Is the Topology of Nature Noncommutative?

0:00 It is our pleasure today to have a special seminar, a special research seminar, by Professor Fred Van Oestein from the University of Antwerp. The title of his talk is very teasing. Is the topology of nature non-commutative? That is the title of the talk, and it's our pleasure to have it. Thank you. Thank you. It's certainly a pleasure being back in London. I know it was 25 years ago. I had a lot of contact with British mathematicians, but more in the north. It's certainly a pleasure to be back. My talk is a little bit abnormal for me, or maybe usual, I don't know. Because maybe the first question is, is there a non-commutative topology? My interest is really in non-commutative algebraic geometry, or just non-commutative geometry, because there's a lot of confusion about all these words. It's different from what Kant does, which I consider as a kind of non-commutative differential geometry, but really I consider it more or less commutative, although the rings are non-commutative. so maybe i just give i mean i'm going to talk this informally you can always interrupt you can ask questions immediately i'm just going to try to give you an idea what i what i try to do so maybe first as a motivation the nature in the title yeah okay what is what is the topology of nature i guess that's already a completely idiotic phrase but uh nature for me is algebra that's kind of narrow definition but let's say like this that the things in nature that you study for several mathematicians appear as rings or as algebraic structures and some geometry associated to that so what kind of algebra as well so my my algebras are usually filtered filtered algebras they have a again a topology if you want but I will only look at this case for very naïve topologies in the sense that z-filtered algebras so I do usually I have a ring or an algebra it doesn't matter at the moment and a filtration is just an ascending chain which I write like this so there is zero somewhere and this a somewhere and in between an ascending chain of additive subgroups which has the usual properties that one must be in degree zero let's say and the

2:30 product rule is the continuity of the of the multiplication so it's contained in f n plus n and usually I will add that the union of all these things is a because if it was not then you just take the union and restrict your attention to that ring or algebra that you get then so this is not really a restrictive Plus, almost always, and in this talk always, or later maybe you know what algebras will appear, it will be about topology, but you can always assume also, for the time being, that they're separated. So this is like a filter of neighborhoods of zero for a house dot of topology on your algebra. filtration and then there are two graded rings I've been working a lot on graded rings and was always eager to use them if they were available so there are two available graded rings the associated graded ring which is the take the consecutive motions of these things and then add them together like this and it has the obvious multiplication and addition and there is the reason which important role in the story, the Rhys ring or blow-up ring. Rhys ring. Rhys introduced this, I think, in commutative algebra. Here the ring is not commutative, it's arbitrary. But on one hand you can view it as the sum of all these parts, f and a. But then I have to say what the multiplication is. So it is maybe more interesting to view it as a subring of the We write a t of a, tt minus one, so this is a central variable that I add. This t commutes with everything, this t commutes with everything. And then this, the grading is given by, take the nth degree bar to be f and a. It's put in the degree tn. So this n is a placeholder if you want, if you look at it as this object. But the advantage of saying that this is a subring here is that you know immediately what the multiplication and the addition is. It comes from here, right? And I don't have to write that down.

5:00 So, there is another magical behavior of the Rieswing integrated ring as follows. That if I can find these two other rings by taking quotients from the Ries... I'm used for a ring, you see, but for the algebra. So what I do, here I send T to 1, here I send T to 0. If I send T to 0 in the Reese ring, I get the associated graded ring, and the grading of the Reese ring determines the grading of the graded ring. If I send T to 1, I get the filtration from the gradation. So what this really tells you is that the filtered ring A is always like a deformation of the graded ring. You see this sort of in algebraic geometrical terms if you add some variable this T is central as I said. You can use algebraic geometrical intuition about this T. So really if I send it to lambda different from zero it also gives you A it's a rescaling it doesn't matter. So to send it to 1 is everything else down 0 so there's only one place where there's something funny happening that is this place where t goes to 0 and in this sense well if you look at the algebraic geometry you will see that this defines the deformation this way even by the parameters t and by the reasoning the reasoning controls everything I'm going to give it if you would just allow me to say a few more words I will I will give examples yes for example I will the Y algebra the Witten algebra some algebra coming from physics okay but there's some general that I just write it down as a metaphysical thing the brush of the Riesling is the speck of A and I glued with or closed by a boundary that is the brush of GA so this is at infinity this you can think of the affine part because it's a speck now these words mean what they mean in commutative algebraic geometry

7:30 but I can do that correctly now it's just words right but I can give these things in non-commutative meaning so that is completely correct schematically as sets as topologies and as sheaves this will be true in my story well I'm not really talking about the non-commutative geometry but it's the basis of what I'm doing is that this is true for the topologies that I'm going to define and this is rather interesting perhaps at least for me because these are usually non-commutative spaces and even if this one is commutative as it is in some very nice examples associated graded ring for example for every ring of differential operators on a variety on a smooth variety or something the i give this as an example if you want then the ga is commutative it is like a either a polynomial or some a fine coordinate ring of variety So this will be commutative and you can use this, this is then used in rings of differential operator theory by, for example, if you define characteristic varieties of modules and mononomic systems, then you use the fact that this is commutative. But first of all, that is not necessary because you can define characteristic varieties immediately here in the non-commutative geometry. This is one of my plans, let's say, for application of non-commutative geometry in, for example, But okay, this in itself is an aim, to have a schematic, topological, set-theoretical meaning for what I'm writing here. Very commutative example, well, first of all, all alphabras are filtered. That's stupid, and perhaps a bad sign, if everything is something, then the something means nothing somebody said if a theorem holds for all objects or for all rings then it's trivial something must be true but okay what is what is the usual situation you have a free algebra maps subjectively to your algebra by selecting a set of generators the free algebra is graded this is this one is graded by the usual degree giving x for example degree 1 every variable these are words not commuting words but they have a degree associated so these are the length of a word would be the degree right if you give everybody degree 1 it is the length counting multiplicities of course so it is the length this defines a

