Silvia Barberina: Reconstruction of classical geometries from their automorphism groups

2:30 So we need to look at structures that have got a very large automorphism, so a high degree of symmetry, and a very natural candidate is the classical mega-categorical structures. There is one of the formulations of the Ritman-Zesky theorem that ensures that we actually get the property more categorical, and the automorphism group of M is what we say oligomorphic, and I'll say in a minute what that means. The automorphism group has induced an action on contagion powers of orbits, and the group is oligomorphic as there are only finitely many orbits from end to the end for each natural number of n.

5:00 This is my favorite formulation of Greenland-Zesky because it links So 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 M 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 isomorphism 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 action on the structure or anything else, that's not very informative because if you take, for instance, a countable set, omega, with no structure, a pure set, then its automorphism group will be the full symmetric group. The graph structure on omega, the complete graph, so A2.7 by an edge, then again, the automorphism group of our complete graph will be sin omega. So we don't really know very much about the two structures. The other extreme is when we have knowledge of the action of Just about as much information as we want in the following sense, so if m and n are categorical, then they have permutation groups, so the action of m and n isomorphic to the action of n, say by an isomorphism that we can call phi, then...

7:30 Images of zero-definable sets in here are zero-definable in here, and vice versa, containing m to the n is zero-definable, and a is n to the n. So, in this sense, knowing the permutation group allows us to recover the structure with the apparatus of zero-definable sets, and we couldn't possibly be asking for more. A very interesting case is provided by looking at the automorphism group of M as a topological group and the standard way of putting topology and topology. The basis of open sets is given by the stabilizers of finite topos in the structure and their cosets.

10:00 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 topology is that if I pick a countable set, omega, in the countable set, some group of the two symmetric groups, This subgroup is closed in topology if and only if it is the automorphism group of some structure leaving omega, which is closed if and only if there is n such that the domain of n is omega and the automorphism group of n is 1, according to the fact that we should say that. And then the remarkable property is the following. All open subgroups of m have countable index. If M and H are open, then there are at most countably many co-index. 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.

12:30 When countable index M and M expect to M. And the small index property has been widely studied in literature on It's been proved for, for instance, all omega-categorical and omega-stable structures in a paper by Hodgkinson, Hodges, Lasker and Scheller. It's been proved for the order rationales 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 small index property, then we can tell the topology for the pure group, because we only need to look at subgroups of countable index in order to get back to basic subgroups. So if we have M, M omega categorical, and M has the small index property, and the automorphism groups of the two structures are isomorphic as pure groups, then the structures are bi-interpretable. And this happens because of the third type, two hominid categorical structures with automorphism groups that are isomorphic as topological groups are by interval, these are equivalent to the two structures being by interval.

15:00 So this is the context in which small index arises from reconstruction. What I've studied in my PhD thesis, though, is a reconstruction method that's a lot less famous, and it was put forth by Mati Rubin, and it's related to AE definability of point stabilizers in the structure. But the definition is quite long, and I'm going to write it in video code. 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 for any two elements in here, I can find an automorphism that takes one to the other. 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 mon-g-m. And then we want to quotient this conjugacy class by an equivalence relation that has certain properties 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.

17:30 And all elements g of the automorphism group. If these two are related, then these two are related if and only if they're conjugates under g. So h is equivalent to if and only if to g. 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. This is an action by conjugation. So, the equivalence class of some H with G will be simply the image of G, the conjugacy class of the conjugate of H. And this is well defined because of this verse here. So that's why we require E to be invariant under conjugacy. And then there's a definability condition on E. We want E to be A-E-definable in the language of groups and this is the pivotal point. So we want a group-theoretic condition that allows us to decide whether two automorphisms in C are 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, But here we pick conjugacy class of automorphisms that have a single fixed point.

20:00 The equivalence relation will identify automorphisms having the same fixed point. And the idea is that in this isomorphism, an element of M will be sent to the equivalence class of isomorphisms that fix it. If we manage to do this, we find E and C. We say that M has a weak A-E interpretation, is Rubin's terminology. And to give you an example, we are going to find such a construction for the order of Russian as our M with M equal to M, and we pick a point in here, so small q and q, then we can certainly find an order preserved in bijection, so an automatism of M that fixes q and is strictly increasing it below and above the fixed point. Now there's a theorem due to Holland about W homogeneous structures, which covers Q, which says that there will be a conjugacy class in this group which will contain all and only new automorphisms in this form, so automorphisms having a single fixed point and being strictly increasing below and above.

