Noncommutative topology in geometry & physics, lecture 1 (contd.)

0:00 All right, so let me go on. I'll double check on that. Let's go on here. So now, so what's the motivation for all these various equivalence relations? So in the community case, you don't really get anything new. But why do you need things like marine equivalence? So let's talk about the case of locally compact grouploids. For grouploids, there's an important equivalence relation which is called similarity or equivalence, which is not the same as isomorphism. So just as a very simple example, if you look at these two grouploids, these grouploids are actually equivalence relations. These equivalence relations are what are called similar, isomorphic. So what does similarity mean? It means that if you think of the groupoid as categories, then these categories are equivalent in the sense of categories. So remember, an equivalence of categories is not the same as an isomorphism of categories. It's much weaker. But it's still a very basic notion. In a locally compact case, correctly, then you need some topological compatibility also, not just the goals and categories. I don't want to go into that, but you can sort of figure out for yourself, I think, what the correct conditions are. The most basic example of similarity of grouploids, maybe I should mention this, is obtained the following way. So suppose you have a locally compact groupoid, so we'll sort of draw this. Here's a picture of G0, which is a set of objects, and then you have these morphisms that connect various objects and so on. So these are the elements of G. Now, one very basic kind of similarity arises as follows. Suppose you take a closed subset in G0. which is saturated in the sense that every point in G0 can be connected by amorphism to something in here.

2:30 In other words, this thing generates the whole group point. So every point over here can be gotten by a morphism starting over there. So then you can consider what's called the cut-down of the groupoid just for this set. So if you take a smaller set here and just consider morphisms that begin and end in this, then you get a much smaller groupoid, but that one will be similar to the big one. And that's the most basic example of the similarity of two points. And if you have similar group oids, then the C-star algebras are not necessarily isomorphic. I mean, this was the, by the way, this example over here was also that type. See here, this is a subset of the objects that generates the whole group oid. And if you just restricted this subset, then you can get that. So that's a similar group oid. So, you have similar group voids, and then the C star algebra is a marine equivalent, but they're not isomorphic. In reference for that, the paper of Jean with Paul Muley and Dana Williams, the equivalence of an isomorphism for the group like C-Steropolis. So here's an example. Suppose the group G acts transitively on the space X. Remember we talked about this group like G cross X. x, and for any gx transitively, for any point of x, you have an arrow that goes to any other point of x. Okay, now, since the action is transitive, the groupoid is actually generated just by a single point in x. So you can replace x by a point, and if you replace x by a point, what do you get? Just get the morphisms that begin and end at this point, so that's just

5:00 This is the stabilizer of g sub x. So that tells you that c star of g sub x is similar to the c star algebra of g cross x whole thing. I think you should assume that g at screen in front of your place. Yeah, you wrote the free statement. I wrote the free statement. This is true in general. This thing, of course, is a G cross product of C0 of x. And this turns out to be actually isomorphic to this thing tends to be a path. So I guess I wrote down a... I should have said and freely, then that would be in this case. But you don't actually need the action to be free. And you generally have that situation right there. There are all sets of misprints in here, so if you point them out, I'll try to fix them. The version you're seeing now is not the definitive version of the dust. I'll just try to write this out very well. Now, since we're in the Institute Poincaré, it would be useful to actually look and see what Poincaré said about the science of topology. So, Planckery's very first paper on topology was in 1892, and here's a little quote from it. So he talks, what's now known by everybody as topology for Planckery was something called Geometry de Situation, a kind of strange term which fortunately went out of existence pretty soon. I don't think it's a very good term, but nobody uses it anymore. When Finca Ray first started studying this, as you see, topology as a subject didn't really

7:30 exist. All there was was geometry. And that geometry was of a very special sort. It was mostly a low dimensional topology connected with complex analysis. So, I mean, Riemann surfaces, of course, existed back then. And that was largely what he was talking about. But he mentions, of course, the possible interest in studying typology in arbitrary dimensions. And one of the questions he poses right at the very beginning is, what are the invariants for classifying spaces in five dimensions? So that is this. The origin of Piper A injector, of course, is in this paper. But before he even gets to that, he mentions this. So what did he say? He said, he asked, do the Betty numbers suffice to determine a closed surface from the point of view of topology? Now, when you say from the point of view topology, you have to see what equivalence relation he was really talking about, and you see his notion of equivalence relation, he says par voie de deformation continue, through continuous deformation, so it seems that his notion of equivalence relation, for him, the sort of natural equivalence relation in topology, of what we would call homotopy equivalence. But the same general kind of point of view could be applied to non-commutative topology also. It's just that you need the right equivalence relation and you need the right set of invariants. The problem is basically to figure out what are the right invariants for studying non-commutative which faces up to certain constellations. All right, so now...

