Classification Theory and the Construction of PAC Fields
Ramsey, Nick
A field K is called pseudo-algebraically closed (PAC) if every absolutely irreducible variety defined over K has a K-rational point. These fields were introduced by Ax in his characterization of pseudo-finite fields and have since become an important object of model-theoretic study. A remarkable theorem of Chatzidakis proves that, in a precise sense, independent amalgamation in a PAC field is controlled by independent amalgamation in the absolute Galois group. We will describe how this theorem and a graph-coding construction of Cherlin, van den Dries, and Macintyre may be combined to construct PAC fields with prescribed model-theoretic properties.
 Nice to be back and thank you to the organizers for organizing it. I also want to make a comment before I get started so Most of this work sort of began During my fourth year of graduate school, so I spent a week visiting Zoe at ENS and we spent many hours in her office each day And so Even though you know, sir, I'm risk I'm writing this many of the ideas came from Zoe and I just wanted to sort of acknowledge that So the plan is to talk about classification theory and the construction of PAC fields so most of the time in model theory we find fields, but this talk is going to be about building them and Here's the map that you know Gabe's gift to the model theory community and The question that we often ask in model theory is sort of which parts of this map have a field in them and If it has a field in it, what kind of field can it be? And of course many of the hardest conjectures in model theory about exactly this question So the staple fields conjecture the super simple fields conjecture stuff like that and I'm interested in building sort of counter examples and Finding new kinds of fields. So here is the SOP in hierarchy just one part of the map Maybe maybe not the most interesting part that it's there So we say T has SOP in if there's some type of PXY or you can make it a formula If you like an indiscernible sequence which is ordered by this type So P a I a J holds if and only if I is less than J and If you take a sort of cyclic union so PX0 through X1 X2 or X1 X2 X2 X3 all the way up to Xn-2 Xn-1 and then you loop back so Xn-1 X0 that's inconsistent Okay, so the idea is you think about this is an instance of the order property and you want to say how close does this get to being An instance of the strict order property and so as in gets higher you're getting closer and closer to transitivity Because transitivity says exactly that you can have no loop of any size. So this says that you can't have small loops Well, it doesn't matter if it's in a if it's in a complete type then by compactness it reduces to a formula So so sometimes it's useful to have this formulation when you want to work with complete types and other times when you want to do compactness arguments It's useful to have a formula so there it goes Yeah, I should have said that in bigger than two and at least three so for one and two it's defined differently Which is going to be something I mentioned many times in the talk. Yeah So I should have said that so so they're They're gonna be many theorems in the well not many theorems There are gonna be a few theorems in the talk that say something for SOP in and when I say that without qualification I really mean it for one two and in greater than or equal to three But of course the argument has to be different for one for two and then for in greater than or equal to three Okay, so this is the in greater than or equal to three case. I'm not gonna define One because young hand did that and I'm not gonna define two because we don't care so Here's the theorem so I'm gonna say the theorem first then I'm gonna spend a lot of time sort of talking about the ingredients So the theorem is just that the SOP in hierarchy is strict in fields So let me say what I mean precisely by that so for all in greater than or equal to three There is a field which is SOP in in SOP in plus one of and in particular It's a PAC field so the SOP in hierarchy is not only strict in fields It's strict in PAC fields or if you like PAC fields of characteristics zero Moreover, there's an open problem. We don't know whether SOP one and SOP two are different whether SOP two and SOP three are different But if they're different that will be witnessed by a field. Okay, so So, you know, maybe maybe someone can prove something by coding them in fields. I'm very doubtful, but okay This is the theorem Now the theorem is nice, but in truth it's much less interesting. I think than the construction So the the whole the whole goal of the project is that we have constructions for coding combinatorial objects and distinct groups You have something like the Meckler construction which takes an arbitrary graph and produces for you a group which is nilpotent step two and Many of the model theoretic properties of the graph are reflected in the group now What I'm interested in is sort of a analogous construction for fields I take a graph I build a field out of it and I want to know what are the model theoretic properties that are preserved by passing from the graph to the field and okay the SOP and hierarchy is just a test case But there there are I suspect many other model theoretic properties for which this construction will be germane Pure fields in the language of rings. Yes Yeah, it's not so hard to find an expansion of a field with this property. Just put a graph on top Yeah, so this is in the language of rings. Should have said that. Okie doke. So Why do we care about PC fields? So of course the starting point is axis 68 paper the elementary theory of finite fields and He characterizes the pseudo finite fields which are defined logically in Algebraic vocabulary. So what he shows is that a Field is an infinite model of the theory of finite fields if it satisfies these three conditions. The first is is that it's perfect Which just says that if it's characteristic p then f to the p is equal to p Sorry is equal to f It has Galois group Z hat And then the last one is this algebraic condition, which is the last missing ingredient Which says that f is pseudo algebraically closed So this was already mentioned by Samaria in her talk But f is pseudo algebraically closed if whenever you take an absolutely reducible variety to find over the field It has a point in the field. Okay, so so Algebraically closed fields are pseudo algebraically closed by the Nostalon thoughts, but you have lots more. So for example ultra products of finite fields Okay, so It became interesting within model theory to just ask how complicated can pseudo finite fields be so there was a question from the you know the famous paper of Chatsodakis van der Dries in Macintay where they characterize definable sets in finite fields about whether or not triangle-free random graphs are interpretable in finite fields and Roshofsky answered this question by showing that the independence theorem is satisfied in in in pseudo finite fields or more generally bounded PAC fields and Using the independence theorem you can show that you have non interpretation of triangle-free random graphs So this is sort of the But Roshofsky works in more generality So recall that a field K is called bounded if K has finitely many degree in extensions for every N and Working in in a general context where he studies definably closed sets in strongly minimal theories satisfying weak elimination of imaginaries and the definable multiplicity property together if that definable Closed set has a certain boundedness property then then the induced structure is super simple. So the Upshot is that if you think about a bounded perfect PAC field you can think about it as a definably close subset of its algebraic closure and That's strongly minimal and it satisfies all the requisite properties. So in particular. He shows that a bounded PAC field is super simple. Oh the perfect assumption was later dropped in this paper of Chatsadakis Pillay on Generic structures for simple theories. So in an appendix to that paper They show that actually if you drop perfect, of course, you won't have super simple But you can still get simple so a bounded PAC field is simple and later Chatsadakis showed this is the Characters this is a characterization. So an unbounded PAC field has TP2 So a PAC field is simple if and only if it's bounded. So we know exactly Which PAC fields are bounded and that's a consequence of a certain Galois theoretic condition? And so there's a sort of emerging theme Which which is already very very clear in the work of Trillian van der Dries Macintyre Which really sort of opened this up opened up the model theory of PAC fields, which is that it you can think about a PAC field Any question about a piece of PC field typically breaks up into two questions One is about classical algebraic information Either about the algebraic closure or the separable closure in the imperfect case and then you ask a question about the Galois group and Normally if you answer those two questions, you can amalgamate them and answer whatever question you might ask about the field so in particular this shows up in sort of the characterization of types that Trillian van der Dries Macintyre show in in their monster paper and in order to do to Give precise meaning to what it means to ask a model theoretic question about the Galois group. They introduced this inverse system so of course the absolute Galois group of a of a Field is a profite group and that's not really a good object. I mean a good model theoretic object For example, if you take an elementary equivalent Group there's no reason that that should have an interesting topology on it or be profite in a meaningful way but you can think about the you can think duly about the system of finite quotients and the maps between them and encode that in some first order structure so that if you take an elementary Equivalent structure then you can take the inverse limit of the associated system and get another profite group So this is really a way of thinking fitting Profite groups into a first order framework. So so here's the definition So you start with a profite group and you look at the set of open normal subgroups And so the domain of the structure will be all of the finite quotients by open normal subgroups Now we're going to have a sort of strange multi-sorted Presentation to this so we won't have disjoint sorts. This is what's a little weird about it but we'll have a sort for every positive integer and a binary relation less than and C which is I Guess binary relation. So sorry, we have two binary relations less than and see an eternal relation p So I don't know what these letters were chosen or why this is the language, but this is the traditional approach so s of g is in lg structure in the following way so the Coset I mean the elements of the sort x in are exactly the cosets of Open normal subgroups of index at most in So notice that the sorts are growing right? Because you get more and more cosets Then you have a comparison G of n is less than or equal to h of m if the Groups I mean the subgroups are have containment like this then This c relation codes the projection So if if in is a subset of m then whenever I take cosets g in I can project it and get a coset of gm and The graph of that projection is coded by this relation c and then on each finite quotient I have multiplication and the graph of multiplication is coded by this relation p So if I have in one into in three and they're all the same So these are all cosets of the same group the same open normal subgroup Then the relation just holds if and only if I have the appropriate multiplication Okay, now this is a pre-order this this partial order symbol so I can take equivalence classes and This is going to be the notation I use for this equivalence class The less than relation the less than or equal to relation is a pre-order Well, so so what this tells you is that if I look at G in and then I look at its image under the projection from G mod in to g mod m Yeah, I mean Yeah, yeah, it's the same as contain into the coset but I mean I prefer to think about it in terms of the maps, but sure it's the same Technically yeah, so okay if we were to do this in the normal presentation You would have district sorts and you would have bijections that sort of tell you the Identifications between the district ones and then you would have a different relation for each sort So so even though you have a sort of globally defined partial order It turns out that this can be like an omega. I mean for certain groups. It'll be Omega stable Because we're cheating here. I mean we're really thinking about this is only being sort what sort by sort But it's convenient to write it as a globally defined pre-order because we all know how multi-sorted logic works You know we can translate if we like if this yeah, so here the sorts are not just joined So but we sort of pretend like they are It works. I promise Okay, so this is a certain structure And I want to mention two sort of notions that are important in talking about the structure So these are duals and subsystems So suppose we're given l k fields and an embedding of the separable closure of k into the separable closure of l So that L is a regular extension of the image of Phi of k so This is confusing but imagine for a moment that this