Reconstruction of classical geometries from their automorphism groups

0:00 A great pleasure to welcome Sylvia Barbino, today from Leeds, a student of the person there, and she's going to talk to us today on reconstructing classical geomachers from their orders and mortals, please. Thank you very much. So I'd like to thank you very much for inviting me. It's very, very nice to meet you. And let me talk about today a participant's thesis with writing and explanation of the first question. So, I'm looking at the structure and motivating question in the following, if you take a first service structure and look at it, just by looking at the group, how much information you get about the structure, this is a very general question, the first thing to note So the question does not always make sense, and we need to restrict the context to highly symmetric contexts, so we need to look at structures that have got a very large automatism, so a high degree of symmetry. And a very natural candidate is the class of omega categorical There's one of the formulations of the Rittman-Zesky theorem ensures that we actually get the property we want. So this says that if M is a countable structure, the flow in our equivalent, M is megacategorical. And the automorphism group of M is what we say oligomorphic. And I'll say in a minute what that means. I.e., the automorphism group has induced an action on contingent powers of m. And the group is oligomorphic is that there are only finitely many orbits

2:30 for m to the n, for each natural number of m. So, for m in omega, we'll put m. This is my favorite formulation of Green Lanczewski because it links a modern theoretic condition to a very natural group-theoretic one. And in particular, the group is a translating dictionary between modern theory and group theory and permutation group theory. So what What this is saying is that the automorphism group is very rich. Lots of automorphisms correspond to very few orbits. So throughout the talk we're going to be concerned with countable omega-categorical structures. And of course we need to restrict the question further because we have to specify what we mean by knowing about M. We might be wanting to get back up to isomorphism, which is of course the most desirable condition, or we might be happy getting it back up to my interpretability. And again, there are many ways in which we can be given the automorphism group as well. In general, if we look at the automorphism group as a pure group, so just look at the group by itself with no reference to its actual structure or anything else, that's not very informative because if you take, for instance, countable set, omega, with no structure, a pure set, any total morphism group would be the full symmetric group. If we decide to put graph structure on omega, the complete graph, so any two points, then again, your morphism group of our complete graph would So we don't really know very much about the two structures. The other extreme is when we have the knowledge of the action of the automorphism group on the structure. And this gives just about as much information as we want in the following sense.

5:00 If M and N are made a categorical, then they have isomorphic permutation groups. So the the action of M on N is isomorphic to the action of what's M on N, saved by an isomorphism that we can call phi. Then images of zero-definable sets in here are zero-definable in here and and vice versa. A containing M to the N is irrefinable, implies phi of A in N to the N is irrefinable. So, in this sense, knowing the permutation group allows us to recover the structure with the apparatus of the human fineball sets, and we couldn't possibly be asking for more. An intermediate and very interesting case is provided by looking at the automorphism group of them as a topological group. And there's another way to bring original terms. The basis of open sets is given by the stabilizers of the final two posts in the structure and the cosine. So, that's in the subgroups of the form G node M, which that G fixes

7:30 So this topology makes four-time into a topological group that incidentally has not topological properties because it's, for instance, completely triangle in the Polish state. And the topology is important for various reasons. One little fact that we will need later is, if we're not topology, we have a criterion for deciding whether we're dealing with an automorphism group or not. And the following is, if I pick a countable set of omega, it will be countable set. A subgroup of a full semantic group, then the subgroup is closed in topology, it is the automorphism group of some structurally living omega. So H is close if you live. There is N such that our domain of N is omega. and the automorphism group of m in h. This is one of the most effective. Now the remarkable The countable property is the following. All open subgroups of what M have countable index. So if h is a group of what m and h is open, then there at most can't be many constants.

10:00 This always holds. When Congress holds, i.e., when subgroups of countable index are open, we are in the presence of a very interesting property that's called the small index property. So when countable index implies open, end, end, respect to end. And the Mon index property has been widely studied in the literature at all. It's been proved for, for instance, all omega-pategoric and omega-stable structures in a paper by Hodkinson, Hodges, Lasker, and Shella. It's been proved for the other rationals by John Truss. And there's also work of Lasker outside the omega-categorical context. And obviously it is highly relevant to reconstruction in the following sense. If a structure has the nismole index property, then we can tell the topology for the pure group because we only of countable index in order to get back to basic subgroups. So, if we have M and omega categorical, and M has the small index property, and the automorphism groups of the two structures isomorphic is pure groups, then the structures are biinterpretable. And this happens because of the third size of the genes. Two omega-historical structures with with automorphism groups, that are isomorphic, isopological groups, I find that. So, M isomorphic

