An Algebraic Model of Curved Quantum Causality

0:00 Thank you. Alexander's idea, well, this one's idea is perhaps one could give an alternative We know that Minkowski space is a metric space, I mean, it's a pseudo-metric space, but topologically, it's locally Euclidean, so locally, and globally, locally is R4, right, Minkowski space. This fellow here says, well, one should not base a physical topology. A physical topology is not one based on open balls, covered by open balls, right? But one based on intervals of the following sorts. A, X, Y, between two events. A set of all Z that mediate between X, two. and he showed actually that a topology for minkowski space based on such intervals such open sets is homeomorphic to the euclidean manifold topology yet yet one must be careful here and not take this at face value because because here this this passion order which derives from the light cones, and Kronheim and Enros people have almost the same idea. This is not the same arrow as you have before. No, this, for instance, here, they derive from the light cones.

2:30 So here, what would x, y mean? Well, x and y means that, say, the norm of delta The norm of delta r is less than or equal to 0. I'm assuming a metric minus 1. And delta r is 0. So, it is an arrow which shows causal efficacy relative to the men. Look how deep the idea of Roeb was. Look, rho derives the metric, the Minkowski metric, by assuming a relation between events to be x, y, to be a partial order. Here, we go the other way around, we assume that spacetime is a pseudometric, in the sense of Minkowski space, and the causal relation is derived from the metric. Look, it is perhaps more physical to think that metric, the space-time metric, is derived from a partial order, from causality, rather than the other way around. because, after all, the matrix structure of the world is a higher-level structure than the topological structure. Look how ad hoc the structure of a Laurentian manifold. First, where God had to create set theory, with respect to which one should define a topology. then of the topology which is the continuity relations of the manual then one imposed an extra structure of differentiability of smoothness and on top

5:00 of this assume that there is a metric field defined oh, oh, you would be too tired by day who knows this it is perhaps it is perhaps more fundamental to assume a causal contiguity relation between events and from it derive C0, topological, C-tinty, differential, and the metric a la Rob. There is some reduction in pro-reductionist argument here. I mean if we have Ockham-Razor, if we can do things with less we'll adopt it, you know, rather than ad hoc. So also this motivation of Sorkin because he says because he says in the papers in the paper 1987 with Bombay Lee he says well he says seems from from since from a causality that's more than a model after partial order we can derive the topology the differential structure on the metric structure namely the signature and i mean and dimensionality yes i didn't mention that and the four dimensionality of a laurentian manifold then why i assume your engine manifold so okay that's it so i i want to i want to i want to show you some now i do not abide i do not abide so much look the the other is not a question please no it's just i'm not sure i should ask it at this Sorry, because I was just wondering, how is this approach related to the Delfand approach? Ooh, yes, very good question. It might take you off the story. Exactly, this is one of the, me approaching it as a mathematician. I will tell you, I will tell you, what is my primitive intuition. Okay, why I'm interested in toposes and applying topos theory? Because it seems, as I opened the discussion, I remember I was talking to you, but my original motivation, it seems to be a bit short-sighted, a bit provincial,

7:30 hawk to think that on Planck scales the geometry should be subjected to some sort of quantization therefore be modeled say by non-commutative mathematics and we have a nice tool for working with for working out a non-commutative geometry of course cons work and not to say because I think one does not do justice to earlier mathematicians mathematicians are quite clever although I think what limits mathematicians is that the original motivation does not come from nature this is to their to their limitation I'm not sure you're right there let's not get into that two things Yes, of course, of course, but so it seems a bit ad hoc to have a non-commutative geometry, yet not to have a non-commutative topology, and my work on non-commutative topology, I would like to apply it also to the problem of quantum gravity, because even Wheeler has this foam conception of over space, over space time. who's topology is subject to quantization, fluctuations. He calls that foam. So, mathematicians have addressed the problem of non-commutative or non-commutative topology. The problem that haunts us now is to put it in the right physical terms. Topology, again, again, there are many... I mean, my motivation comes from many areas of mathematics and theoretical physics. But one, for instance, motivation would be that, again, the notion of topology as, you know, a theory for space does not fit well with quantum mechanics. We do not know what space means. Perhaps, perhaps what we mean by space, and this I've had quite a, quite a, quite a, quite a strong debate with Alain Cohen. What we mean space should be extracted from an algebra.

10:00 You as physicists should have more, more, more, this statement should be more up to you because you as physicists have always wondered what does a structure out there mean. Perhaps it is dynamical actions that can be conveniently organized into an algebra generate space. There's no space out there. What kind of, you know, I've reached the limits of my imagination up to what Wittgenstein calls a mental block. I mean, I will repeat myself, like Wittgenstein said, you know, the most important problem of philosophy is, he said, is why philosophers tend to point to an object and repeat its name. For instance, they say, glass, glass, glass. You know, it is the same here. We have reached the limit of theory when we come to ask, okay, space, who gave it to us? So, the primitive intuition is extract space from an algebra. And here we have Gelfand's motto, which Gelfand said, well, Gelfand said, you want geometry? By the way, geometry. He made space. We want geometry or space. Space. go to the spectrum go to the spectrum to the spectrum of course of course mathematicians mathematicians they have that luxury all the terms are defined you have an algebra Gelfand was working with commutative c-star algebras and he he says how can i i i extract from this uh from this from such algebraic structures information about space and what how can i construct space well he the the prescription was the following

12:30 you start with an algebra you go to the to the to its spectrum system which is the maxi the space of primitive ideals of the algebra, and then you impose a topology on the space, on that space. Ideals, ah, ideals, again, again, here, ideals, ideals, ideals are closest to points. essentially, here, again, where the notion of space and geometry comes here, because the maximal ideas in the algebra are the kernels of equivalence classes or irreducible representations of the algebra. These are the maximal ideas. In the commutative case, it's fine. In the commutative case, actually, incidentally, this, the topology on this set, is a locale. It's a complete, distributive, suplex of locales. A mathematician in Brighton, Christopher Moldy, has been working with a structure called quantiles. And the quantiles are supposed to be the quantal analogs of locales, quantal topological spaces. Essentially, quantiles were motivated by the desire of mathematicians to find the notion of a spectrum, a convenient notion of spectrum, for non-commutative C-star algebras. Right? So, commutative C-star algebra, you have that prescription. Commutative C-star algebra says. then you could match my special you impose the girlfriend topology every everything seems nice and neat you go to the non-commutative case there is no natural topology there is no natural analog of marks of a right algebraic The geometries have broken their faces in that problem. I mean, one does not, cannot define sheaves of non-commutative rings over a topological space. Well, for commutative rings, the problem is pretty straightforward.

15:00 Namely, the organization of sheaves of commutative rings over topological spaces has the natural structure of a topos, or a topos. So, the question is the following, topos to locales, was what to quantal? That is to say, what is the non-commutative analog for a topological space? of a topological space. Quantiles are structures, again, are distributed sub-lattices that derive from non-commutative system algebras. But yet, we do not have means of imposing a natural topology on them. Okay. All right. What is the problem? What is the problem? What is the problem? there's no natural definition of a spectrum for finite with Zapata we have encountered the same I will show you what is the problem we have encountered with Zapata because it relates to what I was saying with finite pathological space so could you just repeat the last remark and catch it the problem you've encountered with that you're choosing? In formulate, we do not know a natural definition of a spectrum. I understand. Yeah. The following remark, it didn't quite catch. With the... Spectrum, you said, the uniformity. Sorry? What do you mean by spectrum? Spectrum? Spectrum is the... A structural... A set of ideals, say, over on one algebra. Well, there are many ideas. ideal, maximal ideal. In the commutative case, all of these collapsed in one, actually. That's good. But in the non-commutative case, I will answer to you, especially with my work with Zapatin, how we have broken our phase. This will also give me a reason to show you

17:30 what a rot algebra is okay with it we we witnessed before that we with locally finite open covers of a bounded region of a space-time we have we extracted poses right now with the poses with the partner of the set I will associate some algebras, they're called Rota algebras, after Giancarlo Rota, first found them in the mid-60s omega, right, which are defined, span over, say, a field, in which you say, and C of I will introduce a notation, let's see, let's, the rat, the rat notation, the rat notation, right subject to the port this is an algebra subject to the multiplication You wanted to write y there or here? Where where am I? Ah, x1. Sorry, sorry. Sorry, sorry, sorry. Alright, x1 below. These are the arrows. The formula, the linear span over the say, the complex numbers of the arrows. So, with the multiplication, we must define what the associative product is. Yes, yes. I have a problem with that, but I think I have the solution also. Essentially, it is associative. It is associative because the partial order is transitive. Look, because the product will be fine.