is just an elementary embedding of K into L and then you extend that to a map on the separable closures. So that that That case won't will be the important one for us. So we have this embedding From Ks into LS and then this induces a map from G of L to G of k So if you if you take an automorphism here That is the same thing as an automorphism of the image of this by the isomorphism so you look at the image of the separable closure of k sort of in inside L and then this Tells you that that extends because of regularity. So this is an epimorphism this map here And then anytime I take a quotient of this that's the same thing as an epimorphism from this to a finite group so by composition any time I have a quotient of G of k that gives me a quotient of G of L So then I get an embedding from the inverse system of G of s of G of k to s of G of L So this is an embedding which is called the double dual to Phi to Phi so Each time the direction of the arrow switches. So if I do it twice then I get an inclusion of the inverse systems This one so the way you think about this map is suppose I have a finite quotient of G of k So that's the same thing as an epimorphism from G of k to a finite group But then by composition I get an epimorphism from G of L to a finite group And so that tells me how to how to map a finite quotient here to a finite quotient here or more generally an open normal subgroup here to Okay, does that make sense? Okay now Subsystem this would be the appropriate notion of substructure if the language was different now a subsystem s is is Should I mean you should think about this as a substructure Up to an appropriate choice of the language So what it means is that it's downwards directed and upwards closed in the in the pre-order in Particularly it's closed on their equivalence because this says less than or equal to so you close under all the equivalence classes And then you you take you make sure that you're closed you're directed downward Oh, yes, sorry, I should have said that so to the image Yeah, I mean I guess this is implied by being a regular extension, but but yeah, yes That is my intention. Yeah, so it's a really the example that you should think about is just imagine that L is a regular extension of k So I mean forget forget about k and think about phi of k and then this is the inclusion Or this is rather the identity map. This is restriction and this is dual to the restriction that that's the I mean You don't lose anything by thinking about that case Okay, so let me tell you about this theorem which which I'm very enthusiastic about and it's a monster So I'm gonna I'm gonna draw a picture and then everything will be clear. I promise so so Here's here's a field f. I should not wait yeah, okay, so f and Here's s of G of f and we suppose that we have an instance of the hypotheses of the of the independence theorem in the field So so what does it mean to have the instance of the hypotheses of the independence there? You have a substructure when in this case this elementary substructure e you have a and b which are independent over e And you have c0 and c1 which have the same type over e and c0 is independent with a and c1 is independent with b These are the hypotheses of the independence there and then by by taking double dual you you get an associated map. I Mean an associated picture In The inverse system of the Galois group So here's s of g of c0 s of g of c1 and And suppose that not only do you have the hypotheses of the independence theorem here, but you also have a solution So this gives you S zero which amalgamates these two things Yeah, so so all of this is assumptions so I start with the hypotheses of the independence theorem in in the field f I look at the associated instance of the independence theorem in the inverse system and I assume that I have a solution Then then that means that there's a solution here so that s0 you can think of as sg of C so so we are starting the existence of this so The picture here. I mean the way it's formulated is a bit complicated, but this is really the idea so you start with an instance of the Hypothesis of the independence theorem in the field This gives rise to a similar instance in the inverse system if you can solve it in the inverse system Then you could solve it in the field So this tells you that amalgamation problems in the field reduced completely to the question of amalgamation in the Galois group Yeah, so so everything here the way it's written is they're all algebraically closed subsets so in particular the sub fields So so so they have Galois groups So so yeah, so so this is why it's a bit difficult to state So you want to you want to make sure everything is algebraically closed so it makes sense to take the Galois group, but But yeah, don't don't read this just look at just look at the picture Everything will be clear. So I didn't want to lie to you, but Yeah, this is what it says. Okay, so here's a cool corollary so Zoe proved in the paper where she proves this paper that if you have a PAC field and the theory of the inverse system of that field is In SOP in for in greater than or equal to three You get that the field itself is in a sop in and then my thesis I sort of did the Requisite work to extend that to one and two so in in the case of in greater than or equal to three there's a sort of direct Translation of SOP in into an amalgamation problem because you want to show that this type which is supposed to be inconsistent becomes consistent Now for n equals one two for n equals one you use chem independence in the Galois group and translate that and characterize chem independence in the field so a p a c field is in a sop in if and only if the Galois group is in a sop in and Or if if the Galois group is in a sop in and then for two you use strongly indiscernible trees So this is a somewhat different approach for the other two But it it's a corollary of the theorem. I said before in a very direct way. So there's nothing deep about this extent Okay, so this tells us that the question of whether or not we can build p a c fields With prescribed model theoretic properties at least for the question of SOP in reduces to a question about the absolute Galois group So here's a strategy. So we know that we reduced this to the absolute Galois group now What we want to do is we want to build groups. We want to build profanate groups that code graphs and then Because we can find enough graphs We can find enough groups and then realizing those is the absolute Galois group of p a c fields We went we find p a c fields that are SOP in in a sop in plus one because this is this is the strategy So what time do I need to finish by the way? Do you remember? I? Mean I can talk for half an hour. Okay. Good. Good. So I'm on schedule. So I'll stop it 10 53 Okay, 10 10 to 12 Yeah, well now now we just lost two minutes. Yeah, okay Okay, okay, so All right, well, I don't know I'll probably just talk and then someone will tell me to show up and that's what I Okay, so um, so the idea here is to make make use of a certain coding construction Which was developed by trillion van der Dries and McIntyre in their unpublished long paper on On Psc fields now there their question was just about decidability So they they were using this coding construction for a very coarse kind of problem They want to show that you can code graphs and then you get some undecidable Stuff in in a p a c field and then they were happy now I want to just sort of reanalyze that construction with a more fine-grained approach and understand which sort of sort of find structural model theoretic properties are preserved by by this construction So the this construction is theirs But but the analysis will be a bit more refined. So so here's the idea. So This this slide will just set up notation. So we start by fixing From now on to distinct primes odd primes P and Q now CP will be my notation for the the Z mod PZ but thought of as a multiplicative group which would be important just for notation and And okay DP is the dihedral group. So this has two P elements. It's generated by an alpha beta so alpha is an evolution and If you conjugate beta by alpha. Oh, sorry, that's a typo. They should say Alpha beta alpha beta inverse so if you are alpha beta alpha inverse beta maybe okay anyway But what you want to say is that alpha beta alpha inverse equals beta inverse. That's what it should say So that should that's the same thing as the relation alpha beta inverse beta Yeah, and there should be an inverse so so Yeah, so there are a lot of things missing here. Yeah, so okay, so So okay, so you want these relations so alpha squared So so here are the relations alpha squared alpha beta alpha inverse beta and beta to the P and That's it. Okay, so so a lot of things missing anyway, you know what the dihedral group is so Okay, okay Okay, so the relations are written here anyway, so You can you can? Okay, so there's a Z mod PZ sitting inside generated by a Generating by beta and the quotient by that gives you an epimorphism onto the cyclic group of order two plus or minus one Right, so let's call that tau Now using that you can define an action from the product dp times dp to I mean acting on CQ by The following rule so if I take a pair of elements from dp cross dp xy now I look at whether their image in C2 is plus or minus one and I raise a to the power the product tau x times tau y so that means that if tau of x equals tau of y Then I do nothing and if tau x is not equal to tau y then I invert That's the rule now because I have an action here I can define a semi direct product and W is going to be the semi direct product with the action defined here So that's W and of course this fits into a short exact sequence where W is going to be the semi direct product of dp I'm dp by Cura and just to give it a name lambda is going to be this map that Takes me from W to dp times dp. I just forget about the CQ part Okay so now we want to build the graph so first start with a graph on vertices a and with edge relation r and Form the product Where you take a copy of dp for every vertex and the copy of W for every relation r for every edge r then of course you can take any elements of this group should be called h as D bar times W bar where this is a vector in dp to the a and W bar is a vector in W to the r and We have the associated projections, so I'm just setting up notation here so pi of a is the projection under the coordinate a and Pi r is the projection under the coordinate r and Then we define a group g gamma as follows so I look at the elements in this product I Look at the elements of this product Yeah, so W Is here so it's the semi-direct product so so maybe I should just write this because I think it's worth remembering So W is going to be CQ semi-direct product Dp times dp Okay, so Yeah, yeah, sure sure sure so so we have tau Which goes from dp to C2 which we're thinking of is plus or minus one No C2 and then and then dp times dp acts on CQ by X Y acts on alpha by Alpha to the tau x tau y So again you take x and y you look at their images under tau if those are the same then you do nothing if they're different than you invert That's the rule Sorry Yeah, yeah, so I said that I had a typo so H is what I'm calling d to the A times W to the R Okay, but I won't use H ever again, so you don't need to worry. Okay, so G game is this group so Here's here's the way it works so So maybe one more piece of notation which is worth remembering is you have lambda which goes from W to Dp times Dp which is the thing that just forgets about the CQ coordinate, right? So so lambda here came from this oops it came from the sequence. So here's lambda which which I'll remember here So it's the projection from W to Dp times Dp that forgets about the CQ coordinates Okay, so so here's the definition of the group so I look at The vectors d bar w bar so that if I if I have an edge in my Graph which goes from a to b then I look at the elements in the R coordinate and I take its projection in Dp times Dp so W are it's it's something here So it's in a copy of W so I can apply lambda I can forget about the CQ part Then I want that to match the Dp parts so if I have an edge from a and b and then I look at the Like the coordinate that is associated with the vertex Then that's Then they're sorry if I look at the coordinate that they come from the edge And I apply lambda then I get the coordinates that come from the vertices vertices they match so So this is the definition and then just as a trivial remark As defined if the graph is empty then you get the empty set here Yeah Yeah, I mean This is a homomorphism. So it's it's fairly trivial to just check that it's close to their multiplication It's also profite because this condition is closed So here you have a big product of finite groups, which is Which is obviously profite because it's by T. Knobs theorem. It's compact separable I mean compact totally disconnected topological group and Then this is close to it's a close-up group of profite groups. It's profiting. So this is a profite group and This is just a sort of side comment if a is empty or sorry I mean if the graph is empty then in principle this definition gives you the empty set which we don't want So if it's an empty graph, then you just define the group to be the product of the vertices and this this coheres Okay, so that's a that's a group now Now I want to tell you an