12:30 and as topological groups, this condition is equivalent to the two structures made by a term of the board. So this is the context in which small index rise in public destruction. What I'm studying in my PhD thesis though is a reconstruction method that's a lot less famous and it was put forth by Matthew Rubin and it's related to AE definability of poly stabilizers in the structure. But the definition is quite long and I'm going to write it in So again, we pick our favorite omega-thegoic structure. And for the time being, we suppose that M is transitive, by which I mean that the automorphism group has got a single orbital structure, and if you put any two elements in here, I can find an automorphism that takes one to the other. Okay? So Rubin says the following, we should look for First of all, a conjugacy class in the automorphism group. And I'll write conjugacy class as more than G to be equal to M. And then we want to quotient this conjugacy class by a an equivalent relation that has certain properties, so equivalence relation E and C, search next.

15:00 First of all, we want E to be respected by the action of the automorphism group on C by conjugation, by which I mean that for all H, K in the conjugacy class and all elements is g of the automorphism group. If these two are related, then these two are related if and only the conjugates under g. So h is equivalent to k. If and only h to the g is equivalent to k to the g. This is a separate condition. And the third one is that we want to choose c and e in such a way that the action of the group on the structure is isomorphic to the action in the portion conjugacy. So I call this now. This is an action by conjugation, So, prevalence class of some H, the G would be simply the image of G, the conjugacy class conjugate of H. And this is what we find because of this verse here. So that's why we require E to the invariant of the conjugates here. Okay. All right, run from this B. And then And with the definability condition of E, we want E to be AE definable in the language of groups, and this is the pivotal point. So we want a group-theoretic condition that

17:30 that allows us to decide if there are two automorphisms in theory equivalent or not. So we are not allowed to make any reference to the action on the structure. Now, what happens generally, but not necessarily, that here we pick conjugacy class of automorphisms that have a single fixed point. The equivalence relation will identify automorphisms having the same fix for it. And the idea is that in this isomorphism, an element of M will be sent to the equivalence class of isomorphism that fix it. If we manage to do this, so if we find E and C, we say that M has a weak AE interpretation. This is Rubin's Tuminoschi. And to give you an example, we're going to find such a abstraction for the order of So, as our n, we take q, that's going to be equal then, and if we pick a point in here, so small q in q, then we can certainly find an order to preserve the bijection, so an automotorism of n that fixes q and is strictly using below and above the fixed points. to say g in oxygen, such that when q is fixed by g, and 2, g is strictly increasing on minus infinity q, q plus infinity. Now, there's a theorem due to a column about

20:00 W homogeneous structures, which consequently covers q, which says that there will be a conjugacy class in this group which will contain all and only new automorphisms of this form. So, although it isn't having a single peak, it's going to be strictly increasing below and above. So we can define C to be G to the point. So, I mean, I haven't said this explicitly, but here we're always allowing definitions with parameters. So in this case, RC is defined by a parameter from G. And I claim that two two automorphisms in here have the same fixed point if and only if the product is again in the conjugacy class. So, this H equals to XK, so they have the same fixed point if and only if the product HK is again in C. And why is this true? Well, this direction, if they have the same fixed point, the product will again have the same fixed point and it will be strictly increasing below and above. Hollands theorem assures us that then the product will be scogelous in last, so this direction is fixed, HK equals fixed damage, etc. Conversely, if the fixed points are different, the product will be strictly increasing.

22:30 And hence, again, by Holland's theorem, it doesn't belong to C. So, I have found group theoretic property that allows me to express this fact. So, what I'm saying is that there is a weak interpretation for Q. Based on the equivalence relation expressed by this formula, to y, if and only h, there is a z conjugated element such that x, y to the z is g. So this equivalent relation is expressed by existence of conjugating element g and this will be parameter. This is an existential formula, so the quantifier complexity is no capital. So this is an example that we came to the definition. Given the definition for a transitive structure, So if there's more, if the, if M is partitioned into more than one orbit, a weak interpretation for the whole structure consists simply of weak interpretation for each single orbit. And so in his paper Rubin gives a definition, defines weak interpretation, and then he proves a very, very nice reconstruction plan, which said the following. So, if we work with omega categorical structure, and his original paper only looks at structures without algebraicity, so every set has got treatment algebraic closure. And M has a weaker interpretation, and we picked another, we picked another structure that's omega categorical, without algebraicity, and such that the two otomorphism groups are isomorphic as pure groups, so no topology or no fermentation group. Then the existence of the weakening interpretation ensures that the two structures of the otomorphic fermentation group hence and can be recovered with the apparatus of steer and final sets. And David Evans pointed out that if we need this condition