10:00 gradation but also a filtration every gradation if you have a graded ring any ring with gradation it has a grading filtration by taking fnr everything the tail before rn everything before and then it grows and it satisfies these rules so from a gradation you also get the grading filtration if you want so gradings are very nice filtration in a sense okay so this grading filtration defines by taking just a quotient taking the images a grating does not define a grating on a quotient because the kernel is not necessarily homogeneous but for every kernel you do have a filtration so this grating filtration did you have A just an image filtration for FN you take the image of the grating filtration so these tails there's a very classical geometry that everybody learns in the secondary school where you use this trick. You say for example k xy and you divide by some equation of a of a plane curve. So k xy are the polynomials in two variables, some equation, maybe I'll write a small f, it's not homogeneous equation, some relation this is our ring a it has this grading filtration the polynomial ring has this grading and this comes from the free algebra model of the connotation relations already but okay because this grading this has the filtration defined by the gradation just like what I explained here polynomial ring is graded this quotient is filtered in that way okay so what is Let's take something that everybody knows, I don't know, if you have your favorite curve maybe How do you find the reasoning? You're homogenized. This is, you know, class Z. Oh, let's write T in here.

12:30 This is the reasoning of this field. So the filtration coming from that gradation of the polynomial is what it's called the standard filtration. The reasoning for this standard filtration is this. And what is the associated drainage ring? So if you send t to 1, you get that ring, if you send t to 0, you get the associated train ring. Right? So we get some strange kind of c-plotomic points. It all depends what k is, and maybe you're used to using the complex numbers of the real inertia. And you call it the blow-up. Right, it is the blow-up, yeah. It is the blow-up. To homogenize the equations, well, okay, be careful. made. I don't say that all resprings that you cannot ever make will be just obtained by a homogenization of equations. If some ring is defined by nice equations, then the respring will be defined as this homogenization. On the other hand, the GA is almost always like what I said, the D-homogenization. So you can do it in a hyperbolic sense as well? Oh yes. I'm going to give you another example in a moment, non-computative if you want. This goes for every affine algebra, this is absolutely general, but this is a very basic example. I want to show this to recall that you know from secondary school that you know this thing, because you know how do you find the projective variety? You homogenize. You find projective variety where the curve is an affine part of something. How do you find the points of infinity? You homogenize and send the variable to zero. intersect with a so-called hyperplane at infinity. So this is at infinity of the circle. These are the, what we call it, signatomic asymptotes or points. Depends on your field, R or C, or whatever. OK, so all algorithms are filtered, all the classical ones, all the affine coordinate rings of varieties. And you see this thing has a meaning, this classical geometry that you get here. Right, let me also give a non-computer example. To start with, let me take the while algebra. So I take the free algebra and two generators. And I take the two-sided ideal. Two-sided ideal. When I say ideal, I mean two-sided, but sometimes it is nice to stress.

15:00 So the ideal generator on the left hand right by one element. both sides. So this is the first Y-algebra, which of course is well known for the people in physics I suppose. This is coming from the insertive principle. This is the algebra generated by the operators, the place vector and the impulse. xp minus px is what you have an I, Planck, maybe a C there, it depends on the system, but otherwise you divide, anyway you divide by all the constants, and you go back to this kind of algebraic relation. This is just this, this, this, you say natural unit, okay, yeah, so, now, you can give, there are two interesting filtrations. First, the Bernstein filtration. Now there are more filtrations of course, but there are two that are used well. Bernstein filtration is by giving x and y degree one. Now when I say degree, that has to be understood in some way, some correct way. Of course the filtration is not really a degree, it has some kind of degree, filtration degree, but not the gradation degree. But you know what I mean, you just put x and y in degree 1, so this means that F0A is C, F1A is C, X plus CY, and so on. F2, all the length 2 words, length 3, and so on. If I take the reasoning with respect to this then I get CXY T so I can go to the long commuting variable T and then divide by the commutation relations but I will immediately assume that I have a central T and divide only the homogenized relation So this is the reswing, and the GA is easier to see because this element has degree 2, degree 2 is degree 2, it is 0, so you have a degree 2 element, it wants to be of degree 2, but it is of degree 0, means that in my quotient here, it goes to 0, because I divide

17:30 by one before so if it's too low what that means that my associated grade ring is commutative and in fact with some idea about Poincare and Birkov grade or something you can need easy proof that is the polynomial there's another interesting filtration the operation filtration of the operator filtration for fuchsium this is a fuchsium filtration that you get by giving degree x zero 10 degree y, for example, 1, okay, so you read y is an operator, maybe it should be minus 1 because it's kind of a derivation, but okay. Then, the ga is the same, so the, let's put an O here for operator, the reswing is different, because the reswing separates these cases, b y x minus x y, homogeneous but y x is now degree 1 in x y, so there is only one t here. And the g in this filtration, operator filtration, now this is dangerous, this is just again the polynomial, created in another way where x is degree 0 and y is degree 1, but the same ring. So the associated graded ring does not really discern the two filtrations. The Ries ring does. The Ries ring is really different. This one, if you think for five seconds right now, you see this is the enveloping algebra of the Heisenberg, the algebra. Basis x, y, t, this is the commutator and the rest commutes, so that is the Heisenberg relations. So this restring is the Heisenberg, well, is the enveloping algebra over c of the Heisenberg algebra. This one is a quadratic extension. It's funny, I mean, it looks like it's small of it, but it's a quadratic extension. So the Berenstein filtration is in a sense the most natural one to give X and Y the same role

20:00 but in a funny way it is not so nice if you take the reason it is not a double quadratic extension if you look at algebraic structure where for the operator filtration it is very nice Okay, so I can also show you some application from gauge theory. Well, let's see, maybe if you want I can define these things later. First look at the example, that's the reason now. example this is the Witten algebra so I have three generators again and this comes from this churn coupling constant engaged here I should immediately say I'm certainly not a specialist engaged here but it's very interesting that these algebraic structures that appear fall very neatly in our picture so we have relations for the Witten ring here if you know them fine if you don't know them so is some sense that q deformed heisenberg out yes well well there isn't there isn't there is a thing but there is something which is not really only q but it is how shall i say it's a deformation of some quadratic algebra that is regular that has finite dimension that is a quantum space in the sense of money so it's a deformation of space, that's true. In fact, that's what I'm going to see when I calculate the associated with the ring, by my magic triangle here. I usually call this mathematics, the magic triangle. But anyway, so what these elements are for me is not much... very important of course that there is nice central element, which is the deformation of the cashmere. Okay, so I have to invert at a certain moment, because about that, it's not going too far in the algebra, but there is a theory of birationality that makes this ring geometrically and algebraically birational to quantum SL2. So that the representation theory is the same except on some locus where there's bad representations for the Witten algebra and they vanish in the quantum SL2. And the reason is this element A,