20:00 All right, and of course one can check that this is associative product exactly because the underlying, the posit from which these algebras derive from is a transitive structure. Ah, but transitive structure, what does it mean transitive structure? For instance, where would we have a problem with transitive structures? My contention, I have not yet been able to prove it, but my contention is that a transitive causality will be of problem when we consider gravity because now when we think of this i mean essentially curvature measures the tilting of causality right from here right so we have here here x y z we have x causes y and y causes z but that does not imply of course of course one one can easily think about I mean if one just that's why I'm saying one does not take should not take mathematicians on face value I mean the mathematics of face value mathematics to a physicist should mean nothing one should start from the correct physical question. And the physical question here is, look, start from the very naive expression of the equivalence principle. The equivalence principle tells you the curved manifold, the curved manifold of general relativity is locally Minkowski. Locally Minkowski. Of course, as we said earlier, Rob and the rest of the gang showed that Minkowski space, Minkowski space can be modeled after a partial order here here effectively what we're saying yes the current manifold is only locally other way to say Minkov and a fancy way

22:30 to say Minkovsky space is flat but precisely because causality is a transitive relation causality covers covers events as far apart from each other as the birth and death of the universe causality as a partial order is a non-local relation between events gravity which is a local theory for causality namely gravity is a measure of how the light called local causality tilts at every event does not allow you for a global definition of causality it does not mean that event and event x will cause in event y if it's mediated by z no what what determines what causes what is the gravitational field again causality modeled after a global due to transitivity is not a good relation for causality for a curved space-time. So, here, in some sense, okay, so we have associated, we have associated with such partial orders, with such partial order, a formal structure which is called the Rota algebra, which is the Rota algebra, and we would like some categorical result, that is to say, ah, when is the topology encoded when is the topology encoded in the partial order the same from your mark indistinguishable from a topology encoded in the algebra and our result with Zapata is the following consider the following ideals in the algebra ideals

25:00 These are the ideals. One can check that it's an ideal. Actually, it's the span of all the arrows, excluding the reflexivity relation at every point, at every vertex of the pose. Okay. You can check out this idea. Okay. Actually, these ideals are obtained from us kernels of irreducible representations of the Rota algebra. Okay. And then impose between these, these are now my points, huh? These are points in the spectrum of the algebra. and impose the following topology, which is the Rota topology. The Rota topology is generated by the following relation. We say that ideal P is raw related to ideal Q if and only if the product ideal is strictly contained, but not equal. In the commutative case, this would not mean. But these rote algebras are not commutative. It is strictly included in the intersection ideal. Now, this is non-commutative algebra. The Rota algebra obtained from a whole set is a non-commutative algebra, because you can verify it. all right um if one tried blindfolded to impose the gelfand topology on the primitive spectrum of the rota algebras one would end up with a trivial topology that is the completely disconnected spectrum completely disconnected cell the same would would they would the amount if one tried to to impose what is called the zarizki topology three these are fancy names there are not so so many so many topologies you can impose

27:30 there are not much not many i mean but the only non-trivial that is to say none not completely disconnected topology that you can impose is the rota generated by discrete so it is essentially the European sets are these points as the error in nature sorry sorry now now my original question that I posed is when is when is the finite a substitute the topology encoded in the finitary substitute fn the same as as the as the topology of the algebra the rota topology of the algebra the answer is when when the poset structure is obtained as the transitive closure transitive closure of the rota relation so rota relation would say p if and only I'll tell you what size Rho I Q star Q means P star Q means means and there does not exist as z that mediates between the two such as p z is q all right the the topology obtained as the transitive closure of this relation other way to say other way to say the topology of the rot algorithm obtained from a finite age substitute is the same as the topology of the finite age substitute if one identifies the transitive reduction of the partial order topology of the substitute with a generating relation between the primitive ideals in the spectrum of the natalgie now now this is this if there is a this case that there is no such a certain idea what does it mean for this product

30:00 does it mean that it's always zero in some sense in some sense this tells you the following it tells you it tells you i understood i understood your question let me let me try to enlighten you morphe in a more physical way it tells this is wrong way it's a wrong matter like only okay it means formally it means that exactly it means it means exactly but not immediately between p and q but what is important again it is important what is important to see here is that it look indeed the germ the germs of the topology of the of the rota topology in the rota algebra associated with the germs the generating relation for the rota topology are exactly the immediate arrows called covering relations but what does that mean in some sense these relations are the contiguity relations the locality relations the immediate neighborhood relations You do not care what intervenes between any two, because intervention relations are local relations, as I tried to illustrate in the case of gravity before. But here we have a nice example of the proof of a theorem, actually, that exactly highlights the importance of these germ relations. Namely, again, the poset topology of the finitary substitute is homeomorphic, is the same, is equivalent to indistinguishable, to the one imposed on the spectrum of the rota-alget associated with it, only when the immediate arrows in the poset, in the underlying poset, the continuity of this, the covering relations, are identified with the germ between, of the topology between the the elements of the primitive spectrum of the algebra. So essentially, here we have the germs for locality.

32:30 Another way to say, again, this is my second objection to the notion of topology, apart from the fact that I said, what does mean topology as of a pre-existing space ? Also, here we question the very notion of topology in the following sense. topology is understood as the study of the global features of space here and here perhaps as I will give shortly the definition of a sheath here what we care about is local topological properties local topological properties how this how does this sound to you topology any topology we say that's local topological topology deals with shapes you know how how how space looks at large handles you know worm poles figures figures what does it mean i mean topology perhaps tells you is a classification think that you know topologically this is in this thing you should be from this what did you find a circle in quantum mechanics where did you find what are these these are these look mathematics of course as basil tried to object of course of course mathematics is it is greatly influenced and motivated by physics but it may just may be the case Indeed, so much the worse for us that mathematics was developed chiefly from macroscopic experiments. And this gives also quite, justifies as well some ideas that our friend David Finkelstein has for saying, yes indeed indeed if there is such a thing as quantum topology then presumably as topology is based on a classical set theory to to be is founded on a classical set presumably for quantum topology whatever that ufo means one has to develop a quantum so he goes even more but the base he doesn't even ask what is a quantum topology that i have problems defining

35:00 but he even asked what is a quantum set I don't know and he models of course after a grasp of algebra okay all we can do is model still though it seems natural at least question the principal idea behind topology which is the study of the global features we do not I mean, apart from space being a static thing out there that does not participate in the dynamics of things. Here, of course, illustrates another interesting thing that from abstract ROTA algebra, this is ROTA algebras associated with concrete covers of the manifold, okay? But there are also abstracts which are not defined with the records through a background posep. And even more background space from which these posep are derived from. This illustrates exactly the tendency of what Gelfand said. You want a topology, what does it mean? You start with an algebra. But the limit, what are your points in your structure, improves an algebraic relation, because these relations are algebraic relations. And then, fingers crossed, perhaps you can do something. But at front, to say such a radical term like non-commutative geometry, I have a big problem. Geometry, geometry, if we take Bohr's, correspondence principle quite seriously geometry is always commutative and geometry is bosonic namely we record we measure commutative numbers geometry since I think I don't want to imply any racial distinction but since I'm Greek I have a feeling for the word geometry geometry is a theory of measurements of space geometry non-co there's no non-commutative geometry geometry geometry geometry geometry