25:00 and veracity, then we get that M and N are by interpretable. So we still have got a very strong reconstruction result. Now, in his paper, Rubin has got proof of the main theorem and then a series of applications. So he finds we came to petitions for mostly only combinatorial structure. So various classes of trees. We've got the random graph, the pure set. And two quite important examples that are not covered by small mixed property are the random tonal length. So a tonal length is a direct, complete direct graph between the two edges that we have on. And the countable homogeneous partial, countable universal homogeneous partial order. And in the beginning, we thought it might be interesting to look at structures that have functions lying around. And they're either functions or relations of higher arbitrary because Rubin only treats binary structures. So the first obvious choice was two sets. Infinite dimensional vector spaces are required for this. So we let V be the of dimension omega over a finite p We also learn to break reasons this is omega categorical this object is uniquely determined that I mentioned in the field. And there are various groups acting moving, the general linear group, and then a bigger group called gamma LV, which is the group of semi-linear maps. These behave like linear maps that will be allowed to twist, that will allow scalars to be twist by an automorphism of the field. So these will be things of the core into the

27:30 objective, and such that there is sigma, an automorphism of B, such that for So, vectors V in W in scalar AB, image of the linear combination AB plus VW would be A, twisted by sigma, V, F, V, plus V, twisted by sigma, that name. These are well known objects. They are the countable dimension analogues of the finite countenbales. And what I did was to find the interpretations for a range of structures. structures. So this is the statement that's made here. We look at structures leading not exactly on the vector space but on the projective space, so PgV. So this is the sightable one dimensional subspaces of this. And we're not working directly with V for reasons that I will explain later. OK? So one-dimensional superfaces. OK. Superfaces to be able to make. And we equip our vector space with bilinear force. So we're looking at the projected versions of the following geometries, this inflective space, the unitary space, in the projective guise. And it so happens that all of these are omega-categorical. Categoricity is not preserved here in a multiple dimension. This is totally categorical. These aren't, but... Which is you, sorry? You? Unitary. Unitary. It's got an affirmation. And what we do is we look at the projected

30:00 general linear group acting on PGB and the projected group of semi-linear maps acting on PGB. And we look at all structures with domain PGB whose automorphism group lives between these two. And likewise, we look at analogs for spaces with forms. And we find weak interpretations for each piece. Now, there are a few things that I need to say. First of all, by saying that we look at an omega categorical with omega Pgb such that the following holds, we're implicitly considering just those groups in between these two that are closed in the topology. To remember earlier I said, group is an automorphism group, if and if it's closed in the topology. And incidentally, because we're working with omega-categorical structures, it allows us to avoid space-finding language for these structures, because in the case of omega-categorical structures, the zero-definable sets are really finite that you know of orbits. So, the structure is really given by the action of the automorphism group. As soon as you know the orbits, you know what's definable and what isn't. So, this is what I mean when I say that I'm, you know, I get theoretical and it's automorphism groups between these two. This automorphism group is given as a topological group? Uh, yes. I mean, we're really looking at the pure groups here. But I thought I could do away with mentioning what the structure really is by saying, you know, we look at whatever is between these two ends. So I'm assuming it is closed. Yes, but when you recover this structure, you know what are the closed subgroups of this autophysic group?

32:30 Yes, yes. And it turns out that it's enough to look at the bottom groups. We don't really need bother about examining all of these because as soon as we've got a weaker interpretation for each of these, we've got a weaker interpretation for the rest of them. And this is my knowledge the following time. So, if I've got m, we made it hierarchical, and H enormants a group of all of them, such that H is oligomorphic, When H is essentially definable, then we want to be transited in a lot of conditions. and H is closed, then if H acting on H has a weak age of position, It's well known that each of the bottom groups is normal in the top groups, so by virtue So they expect a weak interpretation for the bottom ones kills off the remaining structures.