22:30 up to inverting this I can go from written algebra to quantum SR2 so in terms of let's say so it has a non-vanishing first term class yes but I really would like to know I don't know whether physics to know what some of the things I'm going to talk about actually mean for the physics that's maybe your touch I guess I guess I was just trying to really ask is if I in terms of so in terms of like some kind of the algebra does this have a representation that you know so so with what Q be some some sort of element of I really don't know where this has a meaning but the point is that so the representation theory of this is not so trivial there is what Witten calls bad representations the point is that you can remove them at least for me by inverting this element you remove them but that is by using it by rationality which is algebraic and also geometric in the non-commutative sense if you have a non-commutative geometry I can tell you that when you associate the non-commutative geometry I want to associate to this you do that with the topology and everything it becomes by rational to the quantum method too but exactly by rational means on two open sets they are the same open sets in this non-commutative topology that the localizations become isomorphic but the localizations are also are non-commutative, of course, the rings are. So the definition of birrational is intuitively the same, but on sets in the non-commutative biology. And they are more than what you expect, maybe. We'll see, we'll see later. But this element certainly has to be inverted to get rid of the problems. Anyway, I calculated here the resring by homogenizing. And when you go down to the associated graded, the Witten algebra, or generalized gauges,

25:00 these parameters can be general here, in Witten's case he has one specific algebra, but they're all at the same time. The associated graded is always quadratic, see? It's quadratic, it's a three-dimensional quantum space. that is quantum space in the sense of 9 and that means a fine positively graded non-commutative C algebra so positively graded it's quadratic so the relations are quadratic defined by quadratic relations and it's regular in any sense you can invent there are at least two that are in this case or one I like more it's older regularity condition A I think he got. So for us, I'm not going to talk about it, but since maybe some people are interested in physics here, generalized gauge algebras are just positively filtered algebras, and by definition that follows, it is going to be a final global dimension, homological dimension is final, and the associated, the definition of a generalized gauge algebra is such that it's nicely filtered, the associated gradient is quantum space. So there's lots of examples rings of differential operators enveloping algebras also quantized versions they're of the almost commutative type when the associated gradient is commutative but you see in the Witten algebra and that's why i mentioned this example in particular the associated gradient is not commutative so there is here many things to say but that's not this is only as a motivation here at the moment. This deformation here, when this ring is commutative, then you describe this by the Poisson structure, Poisson brackets on this GA here. The Poisson brackets define this deformation, in fact. So to give this deformation is the same as Poisson structure. this is in the commutative case if commutative for example the Weyl algebra you get the filtered ring the Weyl algebra and the deformation of the polynomial ring by the usual Poisson bracket this is certainly known but interesting here this G is not commutative in fact you can define a non-commutative Poisson structure that makes this deformation so in other words up to defining a non-commutative Poisson structure I don't have the most general definition I think but I have a definition that works in the twisted case because this is an Heure extension, automorphisms and derivations only so there you can do

27:30 non-commutative Poisson structure and deform the quantum plane, quantum space sorry, 3D meso-quantium space, you deform it into the Witten algebra by this non-commutative Poisson structure. This is, for example, a construction in non-commutative geometry, in my version of non-commutative geometry. And this is really non-commutative. It uses, if you want to express this geometry in scheme terms, scheme theory, you use a real non-commutative topology, which you don't have in Kant's theory or in other existing theories as far as I know. But there are many things to know. The topology. The topology is very non-commutative. In Consphere, the topology is commutative. It's classical, yes. So we have a real non-commutative topology. How good is this, I don't know. But it allows you to do certain things. That's what I want. Okay, so this is sort of an introduction. I spent a lot of time, but maybe it's important to have some motivation to see why am I doing this. so this algebraic geometry I want to, first of all it's algebraic but it can be it can be made analytical of course but I use completions in it all the time so what I want I want is of course I can embed everything in a projective space I will do the projective geometry the fine geometry is embedded so in order to catch two flies in one flow I'll do it immediately graded and projective so we want a Grotendieck topology. There is topology, but there is also Grotendieck topology. There are different notions. So I want to generalize not only topology, analytical topology, but I also want to be able to generalize, let's say, etal topology to n-combology, in fact, later. I want also to generalize the etal topology and that kind of coverings that you use to the non-committative situation. So there will also be a non-committative Grotendieck topology. It must be something affine in the story. If you define varieties, you use coverings by nice, affine varieties, right? In the computer theory, you do that. So I want also to have this in the non-computer theory. It should be a basis of the topology. An open affine should be affine. It should be the speck of something. So that's the localization theorem. This really tells you that some open affine is the speck of the corresponding localization.

30:00 I'm going to explain that if I have time but anyway my talk will perhaps be brutal at some points and not all the detail but if you want to know first of all I'm willing to go on and also to everybody interested I'm here for another day so I can give more detail later if somebody really wants to know but this is a kind of localization theorem it says you that there is open a fine neighborhood and they look like it's a real speck again Okay, this is a growth in D. This is true, not very interesting. Ah, this may be interesting that you can use Czech homology in order to calculate the load, the sheaf, the sheaf homology, yeah, load homology, the sheaf homology. Use Czech homology on the long-commute-lift topology. That is interesting, and I have seen some recent applications in the determination of certain volumized spaces. Certainly I also want the global section theorem, because I want, if you study a functions or vice versa a certain ring of functions associated to a geometric object and you want a nice correspondence between the functions and the geometrical data so this is classically given by Serge's global section theorem that if you take coherent shapes over your geometric of your scheme or over your geometric object that the global section of these schemes correspond to the modules so there's so we do have it but this is very very restrictive on the topology i mean it really forces you to have enough open sensors as one can see if you're looking at but there will be such a global security there is also a shared duality there is a theory of formal schemes and completion along closed sub schemes so i just list here some properties of this geometry that I'm going to construct that makes you perhaps believe that it is a geometry. There is a good theory of regular algebras in terms of homological algebra. There is dimension, the schematic dimension is a new dimension that works very well. Then for domains there is even a relation with valuations and you can develop a divisor theory. I have another talk with about this and you have even a Riemann-Roch theorem, for example, for curves and for the higher dimensions you have to go to the Komlodzka Riemann-Roch theorem. So, oh yes, points. There are points. There are several kinds of points even.

