Noncommutative topology in geometry & physics, lecture 2 (contd.)

0:00 Okay, should we continue to go ahead? Okay. Formal announcement? Okay, this is an informal announcement. We are thinking to begin one more seminar series, which will go throughout for people who are here for the three months of trimester. So it's an informal seminar, which we'll meet in the late afternoon after the talks. Today, I think we'll be at 5 o'clock. And Christian Bowman will discuss his work on SAC. I'm volunteering. Is Christian here? He agreed to speak yesterday. So today, if you're interested, we'll have a presentation down in the tea room on the second floor. And we'll have some wine. And it will be very informal. And that's today's schedule. Other days, we will announce it as people become more confused and want to talk about things. The whole idea is to have informal discussions with some wine. If the discussions get good, the wine might get a little more expensive. Let's start with two euros a day. But that's today at 5 o'clock. Okay, so in the first half of the talk today, I introduced the general subject of equivariant just to review, we introduced the general subject of equivariant topology, we introduced the cross-product algebra. I would explain how equivariant k-theory in the sort of traditional world of equivariant topology is related to the k-theory of the cross-product and mention some of the standard equivalence relations in equivariant topology. And now we want to actually use techniques of non-commutative topology, non-commutative geometry to come to bear

2:30 In fact, they give interesting information, and I mentioned a few sort of key questions here to get started. Now, these questions, even though they're rather elementary, are actually not so easy to answer, and we'll see why in a moment. So, first of all, all those questions are closely related, so let's review a few things. First of all, if G is compact, the equivariant k-theory is the same as the k-theory cross-product. And if A is contractible, then certainly its k-theory is zero. So now let me explain what the connection is between those two questions. There were two questions. There's part one and part two. So what's the connection between part two and part one? Suppose you have a homotopy of g actions on a c star algebra. So you can make that into a single action on a different algebra. Simply take the contractable algebra. So this algebra contracts just by sliding things back down to zero. Take the contractable algebra of functions on the unit interval that vanish at zero. with values in A. And at the level T, you act by the T-th action of G. So if you have a homotopy of G actions, certainly you get a G action like this on a contractible algebra. So therefore, answering question two, in other words, question two had to do with, is the equivariant k-theory, the k-theory of the cross-product, constant in T? Well, the answer would be yes, if you know that the k-theory, equivariant k-theory for action on a contractible algebra is zero, because then this thing would be zero, and Then by using the exact sequences in k-theory, you can see that all the various values of t get the same answer. So question 2 is reduced down to question 1, basically. Now, as far as question 1 is concerned, there's one case that's easy, but now you have to go out of the category of compact groups. That's why I said I wanted to temporarily drop that assumption.

5:00 So suppose g is equal to the reals. Then, if the k-theory of A is zero, even non-equivariantly, I mean, equivariantly, this would be easy, but I'm assuming it's zero, but not in any kind of equivariant way, then the k-theory of cross-product is also zero. So that's because of the so-called Kahn isomorphism there, in fact there's nothing to do with the usual Kahn isomorphism there, it's just a formal analog of Kahn. So this is a very beautiful paper, advances to 1981. Compared to many of Kahn's other papers, this one is very elementary, so this is one that you can easily read. So you can answer question one affirmatively if g is equal to r and then by this trick over here you can also answer question two so therefore the answers to all those questions in other words for g equal to r So that's a case where all those questions have an affirmative answer. We have to go out of the category of frontback groups. But similar questions for other groups G are in fact related to the bound front conjecture. So, it's actually a rather amazing paper here. Yeah. Paper of Mayer and Nesk, Bound-Pon Conjecture via Localization of Categories, who basically points out an amazing connection between the bound-conjecture and those simple-minded questions that I was just talking about. That's why these questions are not so elementary. Okay, but now we want to come back to compact groups.

7:30 So this thing about actions of the reels was nice, but does really help you with compact groups. And we'll even look at finite groups now. And let's look at the commutative case. So suppose we consider what would seem to be the very simplest possible situation. You have a cyclic group of prime order acting on a finite CW complex. So what could be easier than that? Now, there's something known as Smith theory, which was originally due to P. Smith of Columbia University back in the middle of the last century. And he showed that in this situation, basically, where you have a cyclic group of prime order acting on, suppose it acts on a contractible finite CW class, then the reduced homology of the Hick set with z mod p coefficient, so fp here is a field of p elements, has to vanish. And this is sort of, you might wonder how you prove something like this, but it's a sort of very explicit proof. I mean, you just write out the cellular chain complex and just think about how G acts on the cellular chain complex when this sort of immediately comes out. So now, let's go back to our questions and try to answer them using the localization theorem. So remember, the localization theorem says we're taking G cyclic of prime order, so it's a billion. That makes it a lot easier. So the equivariant, what is the representation range for a cyclic group of prime order, by the way? So if g is z mod p, then r of g, the representation ring, is of course the group algebra of the

