. No, no, not at all, that would be no problem. It would be easy. Then we can just go fairly early and... Yeah, I'll just check what the... because the Sunday trains are different. I was told that the Sunday trains are very cheap if you take the... That should be good. The trains in Belgium are pretty cheap anyway, at any time, but it is not far to Mechelen, so no, no, it should be fine. It shouldn't be expensive. Yeah, it should be extremely nice. Yes, okay, let's do that then. I'll just leave a note later on after I check the train times. I've got a note, but it's only the weekday ones. Okay. Yeah, that'd be fine by me. 9.30 would be perfect. 9, 9.30, that'd be great. And then we'd have the whole day to relax. You must have already moved from room 2, yeah, except for a key there. Yeah, yes, I already... Well, no, I haven't actually moved, but they told me I could move to room 34. And I just left my stuff downstairs in the... just behind the desk in the breakfast room. And they said when they'd serviced the room, they'd put it in there tonight, so I'll have somewhere to stay. Oh, we're very nice about it, actually. A nice little hotel. I don't know. Hotel Aristotle. I don't know why I stopped. But no. Yeah.

15:00 So when Beck and Chauvali and you were working on these, I mean, I was fascinated to see how profound the kind of conceptual background was, but all of these ideas which later became fully formalized in the theory of doctrines. You mentioned that Bishop Beck was in touch with Grothendieck at that time, that they were corresponding, did I miss? No, no, no. No, I misunderstood. No, no, just that he was reading. Oh, he'd just been studying it, okay. I don't even know, I presume he was reading Chevrolet, actually. Oh, okay, right, right. But whoever it was, I mean, these were certainly commonly available. They were commonly available ideas, but they... But there was never any kind of direct, sir, but there was any, was there any kind of direct input from the Grotendieck, Grotendieck himself from, from, you know, given what he's, what he was doing in descent theory at that time, his connection with these ideals that you could see it as a, connecting with his own, okay. So the connection was worked out entirely from your side. The theory is a category with some structure, again, but this is a second order. So like this category with products is one thing. So you don't see like sketches, kind of, how you would sketch some doctrines. If you're presenting theories as some kind of category, then some kind means, well, there's some additional structure on the category itself, and that additional structure can be presented by a sketch. In other words, in many ways, you can say, well, this arrow is actually product projection, or this arrow is actually an evaluation map, or even high-order structure. Not just first order, not just algebraic, but very, very complicated kinds of structures could be imagined as... And usually, once you test, that's the thing again, you see, if you don't specify the doctrine, then there's no meaning to the free theory.

17:30 All you're going to present, this is the mistake logicians make, they start with the syntax, and they actually secretly know, as Baez likes to say, they secretly know what the doctrine is, you see. They think, maybe they have some wrong idea, they think it's a cylindric algebra, but they never say that, they start off in presenting it, and then maybe after a few weeks they say, oh my, we can make an abstraction and find this idea of cylindric algebra, or even Boolean algebra, or whatever. But they're always starting from the syntax, as if it was something completely… They actually secretly know what this doctrine is, I claim. They know it maybe only in a rough sense. And then the precise calculations would help them to make it more precise. But there is a... in other words... Propositional calculus would never have been invented if somebody already didn't have the idea of Boolean algebra. They may not say Boolean algebra, it may not even be fully conscious, but somehow they have the idea that there is such a thing. Historically they knew already Boolean algebra. Is that true? I think so. But to call it, I think that the things are... What people called logic in the Middle Ages, and so if you formalize it, it's some kind of syntactical... Yes, but the idea that syntax comes first, which was always the formal logician's viewpoint, that's of course the point that is completely distorted and which we're understanding where presentations come from. And that's where the point that in fact there is almost invariably some kind of algebraic structure in the background, which may not have been formalized or apprehended. The algebra has actually formalized as an algebra, but clearly is an algebraic structure that's there, you know, in the background in the case of, well, natural, in the case of logic, it's obvious because that's what Thomas Thierry makes clear. A single presentation. A ring could have variables and equations in many different ways. Algebra doesn't have a presentation. But the doctrine of algebra determines what kind of a thing a presentation is. Yes, yes. Well, that makes sense. It determines what kind of a thing a presentation is. In other words, a particular syntactical system is certainly not determined, but the kind of thing that it's required to be is.