32:30 Because people, at the end of my talk, they will always ask, are there some points? Yes, there are points. Why do I not go into detail about this? There are many kinds of points. There are these so-called point-modules, line-modules, which I don't, it's not really my cup of tea. There are points which are determined also as torture theories, as localizations, which is like incognitive cases. so I will mention something about this and the project is of course the big project is the singularity theory for non-connective varieties which is a very open question okay all this being said we have to now say we want to do this at least this and we want to have topology and a skew structure on it and sheaf theory on it completely non-commutative. No pi rings, no fineness over the center, absolutely non-commutative. Some people cheat, they say quantum groups are truth of unity, but then these are pi rings and they're finite over the center. It's too commutative. Okay, so if you think about the commutative case for a moment, you have some, let's look I find the situation in the speck of the ring R. For every ideal you have the Zarischke topology, the Zarischke open sets. So these are all the prime ideals not containing I. So P prime ideal. This is already annoying because a prime ideal suggests some two-sided which commuted it but later we have some problem we have to avoid this but okay just to remind you what is the the thing with this topology if I take the intersection of two open sets I get an open set and it corresponds to the to the product or the intersection of the ideals and for the union I get the sum right now suppose I play this game with a non-commutative ring the right on the right hand side on the first line is the product ideal the product here this is also equal to it's also equal to this because the open set is the determined by the prime by the radical right and the radical of the of an intersection of ideal and the product is the same in the

35:00 commutative case so it is this this is interesting of course because if the ring is not commutative here I have no problem the Sun is commutative and so So the union of my topology, even in the non-commutative case, if I write down exactly the same thing, it would not be changing. Well, it would be commutative, but not the intersection. It would be non-commutative, because ij and ji are not the same now. So non-commutative topology, sorry to pause you, but perhaps... So non-commutative topology pertains to the fact that while in the usual topology we have a commutative intersection, say, of open sets, whatever we mean by that. That's the starting point, yeah. That's the starting point. But I made also the, I will show you the axioms. So I start with some, I can give you an axiomatic definition. It's axiomatic, you do with it what you want. And then I do the growth and the aversion, and they both have to live together in practice. That's the plan of the talk now. indeed this is a starting point if you have a once you see that your ring is non-commuted first of all prime ideal what do you mean left ideal right ideal no idea at all i will change that completely and the change comes from the fact that what is really important i showed you all these properties what is important is that on for example let's take a basic open set basic means given by one element then you know that this is the spec of the localization this is basic open set is affine that's given by taking the spectrum of the localization so really and that's the basis of the topology affines must be a basis of the topology of your abstract variety so affine sets are particularly nice they are specs of some localizations so is you see the topology as I already said depends on the radical not really on the ideals so the same open set so there is some bad correspondence between ideals or a not so good correspondence between ideals and open sets you need radical but the localization functor also only depends on the radical so it is in a sense better to think that you have to an open set the localization functor which is in fact the tensor product over R is general here so A of R whatever you want so the localization

37:30 function of a module is the tensor product of the localization of the ring so okay so this property that is a tensor product in the non-commutative case is called perfectness or exactness so these are exact localization functions exact left and left and right exact completely exact exact localizations okay so so now my philosophy is that well really this topological space I'm interested in I can also define a localization for every ideal if you want I can write down the definition as a limb and something like this but it doesn't matter I mean you can define the localization which in this special case for a principal ideal is exactly the usual inverting this is I think the So you can think of a topology as a bunch of functors, if this opensets and to an open set is given a functor, a localization functor, and this bunch of functors on this topology must behave well. So that's the sort of starting point. whereas then for any ring R you can always define what is called the localization, abstract localizations in the sense of KML for example, torsion theories and so I'm going to use these localizations really as the as generating the topology so what do you see here also XF for basic open sets this is fg so and the qfg if I look at this functor this tensor functor this is just the product of the two this is the composition of the two localizations and again because of the commutativity the product of the localizations commutes this will be wrong in the non-commutative

40:00 theory so let me remind you before I do the abstract topology the case where it's really applicable an S in R is probably, I say ur is set but the Scandinavians tell me to say ure, it's what I find hard to do I think everything here, everybody in Britain also closed this on ur is set but okay so this is a multiplicatively closed set let's say one is in it it's multiplicatively closed. If the second error condition is two error conditions, the second error condition says if rs is zero then there exists an s-accent, r is zero. Something annihilated on the left, on the right, can also be annihilated on the left, by another element maybe, so s must not be the same. otherwise it would be a kind of commutative now it's also a kind of commutative and the first is that for all r and s in rs respectively there exists r prime s prime such that s prime r equal r prime s also the change from right to left in some sense this is the two conditions that are necessary and sufficient fraction you can make you can make this ring S minus 1 R as the fractions you can divide by the we can invert the elements of S this has a meaning now otherwise two things are equivalent you have to say what it mean but then you have to you know normally you say okay the product of the middle and the exterior has to be equal but now the product is left and right but using this other condition you can give a meaning to fractions. So there exists a meaning of fractions as a concept. So this localization functor Qs, that's an exact functor from modules to modules if you by tensoring, tensoring with m-m, this one is one of r, your tensor, that's your functor the localization functor, expressed on modules is just a tensor with the localization of the ring this is an exact functor