10:00 dual group, right, because g is abelian, so every irreducible representation is direct characters, and the dual group is also Zp, so therefore this is just that seemingly very innocuous commutative ring, that's an ethereal commutative ring. And the equivariant k-theory localized at P is, by the localization theorem, the same thing as the equivariant k-theory of Y. Y is a fixed set localized at P, provided that P has all of G as its support. Remember, prime ideal in the representation ring has a support, which is a cyclic subgroup, so either that's the whole group or it's just the identity. And p having all of g as its support is equivalent to saying the p does not contain the augmentation ideal. The augmentation ideal is the ideal generated by t minus 1. So if P does not contain the ideal generated by T minus 1, that means that the support of P is all of G. So let's use this, and we'll do it in a very concrete... Oh, so Smith theory, so Smith theory, original Smith theory, said that if X is a contractible finite GCW complex, and z on p acts on x, then they reduce the molyb to fix that and fp coefficients to zero. So that was Smith's theorem, which is very elementary. But what's more interesting is that Lowell Jones proved the converse to that. So this is actually an if and only a statement. The converse actually is a hard theorem, even though the Smith's theorem is relatively elementary. This was Jones's thesis. This is a key paper in equimariant topology. So there's this one over here. Converse to the fixed point here of Paul Smith, analysis of math in 1971.

12:30 So that's an internolea statement. For the converse, given the y, you have to construct the x. Basically, the idea is to use the method of killing Hamatake groups, but you have to do a very careful exercise in the structure theory to do that. So let's take a very concrete example. Plus p is equal to 2. What could be easier than that? people are smiling because any topologist knows that actually prime 2 is always the hardest prime ideal, not the easiest and suppose L is an odd prime and we'll take P to be the prime ideal generated by T plus 1 this is the group ring the representation ring also. So here p is equal to 2. So if you divide out by t plus 1, that means you're essentially setting t equal to minus 1. So you're looking at the character where the generator goes to minus 1. So this ideal doesn't contain the argumentation ideal. And when you divide L by P, you get the field of L elements, so that's certainly a maximal ideal. And if you localize the representation ring of that prime ideal, it's easy to see what you get is the integers localized to L. So that means that's the sub-ring of the the rationals, where the denominators are not allowed to contain the time l. Okay, now suppose you construct a y which is acyclic for f mod p coefficients but not acyclic for f mod l coefficients. So an example of such a thing would be a z mod l Moore's case. So this has only one non-zero homology group, which is z mod L, and all its Betty numbers are trivial. So this thing is acyclic for z mod peak homology, but not for z mod L homology.

15:00 All right, then, since there's only one non-zero homology group, the k-theory is easy to compute. theory also has a z-modell in it. And now we can compute, so now we use Lowell Jones' theorem to embed the y into an x, where x is contractible. And now the equimariant k-thory of x, localized to g, is the same as the equimariant k-thory of y, localized to g. But g is acting trivially on y, because y is the fixed error. So the equimariant k-thory of y is just the ordinary K-theory tensors of the representation range. So therefore, I'd be localized if you get this. So k star of y, we know we computed this with Z mod L. Now we're tensoring over Z with RG localized at B, which is Z localized at L. So Z mod L tensors Z localized at L doesn't do anything. Just gives you Z mod L back, and that's non-zero. So now we've arranged for this thing to be non-zero. It has L-tortions in it. That means that we have a contractible finite GCW complex whose equivariant t theory is non-zero. That means it gave a negative answer to question number one. Going back to the question, it turns out, from that example, you can see that the answers to all of these questions when G is the cyclic group of order two have to be zero, have to be no. So the answers to all these questions are no. Just from that simple example. Well, for connected compact regroups, it turns out it fails also basically by sort of souping up this example. That was the simplest example I could pick up. You already see this is a rather non-trivial thing because we used Jones' theorem, which is actually quite difficult. Another thing that comes out of this is that you see that if you change the question just a little bit, instead of requiring that the k-theory of the cross-product be 0, we just require that the k-theory of the cross-product