20:00 Yes, exactly. You know, modulo, making precise, and conventions, and so on. Essentially, it is... It's odd. You see, all these things are sort of half-known, sort of half-understood. They're not made explicit. One can see it even in the history of formal logic, I mean, because there was always that Buhl, Schroeder, Scholem, Tarski tradition that did, of course, retain close touch with algebra, with algebraic understanding, with the algebraic point of view on the structure, and it was actually turned out to be the, actually, the really scientifically fruitful approach. And, well, and of course now you've made the point yourself that... What we're again learning to call narrow sense logic is in fact, of course, deeply connected with these insights of Grassman about intensive and extensive quantity and the way that they behave as covariant and contravariant functors, and that it's the study of the kind of, you know, the roots and supports of, which as you've made yourself, supports are not necessarily... Just the default definition of supports, which was just that it's where the thing is, where the thing is, yes, exactly, vanishes is, you know, of course, I mean, that is a part of it is subsumed in a much more general definition of support. So once you can understand that, then you see how narrow sense logic naturally falls into this more, much more powerful unifying algebraic picture of understanding. The roots and supports of intensive quantity. And, of course, actually extensives as well, but that hasn't been so developed. I think the motivation of some of these people doing sketch theory is exactly like, we don't need this full logic, we're just doing something more geometrical. Yeah, but this, of course, is to make a false opposition between logic and geometry, and not realizing that geometry is itself, in the extended sense, is already the source of logical constructions. That's the point. Right. In other words, depending on the nature of your doctrine, the extra ingredients in the sketch might or might not involve things like implication, conjunction, and so forth. No reason why it should. In fact, I often pointed out that Grotendieck didn't need logic in the following sense. He didn't need the narrow sense logic.

22:30 Which is the tradition of model theory and logic, you see, is that every kind of structure is reduced to relations. Maybe it isn't really relations, but you can always translate it and take products and sub-objects of products and then have substitutions. So you have a whole technology which they've developed for presenting this kind of first-order algebra. Which is precisely why this false idea that there is some absolutely general, all-encompassing notion of structure comes from, I think. But anyway, the point is that you don't really need to make that, you don't have to use that technology. The kind of models that are classified by topos is by going to topos. He said, well, any kind of structure that can be described in terms of co-limits and finite limits, because that's what's preserved by inverse image of geometric morphism. But so conversely, any sort of structure that can be described in those terms has classifying topos. And to see that and to calculate even with an example and so forth. You could just deal with it directly. Well, I know what co-limits are, I know what finite limits are, how they might commute the induced maps. I can make all kinds of calculations there. I could even develop a computer program in technology. Maybe it's dreams. To deal with this and never ever descend into the sub-objects insisting on the description in terms of sub-objects of products of those even though since topos logic is rich enough you could do that that would drop out of it for free you could do that but but in a way i mean so if you are if you are well versed in the logistic technology It would be helpful for you to make this detour, but if you're not over it, it's silly that it is just a detour. You deal directly with co-limits and finite limits from start to finish, rather than from start to finish. Yes, yes. And this is the idea that you're thinking, well... He was sketching out with that, I'm sorry, I must be very careful, I mustn't describe it as a chart, but, you know, that little, actually quite large as I understand it, diagram that he drew on the tablecloths when he was at dinner in Buffalo in 73, that you've told me is still in Jack Duskin's office.