42:30 okay, so that is good, that is like here, RQF in the commutative case the role is going to be played by QS. Okay, so I now treat these functors as if they were the open sets. I view the topology, in fact, on these functors. There is very important observation that if I have two such ERVIS sets, then they don't compute. The localizations do not compute. so what do you do I look at all the onus sets I'm learning R and there is one condition in the whole theory later there must exist enough there's only one condition for the whole theory to work but there are some covering in fact there is a covering then everything will work all the geometry will work this is very minimal and I verified this in in all the interesting classes of algebra you can you can probably meet well maybe not all but as much as I've met so then here then you take wr this is the eroset non-trivial I mean not for example the element one I don't want it not eroset is not consisting of invertible elements okay then I look at all the words so this is things like S T S V T S so the words in other sets and this is really the basis for my topology so to such a word there corresponds a functor QHQT in the same but in this way QVT SQT QS so you look at the words in other sets so a word in other sets corresponds to a functor that is a word in the localization functor but by choice of convention and I write it in the other order for that has to do with the sheaf theory later where you define the sheaf on the opposite of the topology covariance that's the reason but okay so these are these things builds is going to be the this is the name of an open set you can think like that this represents an open set for example commutative case F F G right but they commuted but here they don't commute so you have even for Notice that you may have infinitely many things in your topology, whereas in

45:00 computer, if it would be finite, a finitary approach to some compact manifolds and so on, well here you get infinite, even for a finite number, but because of the repetitions you may end up with infinite anyway. Okay, let me now just forget this for a moment, this is just to be able to tell you at the a certain moment okay look back now you have an example where everything what I said will work it will everything I'm going to do abstract applies to a class of algebras that includes all the algebras you think of schematic algebras they are and the topology will be phrased in these words but so it remains to do the general theory and the abstract definition you told me one One hour and a half, was that correct, or is it one hour? One hour, one hour. So you have a quarter? A quarter or something. Eight. Yeah, okay, I'll try. I'll do what I can. I can be very short in that you don't have to be thinking out about these technicalities. So you can phrase everything in post-its, partially on the set. I don't know if this notion existed. I didn't find it, but I'm not a specialist uptight the lattice theory or positive theory, but I define cover by this way, that a lambda is covered if it's bigger than all of them, and if another one is bigger than all of them, then it is bigger than the lambda. That is a natural definition of cover. It must exist, but I don't know. Okay. So if it is a lattice, then it's exactly what you expect as a definition of our cover. A global cover is that the one is in the lattice. I usually have lattices with 0 and 1. That's the empty set and the whole set later. And one must be covered. See the existence of a global cover is exactly what I meant here. There must be enough ursets, if this is my topology, there must be enough ursets to cover. And we can write down algebraically what that means and then verify that in all rings that I tried, except the free algebra, the free algebra is not true. But it should not be but in all other algebras it is true that I checked directed are not ok let's go on I want to go as quickly as possible to the definition of the topology

47:30 so let's for the moment so what I define is by taking these directed so these things that have a descending direction you take the limits and then with some definition you can extend the operations so this is like adding the points topology then you make the lattice of the topology by adding all the conversion things to the directed sets so you sort of add the limits to your set it is like adding the points to the topology which makes it a very strange thing but it's still a lattice it's not a topology so I talk in terms of lattices instead of topologies that allows me to add the points if I like so under this construction which I'm not going to talk too much about the usual properties of lattices, modular, braver, complete, distributive are inhabited. So you can go to the limit space to add the points to keep all the all the properties that you would be interested in and also the operations of the meat and the joint can be lifted. That is it is useful because if you go in detail I need it but for now let's not worry. ok, well here are the actions I want to define a topology I begin by taking a potent with 0 and 1 and there must be an intersection and it must be smaller than y only y, but later x will follow for the 1 and the 0, I have the usual rules but this is an interesting no new potents A repeated intersection is zero, then the element is already zero. This is an observation that is new. Well, of course, you have it in a classical situation, but you have to keep it. Then, putting brackets, okay, associatively, no problem. The continuity, if I have two elements and a relation, then the relation is carried over by multiplying on the left and on the right. This is the continuity of the intersection of this product or whatever you want to call it. You can read, so these actions are perhaps not the most weak, I don't know, but these are weak enough so that you find it. Okay, the example satisfies all these things and maybe a bit more, but okay. So there is an interesting thing.

50:00 first I tried to define such topologies like saying, okay, let's do something like A meet B if not B meet A and that's all that should go around like here, like in my non-committal version but when you do that, what about C inter C, maybe, and I first believed I should take here C, why not well if you do that, then very quickly because then A into B A into B is A into B and you can begin to play and it turns out and I can even prove this on a very in a very elementary way on the abstract level if you do this then your topology will be commutative so if you assume that all elements are idempotent you cannot get a known from your drift topology so I am forced I am forced to view one idempotent so I call the element idempotent if it is, and now we call this joint idempotent. These are the commutative things. You can think of this as the commutative shadows of your non-commutative topology. So, and this is what people in physics and mathematics are doing their topology on. The non-commutative things, for example, here is a sort of minimal version, when you begin to make these words. You start with one, so for example, A, in my story here, can be idempotent, A intersection B is not kind of potent, right? And then you have A, B intersection A intersection B, and so on, and you can begin to make a whole word. All these words are now non-commutative sets corresponding to the same commutative set. So you blow up your commutative space into a big non-commutative topology. Space is another word, but okay, we can come to that. I have, I think I have told you when we met that the self-physicist David Finkelstein that he has thought about he calls his scheme quantum set theory and he builds a quantum topology on it, whatever that means which the first condition is that okay, it's non-commutative it is anti-commutative so it's like Raston numbers and being anti-commutative, the second would mean that it is mill potent although I see that you exclude mill potent yeah if you take anti commutative that's very very interesting that is the anti-version