interpretation of graphs in profite groups so Given a profite group G which you know any one at all I mean this might give you nonsense but but for any profite group at all of you can define a graph gamma G AR so the vertices consist of open normal subgroups such that the quotient by them is DP okay, and You you say that there's an edge between in one and in two If they're distinct and there's an open normal subgroup G so that the quotient of G by M is W and M is a subset of the intersection of these two open normal subgroups so this is a fact that if you Take the graph you build the group and then you interpret the graph again you get the graph back So this doesn't do anything and there's a this is sort of nice picture of what this looks like in the inverse system So the inverse system you think of the sort Okay Okay, so in the inverse system you have the sorts let's say one P or sorry one two P and four P squared Q and I'm gonna have several Elements here or really equivalence classes which correspond to copies of DP so so these are like in two so these are elements of sort two P in the in the inverse system, so this is S of G and I have an edge between these two things if I can find some in here So this picture tells me how to interpret this Infer system so this interpretation although I've described it in this language is really an interpretation in the in the inverse system Okay, so this tells us how to build a group but the the real question is can I possibly have this group as an absolute Galois group of a PAC field if not then it's not so helpful so in order to talk about that I'll have to tell you what a profite group is Or say a projective group so a profite group G is called projective if it satisfies this following is this property so I take an epimorphism from G to a and from B to a and and What I want is I want to be able to complete the diagram with a homomorphism Yeah, it's the same thing as being a projective object in the category of profite groups Oh Sorry, sorry. No, I really meant to say that a and B are finite. I mean they could also be profite But yeah, so yeah, sorry, so a and B are funny. Thank you Yeah, so It's known that you can't always complete this diagram with an epimorphism that would be the natural The natural guests for this and notice that that would be dual to the extension property that you're familiar with for say for say classes So when you have the stronger property that you can complete the diagram with an epimorphism that that's called super projectivity or sometimes the Iwasawa property and Zoe showed in a thesis that the inverse system of a group with the stronger property is Omega stable And so these are very very nice groups and the PAC fields with those as their absolute Galois groups are NSP one Which which is a consequence of the theorem I mentioned earlier, but it was known much before this is something that you tie and I worked out Okay, so Oh, then I oh shoot. No, I just I just use the wrong arrow. So Okay, many typos here. So a and B are finite and this The text is correct. Although the diagram is wrong. So there's a continuous homomorphism Here Yeah Okay, is that good cool, so We need to know the answer to the following question. What are the absolute Galois groups of PAC fields? so the Is this sort of striking theorem of van der Dries and the Botsky which shows that the class of absolute Galois groups of PAC fields is Exactly the class of projective Prophetic groups so secretly here. There's there's one direction which was known already to ax which is that if you have a PC field Then it's Galois group is projective that occurs already in the 68 paper so the the thing that Lobotsky and van der Dries do is they go the other direction they show that if you have a Projective profanity group you can build a PAC field that has that as its Absolute Galois group. So what we want now is a construction not to just take a graph and build a Profanity group, but we want a construction that takes a graph and builds a projective profanity group such that I can recover the graph from that group so How do we build projective groups so if you have a profanity group, there's a certain subgroup called the Frittini subgroup which is Denoted capital phi of H and this is generated by the maximal open proper subgroups of H so From the definition you can tell automatically that it's a characteristic subgroup. So it's normal and a Frittini cover is an epimorphism Such that the kernel is that should say phi of G. Yeah, so the kernels contain in phi of G Now every profanity group has a universal Frittini cover, which is the same thing as the smallest projective group that surjects onto H So so this is defined. This is unique up to isomorphism and Here's the second key fact if you take the graph You build the group Then you pass the universal Frittini cover and then you try to interpret the graph again using the interpretation I described before you get the graph back So actually this process of replacing G with its universal Frittini cover does not affect the graph that we interpret in the inverse system Okay, so this is wonderful because what this means is that if we take this graph We we build the universal Frittini cover then we can realize that at the absolute gaoual group of a PC field and then we get what we want No, no, I clarified subgroup of G. Yeah, it's just a typo is subgroup of G Of 5g yeah, well in particular the subgroup of G and it's also a subgroup of 5g that one actually has content Okay, thank you Okay, so now I want to talk about how the analysis of these inverse systems of Universal Frittini covers of G gammas go so we want to understand certain important quotients Of this group so notice for example if I take in copies of DP So I look at DP to the end there's a copy of Z mod PZ to the end sitting inside This is generated by the betas in each copy so Now if I take the quotient of that then I get a big copy of Z mod twos so I can quotient by this group and I get Z mod twos each the end Now sitting inside of Z mod twos each the end as as a normal subgroup. I have the diagonal subgroup So these are the things which are either zeros and all coordinates or one and all coordinates Wait is this what I want? Yeah, I guess so so what I want is the quotient to be Z mod 2z Yeah, so so I'm not sure what I mean Yeah, so there's a quotient of the Z mod 2z which which is okay, so fix whatever normal subgroup makes this true You have a quotient of this diagonal which Exactly, yeah Yeah, okay, so I said nonsense here Yeah, yeah, I'm not sure what I had in mind when I wrote this okay, so so anyway that Let's ignore this for a slight moment so Here's one thing that is certainly true so if So let me let me rewrite the fact in the following way so let me let me write it on the board very quickly So if you have a DP and a DP sitting at level 2p and you take a 4p squared q then there's a unique w sitting on top if there's one at all Okay, so this follows from the fact that in the graph interpretation if that the the the copies of DP you see are always associated with with vertices so every every DP is really the the kernel of a coordinate projection to a DP and And every W that you see is is really coming from the kernel of a pi r So all of the W's and the DPs that you see in the inverse system come from this configuration So that means that from from these DPs. There's a unique W that's sitting on top from the definition of the group now I want to define a system a definition called the graph closure and Here's the way it's defined so you define the graph closure a sub gr to be the smallest Subsystem containing a so that if you take an element and you get a Z mod 2z Then you include all of the all of the vertices that you that you see so So Here at level 2 you might have a DP above Or it might be finitely many DPs above So so if you if you have a Z mod 2z Which is a quotient of a DP then you put the DP in that's what this says so if if I see a element of this structure and it gives me a Z mod 2z then I put in the the the kernel of the vertices that are associated to the C meaning that that Surject onto the Z mod 2z non-trivial now Additionally if I have two things here which look like vertices so which give me Structures isomorphic to DP so if I see two DPs then if there's an edge between them in The subsystem then I put that in the closure so the idea here is The I take the small substructure that sees every piece of the graph so if if I take a Z mod 2z and I I see vertices above it Finally many then I put those inside the graph so they're all algebraic over the Z mod 2z And if I see an edge between the vertices that I put in then I make sure that I put those edges inside this structure So the the whole point of this definition is that the interpretation of the graph Applied to this substructure is an induced subgraph of the gamma that I've defined Okay, so forget about this nonsense that was written before this is really the definition so this is this is just a definition so it's It's a way of getting an induced subgraph from a certain set Which one Yes, sorry so in the in the subsystem I have this partial order right and then you have an equivalence relation and on the on the equivalence class of a particular element Because I have this this relation P which is like the graph of multiplication here This is isomorphic to a finite group So so each element in the inverse system if I take its equivalence class that is a finite group and I can ask What group is it isomorphic to okay? So so here when I say see is isomorphic to Z mod 2z I mean I take an element I look at the Isomorphism type of the finite quotient associated to this class Which is which is given to me by P in the language of the inverse system and if it's Z mod 2z then I look at which DPS are above Yes, okay, and then if there's an edge I make sure that I put the edge in Okay, so here's here's the theorem which sort of tells us how to analyze types in the inverse system so suppose I take two finite tuples from the inverse system and I look at this the graph closed Substructors that they generate so this is a subsystem which is graph closed so it means that the the interpretation of the graph here it's going to be an element you know an induced sub graph of gamma and the The thing here will also be an induced sub graph of gamma And I look at the induced map Which goes from gamma a to gamma b So I first take these substructures I form the interpretation of the graphs and I look at the induced map between those graphs and I ask the question Is that partial elementary? So if that's partial elementary in the theory of the graph Then actually the map that sends a to b is partial elementary in the theory of the inverse system So this tells you exactly how to analyze types in the inverse system all you have to do is understand types in the graph and the quantifier free type in the in the inverse system and Using that you can Well because the quantifier free type is encoded by the isomorphism type of the substructure generated by the tuple Right it's sort of like in for say limits right like two things have the same type if there's an isomorphism of the structures that they generate Okay, but okay, so quantifier free type is not strictly speaking true because the language is not correct right, I mean like the the the Substructures don't correspond to subsystems, but but you think about it really being secretly quantifier free I mean you can you can you can formulate an appropriate language in which it's quantifier free and it's not too bad But okay, really I mean the isomorphism type of the substructure which which we think of as model theorists is coding the quantifier free type But up to an appropriate choice of language Okay, and then using using this analysis of types you can prove that for all positive integers and here I was careful to write it that the theory of the Inverse system of the universal for Chini copper of the group that you build from the graph is Inesopian if and only if the theory of the graph is So this is enough to tell you I mean not really the theorem But really the proof of the theorem tells you how to analyze types and you can you can code the relevant amalgamation problems in terms The graph and then solves it Okay, so now let's just put it all together and Yeah, this is right on time. So for ingrid and they're equal to three there are graphs. There, you know, they're known examples of graphs These loop-free graphs of a certain kind which are SOP in and in a sopian plus one So you just build the the right graph now There's a general fact which is which is easy to show although. I don't know if it's written anywhere Well, I mean I wrote it down, but I don't know if it had been written before anywhere else Is that if SOP 1 and SOP 2 or so be two or not equal to so be three this will be witnessed by a graph gamma and you know to prove this it's very simple you just open up your copy of Sue gows and variant descriptive site 3 you look at the proof that graph isomorphisms broth pleat and what you do is you just code arbitrary structures into graphs you it's very simple to show that SOP n for example is preserved under this operation because essentially definability is not changed What that's the idea yeah, so you Yeah, yeah, but okay. You there's a small check. I mean I I was clear that it's trivial. I mean it's not a deep fact Okay, so then you