Quantales — Noncommutative Theory of Penrose Tilings

0:00 And the same tile at the level N, so right N here, it will end up, so suppose this is tiling at the N level, so after we remove N edges from the original tile, so this thing will say that you allow the particle to go around the plane as much as you like, as long as in the end you end up in this slightly larger tile, which you'll get at the level N plus one. and you write l if the tile where you ended up is a large tile so ln would mean that you ended up inside this tile here and sn would be the same thing except that you ended up inside the small tile then you have a dual pair of measurements this one will succeed if you start from a tile which is large at level n and then you're able to let the particle follow any path provided that it ends up in the level n plus one tile and this is the same thing for small so there's a very concrete motivation for this which is not It's geometrical, it's not really physical, because there's not a physical system involved here. And what's the difference between the dual and the dual and the dual and the dual and the dual and the dual and the dual and the dual and the dual, and what he said is just some... Here, you allow the particle to flow around freely, as long as within this level, endless one tile, it ends up on the large tile at the dual hand. Here, first, you have to have the particle at the large tile at the dual hand, and then you'll let it wander around freely. So this is a temporal dual of this one. I mean, it's at, you know, that same large time as itself at the beginning, but do you mean it's the same as where it will be at the end? Sorry if I'm not getting this. I'm sorry if it's obvious. I'm sorry if it's... It would be clearer if what is now the end point, in other words, the dot somewhere to the right, actually was over at the left. Just reverse the path. So if you had a path starting close, put it this way. the particles first in the large top, then you have a path in this direction, and it ended up ending up here. So you said, okay, I have actually observed this.

2:30 Now, if you start from this point and go in the opposite direction, then you end up here in the large top. So this is going to be... So you'll be starting from this position, which is a small tile, which you write like this. I start it on a small place, and then I let my particle move. So here is, I let my particle move, and I end up in a small tile. Here is, I'm in a small tile, and I let my particle move. Okay, and all four of these have the precondition that they belong to the n plus one, the same one, right? Yeah, that's right. What kind of powers do you allow? Are they just polyhedral paths? Any continuous paths? Any continuous paths. This is going to be, this is really for motivation because this is going to disappear. I'll be writing down some axioms which are very easily motivated. like this. So you can't compose to experiments by multiplying. This means that you let your particle move and you end up in a small tile. And this means that you're starting on a large tile and then you let your particle move. Now, of course, these things are incompatible. If you end up in a small tile, you're on a large tile. Therefore, you'd say that this entails pulse it's an impossible experiment another thing you can do is if you do something that ends up in the small time then certainly if you did the same experiment but one level up you would discover that you're in a large time because the level above at the level of all these skills a large time So you can write down a lot of things like this, always following this motivation that you'd have the particle fall in the past. I'm sorry I have slightly slid things down by asking something that's meant to be obvious, but the one that you've written is obviously false. Why is it you can't start in a large of level M and finish in a small of level M? Because after all, that's what you've done in your example. Because this one has the post condition that after the experiment is performed, the particle will be in a small tile. But this one has the precondition that in order for the translation to be performed, the particle has to start in a large tile.

5:00 So the post condition here is the negation of the precondition of this one. But I thought it meant that, you know, it starts there and it finishes there, and you don't ask yourself what in between, but it's trepidly allowed to say, I want it to start here and I want it to finish here. I mean, that's... Do you mean one path and then another path? Or are you talking about a single path? If after trying the first experiment there, you then try the second one, it will fail. Oh, it's because by coincidence that happens to look a bit like D-Racket. You mean you read it from left to right? Yeah. Right, okay, not from right to left. Did that help? Yeah, I meant this to look like Brasley has been together in a metaphor. You're quoted by choosing start as the right animal and finish You have this inner product between the things which these two things are orthogonal The Quantale product sorry, I was not here The Quantale product, you read it and then Yeah, do something and then you think something else Okay, so It's a very peculiar looking axiom, for instance, which is, I want to short it this way. Now, one will define true as a very special experiment, an experiment that can always be performed because you get absolutely no information from it and you can't consider any pass provided at this target and at the same place so what what one is saying here is that this kind of experiment implies that when that's a particular way of performing this one now what is this one this one means that you do something and you end up on a large type of event and then this means reverse it can go back to where you started from. So this is saying that wherever you are in a tiling, whether it's in a small tile or in a large tile, you'll always be able to move to a large tile at level n and then move back. So that's how this is a particular way of doing this. But it also implies something very non-local about tiling. It says that no matter where you are in a tile, you'll be able to find this small this finite pattern so this is a really strong accent and it's the one

7:30 that ultimately was going to distinguish this non-committal theory from something you would get in using the house Is this meant to say that the truth implies that transitional, it means that the experiment always goes? Yeah, it always succeeds, and one way of doing it is that you start and end at the same now this this is a bit this really doesn't work if you look at it here because this would say that any path that's towards and ends at the same place it would just null path is the particular way of doing that whereas here you require that your path has to visit some large tile even if it returns so in this case this is an axiom that is actually all in respect it up to homotopic, because in the plane, then everything is now homotopic to everything else. And that is the reason why I was talking about homotopic, because in the plane, then your, your homotile associated to the local compact, your, to the, to the fundamental root part is just the front of binary relations on, on the planet X. And this also means that in terms of these accident that we came up with, the only way to really satisfy everything, and everything down here was okay, we'd pass. But this one requires up to homotopy, and therefore, you're actually thinking of the experiments as just binary relations on the plane. So this, in Presumably, it's also true that you'd exhale massive, so long as possible. Sure. Well, for the same reason, because you won the front enough, you won't pick it on the map if you won. There's a little, yeah, I mean, up to some technicalities, because... But, yeah, I mean, you can do something if you start with an L, you do something like that. And if you wrap the small tiles in the large tiles, this is because of this difficulty that you can have repeated ones.