52:30 this is the anti-finkelstein thing yes but it's interesting no it's there's no new potence maybe in another version you could do that no idempotence in some sense the idempotence are the classical things in your sense they become zero because they correspond to the classical thing So I have a similar thing for, well first some observation that, yes, this is interesting, see, if, and this is really very easy to prove, if both xy and yx are idempotent, then your topology is commutative. If only one is, then it's almost commutative and I thought that maybe I could get there, but this is such stupid mathematics, you know, we have to run around here and compute it, or could not do it, depending, but get this strange rule, which I can almost not believe, but I couldn't get any further. but with both things you can get further so that's what you get if both intersections have to be idempotent so if everything is idempotent then both intersections are idempotent again so you get that so you have similar actions for the union completely similar also with this idempotent story here if xxx is the union of x a number of times is sounds logic same thing so in fact first I was unhappy as I said I think oh god these things this is not true later I was very happy because exactly here what I could prove in this this was in localization TV not noticed in this a very easy thing once you begin to think about it that yes if this so there is a idempotent localization, let me not explain it now, if you want to know, I can explain it. But in fact, idempotency of this localization corresponding to these things, these Gabriel filters, that actually means the commutativity of this relation. So in localization theory, in fact, it is true that idempotency of the product means the commutativity. So let me just write this idempotent, and to be correct, both sides are LDS, if they are idempotent kernel functus, or torsion theories as they are called, then they commute.

55:00 These localizations commute. And that is exactly the same as in that lemma here. Well, this is exactly the same proof, but I never saw this resulting locally in ring theory. and it comes out of this lattice analysis as a general fact and then I checked it here and it is true and nobody had seen that it's much detail but it's interesting that even as a kind of confirmation of my strange dream really that it makes sense that the idempotent to demand idempotency so to demand this A intersection A is A or whatever is in fact forcing the commutativity that is localization are kind of totally fingers. Okay. Just for the fun of it, we need some kind of modularity. I put here the weakest form that I could live with. You can put stronger in fact my example here in non-community geometry has a stronger modularity. But at the moment I was writing these actions I thought, well, sometimes you have to be more general than what you really need. I wrote down the weakest version. And then I need only a modularity where I have two out of three elements. So normally you have a relation with three, right? If one of the three equals the other one, then you have this weak version of modularity. This is enough for all the rest I'm going to say. But in fact, in the example, I have a stronger modularity. I have much, I have almost, I will write it a bit later here, I think. It will come up a bit later. For the distributivity, you have a similar thing for the modernity. Distributivity, I don't want to put new definitions, you just derive from all the rest. This kind of relations. And then, you see, if you drop these things, then you would be in the classical case. But it is enough, in a sense, this is up to radical. After some powers for the union in the intersection, everything behaves well. so this may not be bigger than this but it's bigger than the intersection with itself this without this is not bigger than that maybe but when i take the union with itself

57:30 so up to the trivial in the sense of commutative the trivial powers that you take in the also in the sense of these words then these relations hold i wonder if that distributed stupidity i can't say it yeah i wonder if that ties up with finkelstein's version where he shows if you've got quantum logic yes that distributivity is actually violated yes yes is that related to what you're saying it probably is especially i don't know this works so well quantum logic yes well essentially in quantum logic what you By pantalogy, you mean the closed subspaces of a human space? The lattice of closed subspaces of a human space? It's non-distributive. It's non-distributive because... Yes, it is non-distributive, but it is up to this non-commutativity. I mean, you have this trivial way of repairing things by intersecting with itself or adding with itself, right? It should not make a difference classically, but now it does. and now it repairs the distributivity so it is up to up to the trivial thing in the non-commutative sense that it does hold but it doesn't write so it doesn't hold but up to that kind of reparation it holds that's why i didn't put any new distributivity relation because you get them up to what you actually want so it's interesting to point out that what i call here the quantized case is all these quantum algebraes and really they correspond to this thing as I wrote it here it is still almost visible this is through non-commutative also yes because the sum is commutative so for my union I don't really need all these my union will be commutative in the example in the example but I didn't want to put it as the general actions because well it's obvious you want to do symmetrical to the to the joint but really in the example i can i can do this non-commutative geometry the only thing non-commutative is the intersection the union is okay yes even in quantum logica this is the non-commutative thing is the intersection of the closed subsets otherwise otherwise for the union you have the closure of the linear span yeah which is yes okay it is commutative the union the phantom logical union is the closure of the linear spine of course it's

1:00:00 but for the symmetry of the theory I could not make the union play such a special role in this general definition of a non-commutative topology I wanted it to be also non-commutative I think all theories that I have seen that claim to do a non-trivial non-commutative extension of the usual topology or set theory say that what has to be made non-commutative is a theoretic intersection yes not the union the union seems always to be commutative yeah but i think this is the next step only i guess the more physics come closer can you imagine uh i think more and more chaos will come in our mathematical formalism and so So we will have to give up commutativity, we have to give up associativity, we have to give up everywhere definedness. So 500 years later, because of physics, maybe near rings with our partially defined structures will be most important. I think we will have to give up all our pre-concepts. We always think nature has converse. So I think we go more and more into chaos with other theorems about these more chaotic structures. But, I mean you can have equally deep terms for near rings, that's for rings, but then people, if you say that now, they will say, oh, well, okay, there's enough problems with rings, why look at nearings? There are enough problems in commutative geometry, why look at non-commutative geometry? But I think nature, and that's the title, will force us to adjust our opinion, because nature is not like we think it is. We just approximate, and more and more chaos will have to be in our definitions. So I really believe, you know, also the commutativity of the union will have to go up. Everywhere definedness will have to go up. But not now. For my example. So then there's a lot of, so I finally get a number 10 action. But I don't have much time, let me just, there's a kind of focused distributivity. But really, it's the 10, this I don't use A10 prime. This holds in the other case also. But I only use A10. global cover induces covers. I have a global cover, so the definition of schematic algebra is just given by assuming that there is a global cover. Action A10, if there is a global cover, it induces covers on every element. That's normal. If you have a global cover, you take intersections with something, it's a cover.