build g gamma from that graph and then you pass the universal for a teeny cover and we know that that operation preserves the graph Then you use banded Dries-Libotsky the inverse gawa problem for PC fields to realize G tell the gamma as the absolute gawa group of a PC field and then Gamma is interpretable in the field. So this is something which Well, this is the universal for teeny cover of the group g gamma then gamma is interpretable in the field so this is a somewhat delicate point in the sense that sg of gamma is not actually interpretable in the field Gamma because it's the the the gawa groups are really only interpreted up to conjugacy So sg of gamma is is is interpretable in the pair the field together with its algebraic closure the field together with separable closure But not necessarily in the field itself However, the graph is interpretable because you can write polynomials that say I'm separable and my gawa group is DP and I'm separable And my gawa group is W and using that you can code the graph even if you can't get your hands on the gawa group itself and So this gamma is interpretable in the field and so f is SOP in But then also it's in a sop in by this theorem of Zoe Which shows you that if the if the group itself is in a sop and then that passes to the field Oh shoot, yeah, in this op in plus one. Yeah, this is a remarkable theorem, right? So Which which one Zoe's there? Here I Mean so for this one this this is me and then this one is Zoe So so so look so there's there's two steps, right? So one is that you have a graph then you get a group and then you have the group and you get a field and then You want to prove that in a sop in is preserved at each step of the way So I start with a graph and I build a group I show that if you start with an in a sop in graph Then the in the in a sop then you get an in a sop in group So it shows that if you start with an in a sop in group and you build an in a field you get an in a sop in field Yeah, yeah, so so that so so Yeah, so so this theorem is a corollary of this there Because you code sop in as a as an amalgamation problem They use it for all three So so okay, so you have to use strongly indiscernible trees or came independence to do the one one in two case It's not such a direct translation, but but you use but you translate everything into an amalgamation problem And you cite this theorem Okay, so Enrique standing up so I should stop thank you. I
 Nice to be back and thank you to the organizers for organizing it. I also want to make a comment before I get started so  Most of this work sort of began  During my fourth year of graduate school, so I spent a week visiting Zoe at ENS and we spent many hours in her office each day  And so  Even though you know, sir, I'm risk I'm writing this many of the ideas came from Zoe and I just wanted to sort of acknowledge that  So the plan is to talk about classification theory and the construction of PAC fields  so most of the time in model theory we find fields, but this talk is going to be about building them and  Here's the map that you know Gabe's gift to the model theory community and  The question that we often ask in model theory is sort of which parts of this map have a field in them and  If it has a field in it, what kind of field can it be?  And of course many of the hardest conjectures in model theory about exactly this question  So the staple fields conjecture the super simple fields conjecture stuff like that  and I'm interested in building sort of counter examples and  Finding new kinds of fields. So here is the SOP in hierarchy just one part of the map  Maybe maybe not the most interesting part that it's there  So we say T has SOP in if there's some type of PXY or you can make it a formula  If you like an indiscernible sequence which is ordered by this type  So P a I a J holds if and only if I is less than J and  If you take a sort of cyclic union so PX0 through X1 X2 or X1 X2 X2 X3 all the way up to  Xn-2 Xn-1 and then you loop back so Xn-1 X0 that's inconsistent  Okay, so the idea is you think about this is an instance of the order property and you want to say how close does this get to being  An instance of the strict order property and so as in gets higher you're getting closer and closer to transitivity  Because transitivity says exactly that you can have no loop of any size. So this says that you can't have small loops  Well, it doesn't matter if it's in a if it's in a complete type then by compactness it reduces to a formula  So so sometimes it's useful to have this formulation when you want to work with complete types and other times when you want to do compactness arguments  It's useful to have a formula so there it goes  Yeah, I should have said that in bigger than two and at least three so for one and two it's defined differently  Which is going to be something I mentioned many times in the talk. Yeah  So I should have said that so so they're  They're gonna be many theorems in the well not many theorems  There are gonna be a few theorems in the talk that say something for SOP in and when I say that without qualification  I really mean it for one two and in greater than or equal to three  But of course the argument has to be different for one for two and then for in greater than or equal to three  Okay, so this is the in greater than or equal to three case. I'm not gonna define  One because young hand did that and I'm not gonna define two because we don't care so  Here's the theorem so I'm gonna say the theorem first then I'm gonna spend a lot of time sort of talking about the ingredients  So the theorem is just that the SOP in hierarchy is strict in fields  So let me say what I mean precisely by that so for all in greater than or equal to three  There is a field which is SOP in in SOP in plus one of and in particular  It's a PAC field so the SOP in hierarchy is not only strict in fields  It's strict in PAC fields or if you like PAC fields of characteristics zero  Moreover, there's an open problem. We don't know whether SOP one and SOP two are different whether SOP two and SOP three are different  But if they're different that will be witnessed by a field. Okay, so  So, you know, maybe maybe someone can prove something by coding them in fields. I'm very doubtful, but okay  This is the theorem  Now the theorem is nice, but in truth it's much less interesting. I think than the construction  So the the whole the whole goal of the project is that we have constructions for coding combinatorial objects and distinct groups  You have something like the Meckler construction which takes an arbitrary graph and produces for you a group which is nilpotent step two  and  Many of the model theoretic properties of the graph are reflected in the group now  What I'm interested in is sort of a analogous construction for fields  I take a graph I build a field out of it and I want to know what are the model theoretic properties that are preserved  by passing from the graph to the field and okay the SOP and hierarchy is just a test case  But there there are I suspect many other model theoretic properties for which this construction will be germane  Pure fields in the language of rings. Yes  Yeah, it's not so hard to find an expansion of a field with this property. Just put a graph on top  Yeah, so this is in the language of rings. Should have said that. Okie doke. So  Why do we care about PC fields? So of course the starting point is axis 68 paper the elementary theory of finite fields and  He characterizes the pseudo finite fields which are defined logically  in  Algebraic vocabulary. So what he shows is that a  Field is an infinite model of the theory of finite fields if it satisfies these three conditions. The first is is that it's perfect  Which just says that if it's characteristic p then f to the p is equal to p  Sorry is equal to f  It has Galois group Z hat  And then the last one is this algebraic condition, which is the last missing ingredient  Which says that f is pseudo algebraically closed  So this was already mentioned by Samaria in her talk  But f is pseudo algebraically closed if whenever you take an absolutely reducible variety to find over the field  It has a point in the field. Okay, so so  Algebraically closed fields are pseudo algebraically closed by the Nostalon thoughts, but you have lots more. So for example ultra products of finite fields  Okay, so  It became interesting within model theory to just ask how complicated can  pseudo finite fields be so there was a question from the you know the famous paper of  Chatsodakis van der Dries in Macintay where they characterize definable sets in finite fields about whether or not  triangle-free random graphs are interpretable in finite fields and  Roshofsky answered this question by showing that the independence theorem is satisfied in  in in  pseudo finite fields or more generally bounded PAC fields and  Using the independence theorem you can show that you have non interpretation of triangle-free random graphs  So this is sort of the  But Roshofsky works in more generality  So recall that a field K is called bounded if K has finitely many degree in extensions for every N and  Working in in a general context where he studies definably closed sets in strongly minimal theories satisfying  weak elimination of imaginaries and the definable multiplicity property together if that definable  Closed set has a certain boundedness property then then the induced structure is super simple. So the  Upshot is that if you think about a bounded perfect PAC field you can think about it as a  definably close subset of its algebraic closure and  That's strongly minimal and it satisfies all the requisite properties. So in particular. He shows that a bounded PAC field is super simple. Oh  the perfect assumption was later dropped in this paper of  Chatsadakis Pillay on  Generic structures for simple theories. So in an appendix to that paper  They show that actually if you drop perfect, of course, you won't have super simple  But you can still get simple so a bounded PAC field is simple and later Chatsadakis showed this is the  Characters this is a characterization. So an unbounded PAC field has TP2  So a PAC field is simple if and only if it's bounded. So we know exactly  Which PAC fields are bounded and that's a consequence of a certain Galois theoretic condition?  And so there's a sort of emerging theme  Which which is already very very clear in the work of Trillian van der Dries Macintyre  Which really sort of opened this up opened up the model theory of PAC fields, which is that it you can think about a PAC field  Any question about a piece of PC field typically breaks up into two questions  One is about classical algebraic information  Either about the algebraic closure or the separable closure in the imperfect case and then you ask a question about the Galois group and  Normally if you answer those two questions, you can amalgamate them and answer whatever question you might ask about the field  so  in particular this shows up in sort of the characterization of types that Trillian van der Dries Macintyre show in in their monster paper and  in order to do to  Give precise meaning to what it means to ask a model theoretic question about the Galois group. They introduced this inverse system  so of course the absolute Galois group of a of a  Field is a profite group and that's not really a good object. I mean a good model theoretic object  For example, if you take an elementary equivalent  Group there's no reason that that should have an interesting topology on it or be profite in a meaningful way  but you can think about the you can think duly about the system of finite quotients and  the maps between them and  encode that in some first order structure so that if you take an elementary  Equivalent structure then you can take the inverse limit of the associated system and get another profite group  So this is really a way of thinking fitting  Profite groups into a first order framework. So so here's the definition  So you start with a profite group and you look at the set of open normal subgroups  And so the domain of the structure will be all of the finite quotients by open normal subgroups  Now we're going to have a sort of strange multi-sorted  Presentation to this so we won't have disjoint sorts. This is what's a little weird about it  but we'll have a sort for every positive integer and a binary relation less than and  C which is I  Guess binary relation. So sorry, we have two binary relations less than and see an eternal relation p  So I don't know what these letters were chosen or why this is the language, but this is the traditional approach  so  s of g is in lg structure in the following way so the  Coset I mean the elements of the sort x in are exactly the cosets of  Open normal subgroups of index at most in  So notice that the sorts are growing right?  