37:30 is always commutative and this exemplifies to to why geometry is commutative because you go to the spectrum where things are key more appropriately would be to say to the center of the algebra but between the spectrum center center after they're not that different but still again still again I think you want to see but all right I do not it sounds it sounds fancy non commutative geometry it sounds fancy and perhaps caught only on merit of the title has sold many copies but in what his book but it's not a Conceptually, it's not the right to say. It's not the right word to use. Also, another conception that I have a problem with is Wheeler's. He says, one of the cheap models, also required from Bohr. He says, no phenomenon is a phenomenon unless it is an observed phenomenon. that's a lovely tautology that's a lovely tautology but to me to me it sounds not only because it is tautology it sounds to me a bit trivial but also i was i was i would like to ask are there non-observable phenomena by etymology the word phenomenon is always theory and observation independent. I mean, there's no unobservable phenomenon. It's absurd to use the word phenomenon. No phenomenon is a phenomenon unless it is an abstract phenomenon. There's no unobservable phenomenon. Unfortunately, he took an example which, if you read Bohr, actually says he should not be doing what he's doing, but he comes to that conclusion. Aha. And yet Wheeler claims that he was just doing Borg. But in fact, he was trying to analyze one of these experiments in a way Borg was like, oh, no, then. He didn't make a choice, he means. Yes, he didn't make a choice. Actually, I've got a quote from Borg in one of his papers where he says, don't do that, you'll be led to competition. Yes, and then we have the, you know, the smile of the fuzzy driver.

40:00 Yeah, yeah, yeah. Bites its tail. He's a poet, he's a poet. We need poets, but of course. He doesn't understand a poet. Nice, I mean, also, I mean, this is not the boundary of a boundary. it is important i mean but that is that is an interesting but yes back up on mathematics we've known of course of course this is the i mean the epitome of homology but um there he was right what i wanted to hide and to tell you another uh is uh just before we go on to another aspect yes you said this is something you and the patron I've smashed your face against yourself. To try to find a topology for the spectra of Rota algebra associated with finite A substitute, that would be non-trivial, non-disconnected. And you can't. The only way you can find a tribut. We broke our base and we found the Rota topology. I mean, in some sense, we discovered Rota. OK, yes. But it's the only thing we can do, of course, infinite-dimensional roto algebras, and they are involuntary roto algebras, and when you go to system algebras, but we wanted, in the finite case, because the finite roto algebras arise naturally as from finitary substitutes, from the finitary substitutes. What do you learn from the point of view of a physicist in the algebra? What do you learn from your roto typology? Yes, yes, we learn the following, that what Sorkin can do with Poe-sets, we can do with algebras. So, for instance, in the Poe-set, you cannot superpose arius. We can superpose the arius. So to this superposition, we can give a quantum interpretation. And then, what we do actually give, is we give a physical interpretation to the limit theories. Here, this correspondence is omega n, omega n is fn, as I showed you, to omega n. Then again, you form an inverse limit, an inverse system of Rota algebras that possesses an inverse limit,

42:30 and that inverse limit is homeomorphic to the original manifold that you started with, the x. Now, now, what was the physical interpretation, as we said before, for the inverse scheme? Ah, it is exactly the structure that you obtain at the ideal limit of infinite resolution of your original mark into its point events. Here we have certain bigger, fatter neighborhoods about these points. So, in some sense, our scheme is infinitely pointless. less. But the pointed structure we obtain at the physically implausible limit of infinite resolution of this. And with Zapaterin, we were able to interpret this as a correspondence principle. How? The Rota, and this also gives more weight to the Rota algebra. The Rota Algebra is a discrete differential manual. That is to say that with Rota Algebra you can define a differential. The Rota Algebra is naturally graded, if you like to use the graded. Can I just ask one question about the coverings in the Rota Algebra, the condition you've just described. No, no, no, it's okay, it's all right, I remember what you're doing now. What's the condition on the localization of the coverings? I mean, is there a condition that every object has a smallest covering? The condition on the coverings is the original space that you started with admits a locally finite open cover. Right. That's the law. And by locally final, because you came up a bit later, you didn't hear us. I'm sorry. It's okay. The condition is that every point of the original manifold that you started with has a neighborhood about it that meets the covering cells only at a finite number of these covering cells. That is the definition of the locally final covering. So, it would be interesting to show you, let me say, if that's the cover, this would be one equivalence plus P, Q, R, S, T, U, U.

45:00 Is that a condition which can naturally be expressed in terms of the topos-theoretic machinery? Oh, yes, yes. And that's what I was interested in. I'm trying to get my hand around the distinction between the rotor, the condition on the coverings in the rotor algebra, and in the case of the Zariski topology. And particularly, what does it do to the structure of the topos? For instance, does it affect what's technically known as the weakly decidable sub-object condition that you've got localization? You mean with the sub-object classifier in their respective topos? Yes, yes, yes. There is a lot to be said. Okay. To enhance your appetite about this thing, we have speculated with another collaborator that perhaps in the topos of... I define with every root algebra what I call a fanatary spacetime sheath. And you know that the topos, the classical topos of sheaves, of sets, over topological space, has the natural structure of being locally a locale. Yes. Locally a locale. That is, say, some object classifier is a complete hiding algebra, object-wise. Okay, here, for the Rota algebras associated, we have given a causal interpretation to the structure to the Arus, in there, in the topos of finitary space-time sheeps of Rota algebras over the causal sets of Sorkin et al., and all the collaborators, the natural sublative classifier is the Lorentz group. So, sounds very far-fetched, but essentially, in a fancy way, but in the topos of sets, the inclusion relation, you can think of it as an injective arrow. Exactly, yes. Injective arrow x to y. In the other topos of fine-tested space-tested, so causal sets, this has the causality, and it is, in some sense, the local symmetry of the inclusion.

47:30 relations which in the topos is encoded in the sub-object classifier, here the local symmetries in that topos is a group, is an object having a group structure, and it is isomotic to the Lorentz group. And group action? Yes, group action. Absolutely fascinating. So what is between set, what is classically regarded as set-theoretic inclusion with a causal So you have to look at the G action to look at the G action exactly so this is all very naturally connected up with the sort of G set approach to Okay, that's the first one. That would be the offset that came from this. What did I want to say? Sorry, don't rub it off yet. No, no, no. We were going to talk about sheeps. We were going to talk about differentials. Ah, yes, yes, yes. So, with the... It's naturally graded. It's naturally graded. Omega and 0, which is the span over C over the dp. Oh, okay. For instance, define degree of any element in the rota algebra. to be the number of z's such as tzq. Okay. Right? So here you see omega 0 is nothing mediates. Omega 1, 1 mediates. Omega 2, 2. So also, you have to go to the papers.

50:00 Very quickly, there is also a Keller-Kartan differential defined from omega n to omega n plus 1. So at a discrete level, there is a view of the boundary operator. Because essentially, I didn't tell you, there are so many things to tell you, but these Roth algebras are simplicial complexes. So, Alexandrov actually thought of them as simplicial complexes. And his construction, his original construction, is what he called the nerve of a covering. So you have a covering, the nerve, that is to say the mutual intersection of the covering sets, and this is a sibilatio-complicola. Right. And the dual, of course, would be something like, you know, going from a symbol to a volante. That's right. Right, that's nice. Yeah. Right. They're called, this is, well, this is the co-boundary. Yeah. All right. So we have these discrete differential manifolds. Why I'm using discrete differential manifolds is in the literature, these are very, these are structures very well defined. they are they were defined by a greek fellow the marquis and miller hoyssen quite recently no miller hoyssen no no miller hoyssen interesting he's a general he's german German Turk. German Turk. Not far away from Greece. We are in the Greek. Between East and West. The spell of Greece over Germany. There was a better book of that one. They define what is discrete differential. Now, but that also motivated us to see, ah, well, these discrete structures are discrete differential manifold so so some notion of some reticular notion of differentiability is inherent in them let me let us see we are able from the continuity relations at the limit of infinite resolution to obtain the topological manifold structure of the original manifold at the infinite

52:30 refinement perhaps perhaps at the limit we can also we we can also derive smoothness you can you can prove that look this omega 0 at the limit of infinite refinement will appear as the points this as the module for differential forms on it so the cotangent vectors and this and this higher order forms oh my god of course of course since since you can show that actually d0 d squared equals zero so it is the keller i mean this this one you said is one form so differential points in general the one forms look not forms though not completely anti-symmetric tensors not completely anti-symmetric tensors although what what acts on them is the cartand differential they are not completely anti-symmetric i mean i mean you can you can get um so why are you objective we do not object you see i'm sorry you make a comment again again basil again does you see what or the glory of being a physicist he didn't have to know he didn't have to know he said okay my intuition is upon himself is a grass and i and we we after all this circle of sheaves and blah blah blah define number two I think ours goes in some I'm not advertised in here anything but I think it's more well found no it's quite clear what you're doing, you're building up the whole structure which this one particular idea is for the strength that's what you'll do, you're showing it's much bigger structure, much richer structure So we get the smoothness. So Sorkin, Sorkin was probably right in saying, well, assume at your basis, assume a Po set. Yet, if you do, if you, if one limits oneself to Po sets, where there's no natural operation of subtraction, because, mind you, mind you, we're using the algebraic structure of the Rota algebra.