1:02:30 that's an action and now with this I can I can begin to make this Q topology but I don't have time but I want to show you in in two minutes more or less the Grotendieck situation very quickly because there's no time Grotendieck topology is given by a set of arrows well you begin in any category if alike you give covers these covers have to satisfy the three axioms of growth and make and two are very easy a set of covers itself you can say the identity is a cover if something is a cover and the covers are and these elements are also covered then the composition this is the induced cover if you cover something and then you cover every part then that smaller things here also cover the whole thing. This is G2. G3 is the interesting thing that's the pullback and that's the intersection here. The distance of a fiber product. If I have a covering given and I have a map in my category to this one then I have a sort of induced cover. Just like my A10 action a little bit. This induced cover in fact comes from the intersection but that's given here in categorical notion as a fiber product pullback. Okay, so this is the usual Brotendig topology. Now I wanted to adjust that to the non-community case so that my original topology also fits in it as examples. So I'm not going to give you too much detail, but you can always make copies of the slides. It's also in my book. You know, the pullback is replaced by this kind of thing. Normally, you want this. You want to have an element that maps to these two. It factorizes via the pullback. That's your definition for the Grotendiecter project. Now what do I have? I phrase it for two elements that are for symmetries. If I have two elements that map these two, then over both intersections, it factorizes. So for one element only, if you're not interested in two, for one element only, you have a map here and a map there.

1:05:00 Why should you have a map here? That is not logical in the sense of what I call generic relations. But now I really have no time to explain this. Generic relations are the ones that you can see. For example, A intersection B, smaller than A, is a generic relation. You see it on the form. but if A is smaller than B for some reason, I don't see it it's not a generic relation this one is generic, or for example A is in unit B intersection B intersection A or something you see on the form that this is generic but I let not go into this but really, why should it be a map from here to here it is not on the same level, this uses two letters so in my philosophy of words these things are words if I have two letters two letters not from one to two and indeed the right definition seems to be it factorizes in via this way via the self-intersection of course if the self-intersection is t then i have the classical definition right if the intersection t is t fine then you get classical pullback so the non-commutative situation is just that i go here i go there i want to go there but yes there is a small price I can only go there from here because I have to have a letter here see t goes to x prime t goes to x i so it's natural t goes here t goes here so this goes here this is natural this would not be natural Grotendeech's original definition is not symmetric when you make the definition symmetric like upstairs it is not commutative it's strange that the commutative theory is not symmetric because the fact that it commutes destroys the symmetry you don't see it so when you write the symmetric form and that would be this then that is the non-commutative form so this is the non-commutative G3 the Grotendig action then I played around with it and then we using these generic relations I can define things on this topology and without going into so a very important example for me is is this or even more general instead of using oversets general localizations have time to go in this you have a larger topology here if you want but using the exact localization the exact so in this in this theory why is this exactness important because here if you if you look at the proof of this that's the exactness and also the fact that the ring after localization becomes so-called

1:07:30 strongly graded so in my example I have a ring that is generated in degree one is graded and so on okay fine no details but here a fine business so it pays to look at the exact localizations the exact factors so a topology is words of exact factors and they are satisfying a number of actions that are mentioned there and that is exactly what i find in non-commutative algebraic geometry for all these quantized algebras and so on and then i i'm not going to talk about it but I can at least say that I begin to do the sheaf theory and so on separated pre-sheaf stocks stocks and points if you want to you can define points you can define stocks you can go to the limits to the by certain constructions but what I want to show is the Sear theorem if I find it here so I do this with essential factors I'm looking for certain otherwise I would just say that well here is the notion of spectrum if you want to see what the structure scheme is structure scheme in a spectrum the shear is somewhere between my pages now okay let me just say when you take the sheaf theory on this on this kind of non-commutative topology even that with the union non-commutative and then you do everything to find the structure sheaves define the sheaves coherent sheaves if possible then you want to do the Sear theorem it is true so this is interesting here also that this topology looks completely ad hoc this construction at first of course you can see that it generalizes the commutative case in some sense. But if people like Manin and Artin are doing, is saying, okay, and most people doing non-commutative geometry will say, this is a ring of functions on a vertebral variety that I don't know, and I look at the coherent sheaves over this ring as the modules. So they pre-assume the existence of a SEER theorem. They just say that if I study the modules, finally generated modules over this non-commutative ring, the quasi-coherent or the coherent if it's fine to generate the sheaves over the virtual space that I don't know so what I did is I defined the topological space

1:10:00 with the sheave theory so that really if you take the coherence sheaves over this topology and you take the global section you get indeed the certain that it relates to the to the modest or finite length over your so I posted early this is a kind of and I it's really a very narrow margin I find it's a kind of justification of this construction I don't think you can do this in many ways because you really are fixed by it if you really want to share here the coherent the modules of a given ring is fixed you cannot change this category that's there up to Marita and Netherlands perhaps then your topology in a sense is also fixed I don't know to what extent but so that I was really surprised also Michael Artin said that it's a sort of miracle that you know you want you want CRTM to be true to work without the virtual variety without using the real geometry which you do not exist perhaps so I define one and then I have a theorem so that it brings you back to to this interpretation of the of the coherent and quasi-coherent shapes well I have to stop I think but I have this book algebraic geometry of associative algebras it appeared in Marcel Decker's monographs and I also have these notes and if somebody wants to talk I'm here so there is of course in this work there was a lot of abstract nonsense maybe but on the other hand I always feel that this is the basis of something which I cannot touch really because I don't know enough physics in a sense it makes for example the wild algebra commutative if you look at the wild algebra in a commutative idea it is non-commutative so you cannot diagonalize operators and then you say I cannot measure them but really the one is very commutative in the non-commutative topology and you said something that somebody said that in some topos everything can be made this is the cliche so in a sense so if you put the right topology some at least I will not go that far but some non-commutativity is in fact a commutativity in another topology yes it's actually in the bi-categories that it can all be made commutative any topos. I mean, but you're already building in so much. But in any, not even. In an appropriate topos. In an appropriate topos. I mean, topos, but one might argue that's because topos is already so set-like that, you know, they're not the appropriate category for understanding the non-commutativity.