22:30 So we can define C to be G to the... I mean, I haven't said this explicitly, but here we're always allowing definitions with parameters, so in this case our C is defined by a parameter G. And I claim that two automorphisms in here have the same fixed point, even if the product is again in the conjugacy class. K, so the product H, K. 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. Holland's theorem assures us that then the product will be misogynistic, thus its direction is different. Conversely, if the fixed points are different, the product will be strictly increasing. I have found a group theoretic property that allows me to express this fact, so what I am saying is that there is a weak interpretation for Q, based on the equivalence relation expressed by this formula, X is equivalent to Y if and only if there is a Z conjugation tournament such that XY to the Z is G.

25:00 This equivalence relation is expressed by existence of a conjugating element in G, and this will be our parameter. This is an existential formula, so the quantifier complexity is looked after. So this is an example that we came to the definition. Given the definition for transitive structure, So if M is partitioned into more than one orbit, a Wicca interpretation for the whole structure consists simply of Wicca interpretations for each single orbit. So in his paper, Rubin gives a definition, defines Wicca interpretation, and then he proves a very, very nice reconstruction theorem, 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 two algebraic versions, and has a weaker interpretation, and we pick another structure that's omega-categorical, without algebraicity, and such that the two automorphism groups are isomorphic as pure groups, so no topology group, no fermentation group at all. Then, the existence of a weakly-interpretation ensures that the two structures involve a small permutation group, hence, M can be recovered with the apparatus of still-refinable sets. And David Evans pointed out that if we leave these conditions, so if we allow antiparasity, then we get that M and N are bi-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 weak interpretations for mostly combinatorial, only combinatorial structures, so various classes of trees, he's got the random graph, the pure set, and two quite important examples that are not covered by Smolnik's property are the

27:30 A random tournament, so a tournament is a complete dielectric graph between two edges, it doesn't matter. 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 are either functions or relations of higher arity, because Rubin only treats binary structures. So, the first obvious choice was to look at infinite-dimensional vector spaces of a finite field. So we let V be a vector space of dimension omega over a finite field A. We don't want to break reasons because this is omega-categorical. This object is uniquely determined at the left of this dimension. And there are various groups acting on V. We've got the general linear group, and then a bigger group of gamma and V, which is the group of semilinear maps, being behaved like linear maps except that we're allowed to twist, that we're allowed scalars to be twisted by an automorphism of the field, so these will be things of the form V to V objective, and such that sigma is an automorphism of the field. Such that for vectors V and W and scalars A and B, the image of the linear combination A equals BW will be A twisted at sigma.

30:00 These are well-known objects. They are the countable dimension analogues of their finite counterparts. And what I did was to find weakly interpretations for a range of structures. This is the statement of the main theory. We look at structures living not exactly in the vector space, but in the projective space, so P, G, B. So this is the second one-dimensional subspace of this. And we're not working directly with B for reasons that I will explain later. The projective versions of the following geometries, the symplectic space, the orthogonal space, in their 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. Which is U, sorry? U? Unitary. Unitary. And what we do is we look at the projective general linear group acting on PGB and the projective group of semi-linear maps acting on PGB. And we look at all structures with domain PGB whose automorphism group lives between these two.

32:30 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-PGV such that the following holds, we're implicitly considering just those groups in between these two that are closed in the topology. Remember earlier I said a group is an automorphism group, if and only if it's closed in the topology. And incidentally, because we're working with omega-categorical structures, it allows us to avoid specifying language for these structures, because in the case of omega-categorical structures, the zero-definable sets are really finite units of orbits, so the structure is really given by the action of the opomorphism 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 anomegastegorical and it's home to the groups between these two. This group is given as a topological group? 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 of anomegastegorical. So I'm assuming physics closed. Yes, but when you recover this structure, you know what are the closed suburbs of this autobiography. Yes, yes. And it turns out that it's enough to look at the bottom groups. We don't really need to bother. I'm going to talk about examining all of these, because as soon as we've got a weak interpretation for each of these, we've got a weak interpretation for the rest of them, and this is when we're just following that.