Because you get more and more cosets  Then you have a comparison  G of n is less than or equal to h of m if the  Groups I mean the subgroups are have containment like this  then  This c relation codes the projection  So if if in is a subset of m then whenever I take cosets g in I can project it and get a coset of gm and  The graph of that projection is coded by this relation c and then  on each finite quotient I have multiplication and the graph of multiplication is coded by this relation p  So if I have in one into in three and they're all the same  So these are all cosets of the same group the same open normal subgroup  Then the relation just holds if and only if I have the appropriate multiplication  Okay, now this is a pre-order this this partial order symbol so I can take equivalence classes and  This is going to be the notation I use for this equivalence class  The less than relation the less than or equal to relation is a pre-order  Well, so so what this tells you is that if I look at  G in and then I look at its image under the projection from  G mod in to g mod m  Yeah, I mean  Yeah, yeah, it's the same as contain into the coset but I mean I prefer to think about it in terms of the maps, but sure it's the same  Technically yeah, so okay if we were to do this in the normal presentation  You would have district sorts and you would have bijections that sort of tell you the  Identifications between the district ones and then you would have a different relation for each sort  So so even though you have a sort of globally defined partial order  It turns out that this can be like an omega. I mean for certain groups. It'll be Omega stable  Because we're cheating here. I mean we're really thinking about this is only being sort what sort by sort  But it's convenient to write it as a globally defined pre-order because we all know how multi-sorted logic works  You know we can translate if we like if this yeah, so here the sorts are not just joined  So but we sort of pretend like they are  It works. I promise  Okay, so this is a certain structure  And I want to mention two sort of notions that are important in talking about the structure  So these are duals and subsystems  So suppose we're given l k fields and an embedding of the separable closure of k into the separable closure of l  So that L is a regular extension of the image of Phi of k so  This is confusing but imagine for a moment that this is just an elementary embedding of  K into L and then you extend that to a map on the separable closures. So that that  That case won't will be the important one for us. So we have this embedding  From Ks into LS and then this induces a map from G of L to G of k  So if you if you take an automorphism  here  That is the same thing as an automorphism of the image of this by the isomorphism  so you look at the image of the  separable closure of k sort of in inside L and  then this  Tells you that that extends because of regularity. So this is an epimorphism this map here  And then anytime I take a quotient of this that's the same thing as an epimorphism from this to a finite group  so by composition any time I have a quotient of  G of k that gives me a quotient of G of L  So then I get an embedding from the inverse system of  G of s of G of k to s of G of L  So this is an embedding which is called the double dual to Phi to Phi so  Each time the direction of the arrow switches. So if I do it twice then I get an inclusion of the inverse systems  This one so the way you think about this map is suppose I have a finite quotient of G of k  So that's the same thing as an epimorphism from G of k to a finite group  But then by composition I get an epimorphism from G of L to a finite group  And so that tells me how to how to map a finite quotient here to a finite quotient here or more generally an open normal subgroup here to  Okay, does that make sense?  Okay now  Subsystem this would be the appropriate notion of substructure if the language was different  now a subsystem s is is  Should I mean you should think about this as a substructure  Up to an appropriate choice of the language  So what it means is that it's downwards directed and upwards closed in the in the pre-order in  Particularly it's closed on their equivalence because this says less than or equal to so you close under all the equivalence classes  And then you you take you make sure that you're closed you're directed downward  Oh, yes, sorry, I should have said that so to the image  Yeah, I mean I guess this is implied by being a regular extension, but but yeah, yes  That is my intention. Yeah, so it's a really the example that you should think about is just imagine that L is a regular extension of k  So I mean forget forget about k and think about phi of k and then this is the inclusion  Or this is rather the identity map. This is restriction and this is dual to the restriction that that's the I mean  You don't lose anything by thinking about that case  Okay, so let me tell you about this theorem which which I'm very enthusiastic about and it's a monster  So I'm gonna I'm gonna draw a picture and then everything will be clear. I promise so so  Here's here's a field f. I  should not wait yeah, okay, so f and  Here's s of  G of f and we suppose that we have an instance of the hypotheses of the of the independence theorem in the field  So so what does it mean to have the instance of the hypotheses of the independence there?  You have a substructure when in this case this elementary substructure e you have a and b which are independent over e  And you have c0 and c1 which have the same type over e and c0 is independent with a and c1 is independent with b  These are the hypotheses of the independence there and then by by taking double dual you you get an associated map. I  Mean an associated picture  In  The inverse system of the Galois group  So here's s of g of c0  s of g of c1 and  And suppose that not only do you have the hypotheses of the independence theorem here, but you also have a solution  So this gives you  S zero which amalgamates these two things  Yeah, so so all of this is assumptions so I start with the hypotheses of the independence theorem in in the field f  I look at the associated  instance of the independence theorem in the inverse system and I assume that I have a solution  Then then that means that there's a solution here  so that s0 you can think of as  sg of  C so so we are starting the existence of this so  The picture here. I mean the way it's formulated is a bit complicated, but this is really the idea  so you start with an instance of the  Hypothesis of the independence theorem in the field  This gives rise to a similar instance in the inverse system if you can solve it in the inverse system  Then you could solve it in the field  So this tells you that amalgamation problems in the field reduced completely to the question of amalgamation in the Galois group  Yeah, so so everything here the way it's written is they're all algebraically closed subsets so in particular the sub fields  So so so they have Galois groups  So so yeah, so so this is why it's a bit difficult to state  So you want to you want to make sure everything is algebraically closed so it makes sense to take the Galois group, but  But yeah, don't don't read this just look at just look at the picture  Everything will be clear. So I didn't want to lie to you, but  Yeah, this is what it says. Okay, so here's a cool corollary so  Zoe proved in the paper where she proves this paper that if you have a PAC field and the theory of the inverse system of that field is  In SOP in for in greater than or equal to three  You get that the field itself is in a sop in and then my thesis I sort of did the  Requisite work to extend that to one and two so in in the case of in greater than or equal to three  there's a sort of direct  Translation of SOP in into an amalgamation problem because you want to show that this type which is supposed to be inconsistent becomes consistent  Now for n equals one two for n equals one you use chem independence in the Galois group and translate that and characterize chem independence in the field  so a p a c field is in a sop in if and only if the Galois group is in a sop in and  Or if if the Galois group is in a sop in and then for two you use strongly indiscernible trees  So this is a somewhat different approach for the other two  But it it's a corollary of the theorem. I said before in a very direct way. So there's nothing deep about this extent  Okay, so this tells us that the question of whether or not we can build p a c fields  With prescribed model theoretic properties at least for the question of SOP in reduces to a question about the absolute Galois group  So here's a strategy. So we know that we reduced this to the absolute Galois group now  What we want to do is we want to build groups. We want to build profanate groups that code  graphs and then  Because we can find enough graphs  We can find enough groups and then realizing those is the absolute Galois group of p a c fields  We went we find p a c fields that are SOP in in a sop in plus one because this is this is the strategy  So what time do I need to finish by the way? Do you remember? I?  Mean I can talk for half an hour. Okay. Good. Good. So I'm on schedule. So I'll stop it 10 53  Okay, 10 10 to 12  Yeah, well now now we just lost two minutes. Yeah, okay  Okay, okay, so  All right, well, I don't know I'll probably just talk and then someone will tell me to show up and that's what I  Okay, so um, so the idea here is to make make use of a certain coding construction  Which was developed by trillion van der Dries and McIntyre in their unpublished long paper on  On  Psc fields now there their question was just about decidability  So they they were using this coding construction for a very coarse kind of problem  They want to show that you can code graphs and then you get some undecidable  Stuff in in a p a c field and then they were happy now  I want to just sort of reanalyze that construction with a more fine-grained approach and understand which sort of sort of find  structural model theoretic properties are preserved by by this construction  So the this construction is theirs  But but the analysis will be a bit more refined. So so here's the idea. So  This this slide will just set up notation. So we start by fixing  From now on to distinct primes odd primes P and Q now CP will be my notation for the the  Z mod PZ but thought of as a multiplicative group which would be important just for notation and  And okay DP is the dihedral group. So this has two P elements. It's generated by an alpha beta  so alpha is an evolution and  If you conjugate beta by alpha. Oh, sorry, that's a typo. They should say  Alpha beta alpha beta inverse so if you are alpha beta alpha inverse beta maybe okay anyway  But what you want to say is that alpha beta alpha inverse equals beta inverse. That's what it should say  So that should that's the same thing as the relation alpha beta inverse beta  Yeah, and there should be an inverse so so  Yeah, so there are a lot of things missing here. Yeah, so okay, so  So okay, so you want these relations so alpha squared  So so here are the relations alpha squared alpha beta alpha inverse beta and beta to the P and  That's it. Okay, so so a lot of things missing anyway, you know what the dihedral group is  so  Okay, okay  Okay, so the relations are written here anyway, so  You can you can?  Okay, so there's a Z mod PZ sitting inside generated by a  Generating by beta and the quotient by that gives you an epimorphism onto the cyclic group of order two plus or minus one  Right, so let's call that tau  Now using that you can define an action from the product dp times dp to I mean acting on CQ by  The following rule so if I take a pair of elements from dp cross dp xy now I look at whether their image in C2 is  plus or minus one and  I raise a to the power the product tau x times tau y so that means that if tau of x equals tau of y  Then I do nothing and if tau x is not equal to tau y then I invert  That's the rule  now because I have an action here I can define a semi direct product and  W is going to be the semi direct product with the action defined here  So that's W and of course this fits into a short exact sequence where W is going to be the semi direct product of dp  I'm dp by  Cura and just to give it a name lambda is going to be this map that  Takes me from W to dp times dp. I just forget about the CQ part  Okay  so now we want to build the graph so  first start with a graph on vertices a and with edge relation r and  Form the product  Where you take a copy of dp for every vertex and the copy of W for every relation r  for every edge r then of course you can take any elements of this  group should be called h as  D bar times W bar where this is a vector in dp to the a and W bar is a vector in W to the r  and  We have the associated projections, so I'm just setting up notation here  so pi of a is the projection under the coordinate a and  Pi r is the projection under the coordinate r and  Then we define a group g gamma as follows so I look at  the elements in this product I  Look at the elements of this product  Yeah, so W  Is here so it's the semi-direct product so so maybe I should just write this because I think it's worth remembering  So  W is going to be CQ semi-direct product  Dp times dp  Okay, so  Yeah, yeah, sure sure sure so so we have tau  Which goes from dp to C2 which we're thinking of is plus or minus one  No C2 and then and then dp times  dp acts on  CQ by  X Y  acts on alpha by  Alpha to the tau x tau y  So again you take x and y you look at their images under tau if those are the same then you do nothing if they're different than you invert  That's the rule  Sorry  Yeah, yeah, so I said that I had a typo so H is what I'm calling d to the  A times W to the R  Okay, but I won't use H ever again, so you don't need to worry. Okay, so G game is this group so  Here's here's the way it works so  So maybe one more piece of notation which is worth remembering is you have lambda which goes from  W to  Dp times  Dp which is the thing that just forgets about the CQ coordinate, right?  