17:30 be rationally 0, then the answer would have been yes, in this case. Because since the x had to be z mod p acyclic, are also acyclic for rational homology. So, you see that these questions are very subtle, you just change them a little bit to get a different answer. So if we just asked about this thing vanishing rationally, it would have been true, but with portion it's not true. So it's a very subtle kind of phenomenon that happens here. There's a counter-example of this one also for G finite which is basically based on the example that I gave up there. So, I don't know if you consider that good news or bad news. I mean, it's good news that you can actually figure something out. And it's very interesting mathematics, but it's bad news in the sense that the answers to the questions were no. But it's good news then because it means the K-theory of the cross product is actually very interesting kind of gadget. It can't be computed by a certain simple principle. So now I want to move in a slightly different direction, and for the rest of the hour, talk about equivariant index theory. So now, for this, we need to deal with manifolds. Those questions before were just for arbitrary g-spaces, but now let's consider actions on manifolds. and you want to study invariant elliptic operators. a natural thing, well elliptic operators naturally give elements in the theory that's dual the equivariate K theory, which is equivariate homology. equivariate K homology, you can set up But if you want a good analytic model for it that's sort of nicely compatible with the theory of elliptic operators,

20:00 then it's actually convenient, even though it seems like a lot of extra trouble, to actually work with Equivariant KK theory, which is a bivariant Equivariant K theory that, again, has R of G as its coefficient range, so equivariant kk of the scalars in both variables is R of G. In the second variable, it gives ordinary equivariant k-theory, and in the first variable, it gives the famous dual k-theory. And the nice thing about this bivariant theory is it comes with a bilinear product like this, which, again, is actually bilinear over the coefficient range, which is R of G here. Is G compact? G is compact. Although, actually, Kasparov defined this even for G non-compact. It's just that R of G has to be interpreted in a slightly different sense. I don't know what to explain. And also, you have to modify the definitions a little bit. I mean, basically, everything will work here whether G is compact or not. But let's take G compact. So then the invariant elliptic operators with classes in here. Well, one should be the negative of the other. But since I'm... Oh, for the cut product... Sorry, that should be... This should be a J, and that should be I plus J. Thank you. There are all sorts of little differences like that. All right, so now what are some things about this more general situation? Well, the green Joule theorem holds in K-comology if G is discrete. It holds in K-comology if G is compact. So it holds in both if and only if G is finite. You might think, well, in the case of G finite, it couldn't possibly be interesting The Smith theory, actually, even in the case of a finite secret group, is a very non-trivial thing. That still means that the dual k-theory of the cross product and the dual equivariant k-theory are the same between finite, so that's sometimes useful.

22:30 Also, there's a version of the localization theorem that works in KK. You can find this in a paper line, correct this here. It's this paper over here. But you need some restrictions to make this work. So, for example, suppose X and Y are finite GCW complexes, and you want to look at the equivariant KK of C of X, C of Y, then you can reduce to these sort of G saturations to fix-ups after localizing an equivariant KK. This is proved out of Siegel's theorem. There's also an induction map, which is defined by Kasparov. This is true even for G non-compact, that the equivariant KK goes to the bi-variant KK of the cross-products. And this is related to the Green-Jules map and its dual in a sort of obvious way. You see, so just to mention the relationship here. you have the the equagrammian k-theory of A which is the same thing as kkG of C con A and let's take G-compact for example, so this thing is an isomorphism by green dual over here you also have the Kasparov map which goes from C cross product star of G, G cross bar by A. So this is the cross bar of induction now, which I described over there.

25:00 And, but you see, C is a direct sum and inside C star of G. I mentioned that C star of G is a big direct sum of matrix algebra associated with various irreducible representations. But just take the trivial representation of g, so that gives you a sum n of c inside here, which is actually, so this is a split injection. And so that gives you a split injection from this into that, and this diagram can be used. So when I said these things are compatible, that's what I meant. Okay, now we'll get to some kind of hard equi-variant topology, so I'm going to study group actions on manifolds. This is a huge area of topology. Lots of people have worked on this, including lots of very famous topologists. Even for finite groups. So, for example, take G finite, and X is a compact topological manifold, and suppose G acts by a locally linear action. This is a sort of good category for doing topology of group actions on a manifold, but I should explain what that means. So there are lots of different categories around. One interesting category is a category where the manifold is smooth and the group action is smooth. That's certainly included here as one of the important special cases. But you also have actions where the manifold is maybe even smooth but the group action isn't smooth but it's so called locally linear which means that in the neighborhood of any fixed set you have a sort of a local neighborhood chart that looks like a linear action. So that's actually much weaker than the action being smooth but it's good enough to get good structure to the fixed sets. It implies that the fixed sets are locally flat topological sub-manifolds.