35:00 We have gone M to make it rhetorical, and the normants of these, such that H is oligomorphic, H is essentially definable, transmitted, in a lot of conditions, and H is closed. H acting on M has a weak interpretation, so does the automobility. It's well known that each of the bottom groups is normal in the top groups. So by virtue of this fact, a weak interpretation for the bottom ones kills off the remaining structures. These conditions apply to the bottom groups. I'm cheating slightly in the case of the unitary and orthogonal geometries because those groups are not really transitive on the structure but what in reality I've worked with orbits of these in the projected space so I've considered restriction of this action and A lot of work goes into proving this definability condition, and this starts out by the existence of a generic automorphism, which is an automorphism having a unique conjugative class, and the proof is rather long, and it uses some fixed theorem that's a basic effect in the theory of classical computers, and I don't think there's any time to go into that. So, what I want to do next is show

37:30 How to get a weak interpretation for the symplectic space, which is the easiest. First, I'd better explain why we look at Gb. Why the projective space and not the vector space itself. And the reason is quite simple. We cannot find a weak interpretation for B. It's quite easy to see. B is the alternative. Of GLB on B, yeah. Yes, but that will be relevant to a suitable version of these statements, but not to the general definition of the word. So let V be, I'll say, as above, so I get the dimension of the final field. And suppose that the characteristic of the field is different from 2, then there is no weak interpretation. But as you should see, the proof works for any group that has gotten through this answer. So why does this happen? Let's assume for a contradiction that this object here has got a contradiction. So let's tau be an isomorphism, glb be a glb acting on some c-quotient time e.

40:00 We pick H that's really in the center so it computes with all the elements and we know H has the form of a scalar times the identity of G because that's what the center is 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 was taken to be a refinement where the reaction 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, we can suppose we have a weak interpretation law. So let's look at what h does to the inverse image of g over e. So this will be a vector in V. But obviously, b to the h will be alpha b. We've chosen alpha different from 1. So this is different from b. But this is a contradiction because now h fixes this element, but it moves the corresponding element in the structure. And this is a proof that we work in general for a structure having a treated centre. But for an organism proof having a treated centre. So this is the motivation for going out to the project. Now, in Rubin's paper, all his applications have got very long, bare-hand, convoluted constructions of the conjugative class,

42:30 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 heartbreaking examples, we are lucky because the maps that we need are well-studied objects that are found in the literature. So I'm going to define them. All of the retained observations for these factors are based on transactions, which I'm going to define now. And a linear function of u such that, first of all, tau fixes d. Secondly, tau takes a vector to itself plus the vector d multiplied by the image and the beam. And for a transvection, we said tau fixes d, and then of course it will fix the linear function of d in the one-dimensional space. And also, this linear function here will define a hyperplane that an easy calculation shows is fixed pointwise by tau. It defines u tau, we call it u tau, fixed pointwise.

45:00 And we shall call this one-dimensional space the direction of the convection, and we write v tau. So, to each transaction there are associated two fixed spaces. One of them is the direction, which is a one-dimensional space, and one of them is the hyperplane, u tau, and of course this lies in the hyperplane. If it is the image of the projective image of a transvector, i.e. follows the one-dimensional space B, the space spanned by the image of the vector under the transvection. And it turns out that there are just as many projective transvections as there are transvections in GLB. So there's a one-to-one correspondence. In particular, to each projective transvection, there would be associated .

47:30 So these are the maps that we're going to base our week 8 interpretation on. So are there useful facts that I need to point out? That in GLB the transactions will be a complete holistic class, hence the related transactions will be a complete holistic class in the projected general linear group. 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 really interesting, 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 hyperplane or they have the same direction. So bye. Key add here, I mean the conjugative classes, E, G, L, V, containing all transactions. Now what happens in the symplectic space? Things are quite easy. P, S, E, V. We have that the presence of the bilinear form actually forces the relationship between the fixed hyperplane and the fixed direction. So that if we pick the transvection this, as soon as we know one of the two fixed spaces, we know the other.

50:00 Tau hat, d tau hat. But now, because of this very nice property, we can use the key property to define the equivalent relation that identifies projected transvections that have the same direction. Before I do that, we should note that In fact, for each point in the projective space, we can find a transvection that has that point as the big direction. So for big P, you know, in PGD, that is quite important, because eventually, of course, I want my transvections to represent points in PGD, and this ensures that that's quite possible. Okay, so after having noted that, I claim that if we take tau hat, signal hat, projective transvections, PSPB, then the product, the projective transvection, projective-symplectic transvection, if and only if, and this follows directly by the key property, in this case we needn't bother about

52:30 I'm looking at this hyperplane because it's invisibly determined by the direction. And now, I have found a weakly interpretation of PSPB acting. And curiously enough, it's based on the same formula as we used for the order of rationales. So, this would be projective symplectic transactions being in a conjugacy class. So these two have got the same direction, if and only if this is z, such that the product x, y is conjugate to tau by a z, and this will be our defining term. So this is how we get the key interpretation for the syntactic 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 parallel. We're working with parallel transfections rather than single transfections. And in the other spaces there are complications due to the fact that 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 EGP.