35:00 These conditions apply to the bottom groups. I'm cheating lightly in the case of the unitary and orthogonal geometries because those groups are not really transitive on the structure. But in reality, I've worked with orbits of these in the projected space. So, I consider a restriction of this action. And a lot of work goes into proving this definability condition. And this is by the existence of a generic automorphism, which is an automorphism having a unique conjugacy class. And the proof is rather long, and it uses the fixed theory that's a basic effect in the theory of classical groups. And I don't think there's any time So what I want to do next is show how to get a weak interpretation for that symplectic space, which is the easiest. First, I'd better explain why we look at Gb. Why the projective space and not the vector state itself. And the reason is quite simple, we cannot find a re-h interpretation for V. That's quite easy to see. It's also not transitive. V, the actual V is not transitive. Of GLV on V? Yes. Yes, but that would be relevant to a suitable version of this diagram, but not to the general definition of the way it is. So, let V be, I'll say as above, so we get the dimension of the final field. And suppose that the characteristic of a field is different from two, then there is no weak

37:30 So as you should see, the proof works for any group that has gone through the center. So why does this happen? Let's assume for a contradiction that this object here has So let's tau be a nice more person from GLB B to say GLB acting on some C quotient by E. So this is little g to the GLB and this is an equivalence relation. So, we pick H, I mean, the map in GLB, that's really the center, so it combines with all the elements, and we know H has the form of scalar times the identity of GLB, because center list. And we choose alpha to be different from one. And let's look at what H does to equivalence class G over E. So this will take it to be a refining graph. Well, this is the the way the action works, but this is a central element, so the conjugate of G by H will again be G. So H fixes G over E. On the other hand, if I suppose we have a weak interpretation So let's look at what H does to the inverse image of G over E. So this will be a vector in B. But obviously, B to the H will be alpha B, which is an alpha different from one. So this is different for me. But this is a contradiction because now H fixes this element but it moves the corresponding element in the structure. Okay? And this is a proof that

40:00 we work in general for a structure having a trivial centre, or for an algorithmic proof having a trivial centre. So this is the motivation for going up to the objective space. Now in paper, all his applications have got very long, bare-hand convoluted constructions of automorphism class, of the conjugacy class, in the sense that he, in all his examples, he looks for automorphisms that have got a single fixed point, and he works very hard to produce those. In these algebraic examples, we're lucky, because the The maps that we need are well-studied objects and they're ready, they're found ready in the literature. So I'm going to define them. So these structures are based on transcriptions, which I'm going to define now. So the map tau in glv is a transvection. If we can find a vector d different from 0, a linear function of u, such that. First of all, tau fixes d. Secondly, tau takes a vector to itself plus the vector of D multiplied by the image and the beam. And for a transvection, So we set tau fix is D and then of course it will fix the linear span of D. So the one dimensional span is D. And also this linear function here will define a hyperplane that

42:30 fx equals zero defines a hyperplane u tau, we call it u tau, fixed pointwise by tau. And we shall call this one-dimensional space the direction of the transaction, and we write V tau. So, to each transaction there are associated two fixed spaces. One of them is the direction, which is the one-dimensional space, and one of them is the hyperplane, U tau. Now, there are projective analogs of these maps, so if we now work in PGLP, tau x is a projective transaction. If it is the image of the projective image of the transfection, i.e. it behaves as follows from one-dimensional space B to one-dimensional space, and by the image of the vector under a transfection. The tau material is A transfection. And it turns out that there are just as many projected transfections as there are transfections

45:00 in GLP. between transfections and projective transfections. In particular, to each projective transfection, there will be associated with a hyperplane So, these are the maps that we're going to base our weak A interpretation on. So, other useful facts that I need to point out are that in GLB, the transfections will would be a complete controversy class, hence the projected transactions would be a complete controversy class in the projected general linear group. Not so for, not always so for spaces with forms, but that turns out not to be a problem. And then we have this key property here, which I found ready in the detector, and it's actually a very convenient group theoretic condition that allows us to decide whether two transactions have got fixed space in common. So the product of two projected transactions is, again, a projected transaction, if and only if they either share the same fixed type of line or they have the same interaction. So by key had here, I mean the conjugative class is PGLB containing all transrections. Now what happens in the symplectic space, things are quite easy. PSPB, we have that the presence of the bilinear form actually, of course, is a relationship between the fixed hyperplane and the fixed direction. So, of course, is, well, your tau height must be d tau height,