25:00 Surely you... Oh, yeah, yeah, yeah. Yeah, yeah. I know what you mean. Which was all about precisely that using pacifying topos to... That's true. To characterize the relationship between different theories. What are really all the sub-topos of the Ring Classifier? Johnstone made one little step, which is very good, to make that particular step. But Grotendieck, in principle, saw that kind of thing. And a whole lot, you see, so we, well, local rings, we know that, we use the local rings, fields in three different senses and so forth and so on, all kinds of Japanese rings, perfect rings, integral domains, hence salient rings with separably closed residue fields, all these kind of things. So he was writing down this list of these properties of rings, but... In terms, you can say, oh, well, clearly that property could be expressed in terms of co-limits and finite emits, so therefore it's on the list. And again and again, just visualizing how that... The concept is defined. You can see does it or does it not admit a definition in this way without ever descending into the technology. And you get the logic for free, as it were, drops out of it, well, in algebraic form. The fact that it's there in a topos. Yes, it's there in the topos. Probably this fact that I'm explaining as a positive point of genius explains why Omega was the one thing he never... He was very happy with it, but he said he never dreamed there could be such a thing. Interesting, because he was just not thinking about logic. He wasn't thinking of it in that way. No, no, no, he wasn't thinking about it in terms of... Sort of fundamental role of sub-objects. It's a kind of prejudice, you see. In fact, once you start working on algebra... Well, it was still a tremendous discovery. The basic thing is general arrows. And the ones that are restricted to be quotients or sub-objects, well, that's an important mode of analyzing them. But that's not the fundamental... No, no, it's not fundamental. And of course, this was the thing which Groton did, as you say, I think he remarked you, didn't he? He was the biggest. The thing that he missed in his life was the sub-object, that sort of thing. Which he did name the Lorvier object, let's begin with that. Which he kindly, kindly calls...

27:30 Quite rightly, too, but it's... His followers now, some of them, he was calling it that. Even though they have no idea about logic. No, no, of course not, no, no, no. They're probably very good, but they're working on Krillin model structures and so on. Yeah, exactly, all that stuff. And the fact that Omega, when it's connected... I think historically this is a fascinating case for homotopy. Yes, that's right. But they do seem to become completely hung up on that rather than on the other properties of Omega, which, as you say, including, obviously, its role as the subordinated classifier. But Rodendieck also told me in this, I think, that in general he thought topos was the most important thing. Not functional analysis, not algebraic geometry, but in fact he thought it was probably the most important. I think he may have come to this point of view after visiting Buffalo, long before I was there. But, you know, he doesn't say anything quite like that in the earlier, in the Springer version of SGA4, for example, he says, well, we think topos is topology, something like that. Yeah, but the propaganda that he only developed topos just as part of the machinery in his program in algebraic logic, I think is, no, it was always more than that. Always more than that. It's always much more than that. There's one thing Colin, I think, although I have great respect for him, Colin has got wrong in that respect. It tends to be that all the ideas in topos theory that didn't touch everything else in category theory and understanding of limits and co-limits that Groton did were basically just machinery that he gathered along the way as part of what he needed for proving the vague conjectures. And I think that's just completely wrong. No, you never thought that. No, I'm quite sure that's wrong. I'm sure that was it. I'm sure that he had a much more overarching conceptual unifying point of view. It wasn't just machinery. I'm sure he was interested in these deep problems in algebraic geometry and he knew they were important. But that was never the primary motivation, I don't think. I wonder, it's really a tragedy about this, that they've thrown away the opportunity of this meeting in January in his honour, ostensibly in his honour, in Boursouvet at IOTS. I find it's very upsetting. As you said in your writing to me, he obviously hasn't been asked his views on anything. They can actually say that we're not going to have a section on topos theory because we're not going to focus on past achievements.