1:12:30 But this is also my feeling that the non-commutativity that comes into the localization business zone is exactly the one that you need to unravel the non-commutativity on the space. What is absolutely fascinating is your question why the localizations turn out to be even potent functus. So, idempotent, if you localize at the same S, it is idempotent. If you localize twice, QSQS is QS. But QSQT, the composition, if I do that twice, that is not idempotent because of this. That really made me feel so good for three days. That is absolutely fascinating. For three days, I was on the jet i was first i was first really embarrassed by should one really accept that but of course you should accept but i didn't know but but then when i realized that exactly i i knew this all the time without realizing then i was happy then i was really happy said okay fine it's just the computer sets are the idempotent ones and you have a shadow and every idempotent one in a sense can explode in now there's some kind of relations like great relations and so on between these things yeah Basil has that theory, I think he calls shadow manifolds and shadow... I'm coming from physics. In other words, I'm not coming from any serious mathematics. Oh, but this is not serious mathematics. This was a hobby. No, no, really. I never dared to publish this until I saw some paper back on Siebich. But I mean, I think this is touching some very deep points that we physicists are worrying about. I think so. I think so. Immediately, when very vaguely in South Africa, Freddy told me about his theory, and I thought this, indeed, and of course you have explained to me your theory. Yeah, yeah, which is coming up. It's very, very similar. And they've got the same question. It's an absolute question. You've got some answers here. I mean, you're really good for the virtual chair. And if you do this non-commutative topology, but you see when you work with commutative instruments, let's say, the commutative brain, you see the commutative shadow. You see the idempotent sense. Yes, that's exactly right. If you'll forgive my saying so, generalizing growth deep topology to the non-commutative case is not, you know, is mathematics. So I think it's a case of false modesty, if ever I heard it.

1:15:00 Well, it is elementary. Well, it's hard to be. Everything is elementary when you know how to do it. When you get to a sufficient level of abstraction, you can argue that everything is elementary. I think I wrote in the book this is a crossword puzzle in four dimensions when you have things at work it seems to be easy but I worked a long time to make the puzzle but seeing how the non-comitivity is coming out of the as it were the failure of the symmetry in the Grotendieck definition is I think highly non-trivial I mean if you think anyway it's beautiful it's like in logic in fact it does seem to me that there ought to be a pair of interlocking adjoint functors involved in the background. Well, there was a lot of things to think about, I'm sure. But it seems to me also like, you know, we have an uncertainty principle in mathematics. If you really do non-commutative mathematics, you cannot really say at the same time X is in A and X is in B. You can never say that, in fact. You cannot say X is in A and X is in B at the same time. in fact if I can say that just apropos of that last remark back to the point about the localisation the connection between localisation and Eden in topos theory sorry I wanted to cut it you do the localisation of covers condition is equivalent in a topos if all coverings localise and there's a smallest element for any covering it's equivalent to the weakly decidable sub-object condition which says it actually gives you a kind of decidability of identity for the points and I'm just wondering whether this connects up with the different definitions of point that come out both of the Grogan-Dig topology and in your non-commutative generalisation of it. I wish you had time to say more about the definitions of points about the different definitions of points and how they interlock and how they interconnect. But that's in the book, is it? It's in partially everything that's in the book, except that I tied this in with Grothendijk representations now. Yeah. So once you define, you study certain objects, but you first decide what is representing the object. So for rings, for example, you would say modules. Yeah, yeah. So soon you say this.

1:17:30 As soon as you say, okay, modules, then the topology is fixed. Yeah. Because then I go via R-mod, and I have a Grothendijk representation, then I have what I call our top then it's fixed really so it really seems to be a matter also of choosing the representation so you can gain some information by general representation of Grotendieck I mean this something like this every category can do this it has representation space and for every object there is a category here Grotendieck category and then the Grotendieck categories have exact factors between them yeah so you have a arbitrary category if you want even I call a growth in the representation if for every object here you have a growth in the category the morphisms correspond to exact factors between the growth in the categories and there are sets of morphisms for the growth in the categories of course I'm sorry I didn't bring that paper where I do the growth in the representation but really this is you gain a lot of this because here I talk about rings but graded rings, filtered rings, rings algebras, whatever sometimes in the quotient category so really you gain by doing this first on an arbitrary category see what the representation so if you take a ring here, you take a mod for example, this is what the idea is of course, the restriction of scalars is an exact functor and then you have a nice situation where your ring category of rings is represented by the growth in D categories of modules in that sense, in that sense with exact functions between them. Then I define immediately these topologies with exact functions and everything commutes and composes well with exactness and I get a whole system where everything comes from the representation. So really, that's, I think, why is the Zariski topology so canonical? Everybody starts with a Zariski topology. Why? Well, because you represent the ring by modules. Exactly, and because you get the points out so cleanly. So once you decide what you want to represent things by, It follows from that, and it goes via this way, via these torsion theories, localizations, abstract localizations. Yeah, yeah, yeah. And then I can define points which roughly, for one definition, could be via the prime torsion theories, irreducible. For the Neutherian rings, prime irreducible is the same, and everything is semi-prime, just like in this business with the radicals. The semi-prime comes in, but that requires Neutherian assumptions. So really, if you go further and further in the categorical theory,

1:20:00 generality. Which for me was always frustrating because I never knew whether I should want to do that or should do that. And sometimes people tell me yes, yes and then I go on. Because for me this is very abstract. I'm usually used more in the ring theoretical. But of course you can apply now to the Witten algebra. Now you can take your Witten algebra you have these localizations. In the book you find them. Or for enveloping algebras, we know them. topology and you can begin to calculate the czech cohomology so on these things on this topology you can then do czech cohomology and you can calculate the chief cohomology and i know that juri bereste who's now in cornell used this technique to to calculate some modelized spaces for ideals of the wireless branch which i also began with lebrun one result but he did it more general better so I think this is nice that even these calculations because it is actually possible to calculate with these Euricef localization the combinations of them and to do the Czech homology explicitly and you can calculate whereas with chief homology you can usually not sometimes yes not always so I think it presents some methods but I feel a little bit like like a mini version of Moses on a mini mountain and I see the mini promised land but I will not go in there I feel maybe thanks very much