10:00 Since there's a lot of confusion in the literature, I thought maybe I should talk about the difference in similarities between non-commutative psychology and non-commutative geometry. So, non-commutative topology is basically the problem of finding classification of certain kinds of non-commutative spaces up to certain kinds of equivalence relations, like chromatop equivalence, that sort of thing. So you're following the lead of what Poincare did there. So it's therefore natural to consider things like stable homotopy equivalence, since you also, in the regions we saw in the case of groupwise, you want to consider stabilized morphism 2. So then you're led to things like that. And it turns out that once you already start considering stable homotopy equivalence, you end up with something that's relatively close to k-thirty. Not exactly, but has lots of features in time. So people always ask, well, why in non-commutative pathology, non-commutative geometry, does k-thirty keep coming up so often? And I think the answer comes from this. In other words, once you already consider stable homotide equivalence as your basic equivalence relation, you're almost forced to look at things related to k-theory. And there's a history of papers that examine that question. I mean, I started examining that about, I guess, 24 years ago. So this paper over here, the role of k-theory non-computed algebraic topology, and it was since then a very important work of goats and pigs and various other people, mare, nest, and so on. And so now lots of things are known about that, and it's known that a case theory is in some sense forced on you once you build in certain very basic axes. So, the fact that K-Curie keeps coming up is not an asset. Originally, it came up because that was what we could compute.

12:30 I mean, we didn't know how to compute anything else. But now it turns out that the reason why we didn't know how to compute anything else is if you're more or less forced to compute this, because it really isn't. If you choose the right equivalence relation, this really is a separate base relation. No. So to get thought periodicity, you need several things. You need homotopy invariance, You need stabilized isomorphism invariants, which we talked about. And you need one more thing, like half exactness or something like that. But here's an example, an answer to a question very close to Poincare's question. So Quinture asked if the Betty numbers are complete invariants for classifying surfaces up to, let's say, homotopy equivalence. And, well, there's a large category of separable C-star algebras that contains all inductive limits of type 1 C-star algebras. And most of the nuclear C-star algebras that we know about, by the way, if we know anything about their internal structure, they've been shown that the inductive limits of type 1 C-star algebras are things constructed out of those inductive limits. and on that category two altars are kk equivalent and only if they're the same k0 and the same k1. So if you take kk equivalence as your equivalence relation then in some sense Poincare was right and these things are like just looking at many numbers k0 and k1. So those are the complete invariants in that case. But, of course, this equivalence relation, KK equivalence, is very weak. It's very, very much weaker than considering, for example, a stable isomorphism, which is where there's an enormous zoo of algebras and it's completely impossible to get in terms of classification. Now, so that's noncommutative topology in a nutshell, sort of the basic philosophy of the subject and what it's about.

15:00 So what's noncommutative geometry? That's another term you hear a lot these days. So noncommutative geometry, I think from my point of view at least, is the study of analogs of metric structures or connections, sorts of things you have with differential geometry or metric geometry, on noncommutative spaces. and in most cases these involve smooth structures that have dense sub-algebras consisting of so-called smooth elements and then you have analogs of differential operators to find on these smooth sub-algebras and here are a bunch of things that you can study so this is not an exhaustive list but basically non-committative opology described in the previous slides, non-commuted geometry is the sort of thing that's contained in these examples. So here are a few examples that you might look at. One is the theory of connections and curvature, and that's certainly a very big part of differential geometry, and you can do that in the noncommuted world also. So the very first paper to do that, I think in a really interesting way was this paper of Kahn and in 1980. It's a very short paper I guess at one time Kahn was going to write a much larger paper on that same subject and for some reason it never got written so that Kahn and the paper is still the only reference for exactly what he did there. So that's I mean some of this stuff made its way And the book sort of went off in a different direction. This paper is still, I think, very important. So certainly people should read that. Then there's non-commuted Durand theory, which was also studied by Kahn. So the original paper was in the IGS journal. It was a huge paper. and then later Kahn's book and also Tim mentioned this yesterday but one should, I forgot to mention it but one should really also look at the the asterisk volume of