55:00 Last thing I'd like to say is that 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 projected space, but I hadn't read his paper, but he told me that the proof goes through immediately in projected versions. So, in some sense, this is... 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 called Smutir-Froxen-Portes that were studied in a rather long monograph by Czerny and Kruszewski. Some people have attempted to prove a small index for smootly approximable structures that will contain only partial results, so that he approves one index for finite and a finite co-ordination of the geometry, and then they like the enterprise because it seems to lead to daunting. So, one direction in which I'm hoping this could move would be to prove alternative reconstruction results for smootly approximable structures. There is a basic level at which the aim of finding a weak interpretation presumably of proximal 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 incomprehensibility of the structure and its automorphism. If we take that as our reconstruction aim, rather than achieving a fully weakened interpretation, it seems, there is good grounds for hoping that reconstructions versus fractional reconstructions might be a good thing. I'll stop here. Thank you very much indeed. Any questions? I was just wondering about the AE. Do you use it? It's closed on the units of change at any point?

57:30 The AE condition comes from a forcing argument that you've been having. 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 Schoentgen's theorem. And actually, of course, we're happy up to, he called them A.E. It says that A.E. statements are absolute. A.E. statements are absolute. So we're happy with, basically we're just happy with these five concepts, but in reality our definitions are usually the same. They're usually things like the existence of complicated elements. If you don't restrict to some sort of these definitions, if you consider the whole... This was the subject of a series of papers starting from Shellas and then I think that Google also did some work on this on elementary theory of symmetric groups and this case was studied by Tolstik. 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 full second order of... Yeah, yeah, yeah. Alright, I remember results. Yeah, yeah. Because you have, in an automating group, you can fix... Some elements and let others move and then you have characteristic functions. So it is important that it is some restriction on the... Yeah, because A is taken as general facts, etc. Where the quantifiers vary over the sets, the subsets of the domain.

1:00:00 As I say, I don't really understand the argument. I see, that's fine. Not really. And I actually tried quite hard to get rid of the A-E condition, but then Matthew will be coming to give up because he tried it as well, so... You've got more geometric... Yeah, yeah, yeah, sure. So we have a group, I mean, assigned to a categorical structure, well, not any structure, but a meaningful structure. What about? I'm just thinking categorically. I mean, the maps between structures. What does that go over to the maps between the corresponding groups and the functor that... You mean... Yeah, suppose I have an elementary embedding from a local structure, to the south I suppose, it's only categorical, then how does that act on the current ones? Could you recover from that general set-up? I'm not really sure. I mean, it must be a group homomorphism. Presumably is it? It induces a... No, I don't think necessarily. I would hope a continuous homomorphism. I'm not sure. I understand. So if I have to say... Does it not behave well? You would then have to ask that any homomorphism on the smaller structure is extendable.

1:02:30 Is it smaller, right? Yes, I think it must be continuous in one direction, not something that comes across. Say something about the geometries that are instructed. I mean, just from a very human point of view, I mean, can you be more explicit about what these things are? Okay. So, for instance, in practice case, we define bilinear. I've actually got a slide with the definition. So this is the way the Balenian form works. We've got an automorphism of field with scalars and syntactic case, just the identity.

1:05:00 And the form has got these properties. It's automating that we're in the presence of the syntactic case. If it's symmetric, we're dealing with its orbital case, and if it's hermitian, it's the unitary case. And all these groups, so things like people from the military groups, the finite dimensional cases, compounds of all of the mathematics, they're present in, for instance, the classification of finite symbol groups, but, I mean, the infinite dimensional analogues, I haven't really, I've actually asked Google, but I haven't really come across any of that as part of model theory. I'm actually wondering about the significance of general mathematics. This can be considered a continuation of Phoenix Klein's program. That's what I was trying to say. That's more or less what is this program, that you consider geometries as something which is invariant under certain, under action of certain groups. And then you have an inverse problem, that you have an... You want to know what geometry is. It is a divide of what a dimension becomes. Well, I mean, because all these geometries are by definition infinite dimensional, because the field is very small, the field is finer. And this 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. Any other?