55:00 to obtain that. I mean, there's no differential without a difference. Of course. There's so much you can do at the post-it level. You must assume a bit extra structure. And the extra structure that I assume compensates. I mean, it gives us back nice results. plainly evident. I mean, again, Sorkin, which is the epitome of this is, he had the insight to say, yes, a partial order that determines the topological the differential structure of the metric structure of course of course but in some sense here we materialize i mean we show okay it's nice to have after all they are our leaders these brilliant minds but it's sometimes it's nice to to to give proof I mean, I realized that Feynman was the genius. Freeman Dyson worked hard. He was the slobber. Freeman Dyson worked hard. And we owe to him, of course, we owe to him. But anyway. Can you say what this boundary operator does? Yeah, yeah, yeah. It takes you from the simplest course. It's a retroalgebra. In the wrong algebra, and I need vertices. And I need vertices to go from synthesis to phases. And from phases to lower... Well, you cannot go because it's nil-pop. You cannot go only one step down. Well, you're sort of saying what it does in homology. I was asking what... I was thinking... So it's red. I find it hard to visualise in terms of your p sets and the line structure is at our set. These are oriented. It removes mediators. One by one. But you cannot remove mediators. What does the boundary of a boundary equals zero come to?

57:30 I mean, you can do a lot of things at the functional level. I mean, you can define homology. I want to know what d equals 0 means in terms of z. I'm sorry? I want to know what d equals 0 means, even as it's exact, in terms of z. You mean cutting out z? if this is essentially a simplicial complex that you're talking about the sense it just points for instance it gives you with the action of a d of the homology of a reason in one such chain you would get the superposition of all the resulting chains with the mediated events removed. And you can actually interpret that. Well, in terms of Sorkin's causal set. Look, we had the similar... I mean, we wanted to... I think that's what Finkelstein was going to do. Yeah, yeah, yeah, absolutely. With, with, with, um... Actually, we wanted to, to think... Okay, the original... The original... Probably isn't, but it just... Well, no, I think we have very many things in common. But, as you explained to me, your structure is the other thing. Many, many things. Your quest for primitive idempotence in your algebra, it is almost the same at the algebraic level, searching for the primitive ideals. Yeah, yeah, of course. Once you have primitive ideals, you can't be generated primitive ideals. That's right. Yeah, but there's more than that, because there's an essential non-commutivity in one of them. Same here, because there are ordered structures. Yeah, yeah. Of course, if it was undirected, if the graphs were undirected, if there were not directed graphs, of course, the algebra would be a billion, trivially.

1:00:00 But also something coming up in terms of the Jones polynomial. Kauffman, no, Kauffman has, excuse my expression, he has tortured me with trying to relate this work to Nantes. I don't know. I don't know even this structure. I do not understand them perfectly. I have been working in this the last three years. I do not understand Nantes. Nantes. What would be a good reference to this? Our first paper was a pattern. Again, not for advertisement, but our first paper was a pattern in English. Okay, how do you feel? It's IJPT. Volume 1. Page 1. What? You've been around, you haven't been around that long. Hey, I wasn't the same as you did the wrong. You would have had to be to publish it. I'm going to check it out for you. It probably did actually, it was his baby, wasn't it? No he wasn't, there was a guy called Yates. Oh was it? He only took it over later, did he? And then it was Bergman. Bergman? Was it? Bergman? No, Peter Gabriel. No, not P. G. Bergman? I don't think so. He was in the editorial board. He was on the editorial board. I think I was trying to get over as soon as he had started. Now, you were actually interrupted. You were just about to tell us about the work with Ayshan. Sheeps? No, what are you going to talk about? You just started to say something about the work with Chris Ayshan. Maybe I have to go, what is it? About 5 to 8. 5 to 8? No, I have, let me tell you something. You have somebody waiting for dinner with him. No, not at all. He's my little, my stepdaughter takes care of my three little boys.

1:02:30 One is age 4 and the other is age 1. and my first I cancelled a dentist appointment my teeth thank me the other one was that she she has to take it to babysit the two boys but they are not used to her very much. My wife went to a yoga class. Presumably now that we're talking, she's a bit elevated. It's okay, I can go in five minutes. If there is something that you would like to discuss, I'll be glad. If there is some feedback, if you find it interesting. But it was all introduction, I don't know how the rest of it is. It was probably a bit overwhelming for some of them here. It's almost theoretic fascinating. I've sort of skim read some of your papers, but I still find it... I must now go back again and read them in a bit more detail. I'd like you to repeat the sentence about the sub-object classifier. Yes, yes, I'd like that too. because it all seems rather beautifully with the way that you think of the structure in the topos of smooth spaces and actually think of structure and set theory as coming out of that sort of corner. A topos is a particular kind of category. Well, I certainly wouldn't describe myself an expert, but we know the definition of a topos. You don't need to state the definition of a topos. Right. And there is this sub-object pacifier, which is an object in the topos, which is a generalized, it came as a generalization of the truth values of characteristic functions within the category of sets. Sure, sure. Okay. It tells you essentially which... It was in the context of rotor objects and things. Say what you wanted to say. You say it the way you want to say it. All right. Please. All right. So we've got generalised characteristic functions. The Rota algebras associated with a finitary substitute of a topological space, I have interpreted it as what is called a quantum causal set.

1:05:00 Now, a quantum causal set is a locally finite partial order, and associated with it is a Rota algebra. the epithet quantum qualifies it qualifies the Roth algebra in its structure that is to say that the arrows within the Roth algebra can superpose something that was not allowed in the post you just have the arrows from one post to its successor and its successor but not the possibility of superposing arrows you can do that in the Roth algebra the Roth algebra associated with a locally finite post called a causal set the roth algebra is called a quantum causal set now now we we associate with i have associated with with such finiteary substitutes under roth algebras i have associated certain structure called sheaves finite space on sheaves now perhaps you know the definition of a sheave a sheave the the definition that i use more is one i uh according to lazar lazar was a who's a student of Cartan's and now the son of the original Cartan, Enrico Cartan, and co-founder of Schiff and Topos theory with Gotthendick. But Lazar's definition of a Schiff is that a Schiff is a local homeomorphism. A Schiff is a local homeomorphism. Again, we do not care about global aspects of space. What you care is about maps that preserve the generating relation of a topological, the germs. We organize the germs of the topological maps on the topological maps into sheaves. Essentially germs live in the stalks of the sheave and it depends how you patch up, the way you patch up these these talks to obtain the global structure of the sheaf as a global topological space okay now as as the classical topos of sheaves of sets over a topological manifold is an example of a of a generalized topological space generalized meaning here has a particular meaning the generalized. It means