17:30 Karubi and other people put it around here too then there's non-commutative spectral theory papers like this, but I don't know that there's any single one that's fundamental here. But the study of spectral properties of analogs of the classical operators in the noncommutative setting. So there are people doing this sort of thing. So this is also what you call noncommutative geometry. There's noncommutative complex geometry. So, there are various people who have studied this. For example, there was a very interesting study by most of the basically noncommutative the bar equation in this paper on the non-commutative principle and also you can have papers like this so now you see things showing up like this paper Polly Schuch and Schwarz polymorphic vector bundles on a non-commutative two-chart so here basically the idea We think of an irrational rotation algebra, noncommutative two-torus, as being a noncommutative perturbation, so to speak, of an ordinary two-torus. Ordinary two-torus has lots of different complex structures. For each complex structure, you can consider the category of homework effectiveness, and they try to do basically the same thing as a noncommutative one. So there's now fairly substantial literature on this sort of thing. non-commuted complex structures. That's a very interesting direction. And then there's non-commuted Yang-Mills theory. So that was studied, for example, in this paper of Kahn and Riebill on Yang-Nils for non-commutative 2-4i.

20:00 Basically, the idea was that you take a non-commutative torus, on it you have various projected modules which you can think of because of the Sears-Swan theorem as being analogues of vector bundles. On those vector bundles, you have the notion of connections and curvature, And then you can consider the analog of the Yang-Mills equation, in other words, look for extrema, the action functional, which comes from integrating the norm square of the curvature, so to speak. And quantum repo actually solved that problem completely in the case of non-commutative 2.4. One could presumably do that for lots of other non-commutative spaces, and as far as I know, nobody's had any great success in doing it in many other cases, but this case is certainly very interesting. Is there a non-commutative curvature picture? I mean, plenty of duality. Well, the kind of curvature and kind of connection that they're using is the one that's explained in this paper from the New York Times. in there. This theory doesn't quite work in complete generality because this theory is really only set up in a situation where we have a, so to speak, that everything comes for an action of the vector group, and lacking that, you know, sort of completely general theory of connections and curvature is still lacking, so this theory's only set up in a kind of very special situation, so in a situation where this works, then we'll know how to study this, I guess, in general, hasn't been done. That's the situation. OK, so I'm not going to go into most of these for lack of time, but certainly these are very interesting areas of study,

22:30 and you can look at the references. I think I'm going to end early today, because this was just supposed to be an introductory lecture and also there's a lot of time for the afternoon. Are you talking this afternoon? Yeah, I'm talking this afternoon, but that's going to be more technical talk soon. Whereas this was supposed to be a general introduction. Let me just once again mention these references here. This is not an exhaustive list, but these are a few things we're getting started. As Jean-Louis mentioned yesterday, Blackenard's book on K-theory operator algebras. This is really a great book. And the more I look at it, the more I'm impressed. It took quite a lot of effort to put all this stuff into a book that size. Brown-green-reefield paper. This is the reference for the equivalence between stable isomorphism and marine equivalence. So this is the theory of connections and curvature in the non-commutative world. Basic paper that introduced the use of cyclotronology and non-commutative neurometry and so on. I talked about CON-REPOL, talked about ECA-DOD, this is just one of many references on Twisted K-theory. This is just one of many papers on non-commutative complex structures and a few more things. And so I mentioned the newly Renault-Klein's paper that's on, this basically really motivates from the point of view of group voids why you need marine equivalence as a natural equivalence relation on our integrated spaces, because similar group voids from many points of view should be thought of as being the same. And Renault's book, which is the reference on group voids on the C-star Algebers. Marieti Cohen's theory for C-star Algebers was developed by Rieffold. And I mentioned this to people tonight. And so that's

25:00 I think pretty much it. Next time, just to give you a preview, what I was going to do tomorrow is to start talking about non-community topology and extrovariant topology. Tomorrow's lecture, I'm going to concentrate on the case of the groupoid g plus x, the groupoid that comes from a group action on a space, and I'm going to discuss the interplay between the non-commutative topology of this groupoid, attached to it, and equivariance apology, in other words, which is a major area of algebraic apology, the study of the group actions of x, from the point of view, again, various equivalence relations. So I'm going to discuss the various things about equivariance apology as they So thanks for your patience, and I'm happy answering the questions. Oh, it's symplectic genometry, that's a very interesting question, I actually don't studies of the non-commuted symplectic topology, I think because that's not to say that it can't be done, but I think the state of our knowledge is so primitive at this point that we haven't really quite gotten there yet. I would not at all be surprised to see that coming in the future, but as far as I know, there's no literature on it yet. At least that I'm aware of it. Kahler geometry, well that's sort of a blend of simplexive geometry and complex geometry. Non-complex, non-communative complex geometry, as I said, is starting to be developed. But the Kahler aspect, I don't know about that yet.

27:30 Thank you.