47:30 transfection in this group. As soon as we know one of the two fixed spaces, we know the other. But now, because of this very nice property, we can use the key property to define the that identifies projective transactions ahead in the same direction. Before I do that, we should note that, in fact, for each point in the projective we can find a transfection that has that point in this direction. So for the V, we don't I find this important because eventually, of course, I want my transfections to represent points in PGB and this ensures that that's simply possible. Okay, so after having noted that, I claim that if we take our signal and projective transfections in PSPV,

50:00 that the product is a projective transfection, projective symplectic transfection. So, and this follows directly by the key property. In this case, we didn't bother about looking at this type of thing because it's indisibly determined by the direction. And now I have found a weekly interpretation for PSPB acting GB. And curiously enough, it's based on the same formula as we used for the ordered rash months. So, let's see. This is equivalent to one. So, this will be projective symphlexic transrections in the conjugacy class of the core. So these two have got the same direction, even if the missus z, such that the product xy is conjugate to tau by a z, and this would be a refining theorem. So this is how we get the weak interpretation for the synthetic case. And all the other cases are a bit more complicated because in the relative general linear group case, these two spaces are actually unrelated. We need to go up to comparison. We're working with

52:30 pairs of transfections rather than single transfections. And in the other spaces there are computations due to the fact that this, what was it, this property does not hold. So for the unitary and orthogonal cases, it is not the case that we can find a map which has a given direction for each point of the GP. Last thing I'd like to say is that WK interpretations for these spaces, sorry, the small index property for these spaces was actually proved by David Evans. He actually worked with the vector space, with the vector space itself rather than the projective space, but I haven't read his paper, but he told me that the proof goes through immediately in projective versions. So in some sense, this is not a new reconstruction result. What is important is that all these structures here are among the building blocks of a very interesting class of structures was called smoothly-approximable, that were studied in a rather long monograph by Czerny Krzyzewski. And some people have attempted to prove a small index for smoothly-approximable structures that contain only partial results, so that he approves small index for finite and finite cobwebs for big geometry. And then they liked the enterprise because it seemed So, one direction in which I'm hoping this could move would be to prove alternative reconstruction results for smoothly approximable structures. There is a basic level at which the aim of finding a weak aim of rotation for smoothly approximable structures falls through in the sense that there are problems with what we call finite covers, but the interesting feature of weak interpretations is that they give interpretability of the structure in its automorphism group. So if we take that as a reconstruction name rather than achieving a fully weak interpretation, It seems that it's good grounds for hoping that reconstruction versus approximately

55:00 reconstruction might be enough. And if you see anything, I'll stop here. Thank you very much indeed. Any questions? I was just wondering about the AE, I missed, do you use, it's closed on the unions chains at any point or is that a point? The AE condition comes from a forcing argument that you have had in the proof. I have a look at what it said. I actually, I don't understand the proof, I asked him and he told me that it's a miracle, that proof is a miracle. So at some stage he uses Schoenfeld's theorem, and actually, of course, we're happy up to, we call them A-E, but it's good. It says that A-E statements are absolute. Insetary. Yeah. So we're happy with, basically, we're just happy with this . But in reality, our definitions are usually . Well, I noticed that. Yeah. There are usually things like that. If you don't restrict to some sort of these conditions, if you consider the cool elementary theory of then this was the subject of a series of papers starting from Shellace. Right. And then, I think that Google also did some work on this, on elementary theory of symmetric groups. And this case was studied by Tulsteen. And when you allow all sentences, then you can reconstruct the full set theory of M. You just can recover all subsets of M. So the full set theory. Oh, so the whole second-order arithmetic? Yeah, yeah, yeah. Alright, I remember the results. Yeah, yeah. Because, because you have, in automorphism group, you can fix some elements and let us move, and then you have characteristic functions, so to say.

57:30 Okay. So it is important that it is some restriction on the... Right. Yeah, because A statements aren't in general, actually, etc., where the quantifiers vary over the sets, the subsets of the domain. As I say, I don't really understand the argument. Right. And I actually tried quite hard to get rid of the BAB condition, but then Matthew will be coming to give out. I've been trying that as well. Yeah, yeah, sure. So we have a group, I mean, assigned to a longer categorical structure. structure what about just thinking categorically I mean the maps between structures what does that go over to maps between corresponding groups I mean, it's sort of a kind of punter that... You mean... Yeah, suppose I had an elementary embedding from a longer categorical structure. To the south, I suppose, it is a categorical. Then, what... How does that act on the current homomorphism group, and what could you recover from that general set up? I'm not really sure. I mean, it must be a homomorphism, I presume, is it? It induces a stigma.