27:30 If you don't require anything, if you just take a compact topological manifold and you take any old action of G, then the fixed sets can be really horrible. They don't even have to be ANRs, so they could have infinitely generated homology and look nothing at all like manifolds, and then there's not very much you can set, so without some kind of restriction like this and you don't get very much. And here, for technical reasons, we'll assume that G has a G-invariant Lipschitz structure and the action is Lipschitz. So that's not much of a restriction it turns out. Certainly obvious if the action is smooth it's differentiable mass of Lipschitz. It's obviously true if the manifold is PL and the group action is PL. That's a PL map solution. And it turns out that even even with these very weak assumptions here, once you get away from some low dimensional problems it's even true in general. If you don't know what any of these obscure categories of manifolds mean. You can just think about this in the case. So you fix a gene invariant Riemannian metric, and then you have a natural gene invariant elliptic operator, which is the Euler operator D plus D star. This has its usual meaning on smooth manifolds. If you have a Lipschitz manifold, then it turns out, it's important I don't want to tell them that it still makes sense, but if you don't quite understand that, then don't worry about it. So you have the Euler operator, d plus d star, and if x is oriented, the action preserves the orientation, then you also have the signature operator, which is really the same operator as an operator, but the grading on the bundle of forms is different. So there are two gradings on the bundle of forms. And there's the even-odd grading, so we take the even forms and the odd forms, and V obviously changes even forms and odd forms, and so does V star. And then in the oriented case, there's another grading that comes from the Hodge star operator that uses the orientation, and that gives you what's called the signature operator.

30:00 So then you can take the G index of the Euler operator, and you get the G Euler characteristic. That's the G or the other characteristic. That's a sort of obvious invariant of a G-space. And I should have defined that. But it has a sort of obvious meaning. So suppose x is a nice G-space. So nice might mean it has an equivariant homotopy type that would be funny if you see other complex. So then the equivariant Euler characteristic is simply So, please. No, I don't think so. I think what happened was that the screensaver went on, and so it has to be noted. So the equivariant Euler characteristic is just the alternating sum of the Benning numbers been interpreted in the equivariant world. So what I mean is it's just take the homology of x say with rational coefficients, or complex coefficients.

32:30 So if g is acting on x, g acts on all the homology groups. So this thing is a finite dimensional vector space with a g action, so it's just a finite dimensional representation space for g, so this thing lives in a representation. So that's what we mean by the equivariant. And the equivariant, the g signature, is defined in a simpler way. The other product pairing on middle dimensional homology, and so You can split the middle-dimensional cohomology into two pieces, a plus part and a minus part for the product theory, and you take the formal difference of those, and once again, g acts on each piece so that you get a formal difference of representation. So these are useful in variants for a for a manifold group action. And, um, we got it back here. There you go, all right. So we have the G or the characteristic of the G-signature, and it turns out these are actually pseudo-equivalents in there. So I mentioned this sort of strange relation with pseudo-equivalents, so this is the case where it comes up. So how do you know that? Suppose you have a g-map of metapholes, and suppose non-equivariantly that map is a homotop equivalence, but not necessarily equivariantly.

35:00 It's not necessarily a g-map going in the other direction. Well, this homotop equivalence preserves study numbers, so it gives an isomorphism of a homology groups for each eye, so therefore it preserves the equivariant Euler characteristic and similarly G-signature. these things are actually pseudo-equivalence invariants. So that's an example of why pseudo-equivalence is so used for equivalence relations. There comes some such thing here, but you can think of the ethical emerges in this case. The organization comes from ethical at least? It does. I'm mostly interested in the case where g is finite. If g is connected, then the action of g on homology is trivial, and then you don't get anything. If g is finite, yeah, it's a finite dimensional representation by means of matrices that have integer entries. But you get more interesting invariants if you look at the k-inology class of the operator and not just the g-index. In case of the Euler operator, Muc and I actually gave a formula for the K-hymology class of the operator, an explicit formula that was in this paper over here, Equivariant Euler characteristics in the K-hymology Euler classes. And it turns out that there's an explicit formula for the k-homology class of the Euler operator. And you can write it down in terms of something called the universal Euler characteristic, which assembles all possible Euler characteristic data of all the pinstats. So for that reason, the class of the equivariant order operator is an invariant of isovariant on top equals. So that's an example where that funny relationship comes up. So you need the data on all possible fix sets, so that's an invariant of isovariant.