1:07:30 that the sub-object classifier, unlike in the topos, in the classical Boolean topos set of sets, is not anymore the trivial Boolean algebra of 0 and 1, but it becomes a complete hiding algebra. Because the local sub-object classifier is a complete hiding algebra, our interpretation of the topos of sheaves, of sets, of a topological manifold is as a universe of continuously variable sets. They are varying with respect to the base parameter space. Now, my intuition is, look, as the universe of sheaves, of sets, the topos of this, can be thought of as a universe of continuously varying sets over the manifold, shift perhaps the topos well i want to give it to qualify as a quantum topos but the topos of finite space-time shifts of quantum causal sets over the causal sets of sorki et al represents a universe of a variable quantum causal sets now what would be the the the reason not the reason but what would these quantum causal sets vary due to my best bet would be a finitary causal and quantum version of gravity in the same way that in the continuous in the continuum the manifold the gravitational field exactly go encodes information about the dynamics of locality namely local causality the gravitational field g minu x and its dynamics shows you how locally the light computes so in proposing and this is the end statement you're proposing a quantum topos structure for quantum gravity and that i have discussed with precisely um i would like to put front this topos quantum top of finite case perhaps that is the proper universe to to model quantum radical phenomena whatever that means that would be my

1:10:00 at least what i would put on the that's very now it's very hand-waving because i have not developed the motivation is very of course of course another question that i would like to address because chris has has has been led to considering purposes from the point of view of quantum logic and the Cohen's theory, there is something that it is common to, my approach is more geometrical in the sense that I'm trying to understand the small-scale stuff that space-time is made But his is like completely logical. But I think, as Vickers called his book, Topology Biological, I think there may be a very subtle connection between the quantum logic of the world and its chronological structure, as determined by gravity has small scales. So that, Jauch started, I mean Jauch and Piron started, yeah. Has that been developed apart from Christoph? Very much so, very much so. There was a clear one, six months, he was visiting a Belgian... Bob Cooke? Bob Cooke, yes. And he was... Just left, doesn't he? He just left, yes. He went to Montreal, actually. And he told me he was being working on Piron, from the top of the right person, on Jauch Piron for many years now. Actually, he's more versatile. You know, I'm trying to constrain myself. I keep reminding myself... Sorry? We appreciate it. I keep reminding myself that I have to posit the right... Because I think, you know, Bob has that very elegant mathematics at his disposal. But still, I think he has fallen victim to Woody Allen's joke, you know, what was Woody Allen's joke? And this I tell to all mathematicians to shock them, and myself too.

1:12:30 He said, Woody Allen says, I have an answer, can somebody please tell me the question? You know, I think he's so... Yes, yes, yes, you know. Yes, Bob's stuff is very operational. It's a beautiful, beautiful way of doing operational quantum. Yes. But it's not at all clear, you know, there's real physics in the stuff. Yes, where does physics come? I mean, where did it come from a physical question? I have problems. my problem originate from physical question I want to evade the pathological manifold the pathological console I want some finite area such a discrete structure inherently discrete to undermine that background inert ether-like manifold I mean quantum logic essentially came from trying to avoid the problems of Hilbert's place, for example, and von Neumann himself said, please, I don't believe in Hilbert's place anymore, I think it's the wrong structure. And then out came the perk off von Neumann-Lattis, right at the beginning. yes many problems for instance of course i i can see there are there are big problems as we were saying the other day in your office basil that there are problems in trying to unite the quantum mechanics with relativity at the basic level of the hilbert space because for instance you know that there are no finite dimensional unitary reservations of the lorenz group and i mean I mean, one, of course, one has to choose, what do I sacrifice? Do I sacrifice, you know, because the Lorentz group is non-compact, so that's all. So one asks, what am I going to sacrifice? Finiteness. I would like to keep finite. I would like, I like finite. I can work with finite dimensional vector spaces. It's nice. Calculation is neat. You do not have problems with spectroscopy or all this stuff. but then again what do I sacrifice unitarity? if I sacrifice unitarity the probabilistic theory. No absolute spur of the way. No way to find this spur of the way. He doesn't know whether he's thrown unitarity away when you're in crisis. Did you know that? I didn't know that. Yeah, that was Feynman in the early days, that's what Feynman says. I'm not sure whether we throw away unitarity or not. That's why also the paper in that book that you edited

1:15:00 He was thinking, because I'm thinking, I mean, there is a big discord, I mean, on the one hand, what we call space time is an indefinite metric space, and all the Hilbert space for relations, you know, Hilbert space, which is positive, definitely metric. I mean, this you have, for the probabilistic interpretation, of course, the probabilities are positive numbers that add to one, okay, blah, blah, blah, but still, unitarity, operationally, is a bit, is a bit an ify concept, because unitarity, you know how it's defined, it's an integral overall space, okay, definitely a non-local conception, so you integrate over the room to find the particle, yes. But if you give up unitarity, you have no way of defining the square of your probability. Finkelstein had a nice idea. He says, why don't we use Dirac space? Dirac space is a Kilbert space with an indefinite space. But then there are problems. There are problems coming from field theory there. I mean, your ghosts are not... I don't know. There are no problems, I mean why I complain, we'll be all out of jobs. There are no problems anymore in science. No, these are big problems, aren't they? I mean they're really fundamental problems, in the sense that they're stopping us from making any progress whatsoever to solve problems. And they have been around a very long time, and the smartest people, much, much smarter than anybody here, well, probably, certainly a girl a lot smarter than me, I know that, have broken their teeth on them. that's one of the first one has to be careful and the only guide fortunately fortunately for the only guide is nature sounds general but it is it is it is you assume too much if you go to encounter her with your tie and shirt like to take your story further about god you know I'm taking six days to just do your topology. What about re-normality? What does infinity, what does infinity? I saw in a paper by Hawking. It was, I don't remember the year. Definitely it was a journal of mathematical physics. He says, we assume this. He said, we assume that, in the absolute, we assume that space-time is a four-dimensional differential matter, okay? At the end, he breaks his face that there are singularities. You know, essentially, don't use fancy things.

1:17:30 You used calculus. And calculus cannot calculate some things, OK? All right. And therefore, at the conclusion, listen, listen. He says, ah, therefore we infer that nature has singularity. Never. Nature doesn't have singularity. Your theory has singularity, and your theory is of limited applicability and validity. Of course, you assume calculus. Oh, calculus, of course. Try to calculate the field right at its force. It breaks down. You do not jump over the river and say, ah, nature has singularity. Nature has singularity. You have singularity. You don't want to see singularities. Wear another set of glasses. Use another theory. Nature must be renormalizable is another one. otherwise you lose your that's why we've got nine dimensions or whatever it is 24 or whatever it is it's incredible how physicists like that do not realize that it's their theories they somehow identify their theories with reality in some way but this is Hawking has said physics is the model physics is the model the mathematical model so you're asking the following thing if we have the mathematical model what is the physical interpretation you give me the formal system and you don't give me the semantics come on this is not his great he has a very positivist attitude that's clear in the debate with Penrose in that semi-popular book on space-time it's very clear very positivist I suppose he gets it well I'm good for that label but it's pretty clear I think the main influence is Powley it's that towards the end how he was very well how you should have known that it was wrong but probably never went he he maintained that we have reached the limits of rationality

1:20:00 it's okay i'm not and that doesn't mean logical positive that's something that i don't know i don't know if it was never ready to do to jump into the world of dreams oh yeah he's also very sure what was wrong wrong yeah he was a great human being probably yeah but also in his criticism of bones work i think he he was a bit you know he he had too much to defend i mean you you cannot i don't think that you can if you are under the influence of such a great man as war that in some sense like a son I think it's a privilege to be said to be wrong by Pauly, because a number of people, I mean, poor old Abdus Salam did lose one Nobel Prize, because when he was talking about non-parity conservation, Pauly jumped in a taxi with Pauly and said, what do you think of this? And Pauly said, if you publish that, you'll suffer as a theoretical physicist, so good old Abdel Salam put it back in his drawer again and then these two Chinamen came up the coach and speck okay no you have to go I was going to just say about the coach and speck of theorem being in the top of setting because just being a it's just one illustration of the failure stability of the global section function goes. I mean, it's not stable, because you've got the relationship between the local and the global topology. You can't make coverings localised. That's right. In fact, it's a very typical situation in algebraic topology. It's really the generic situation, algebraic topology. It's what Lovia means by a failure of constancy, because you've got a kind of cohesion in the structure which does not show enough in the trivial one point psychological space case I think we're also like Janice I'm sorry if I bombarded you I wish you could be bombarded a lot more we're going to get you over again we're going to invite you over here again later on if you've got the energy of course I have the energy but if I have the energy