10:00 That's not an event even in time, that's facts at higher zoom levels. Yeah, but I didn't want to get into this. But yeah, and the heart of it is yes, you're right. Okay, so I'll just give you the domain result. I won't get into too much detail now. So what you get when you take that set of primitive propositions and those and a few more axioms is get a quantile and then it's homomorphisms to quantiles of binary relations correspond to seeing these things as experiments on paths after homotopy. They are the so-called relational representations of a quantile. from a different part of quantile theory, from this Max A, and from the notion of quantile representation that came from here, which was that quantiles were going to be represented on lattices of endomorphisms of filbert spaces. There was a notion of irreducible, so called algebraically irreducible representation coming from here, and when you apply it to here, you get a classification theorem saying that the algebraically irreducible representations of this continent are precisely the components of the equivalence classes of this quotient. So whenever you have one of these, there is a way of assigning to each element in the set x one of those sequences of zeros and ones. And once you know the sequences of zeros and ones, the whole dynamics, the whole action, is completely determined. That is, these representations are actually classified by the sequences that are assigned to each state in the representation. So it turns out that there is a canonical representation where you take the set K of all the sequences that is automatically split into irreducible components,

12:30 each each of them being an equivalence class here so what you did was you recovered well the fundamental group point of this joint union of all the islands from just from the top one top so that that set of accidents really produced by it this representation theory the whole description of 10 rows times as as equivalents as the quotients have okay there is in fact more that one can say which is I did not mention that I mentioned that K is actually a counter space so because of this you could assign the product ecology to this thing which is not really going to work as we want but well there's another topology which is locally compact here and in terms of which you can define a convolution algebra on this scene as, on this equivalence relation seen as a locally compact groupoid. And you get a Caesar algebra, which is the one that Conn presents in his book. So all the information you need to describe a Caesar algebra is the equivalence relation and the locally compact topology on it. Now it turns out that we don't have this in the print print, but from this point out, you can not only recover the set K, and the equivalence relation can also recover the locally compact topology. And therefore, from this quantile, you recover enough information to reconstruct this whole C-SAR algebra. So in a way, what you have for the quantile of Penrose-Talings is that the reducible representations see these things as experiments that correspond to binary relations on the set of sequences. Whereas for the C-SAR algebra, you have the the same thing, except that these things now have the complex numbers in it. So, each irreducible representation here will correspond to an irreducible representation of the Caesar algebra, where the set of states here now is a Hilbert basis of the representation. So, that's what I meant by mysterious. In some mysterious way, there's a Caesar algebra, which seems to be a lot more than just a Montel. And, well, there you go. I mean, it's all there. I mean, this Montel, which was thought of as something much simpler, in which you just say which paths are possible without assigning any probability amplitudes, ended up giving you the same information that you require to recover the C-SAR algebra. Now, how much of this can be done in general or not is still, well, of course, one of the main things to know.

15:00 But it's a different reconstruction from the one you had here. Here you recovered A, and here you got this A in a somewhat different way. One of the hopes is that somehow there is a map here which can be thought of as the unit of the junction provided you throw away enough morphisms from the category of functus. So it is known that this functor max is not, doesn't have any adjoints or anything, but it's still open to investigation whether throwing away enough morphisms is going to produce an adjunction here. Thank you. I have a question to state his name, to tell us his name, so that we... I was asking... Any questions? My name is... Is there any connection with the work of Kelendonk and Dawson? Oh, you didn't go semi-groups in 10,000 times, I think. The technicalities of it have a connection because old Kelendonk's work is about convolutionalities of some group points. except that there you have one thailand and associate one groupoid to that thailand and then you take its convolutional algebra and from that extract information about that particular thailand whereas here you have the groupoid which is about the sound of old thailand i mean the mathematics of it is pretty much the same um no they i think they they're using well and they're using this to, well, this is solid state physics. You take some crystal, model that as a positive reaction there, which you can because it's probably complicated. So somehow the Schroding operator lives inside this convolution C-SAR algebra that they obtained from that group work. When they study the K-theory of this, they get stochastic information about what they call gap labeling, about the frequencies with which gaps between the lines of the spectrum of a particle flowing freely in this lattice are going to appear. In this case, if you do the same thing, you get this kind of information

17:30 but in a different sense. The theory will give you information about the probability of finding some particular finite pattern inside an arbitrary pattern. But mathematically, it's the same thing. Yeah, Ralph Comperman. Two questions. First, were you using John Mack's representation of non-community C-star L? No, I didn't use it. I haven't, because you said, no, you didn't send me the paper. You gave me his address and I asked him the paper. No, I didn't use it. Okay. All right. Great. Secondly, only if it doesn't take a lot of time, tell me why you resist going in the direction big to small, because I think that's the natural direction, somehow. Big to small. Your tiling, you always increase the size, and when you were asked before, you said, You said, well, yes, sort of, but, and if it's going to take time to explain why you go from a small tile to a larger tiling, with therefore less, fewer tiles. You can do both, I mean, implicitly in these experiments. Can you do, is it a reversible operation? Can you go from big to small as easily as from small to big? yeah okay it's terministic you see when you have when you have a tile like this and if you know so you have the orientations and the colors and you say okay this is a tile at level n then there's a unique way of splitting and you cannot choose it to do it this way or that there's only one solution so in fact this is part of the algorithm that builds a tile from the sequence because will give you numbers at level n, you have a zero. So now you know which tile it is. And if you see the number below, well, you know how to divide the tile. Now this number just tells you, well, it's used to be here or to be here. And you see, look at the numbers. OK, thanks. Any more questions? That sounds like a good question.

20:00 When do we resume? We resume and say, quarter two. Thank you.