So so lambda here came from this oops it came from the sequence. So here's lambda which which I'll remember here  So it's the projection from W to Dp times Dp that forgets about the CQ coordinates  Okay, so so here's the definition of the group so I look at  The vectors d bar w bar so that if I if I have an edge in my  Graph which goes from a to b  then I look at  the  elements in the R coordinate and  I take its projection in Dp times Dp so W are it's it's something here  So it's in a copy of W so I can apply lambda I can forget about the CQ part  Then I want that to match the  Dp parts so if I have an edge from a and b and then I look at the  Like the coordinate that is associated with the vertex  Then that's  Then they're sorry if I look at the coordinate that they come from the edge  And I apply lambda then I get the coordinates that come from the vertices vertices they match  so  So this is the definition and then just as a trivial remark  As defined if the graph is empty then you get the empty set here  Yeah  Yeah, I mean  This is a homomorphism. So it's it's fairly trivial to just check that it's close to their multiplication  It's also profite because this condition is closed  So here you have a big product of finite groups, which is  Which is obviously profite because it's by T. Knobs theorem. It's compact separable  I mean compact totally disconnected topological group  and  Then this is close to it's a close-up group of profite groups. It's profiting. So this is a profite group and  This is just a sort of side comment if a is empty or sorry  I mean if the graph is empty then in principle this definition gives you the empty set which we don't want  So if it's an empty graph, then you just define the group to be the product of the vertices and this this coheres  Okay, so that's a that's a group now  Now I want to tell you an interpretation of graphs in profite groups  so  Given a profite group G which you know any one at all  I mean this might give you nonsense but but for any profite group at all of you can define a graph gamma G  AR so the vertices consist of open normal subgroups such that the quotient by them is DP  okay, and  You you say that there's an edge between in one and in two  If they're distinct and there's an open normal subgroup G so that the quotient  of G by M is W and  M is a subset of the intersection of these two open normal subgroups  so this is a fact that if you  Take the graph you build the group and then you interpret the graph again you get the graph back  So this doesn't do anything and there's a this is sort of nice picture of what this looks like in the inverse system  So the inverse system you think of the sort  Okay  Okay, so in the inverse system you have the sorts let's say one P  or sorry one two P and four P squared Q and  I'm gonna have several  Elements here or really equivalence classes which correspond to copies of DP so so these are like  in two  so these are elements of sort two P in the in the inverse system, so this is S of G  and I have an edge between these two things if I can find some  in here  So this picture tells me how to interpret this  Infer system so this interpretation although I've described it in this language is really an interpretation in the in the inverse system  Okay, so this tells us how to build a group  but the the real question is can I possibly have this group as an absolute Galois group of a  PAC field if not then it's not so helpful so in order to talk about that I'll have to tell you what a profite group is  Or say a projective group so a profite group G is called projective if it satisfies this following is this property  so I take an epimorphism from G to a and from B to a and  and  What I want is I want to be able to complete the diagram with a homomorphism  Yeah, it's the same thing as being a projective object in the category of profite groups  Oh  Sorry, sorry. No, I really meant to say that a and B are finite. I mean they could also be profite  But yeah, so yeah, sorry, so a and B are funny. Thank you  Yeah, so  It's known that you can't always complete this diagram with an epimorphism that would be the natural  The natural guests for this and notice that that would be dual to the extension property that you're familiar with for say for say classes  So when you have the stronger property that you can complete the diagram with an epimorphism that that's called super projectivity or sometimes the  Iwasawa property and Zoe showed in a thesis that the inverse system of a group with the stronger property is Omega stable  And so these are very very nice groups and the PAC fields with those as their  absolute Galois groups are  NSP one  Which which is a consequence of the theorem I mentioned earlier, but it was known much before this is something that you tie and I worked out  Okay, so  Oh, then I oh shoot. No, I just I just use the wrong arrow. So  Okay, many typos here. So a and B are finite and this  The text is correct. Although the diagram is wrong. So there's a continuous homomorphism  Here  Yeah  Okay, is that good cool, so  We need to know the answer to the following question. What are the absolute Galois groups of PAC fields?  so the  Is this sort of striking theorem of van der Dries and the Botsky which shows that the class of absolute Galois groups of PAC fields is  Exactly the class of projective  Prophetic groups so secretly here. There's there's one direction which was known already to ax which is that if you have a PC field  Then it's Galois group is projective that occurs already in the 68 paper  so the the thing that Lobotsky and van der Dries do is they go the other direction they show that if you have a  Projective profanity group you can build a PAC field that has that as its  Absolute Galois group. So what we want now is a construction not to just take a graph and build a  Profanity group, but we want a construction that takes a graph and builds a projective profanity group such that I can recover the graph from that group  so  How do we build  projective groups so if you have a profanity group, there's a certain subgroup called the Frittini subgroup which is  Denoted capital phi of H and this is generated by the maximal open proper subgroups of H  so  From the definition you can tell automatically that it's a characteristic subgroup. So it's normal and a Frittini cover is an epimorphism  Such that the kernel is that should say phi of G. Yeah, so the kernels contain in phi of G  Now every profanity group has a universal Frittini cover, which is the same thing as the smallest projective group that surjects onto H  So so this is defined. This is unique up to isomorphism and  Here's the second key fact  if you take the  graph  You build the group  Then you pass the universal Frittini cover and then you try to interpret the graph again using the interpretation  I described before you get the graph back  So actually this process of replacing G with its universal Frittini cover does not affect the graph that we interpret in the inverse system  Okay, so this is wonderful because what this means is that if we take this graph  We we build the universal Frittini cover then we can realize that at the absolute gaoual group of a PC field and then we get what we want  No, no, I clarified subgroup of G. Yeah, it's just a typo is subgroup of G  Of 5g yeah, well in particular the subgroup of G and it's also a subgroup of 5g that one actually has content  Okay, thank you  Okay, so now I want to talk about how the analysis of these  inverse systems of  Universal Frittini covers of G gammas go so we want to understand certain important quotients  Of this group so notice for example if I take in copies of DP  So I look at DP to the end there's a copy of Z mod PZ to the end sitting inside  This is generated by the betas in each copy so  Now if I take the quotient of that then I get a big copy of Z mod twos so I can quotient by this group and  I get Z mod twos each the end  Now sitting inside of Z mod twos each the end as as a normal subgroup. I have the diagonal subgroup  So these are the things which are either zeros and all coordinates or one and all coordinates  Wait is this what I want? Yeah, I guess so so what I want is the quotient to be Z mod 2z  Yeah, so so I'm not sure what I mean  Yeah, so there's a quotient of the Z mod 2z which which is okay, so fix whatever normal subgroup makes this true  You have a quotient of this diagonal which  Exactly, yeah  Yeah, okay, so I said nonsense here  Yeah, yeah, I'm not sure what I had in mind when I wrote this okay, so so anyway that  Let's ignore this for a slight moment  so  Here's one thing that is certainly true so  if  So let me let me rewrite the fact in the following way so let me let me write it on the board very quickly  So  if you have a  DP and a DP sitting at level 2p and you take a  4p squared q then there's a unique w sitting on top if there's one at all  Okay, so this follows from the fact that in the graph interpretation if  that the the the copies of DP you see are always associated with with  vertices so every every DP is really the the kernel of a coordinate projection to a DP and  And every W that you see is is really coming from the kernel of a pi r  So all of the W's and the DPs that you see in the inverse system come from this configuration  So that means that from from these DPs. There's a unique W that's sitting on top from the definition of the group now  I want to define a system a definition called the graph closure and  Here's the way it's defined so you define the graph closure a sub gr to be the smallest  Subsystem containing a so that if you take an element and you get a Z mod 2z  Then you include all of the all of the vertices that you that you see so  So  Here at level 2 you might have a DP above  Or it might be finitely many DPs above  So so if you if you have a Z mod 2z  Which is a quotient of a DP then you put the DP in that's what this says so if if I see a  element of this structure and it gives me a Z mod 2z then I put in  the the the kernel of the vertices that are associated to the C meaning that that  Surject onto the Z mod 2z non-trivial now  Additionally if I have two things here which look like vertices so which give me  Structures isomorphic to DP so if I see two DPs then if there's an edge between them in  The subsystem then I put that in the closure  so the idea here is  The I take the small substructure that sees every piece of the graph  so if if I take a Z mod 2z and I I see vertices above it  Finally many then I put those inside the graph so they're all algebraic over the Z mod 2z  And if I see an edge between the vertices that I put in then I make sure that I put those edges inside this structure  So the the whole point of this definition is that the interpretation of the graph  Applied to this substructure is an induced subgraph of the gamma that I've defined  Okay, so forget about this nonsense that was written before this is really the definition so this is this is just a definition so it's  It's a way of getting an induced subgraph from a certain set  Which one  Yes, sorry so in the in the subsystem I have this partial order right and  then you have an equivalence relation and on the on the equivalence class of a particular element  Because I have this this relation P which is like the graph of multiplication here  This is isomorphic to a finite group  So so each element in the inverse system if I take its equivalence class that is a finite group and I can ask  What group is it isomorphic to okay? So so here when I say see is isomorphic to Z mod 2z  I mean I take an element I look at the  Isomorphism type of the finite quotient associated to this class  Which is which is given to me by P in the language of the inverse system and if it's Z mod 2z then I look at which DPS are above  Yes, okay, and then if there's an edge I make sure that I put the edge in  Okay, so here's here's the theorem which sort of tells us how to analyze types in the  inverse system so suppose I take two finite tuples from the inverse system and  I look at this the graph closed  Substructors that they generate so this is a subsystem which is graph closed so it means that the the interpretation of the graph here  it's going to be an element you know an induced sub graph of gamma and the  The thing here will also be an induced sub graph of gamma  And I look at the induced map  Which goes from gamma a to gamma b  So I first take these substructures  I form the interpretation of the graphs and I look at the induced map between those graphs and I ask the question  Is that partial elementary?  