1:22:30 I look at the time I'm not from the end of the time. Yeah, it's a relationship. I do have the time. Anyway, thanks very much. We will be talking. Do you want a clap? Thank you. Did I have a chat? What worries me? Oh, yes, I was sitting on it. Did you bring everything out of my room? Yes, I did. I'll give you the junk. Is this your bag as well? That's a bottle of wine, because I have tea today, and I feel happy. I'm going to drink it with my sons. Don't give your sons it. Why not? Get them into bad habits. All the best. Thanks. Really nice to meet you. Thanks a lot. Bye-bye. A lot to chew on there. There's a lot to chew on there. beautiful stuff lindon's not here tonight no he rang up in the middle of the thing before can you give me his number before i disappear tonight is it okay oh good oh by the way there is another right speaking i don't know i didn't want to ask yannick i'll miss this one there's another janet another raptus coming on next monday a raptus as well i said i'm going to be aware in spain well it's not that i i kept he came to see me well you've got to say they started it I don't know what he's like. I know Giannis is great, but I don't know what this guy is like. Where is this? It's here. Here. This is the theoretical physics one. I won't be here, unfortunately. We've never had such a display before, have you? He's actually agreed. The reason why it's up there is because he agreed this afternoon, so I thought I might as well print it out and put the title before I could get. yes it's one of the Tucson you know I have Tucson is home and away I have a two-song conference of Tucson one year and then they have a conference because

1:25:00 Because John Briggs wrote again, he said he hadn't got any answer from me, but he calls me to this conference perhaps because he asked me where I go. Where is this conference? August. Where it is? Shuvda. Yuvda. Yuvda, that's wonderful. Where is that? What's that? That's in Sweden. That's in Sweden. I would have guessed Yuvda. It should be in Israel. I've got to say something there. When is this? In August. I've got to say something. That's because Paro is my friend. I don't know how long he's going to stay my friend. Because I've got a paper that I'm supposed to be proofreading and correcting. So we have a chance to see Morris Pearson's as well. I will see him in two weeks' time or three weeks' time at a mini-workshop. But that's not in Shifter, that's in... But, Carl Skroger. Carl Skroger. He's invited me to write a poem for his book, which I've done, and I'm not sure. I've been able to do it. Yeah, I've got a corrected paper, yes. Talking about corrected papers, you've still got that... Well, you were saying it's Duffin's got lots of mistakes in it. No, I've got a new version. I'd like to have a look at it. Yeah, because unfortunately I'm going to be off from now until God knows when. I might be around for a week or two at the beginning of August, but at the moment I'm just not sure. Sorry, Richard asked this because we want to talk about more physics, but did you have a chance to do anything on the log-in? Yeah. Oh, brilliant. I got in touch with a guy and he is arranging it. Unfortunately, I'm not sure when it will be arranged, but he certainly will be arranged. That's too late for me to do anything. Unfortunately. That's not your fault. Well, I'll give you a ring. Are you going to be around in the week? Okay, well, I'll give you a ring probably tomorrow or Thursday. I'm going to Oxford on Wednesday, and I'm going to be very, very busy tomorrow, but I might, yeah, why? Well, Thursday, bring me home.

1:27:30 What's your home number? 8-6-8-6 8-6-8-6 5-9, no, 9-5-8-7 Sorry, say the whole thing again, 8-6-8 8-6-8-6 Yes, 5-9 I don't understand, that's not a London, this is 0208, that's why I'm getting confused, the 8 is part of the 0208, 689, yes, sorry, that's why I'm getting confused, so it's 0208, 6 8 6 8 6 8 6 9 5 8 7 yeah now that that's a now an intelligible phone number that's okay yeah as you say i live out in the sticks yeah the real sticks will I'm sorry, eh? What do you mean if you use a webmaster one? Is it separate? No, it's not the markets are the same thing. Well that's alright, I mean at the moment there's no problem. They won't because I've got it and you're my help. Well, okay. And they were very sympathetic, they understood that he doesn't think it's going to be... Oh, let's see, you... Do you have one or two thousand dollars? Uh-huh. But from the top? This guy is now in London, it's not in London. Janis is working with... Chris Heysen here in two years. I will drink some water. That's why I know someone's that top of us there. He presumably must know Jeremy Buckingfield as well, if we can get this. Is he teaching? No, he's not, he's got a fellowship, he's got a fellowship, a Madame Curie fellowship, European Curie fellowship. He's just working on doing what he should be doing, which is using his energy to be creative. heard anything from David Peets in this respect nothing no I passed on all the message to Alberto and told him to ring him I told him to ring him PDQ because on that

1:30:00 email that you gave me the copy of he said he's got to speak to the UNESCO people by the end of February but it was actually the last day of the month well something like that well I rang him the same day and he said he would get hold of him what I did was to Alberto's I think Alberto's Yeah, well, I'm right Telling him something different and that the numbers were all wrong but this could be a guy I asked him not to make it public to anybody No, no, well I've now put them in touch I've given Alberto David's phone number and email number and stressed him that it's urgent well actually it's quite a distance from Florence to Paris the Italians have a No, no, but the Italians have this very regional view of their countries, you know. To Alberta, Pari is virtually the Mezzogiorno, it's Africa, even though it's actually in Tuscany, because it's right down. Where's he in? He's in Florence. He's in Florence. But they're just next door to each other. Well, yeah, it's down in Crusader, yeah, to you or me, they're next door. Italians have this very, very city-state view of them even today, especially in Tuscany. But anyway, I told him where to find him and gave him his phone number. well he's a very easy going and amenable guy what I'm worried about David doing is sort of pinching it and putting it up and broadcasting it all over the place before Alberto even realises I've now headed that possibility off by telling him to get in touch with David make sure he doesn't act as a loose cannon but at the same time if he has got connections with the UNESCO Alberto can do to input I think that's important Yeah, well hopefully they all have done so now This was about five, six days ago I'm talking about hounding people Alberto, sorry, you sent me an email I believe I sent you a tax message Right, thank you very much indeed I haven't seen it yet because my computer is still sitting over I'm supposed to get it back tonight But I was told that there was a long email from you on it, so thank you very much. This is the thing you sent to us. Oh, okay. The knots and logic. The knots. No, no, no, this cannot send you, this is over 100 pages. Oh, well this is the thing that, is this the thing you've got?

1:32:30 It's not the thing I've brought, I don't know if I've got that before. I've got the one that's in the actual project. Yeah, it's the one I thought that you... I told you that I will scan bibliography or not. OK, OK, if you could do that. Yeah, I will do that in time. And you wanted me to get, because this is unfortunate, I haven't seen the email you sent. Can you tell me which ones they were? No, I don't remember. It doesn't matter anything. Don't say. Well, if you can remember which ones they were, I'll bring them. I'll send them to you. I'll mail them to you now. You see, I saw them. Well, in that case, I'll have to bring a whole lot. Some papers you've never shown me, but I don't know. I would have to see them. Well, I'll bring the whole lot. I have to say they were not published in time, because otherwise I would see them. There are some that haven't ever seen the light of them. But it's not so important, you know. Okay, well, if you remember which ones they were, otherwise I'll... I was getting all that stuff bound up, and when I've done so, I'll just send it to you. Well, I'll tell you, okay, because it's better, you really don't need nothing, this is huge, this is almost like a book. This is a 100-page paper that you told me about. I would actually know that he copied and sent some cards to his own house. E-mail? No, no, he did, because if I had it as an e-mail, I would... I'm sorry if I got confused, but in fact, what I was going on was, and I've actually got it here, the thing that you sent me was ages ago now. Are you going to be our professor, sir, and what do you collect all this knowledge? No, I'm just a pulsive collector, that's all. Ah, you are a collector, aren't you? A treasurer, possibly, I don't think that is.