37:30 We actually proved it for infinite groups and proper actions, but the case of finite groups is already interesting. The G-signature operator is even more interesting because you get all sorts of other things like the equivariant Obacon conjecture. So that says the following. Suppose you have a G-map from an oriented G-manifold to another space, another G-space, and suppose the second G-space Y is is equivariantly asperical. So that means that all the pick steps for all possible suburbs of G are asperical or they're higher chromatography vanishing. So then the image of the class of the signature operator and the equivariant cohomology of Y is an oriented silo-equalance mirror. So that's the equivariant helicopter projection. It's actually true in some interesting cases. So what's an interesting case? Well, first of all, what's an example of an equivariant in a spherical G-space? How do you describe those things? So here's an interesting example. Suppose you take a complete manifold of non-positive curvature and suppose G acts on Y by isometries. all right by the um since g acts by isometries all the fixed sets are totally geodesic so since they're totally geodesic they themselves they're also complete manifolds of non-positive curvature but by the carton of the martyrum every complete manifold of non-positive curvature is A star, so therefore that means that Y is equivalent to the A star. Well, that's an example of something that satisfies this condition. And if you take this particular case and Y is a complete manifold of non-positive curvature, then this conjecture is actually true. So, this actually follows not from the theorem, but from the proof, with Prosperos proof of the Novakoff conjecture for fundamental proofs of non-positive

40:00 because of manifolds. So you sort of take the same method of proof and just sort of slightly seep it off and you get this. So that's an example of... So here, why do I mention this? The purpose of the course was to show you how non-commutative topology could be used to prove interesting things about ordinary geometry and ordinary topology. So here's an example of a statement that has nothing non-commutative in it. I mean, this is just a statement of ordinary equivariant topology, except for this, but it turns out there's a way you could replace this also with something else that could be defined purely topologically. It's quite amazing that you have something like this, but the proof really requires lots of ingredients in non-commutative topology. I don't know of any way of proving things like this, that you get these sort of complicated equivalency variants just by using sort of standard techniques of equivariant topology without using the non-commutative topology. All right, so here are some other things. The last few minutes I'll mention a few other things you can do. the actual class of the equivariant signature operator is computable rationally once you use the localization theorem in KK. And what happens is that if G is abelian and you localize a finite U of P that has support H, so why, first of all I should mention that for G finite, finite, once you invert the torsion, then it's enough to reduce down to abelian sufferers. Because the equivariant K theory, it turns out, is detected rationally by abelian sufferers. So that enables you to reduce the abelian case. Why is that useful? That's because if you have a, suppose G is finite. So the localization theorem said that for H inside G cyclic subgroup, the equivariant or k homology of s, that this thing localized at p is

42:30 isomorphic to the equivariant theory of x upper h localized at p. So you reduce down to the case of the fixed step. But in general, you have to saturate the fixed step, because the thick step wasn't necessarily G invariant. And then the action on here is non-trivial, and that gets really complicated. But in the abelian case, so after you get rid of the torsion, you can reduce to the case where everything is abelian. When things are abelian, then X upper H is already G invariant, so you don't have to saturate, and then that's actually manifold. It's a much better situation. And not only that, but the G action on X upper H Well, the h is acting trivially, and so the g action on x upper h is basically reduced down just to the study of x upper h together with its equivariant normal bundle. So the action on the normal bundle there, which is something you can compute. So in this case, you actually get a formula for the class of the signature operator on X localized at P in terms of the class of the non-equivariant signature operator on the fixed set. And then it's twisted by some explicit characteristic classes of the equivariant normal operation. The whole problem of the G-signature theorem is only really interesting for G-finite, but then I'm also assuming abelian, but that's because if G is not abelian in this thing, since it's the saturation of X over H, X over H is a manifold, but X over H with parentheses is It's not really a manifold, it's sort of a union of manifolds, and that's sort of a mess. So I didn't really have to deal with that. But after you rationalize, you can always come down there. And what happens if it's not on it, the advantages? Well, if the group is connected, and it acts trivially on the signature complex,