So if that's partial elementary in the theory of the graph  Then actually the map that sends a to b is partial elementary in the theory of the inverse system  So this tells you exactly how to analyze types in the inverse system  all you have to do is understand types in the graph and  the quantifier free type in the in the inverse system and  Using that you can  Well because the quantifier free type is encoded by the isomorphism type of the substructure generated by the tuple  Right it's sort of like in for say limits right like two things have the same type if there's an isomorphism of the structures that they generate  Okay, but okay, so quantifier free type is not strictly speaking true because the language is not correct  right, I mean like the the the  Substructures don't correspond to subsystems, but but you think about it really being secretly quantifier free  I mean you can you can you can formulate an appropriate language in which it's quantifier free and it's not too bad  But okay, really I mean the isomorphism type of the substructure which which we think of as model theorists is coding the quantifier free type  But up to an appropriate choice of language  Okay, and then using using this analysis of types you can prove that for all positive integers and here  I was careful to write it that the theory of the  Inverse system of the universal for Chini copper of the group that you build from the graph is  Inesopian if and only if the theory of the graph is  So this is enough to tell you I mean not really the theorem  But really the proof of the theorem tells you how to analyze types and you can you can code the relevant amalgamation problems in terms  The graph and then solves it  Okay, so now let's just put it all together and  Yeah, this is right on time. So for ingrid and they're equal to three there are graphs. There, you know, they're known examples of graphs  These loop-free graphs of a certain kind which are SOP in and in a sopian plus one  So you just build the the right graph now  There's a general fact which is which is easy to show although. I don't know if it's written anywhere  Well, I mean I wrote it down, but I don't know if it had been written before anywhere else  Is that if SOP 1 and SOP 2 or so be two or not equal to so be three  this will be witnessed by a graph gamma and you know to prove this it's very simple you just open up your copy of  Sue gows and variant descriptive site 3 you look at the proof that graph isomorphisms broth pleat and what you do is you just code  arbitrary structures into graphs you it's very simple to show that  SOP n for example is preserved under this operation because essentially definability is not changed  What that's the idea yeah, so you  Yeah, yeah, but okay. You there's a small check. I mean I I was clear that it's trivial. I mean it's not a deep fact  Okay, so then you build g gamma from that graph and then you pass the universal for a teeny cover and we know that that operation preserves the graph  Then you use banded Dries-Libotsky the inverse gawa problem for PC fields to realize  G tell the gamma as the absolute gawa group of a PC field and then  Gamma is interpretable in the field. So this is something which  Well, this is the universal for teeny cover of the group g gamma then gamma is interpretable in the field  so this is a somewhat delicate point in the sense that sg of gamma is not actually interpretable in the field  Gamma because it's the the the gawa groups are really only  interpreted up to conjugacy  So sg of gamma is is is interpretable in the pair the field together with its algebraic closure the field together with separable closure  But not necessarily in the field itself  However, the graph is interpretable because you can write polynomials that say I'm separable and my gawa group is DP and I'm separable  And my gawa group is W and using that you can code the graph even if you can't get your hands on the gawa group itself and  So this gamma is interpretable in the field and so f is SOP in  But then also it's in a sop in by this theorem of Zoe  Which shows you that if the if the group itself is in a sop and then that passes to the field  Oh shoot, yeah, in this op in plus one. Yeah, this is a remarkable theorem, right?  So  Which which one Zoe's there?  Here I  Mean so for this one this this is me and then this one is Zoe  So so so look so there's there's two steps, right?  So one is that you have a graph then you get a group and then you have the group and you get a field and then  You want to prove that in a sop in is preserved at each step of the way  So I start with a graph and I build a group I show that if you start with an in a sop in graph  Then the in the in a sop then you get an in a sop in group  So it shows that if you start with an in a sop in group and you build an in a field you get an in a sop in  field  Yeah, yeah, so so that so so  Yeah, so so this theorem is a corollary of this there  Because you code sop in as a as an amalgamation problem  They use it for all three  So so okay, so you have to use strongly indiscernible trees or came independence to do the one one in two case  It's not such a direct translation, but but you use but you translate everything into an amalgamation problem  And you cite this theorem  Okay, so Enrique standing up so I should stop thank you. I
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Nice to be back and thank you to the organizers for organizing it. I also want to make a comment before I get started so Most of this work sort of began During my fourth year of graduate school, so I spent a week visiting Zoe at ENS and we spent many hours in her office each day And so Even though you know, sir, I'm risk I'm writing this many of the ideas came from Zoe and I just wanted to sort of acknowledge that So the plan is to talk about classification theory and the construction of PAC fields so most of the time in model theory we find fields, but this talk is going to be about building them and Here's the map that you know Gabe's gift to the model theory community and The question that we often ask in model theory is sort of which parts of this map have a field in them and If it has a field in it, what kind of field can it be? And of course many of the hardest conjectures in model theory about exactly this question So the staple fields conjecture the super simple fields conjecture stuff like that and I'm interested in building sort of counter examples and Finding new kinds of fields. So here is the SOP in hierarchy just one part of the map Maybe maybe not the most interesting part that it's there So we say T has SOP in if there's some type of PXY or you can make it a formula If you like an indiscernible sequence which is ordered by this type So P a I a J holds if and only if I is less than J and If you take a sort of cyclic union so PX0 through X1 X2 or X1 X2 X2 X3 all the way up to Xn-2 Xn-1 and then you loop back so Xn-1 X0 that's inconsistent Okay, so the idea is you think about this is an instance of the order property and you want to say how close does this get to being An instance of the strict order property and so as in gets higher you're getting closer and closer to transitivity Because transitivity says exactly that you can have no loop of any size. So this says that you can't have small loops Well, it doesn't matter if it's in a if it's in a complete type then by compactness it reduces to a formula So so sometimes it's useful to have this formulation when you want to work with complete types and other times when you want to do compactness arguments It's useful to have a formula so there it goes Yeah, I should have said that in bigger than two and at least three so for one and two it's defined differently Which is going to be something I mentioned many times in the talk. Yeah So I should have said that so so they're They're gonna be many theorems in the well not many theorems There are gonna be a few theorems in the talk that say something for SOP in and when I say that without qualification I really mean it for one two and in greater than or equal to three But of course the argument has to be different for one for two and then for in greater than or equal to three Okay, so this is the in greater than or equal to three case. I'm not gonna define One because young hand did that and I'm not gonna define two because we don't care so Here's the theorem so I'm gonna say the theorem first then I'm gonna spend a lot of time sort of talking about the ingredients So the theorem is just that the SOP in hierarchy is strict in fields So let me say what I mean precisely by that so for all in greater than or equal to three There is a field which is SOP in in SOP in plus one of and in particular It's a PAC field so the SOP in hierarchy is not only strict in fields It's strict in PAC fields or if you like PAC fields of characteristics zero Moreover, there's an open problem. We don't know whether SOP one and SOP two are different whether SOP two and SOP three are different But if they're different that will be witnessed by a field. Okay, so So, you know, maybe maybe someone can prove something by coding them in fields. I'm very doubtful, but okay This is the theorem Now the theorem is nice, but in truth it's much less interesting. I think than the construction So the the whole the whole goal of the project is that we have constructions for coding combinatorial objects and distinct groups You have something like the Meckler construction which takes an arbitrary graph and produces for you a group which is nilpotent step two and Many of the model theoretic properties of the graph are reflected in the group now What I'm interested in is sort of a analogous construction for fields I take a graph I build a field out of it and I want to know what are the model theoretic properties that are preserved by passing from the graph to the field and okay the SOP and hierarchy is just a test case But there there are I suspect many other model theoretic properties for which this construction will be germane Pure fields in the language of rings. Yes Yeah, it's not so hard to find an expansion of a field with this property. Just put a graph on top Yeah, so this is in the language of rings. Should have said that. Okie doke. So Why do we care about PC fields? So of course the starting point is axis 68 paper the elementary theory of finite fields and He characterizes the pseudo finite fields which are defined logically in Algebraic vocabulary. So what he shows is that a Field is an infinite model of the theory of finite fields if it satisfies these three conditions. The first is is that it's perfect Which just says that if it's characteristic p then f to the p is equal to p Sorry is equal to f It has Galois group Z hat And then the last one is this algebraic condition, which is the last missing ingredient Which says that f is pseudo algebraically closed So this was already mentioned by Samaria in her talk But f is pseudo algebraically closed if whenever you take an absolutely reducible variety to find over the field It has a point in the field. Okay, so so Algebraically closed fields are pseudo algebraically closed by the Nostalon thoughts, but you have lots more. So for example ultra products of finite fields Okay, so It became interesting within model theory to just ask how complicated can pseudo finite fields be so there was a question from the you know the famous paper of Chatsodakis van der Dries in Macintay where they characterize definable sets in finite fields about whether or not triangle-free random graphs are interpretable in finite fields and Roshofsky answered this question by showing that the independence theorem is satisfied in in in pseudo finite fields or more generally bounded PAC fields and Using the independence theorem you can show that you have non interpretation of triangle-free random graphs So this is sort of the But Roshofsky works in more generality So recall that a field K is called bounded if K has finitely many degree in extensions for every N and Working in in a general context where he studies definably closed sets in strongly minimal theories satisfying weak elimination of imaginaries and the definable multiplicity property together if that definable Closed set has a certain boundedness property then then the induced structure is super simple. So the Upshot is that if you think about a bounded perfect PAC field you can think about it as a definably close subset of its algebraic closure and That's strongly minimal and it satisfies all the requisite properties. So in particular. He shows that a bounded PAC field is super simple. Oh the perfect assumption was later dropped in this paper of Chatsadakis Pillay on Generic structures for simple theories. So in an appendix to that paper They show that actually if you drop perfect, of course, you won't have super simple But you can still get simple so a bounded PAC field is simple and later Chatsadakis showed this is the Characters this is a characterization. So an unbounded PAC field has TP2 So a PAC field is simple if and only if it's bounded. So we know exactly Which PAC fields are bounded and that's a consequence of a certain Galois theoretic condition? And so there's a sort of emerging theme Which which is already very very clear in the work of Trillian van der Dries Macintyre Which really sort of opened this up opened up the model theory of PAC fields, which is that it you can think about a PAC field Any question about a piece of PC field typically breaks up into two questions One is about classical algebraic information Either about the algebraic closure or the separable closure in the imperfect case and then you ask a question about the Galois group and Normally if you answer those two questions, you can amalgamate them and answer whatever question you might ask about the field so in particular this shows up in sort of the characterization of types that Trillian van der Dries Macintyre show in in their monster paper and in order to do to