1:35:00 You could blend that. Yes, of course, that's the idea. What the hell is it? It's a very well-organised collector. No, it's not that bad, it's organised. No, see, it was space and time and computation and discrete physics. Is that the one that you... That's exactly, that's the one that... But that was the one which you said had the long bibliography. That was the one that I was after. And the 100-page article is the not logic, which is in the book. Look out on people who have also all these bibliographies. Daniel, it's quite a cursive method in general. They can sort of copy now of bibliographies. Okay, if I could, that would be. If you belong to an academic library, you could ask about artists who are here. They sent us free, they copied some British library. yeah i do belong to the british library but surely i mean i mean the british library yes yes if i had a yeah if i had some official standing academic standing but if you just copy stuff in the british library it costs about Well, last time I had the British Library copy stuff, it was about £30 a page. Go to your local library. Go to your local library. How are they from British Library? And you pay 30,000. Well, well. And you order... And you order not as large, and you may pay 30,000 for that. Wow, I didn't realise he could do that. That's useful to me. In fact, I'm curious one day about how he did things like that. I simply asked him. He said he goes to the local library, or does it send it to him? Who is this? Dear Peter Marceau, I don't know, but I'm... Once you've met Peter Marceau, you will never forgive me. I just have to give him... As bad as that, I'll never forgive you.

1:37:30 Yes, Freudian slip. No, it's all bad and bad as Peter. I was in the same hotel, we went back five days in Brussels, and we walked down the conference together. We had a new meal together. And one beer session where he spent all his beer on the table, Oh, is that the thing about Scott the bibliography? No. Oh, it's not. Well, you have to Xerox one paper after that. Well, can I do that? Is that OK? No, thanks. No, no. Yeah, this is the one. my education is not good good enough for for that I really appreciate everything, I mean, the details of it. I would like to talk more about this Rokha Archer, perhaps. Yeah, I mean, what he did here is that he didn't know quite what to do, so he gave us a global picture of the sort of areas. Which is very good, but one has to probably read something around it, but I'm not sure. If I go to this paper and... This is for our letters, if I understand that, but still it will be certainly better than at the seminar, because at the seminar it's absolutely impossible, but in the literature it is possible. I won't do that, I won't do that.

1:40:00 It's on, my little bit. It's on. Do you have to...? It's a privilege evening. Do I? It happens to be my 50th Thursday. I only have a century old one, so I thought... I'd buy a couple of drinks, I think. Oh, that's very good. The property shouldn't have... It's just timing, you know... Yeah, I've got to go off to Romney My Bird to get this bloody computer down. If people have got time, just to come for one drink. Thank you. Thank you. Thank you. Thank you.

1:42:30 Thank you. Thank you. Thank you. Thank you. Thank you.

1:45:00 Thank you. Thank you. Thank you. Thank you. Thank you.

1:47:30 Um, let's switch this off. in the sense of I, A, W, I, minus I, A, minus 1, W, I, plus 1, where these Ws are essentially column objects which have zeros everywhere, except the K element, or the Ith element. So you have a 1 there at the i-th element, and then the other one, 0, 0, and a 1 at the i-th plus-1 element. And if you use these as your equilibrium points, then you're going to be able to play that with yourself. So this is the last one. Remember we got into a mess where you get the ij equals ji. Yeah, well, what Lou does is he defines each primitive idempotent in terms of a pair of vectors. It takes this idempotent, this is not idempotent yet, sorry, let me take this. I have my idempotent as equivalent to the other that I use as a projection operator. Yeah. Well, let me call that a K. Then VK, or V-I, sorry. Then V-I, E-Y's like this. Now, if you put these in here, you then get that relationship satisfied, but you also get the relationship E-I, E-I plus or minus Y, E-I, E-I. That's why I forget the tool, it doesn't work out.

1:50:00 Now the interesting thing, you can see why it's working, because this, what zi looks like as a matrix, essentially, is that it zeroes everywhere except for a 2x2 matrix somewhere in the structure. Which is I... Is that on the diagonal? It's down the diagonal. Yeah, you're right. The two elements must be somewhere... Have a look at all these two by two diagonals. And then the ice would be the ice element here. And there would be... I don't know, you can look at that. You've got elements there, so that when you multiply these two together, when you're well away from later, it gets here. Although, lots of them away from there, they won't match up today. When they're together you get this relationship here. And these are chosen in such a way that you're guaranteed to get that relationship. This was a trick that I didn't. Yeah, that we couldn't work out last week. We couldn't work out how Lou had done it. Yeah. Right, now I think I see. Now, what fascinates me is if you look at the paper you've just Xeroxed. Yeah. Yeah. You'll find that the Lorentz transformation... Have you got it? Yes, I have. Erm... Hang on a minute. Yeah, well it's of course in reverse order at the moment. It doesn't matter, I think. Yeah, there you are. No, no, keep going, keep clicking. Keep going, alright. I just want to see the exact relationship. Go on, keep going. Yeah, there will be. Well, how the hell did you do this? It was just for myself. Okay, anybody else? Okay, hang on. yeah is that the one you're looking at yeah yeah if you take you see he has these pairs a b and then if you want to lorenz transform them you have to have a k a and a k minus one b now these k's are just the a's that he's got for some reason or other that this this has got all of them are raised transformation and if you use the lorenz transformation between these or putting parameters that directly relate to the lorenz transformation you actually get this jones you get the j-putt thing yeah you get the you get the braided you get the braided algebra which

1:52:30 from which we go on I don't know why he suddenly puts that except that he knows that Lorentz's transformation is always I mean right now yeah yeah this is where there's a unimodular yeah but but how does it connect that with the diagonal well just that when you multiply these two things together you're going to pick up a diagonal element oh right everything's going to be zero oh yes except for a diagonal or two by two matrix because the structure has been w right right yes and i have to have this a a the minus one for that sorry structure right and then you'll get the it doesn't matter about that but then you've got to you see i'm not putting them off the off diagonal putting them up along the diagonal like that will guarantee this being equal to zero which is what we couldn't yes that's the thing which we couldn't get but now you've got to make sure that when you put it this you get this relationship otherwise yeah you know you're not going to get the power you've solved one problem and lost the other one yeah and if you choose them as this then you get the uh the first relationship that's in the paper that's it here's a paper that's uh but it's rather interesting that he's always got this a a to minus one yeah of course yeah that's what i said it was what you essentially said you've got to make about sending that which made it all rather trivial but if you take a pair of in other words you take a pair of that then it becomes non-trivial and you've got instructions so it's the point of writing every every impotent has a pair of it and so you can begin to see why

1:55:00 I'm excited about what the relationship between these things and these influences I was playing around with. I know this is not there, but this seems to be an essential structure from online and out of it. It seems to be worth to look at the Rotterdam, because it's also finite too. Yes, it's nice and finite. I've never heard about Rotterdam. Because we practice people so, yeah, and you do pathology, if you want, you can put it down, you don't feel bad, I mean. I'll give you a ring probably on Thursday, because unfortunately it won't be here now for the foreseeable. That's okay. Oh, can I just get that paper off? If I borrow and promise to bring it back on Thursday, I'll stick it in the mail soon. I'm coming tomorrow. I might be able to come in tomorrow, I don't think I have some things to do before I leave, that's all right. OK, well, thanks a lot. OK, well, have a good trip. I don't know... Er, no, actually, I'm looking at the time, I thought I, well, I've got to go and take my computer from Bromley-by-Bow, and it's already nearly nine o'clock. Yeah, so anyway, have a good season. That's extremely cloudy. Well, I think I will be, with a bit of luck, I will be around from, you know, from time to time. Right. There will be gaps in between, so hopefully I'll be able to come in on a few Mondays. Right, we'll drop you in a few minutes. I'll certainly just stay in touch, and if we can get in, if I find out that Bill Morvier is coming over again, then we certainly do want to try and arrange for him to spend some time. with you and with precision yeah and indeed with Yanis who I think would be very interested he would like this son now I have heard him talk he doesn't like point house construction at all because he thinks that the non-competitive