30:00 Past achievements are just ridiculous, isn't it? I'm sure that those are not... Here are Cartier's words. They're just what, you know, Conn has laid down as the mode of order. He's passing on the majority verdict of the people who he's clearly... The other members were Cohn, Bourguignon, and... I don't know the two other guys, but I don't know who the other two guys are. I didn't recognize them. Well, one of them was one of the Russians, but no, I don't think one was one of the committee. Konsevich. Actually, Konsevich. Oh, of course, Konsevich. Yes, Konsevich. Concevich, of course. Concevich, yes, Concevich. And one other guy. Concevich, Kohn, Bourguignon, Cartier. And I don't know who the other guy is. Oh dear, what a waste, as I say, if only Grothendieck had been consulted. Okay, sure. That was extremely interesting. Amazing. ...just subsumed as a fragment of the theory of classifying rings. It's not really, I guess, how to put it as a kind of philosophical argument without going through a lot of details. It's going to be a hundred years before you can really explain this stuff to philosophers. I mean, I say that as a kind of shorthand. It may be 20 or 30 years if they're really good enough. But this is not something that you can just go out and explain to philosophers, even very good ones. You can give them a flavour in a responsible way, which is not just using buzzwords, but I don't think for the time being one can do much more. But I think philosophical agreement should be developed kind of autonomously also, you know, of course with... Of course, of course, I'm not suggesting that the whole of... Of course, it shouldn't be just a reflection of what's happening, even in the most conceptually profound mathematics, it never has been. But at the same time, I'd also insist, and this is a point John Mabry's often made, that in fact, even... Even many analytic philosophers and historians of philosophy underestimate the extent to which it is the philosophy of mathematics that has driven the main kind of metaphysical shapes of Western philosophy ever since Plato, well, actually even before, even since the pre-Socratics, that it's not just, as it has tended to be in analytic philosophy, a rather Cinderella subject, a fringe subject, oh, and, you know, how are we going to get, as it is in the kind of...

32:30 It's always been absolutely central to the tradition. It's been, I'd even say it's been the motor principally that's driven metaphysics for most of the last two and a half thousand years. Obviously there have been independence things driving it too, but I think it has been one of the deepest things that's driven it over the years. As one sees listening to these guys. We have to talk to Jean here about René Guittard and about the, you know, the discussions we were having with him about a rencontre in Fougere. Let's do that work, yeah. Oh, you've still got to finish your cigar. No, no, no, I mean... Is it starting? So I'm interested in what you say, the fact that there are no co-fibrations in his sense, or no interesting ones. And you can substantiate that kind of stuff. So, the co-fabrication of the rules in mathematical topology has dual. So, if you can say the homotopy, covering homotopy property, or homotopy distribution property, homotopy extension property, and the one-piece co-fabrication is also a co-size. Co-fabrication, co-fabrication, co-fabrication, co-fabrication. Well, I think we can continue our discussion during the coffee break, and I would like to thank Sandra. How he can define an order on sheaf in this construction.

35:00 One-sided identity, and what is it good for? Commutative topology. His supervisor is this guy, Freddy van Oysteyer, who I've talked about. Very interesting man, who actually, I don't want to say, but he does pursue this very much. He's in this, solidly in this con, notion of map space. Space is in the non-communicative setting. He thinks he's got a much better way of doing it than Kohn has, but... I'm going. Sorry, go on. What do you feel? Yeah, it's here again. Because here is the... I'm very dubious. Hang on, sorry. But I'm...

37:30 Here is an equivalence relation, it can be considered as an interval gap. Then, your previous slide, when you say, I mean, circle cohomology is the same as the cohomology of the corinth, this is the fact that the monad obtained from the Pulsbeck counter and Carthaginian version is the same as the monad obtained from the action of the human oscillation consistent with Penrose. So, we know a lot of this in the Carthaginian way. If you take the induction term, so this makes a natural pair of planets, and then you can take a moment associated to this, or a common one, maybe, and this is exactly the common one. In commutative case it will be exactly Cartesian because turns are positive. And I want just a quick question. Exact sequences are obtained from spectral sequences. No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no.