1:00:00 Not, I do think necessarily. I would hope that continuous homomorphism, but maybe not. Not sure, I understand, so if I had to say... You would then have to ask if there's any automorphism on the smaller structure, extendable, then something would make sense, but that's all this. So you mean you can't have a... Does that induce some kind of homomorphism or continuous homomorphism from, I don't know, my guess is it would go the other way, from all ten to all ten? Yes. I think it must be continuous in one direction. But that's not something. No, that's not something I've come across. Can you say something about the geometries that are reconstructed? I mean, from just from a very even point of view, I mean, can you be more explicit about what these things are? Okay. So, for instance, in the active space, we find the bilinear from the big cross feet to the feet. So these were, um, well, I've actually got a slide with the definition of success.

1:02:30 That takes quite a long time. And so this is the way the Valenian form works. We've got an automobilism of field with scalars and syntactic case that's just the identity. And the form has got these properties. If it's automating that we're in the presence of the syntactic case, if it's symmetric we're And all these groups, the five-dimensional cases are present in, for instance, quite simple groups. But the infinite-dimensional analogs, I haven't really, I've actually asked Google, but I haven't I don't really come across any of that, but in modern learning. I'm actually wondering about the significance of generating mathematics. This can be considered a continuation of Felix Klein's program. That's what I was trying to say. That's what is this program, that you consider geometries as something which is invariant under certain under action of certain groups and then there you have an inverse problem that you have an automaton group and you want to empower what was the geometry that it is an invariant Well, I mean, because all these geometries are, by definition, infinite dimensional, because the field is very small, the field is finite, and the set which this geometry is, is infinite. So you have to, this vector space, or projective space, or vector space is a simpler case to consider, has to be infinite dimensional, otherwise you can have only finite vector space or finite field.

1:05:00 Any other questions? Well, I should thank Sylvia again for a very nice moment. You think exactly what is the... well, the part of this. All right. These are the elements of this. And that's what we more or less have seen in the tradition of this. Why call it geometry? Well, if you hear what the geometrists call geometry, you wouldn't delete them. Well, there are points and lines and planes that are invariant under the group. And you don't have to take all three-dimensional subspaces or two-dimensional subspaces as your planes, or three-dimensional subspaces as your surface, or your solids. You can take certain subsets up and adjust the bearings into a certain group. and you still get meaningful incidents relations applied in sex, three-dimensional space, and all the rest of it, you know.

1:07:30 Right, that is why it's job. Incidents and inspections. So, are the in-dimensional possibilities further than the younger dimensional ones? No, no, this is only subject to additional conditions that you have also to some topology of it. And, well, there is some field which is called C-star Algebra, which more or less... but it is very rather far from this and this we consider this geometry they move from this point of view the point of view of considering so to say coordinate functions on your space If you want to see it as a geometry, you look at what are continuous functions from your space into, say, complex numbers. And because continuous function you can add and multiply, you now have a ring of functions. But they now look at this ring as an abstract thing, and even then they look also on operators on this ring. When you move the space, then it causes some transformations of the functions. And this would be linear transformations. So you get the linear transformations of some ring or some Hilbert space and then it starts becoming very... That's analogous to the transformations on something that we wanted to do. Yes, there are, well, intimate dimensional cases certainly are now in the center of tension, but in the context of hibbert spaces and so with some topology on it.

1:10:00 But the anthropologists are becoming more and more weird, and there is a whole philosophy of it. You can, for example, look up at the introduction to papers and books by Alan Combs. All right, he's a sort of rhinoceros, and we have a really big switch which does a lot of mixture of physics and mathematics, and he also starts with the form of all this, which is from Yalan and Neumann. I like to say to you, but you know that you were a graduate of the United States and you were a graduate of the United States. I can remember she did it. I was a graduate of the United States. She did it. She did it. She did it. She did it. She did it. Well, I won't see. I haven't had some idea. She did it. But she was doing it. It's not a good idea of the person who is in the right place of the place of the country. Come on. Hello. Hello. I want to get everyone to join this very much. Hello. You're welcome. I'm very much. Good evening. Well, I just want to hear. Sorry. Oh, sorry. Thank you. Okay. Thank you. Thank you.

1:12:30 It's absolutely not that you have there, it's still there. Thank you.