1:57:30 is just coming from the fact that it's sandwiched in between two by categories and it's just a an artificial way of looking at the structure which is much naturally understood in the right categorical setting where you know you maintain the cartesianity and agiteness but he does and this is of course the objection that he was making to Kohn's work you know the point of view of the Morita equivalences and which in fact very much aligns with what Janus was saying that the reason he doesn't like the non-comitivity in the geometry the yeah but he will be very interested I think in this good stuff yeah thanks Thanks again. Well, I say, that's beginning to move, that's tying in very nicely, that paper. I hope it's debugged now. OK, well, if I support anything... There may be some bugs in it still, but Clive has actually been through it and made some suggestions. I've changed it to respond to that. And I'd like to go over the Luke Carpenter stuff in connection with what I was going to be just touching on now. That was just going a little bit too fast for me to pick up. It's coming, you know, I'm sure that it's been... It's understanding how you think of the structure of the diagonal, in terms of where you get the space-time geometry out of the process that I need to get my... You see, what's happening here, instead of having a point which has a simple... You see, it might be that these points are actually sort of two... they're more complex points, if you understand what I mean. See, normally a point would just be a 1, 0, 0, 0 down here, so that's quite trivial. But now what Lou is putting in is saying, oh no, but if you want to relate these things in a good way, what we need here is some structure in the point. So it's a sort of a point with structure. Yes, yes, which is also very much Bill's way of looking at things. He's, you know, the idea of getting this. Isn't that fascinating when this point structure now comes into the generation of the von Neubenarship? Yes, yes, yes, that is interesting. Well, at least we see now the role of having to look at two vectors at a time in order to get the structure around where he's getting the zero condition from.

2:00:00 I'd like to understand... You see, I think Owen's method would have worked. What is Owen's method? Well, it's just saying orthogonal. Right. No, it's not going to work because it wouldn't get you this relation. Remember, we were either getting that relation or we got that relation. We couldn't get them both together. The EI, this one, you know, the... Yes, that's right. That was the problem we had last night. Yes, the neighbourhood. And then this one for distant neighbours, as it were. Yes, that's right. We were asking, you know, was there anything we could think of that would be a model of an algebra where you had the relation just to the... When we got this one right, we couldn't get this one right. We couldn't get the next... When we got this one right, we couldn't get that right. Yes, you could get the nearest neighbour relation or the relation to everything other than the... Yes, yes. But now you can get both together, provided you look at the vectors in the impotent. That's the VK equals I As a pair of vectors AWK equals, yes, minus With this particular value WK plus 1 I'm not sure whether that's plus or minus there That's the value for the coefficients in the That's for the, remember these things are like this Okay, I've got a K here So this is VK, VK So this one would be yeah these would be the vectors that you're using you see yeah and now you you take on this side the v5 are you off no chance of a quick drink all right all right you're gonna have to go and get your computer yeah i am okay okay okay well you put those together and uh Bob's your uncle? Yeah. I mean, I know you haven't spotted how I bobs me uncle it, but... No, I haven't yet. That's the problem. That's why I'm irritating the hell out of me. I'll take your word for the Diz Rotary straightforward. It's very trivial. Because essentially what you're doing... Look, suppose you have one zero, and your direct product of Ian with zero one. So you'll get... There's zero everywhere, isn't it? Yes, exactly. The first one's going to be zero, zero, zero.

2:02:30 And then, now this is too trivial, it's not going to work. This gives me an off time now, it's not working this way. I'm sorry, no, I hear this. Cheer our letter, thanks very much for the paper. Well I know I'm getting the tube coming in today. It's amazing the things you can do in the tube. it's that it's that expression on the right hand side of the line a sign that I'm not Now, I'm missing something here. I have to go back and look at what we were doing last week. I could see where it was that we got stuck last week. And what is there in saying about the orthogonality? Well, it's just that these are all equal to zero. That's why they're trivially equal. They're all equal to zero. Right. Minus j is greater than y. Yes, provided, yes. They're all equal to zero. That's orthogonal. That means the two vectors are orthogonal to each other. Oh, because I'm being stupid. I'm being incredibly stupid. I'm being incredibly stupid. That's why it's not working. Because what I need to have is one, no, I've got to do a bigger, if I do one, one, zero, one, one, zero, and that's equal to one, one, one, zero, zero, and this is... Ah, yes, that's what we were missing, you were saying, it was just too simple. I was using, yeah, it was too simple. And now, you see, if you drop it down once, then these zeros will come up here, and so you're going to get the other one. It's going to be zero, zero, zero, zero, zero, one, one, one, one. And then you'll have your diagonal.

2:05:00 And now you've got your diagonal. Sorry, it was just being there. It would have been too simple. Yeah, that's going to work nicely. Did you have a chance to speak to Precision, by the way? No, I haven't. No, it's OK. I'm not tuned. I've been so busy trying to catch up with people chasing me. do get a chance to speak to him and broach this whole project I mean I've already said well I'm prepared to put into it you know this year and next 18 months and all I'm waiting now is to hear from Alberto to see what you know results of his chatting to David Peet have been and to hear from him he's supposed to be speaking to Bill whether Bill still plans to come over here in April and so for how long and how much time he's planning to spend in London as soon as I pin them down I'll let you know. The trouble is I now have just six days in which to get this tied up, and then I'm away until the end of the month. I won't be back in circulation because I've got to do two trips in Spain back-to-back, and I won't be back in circulation until the 31st of March. If he is coming over, it might not give me very little time to get ready, but, you know. No, but one thing, I think I've started to tell you about today, Christopher, are you there? Oh, yes. The people are supposed to be getting their logging game, possibly. to run a half-day conference to celebrate my career here at Birkley. Well, I think that's the very least they should do. I would have thought you could have run a little bit more than just a half-day conference. Well, I mean, I guess I'll be... I think that's the very least they could do. Well, that's what they're actually going to do. Yes, I understand. I see this stuff now. It's quite trivial. Once you've done the three... Once you see these things... Once you see where the diagonal is coming from. And I see now what's going on with your fingernail single vector yeah that's stupid i mean it's it's too that's why it won't work you see that's what we were trying to do yesterday last monday yes that's right that's why it wouldn't work and it wouldn't work well if you're going to get that computer um Thanks very much for that. Yeah, yeah, there's a very interesting stuff, actually, the rest of the stuff in that general

2:07:30 looks very interesting as well. Well, yes, that's nice to see. Was it just that number? There may be a seat or something. Yeah. Thanks for coming. Well, it's nice stuff. Well, that's what I wanted to do. So, we're just a little bit worried. Did you say which library's paper? Oh, the British Library, last time I went in, I had just an oven, because they built a cup, and it's not that hot. Well, they've had that oven. Not yet. I think they've had no idea. No? In that case, I may actually chitly use a pirate, but it could probably have a whole thing sometime. Here we go. Oh, hi there. Oh, great. 4, 3, 2, 1. That takes us. Kathy McGowan. That wasn't the clue. That might have been the clue that was in your mind, but it wasn't the clue that was in my mind. What was the clue that was in your mind? That was strange. On the other hand, Thank you. I'm sorry, do you have a car or are we going to take our way back to the kids? I've got the old second line today. Oh, sorry, it's good to look up.

2:10:00 Are you feeling very tired? What? Are you feeling very tired? No, I think you're good today. I don't know why I have a vegetable and I have a little bit today. So this is why I keep my head straight. Yeah, I get my pillow through. I don't remember. I thought you were just with you. I didn't come, I don't keep my car. I thought it's better to go out anyway. I had to ride down for a few hours. I hope you're sitting on a hundred percent back. Well, I'm very grateful for the first place for this one, and in some time we must sit down and have a serious talk about that paper report here. I mean, you see some of the stuff there, that's supposed to be one thing there. No, the one that you have to do is talk about. Some of the... Anyway, you take care. I'm supposed to be around in April, but at the moment I know I do, it might just be for two days, or it might be for most of the month. Because I'm stuck in the air, whether I'm going to show me where to do a trip in April, or whether I'm going to be around in April. So hopefully we'll see you around in April. Have a great day. Yes, thank you very much indeed. And you are all up there. Get better, alright?

2:12:30 Very good, yeah.