45:00 then there's not really anything interesting going on. So here's another non-elementary application. This part here is just sort of for the topology experts. This part is kind of hard to use for everybody else. But it turns out that in good cases, I want to explain what good cases mean, but it's basically that category that I was talking about before, together with some other condition in equivariant topology, which comes up a lot in the study of deep g actions which is that you need some condition that you don't have any fixed sets that have very small co-dimension in the next higher fixed set because if you have one fixed set which is a co-dimension two inside another one then it can be knotted and then you have to analyze all the possible invariants of the knot and then that It's very, very complicated. So this so-called gap hypothesis to rule that out. So in those cases, if you take the class of the equivariant signature operator and invert 2, then it actually computes all of the normal invariant in the equivariant surgery sequence. And the assembly map, which is one of the other things in the surgery sequence, is given by the equivariant index theorem. So, for people who don't know what this is, surgery theory is a way of computing how many manifolds there are in a fixed homotopy class, and roughly speaking the way it works is that You've got an exact sequence that looks like this in the surgery theory. So S is the structure set of X. It means basically the set of all possible manifold structures on the given homotopy type. And it consists of so-called normal data.

47:30 So this is information. This map over here basically sends a manifold to its characteristic classes. Then there's something called the L group, which is a purely algebraic thing attached to the fundamental group. which has to do with the sort of Planck array duality over the fundamental group. So this is a kind of big group of the quadratic forms over the group ring. And then this thing repeats. And this thing over here is called the ascending map. And there's even an equimariant version of this in the G world. but then the problem is basically in most situations none of the things in this sequence are computable so everything here is you get this nice exact sequence which sometimes is useful for certain things but the things that come in here are usually horribly complicated so the point here is that in this sort of good category of the g spaces for g finite after you invert 2 This thing over here is basically an equivariant k-homology group, which basically consists of classes of signature operators, and this thing over here, this assembly map, which turns out to be the index map for those signature operators. And as a consequence of that, it means that all, because of this exact sequence, all of the structures, in other words, all the possible gene manifolds are after inverting two, they're index theory. So this is a really amazing fact. Think about it. It proves the following theorem. So this is actually an explicit theorem that uses all of that non-commutative machinery, but it's formulated in a way that doesn't, you know, that a regular polygist could understand. So there's no non-commutative language in it. So go through the theorem. So you have to bear with me, and the hypotheses are kind of complicated. So I suppose you have a topological orientation-preserving action of a finite group on a compact, simply connected topological manifold. And you have this gap condition that you never have one fixed set, which is a co-dimension two or less than the next one.

50:00 And each fixed set is locally flat, and all the fixed sets are simply connected. That gets further than the fundamental root data you'd otherwise have to keep track of. We also rule out the case of any fixed sets of dimension 3, so the surgery theory doesn't work in dimension 3 for that case. So then, the set of all possible actions is determined up to find an indeterminacy by knowing the following data. First of all, there's the isovariant homotopy type of the group action. And then, the class is the equivariant signature operator . So, these things together basically determine all the possible key actions. And there's also sort of a converse to this, which is given by the exact sequence. In other words, the exact sequence basically tells you, given an isovariant Hamataki type and given the classes of the equivariant signature operators, can you find a group action that realizes all of that data? So you can answer that question also. So the moral of this is that actually lots of problems in equivariant topology which seemingly have nothing to do with C-Sphere algorithms or analysis or non-commutative geometry or anything like that, can actually be analyzed completely using this sort of machinery. Well, maybe I should say for people who want to know what not to come to, I'll tell you that the sort of tentative plan for the remaining lectures. I apologize for the fact that this is all compressed into a very short period of time, but we're sort of stuck with the schedule. But next Wednesday, so let's see, today is the 12th, so that would be the 17th. I guess I was going to talk about non-commutative topology applied to various classes of similar spaces.

52:30 And then Thursday the 18th, this lecture will be somewhat abbreviated so that it can minimize the conflict with the conferences going on. This will be sort of a short lecture on Quistic K-Theory and a few of its applications. There are many other courses going on that are also talking about this, so if you missed this, you're not missing very much because Jean-Louis, I think, was planning to discuss this in much greater detail in this course. And also, with regard to this, again, I'm not going to say so much about applications to foliations because there's going to be another course later. Now, I forget who's giving it now. Somebody's giving a course on index theory for foliations. So if you want to learn more about that, you can go to his courses. I'm not going to talk about that in so much detail, but I'll just talk about a few other examples. And then Monday, so Monday the 22nd, there'll be some examples of noncomputative pathology geometry. Since this is supposed to be a trimester on grouploids in geometry and physics, I'm sort of obliged to mention those three words, grouploids, geometry, and physics.

55:00 So today I talked about grouploids in geometry, and I guess the physics will be saved for that one. So once again, I'm posting the notes as I produce them for this website, and if you notice more mistakes, please let me know in the front of it for our actions. Thank you.