40:00 This is also called Picard. So we had some email exchange about this. It's also called Picard-Lupowitz. And so then I should maybe mention, so you have this Picard-Lupowitz, let's say. And then you have evaluation, co-evaluation. You can look at categories where there is evaluation and co-evaluation on the identity. I can show every, not by you, but by one of the things I wanted to do. For the moment, but I still have to investigate this. So these are commutative groups, but they come from categories. And these categories are even bi-categories. There is an isomorphism of them. That is what we investigated.

42:30 Actually, what the next thing is that I want, we look here at cohomology with values in this Picard group, which is two-dimensional, a billion group. And there should be a cohomology developed. Pre-dimensional meaning blue stars, so they should be bi-categories, and then the, I think, some more exact sequence in the real, that in the case where you take Amatsuka homology can be linked to the spectral sequences that George mentioned. Did that answer your question? Or at least part of it? I have to say, I mean, you know, that was a very, to me, a very clear, very helpful talk indeed. I mean, I've just learned.

45:00 It's very, very clear. It's a model of what a survey talk should be covering now. I said all the right things about Maclean, too, of course. Very interesting. I certainly learned from it. So I'll give you a call later on when I find out about the train task, Bill. Karine says that even on Sundays there are at least two trains every hour in the Meflin, so it shouldn't be. If for any reason whatever I don't touch base with you, I'll come by the hotel at 9.30 in the morning. Is that okay? Get a good night's rest. That's exactly what I'm going to do too. See you. I think it's better if we begin to communicate, and at some point we maybe decide, okay, let's keep it serious. Do you feel like you're handling him well? Yes, that's the problem of what we're looking for today, what we're trying to solve. Keep it serious. Can you commit? No, I don't think so. If I say goodbye to everybody and then I join you, you're not in a hurry, are you? No, I'm not in a hurry. I do want to get back to my hotel to move my stuff. I'm a little bit worried because it's sitting out in the lobby and I would feel more secure if it was under lock and key in my room. So I didn't want to linger. I'm moving hotels. Well, not actually moving hotels, but I have to move rooms tonight. Because I was sharing with Andre until this morning and now I have a different... But my stuff is sitting... I really wanted to go back and put it under lock and key, but... Where do we see them? Well, no, because you don't want to... No, you certainly don't want to come all the way to Stalingrad. No, that's ridiculous. No, no, no. Well, it's only about... OK, it is quite a longish walk. But then whereabouts would you like to go? Just somewhere in the centre, around the Grand Place? Is there somewhere that's not super expensive for beer? Because the problem is all the bars around there tend to be tourist prices. You should not go to the Grand Place, then. No, no, no, of course. But, I mean, there's lots of places near there, isn't there? Yeah, but you do not know them. That's the problem. Well, no, that's why if you tell me where to go, I could meet you a little bit later. I'm going to leave my hotel first and then come into the centre. I mean, I do know the centre fairly well, if you just give me a rendezvous.

47:30 Do you know Le Coq? Do you know the Burschauburg? I know where the Burs is. Yeah, the Burs is not what I mean. The Burs is the stock market building. If you stand with your back to the Burs, you go into the street, you have three streets, you go into the middle one. The big one? The big one that leads towards the ground floor, I know that. And then halfway to the left there is a small popular cafe which is called Le Coq. I know, I will find it. There. OK, see, what time? There must be about half past six, hang on, let me think, let me look, hang on. No, it's not even that. It's about a quarter past 6. 6.15. Yeah, okay. Well, I need some time to get back. So, 7.30? Yeah. Okay, 7.30. Okay, see you then. See you then. Hang on, I just want to ask you something. He had a long conversation with him in Greece earlier this year. There was a meeting in Patras, Sheaves and Logic seminar. He was one of the speakers there, a very interesting guy. It was in honor of Anders Cox's 70th birthday.