Give precise meaning to what it means to ask a model theoretic question about the Galois group. They introduced this inverse system so of course the absolute Galois group of a of a Field is a profite group and that's not really a good object. I mean a good model theoretic object For example, if you take an elementary equivalent Group there's no reason that that should have an interesting topology on it or be profite in a meaningful way but you can think about the you can think duly about the system of finite quotients and the maps between them and encode that in some first order structure so that if you take an elementary Equivalent structure then you can take the inverse limit of the associated system and get another profite group So this is really a way of thinking fitting Profite groups into a first order framework. So so here's the definition So you start with a profite group and you look at the set of open normal subgroups And so the domain of the structure will be all of the finite quotients by open normal subgroups Now we're going to have a sort of strange multi-sorted Presentation to this so we won't have disjoint sorts. This is what's a little weird about it but we'll have a sort for every positive integer and a binary relation less than and C which is I Guess binary relation. So sorry, we have two binary relations less than and see an eternal relation p So I don't know what these letters were chosen or why this is the language, but this is the traditional approach so s of g is in lg structure in the following way so the Coset I mean the elements of the sort x in are exactly the cosets of Open normal subgroups of index at most in So notice that the sorts are growing right? Because you get more and more cosets Then you have a comparison G of n is less than or equal to h of m if the Groups I mean the subgroups are have containment like this then This c relation codes the projection So if if in is a subset of m then whenever I take cosets g in I can project it and get a coset of gm and The graph of that projection is coded by this relation c and then on each finite quotient I have multiplication and the graph of multiplication is coded by this relation p So if I have in one into in three and they're all the same So these are all cosets of the same group the same open normal subgroup Then the relation just holds if and only if I have the appropriate multiplication Okay, now this is a pre-order this this partial order symbol so I can take equivalence classes and This is going to be the notation I use for this equivalence class The less than relation the less than or equal to relation is a pre-order Well, so so what this tells you is that if I look at G in and then I look at its image under the projection from G mod in to g mod m Yeah, I mean Yeah, yeah, it's the same as contain into the coset but I mean I prefer to think about it in terms of the maps, but sure it's the same Technically yeah, so okay if we were to do this in the normal presentation You would have district sorts and you would have bijections that sort of tell you the Identifications between the district ones and then you would have a different relation for each sort So so even though you have a sort of globally defined partial order It turns out that this can be like an omega. I mean for certain groups. It'll be Omega stable Because we're cheating here. I mean we're really thinking about this is only being sort what sort by sort But it's convenient to write it as a globally defined pre-order because we all know how multi-sorted logic works You know we can translate if we like if this yeah, so here the sorts are not just joined So but we sort of pretend like they are It works. I promise Okay, so this is a certain structure And I want to mention two sort of notions that are important in talking about the structure So these are duals and subsystems So suppose we're given l k fields and an embedding of the separable closure of k into the separable closure of l So that L is a regular extension of the image of Phi of k so This is confusing but imagine for a moment that this is just an elementary embedding of K into L and then you extend that to a map on the separable closures. So that that That case won't will be the important one for us. So we have this embedding From Ks into LS and then this induces a map from G of L to G of k So if you if you take an automorphism here That is the same thing as an automorphism of the image of this by the isomorphism so you look at the image of the separable closure of k sort of in inside L and then this Tells you that that extends because of regularity. So this is an epimorphism this map here And then anytime I take a quotient of this that's the same thing as an epimorphism from this to a finite group so by composition any time I have a quotient of G of k that gives me a quotient of G of L So then I get an embedding from the inverse system of G of s of G of k to s of G of L So this is an embedding which is called the double dual to Phi to Phi so Each time the direction of the arrow switches. So if I do it twice then I get an inclusion of the inverse systems This one so the way you think about this map is suppose I have a finite quotient of G of k So that's the same thing as an epimorphism from G of k to a finite group But then by composition I get an epimorphism from G of L to a finite group And so that tells me how to how to map a finite quotient here to a finite quotient here or more generally an open normal subgroup here to Okay, does that make sense? Okay now Subsystem this would be the appropriate notion of substructure if the language was different now a subsystem s is is Should I mean you should think about this as a substructure Up to an appropriate choice of the language So what it means is that it's downwards directed and upwards closed in the in the pre-order in Particularly it's closed on their equivalence because this says less than or equal to so you close under all the equivalence classes And then you you take you make sure that you're closed you're directed downward Oh, yes, sorry, I should have said that so to the image Yeah, I mean I guess this is implied by being a regular extension, but but yeah, yes That is my intention. Yeah, so it's a really the example that you should think about is just imagine that L is a regular extension of k So I mean forget forget about k and think about phi of k and then this is the inclusion Or this is rather the identity map. This is restriction and this is dual to the restriction that that's the I mean You don't lose anything by thinking about that case Okay, so let me tell you about this theorem which which I'm very enthusiastic about and it's a monster So I'm gonna I'm gonna draw a picture and then everything will be clear. I promise so so Here's here's a field f. I should not wait yeah, okay, so f and Here's s of G of f and we suppose that we have an instance of the hypotheses of the of the independence theorem in the field So so what does it mean to have the instance of the hypotheses of the independence there? You have a substructure when in this case this elementary substructure e you have a and b which are independent over e And you have c0 and c1 which have the same type over e and c0 is independent with a and c1 is independent with b These are the hypotheses of the independence there and then by by taking double dual you you get an associated map. I Mean an associated picture In The inverse system of the Galois group So here's s of g of c0 s of g of c1 and And suppose that not only do you have the hypotheses of the independence theorem here, but you also have a solution So this gives you S zero which amalgamates these two things Yeah, so so all of this is assumptions so I start with the hypotheses of the independence theorem in in the field f I look at the associated instance of the independence theorem in the inverse system and I assume that I have a solution Then then that means that there's a solution here so that s0 you can think of as sg of C so so we are starting the existence of this so The picture here. I mean the way it's formulated is a bit complicated, but this is really the idea so you start with an instance of the Hypothesis of the independence theorem in the field This gives rise to a similar instance in the inverse system if you can solve it in the inverse system Then you could solve it in the field So this tells you that amalgamation problems in the field reduced completely to the question of amalgamation in the Galois group Yeah, so so everything here the way it's written is they're all algebraically closed subsets so in particular the sub fields So so so they have Galois groups So so yeah, so so this is why it's a bit difficult to state So you want to you want to make sure everything is algebraically closed so it makes sense to take the Galois group, but But yeah, don't don't read this just look at just look at the picture Everything will be clear. So I didn't want to lie to you, but Yeah, this is what it says. Okay, so here's a cool corollary so Zoe proved in the paper where she proves this paper that if you have a PAC field and the theory of the inverse system of that field is In SOP in for in greater than or equal to three You get that the field itself is in a sop in and then my thesis I sort of did the Requisite work to extend that to one and two so in in the case of in greater than or equal to three there's a sort of direct Translation of SOP in into an amalgamation problem because you want to show that this type which is supposed to be inconsistent becomes consistent Now for n equals one two for n equals one you use chem independence in the Galois group and translate that and characterize chem independence in the field so a p a c field is in a sop in if and only if the Galois group is in a sop in and Or if if the Galois group is in a sop in and then for two you use strongly indiscernible trees So this is a somewhat different approach for the other two But it it's a corollary of the theorem. I said before in a very direct way. So there's nothing deep about this extent Okay, so this tells us that the question of whether or not we can build p a c fields With prescribed model theoretic properties at least for the question of SOP in reduces to a question about the absolute Galois group So here's a strategy. So we know that we reduced this to the absolute Galois group now What we want to do is we want to build groups. We want to build profanate groups that code graphs and then Because we can find enough graphs We can find enough groups and then realizing those is the absolute Galois group of p a c fields We went we find p a c fields that are SOP in in a sop in plus one because this is this is the strategy So what time do I need to finish by the way? Do you remember? I? Mean I can talk for half an hour. Okay. Good. Good. So I'm on schedule. So I'll stop it 10 53 Okay, 10 10 to 12 Yeah, well now now we just lost two minutes. Yeah, okay Okay, okay, so All right, well, I don't know I'll probably just talk and then someone will tell me to show up and that's what I Okay, so um, so the idea here is to make make use of a certain coding construction Which was developed by trillion van der Dries and McIntyre in their unpublished long paper on On Psc fields now there their question was just about decidability So they they were using this coding construction for a very coarse kind of problem They want to show that you can code graphs and then you get some undecidable Stuff in in a p a c field and then they were happy now I want to just sort of reanalyze that construction with a more fine-grained approach and understand which sort of sort of find structural model theoretic properties are preserved by by this construction So the this construction is theirs But but the analysis will be a bit more refined. So so here's the idea. So This this slide will just set up notation. So we start by fixing From now on to distinct primes odd primes P and Q now CP will be my notation for the the Z mod PZ but thought of as a multiplicative group which would be important just for notation and And okay DP is the dihedral group. So this has two P elements. It's generated by an alpha beta so alpha is an evolution and If you conjugate beta by alpha. Oh, sorry, that's a typo. They should say Alpha beta alpha beta inverse so if you are alpha beta alpha inverse beta maybe okay anyway But what you want to say is that alpha beta alpha inverse equals beta inverse. That's what it should say So that should that's the same thing as the relation alpha beta inverse beta Yeah, and there should be an inverse so so Yeah, so there are a lot of things missing here. Yeah, so okay, so So okay, so you want these relations so alpha squared So so here are the relations alpha squared alpha beta alpha inverse beta and beta to the P and That's it. Okay, so so a lot of things missing anyway, you know what the dihedral group is so Okay, okay Okay, so the relations are written here anyway, so You can you can? Okay, so there's a Z mod PZ sitting inside generated by a Generating by beta and the quotient by that gives you an epimorphism onto the cyclic group of order two plus or minus one Right, so let's call that tau Now using that you can define an action from the product dp times dp to I mean acting on CQ by The following rule so if I take a pair of elements from dp cross dp xy now I look at whether their image in C2 is plus or minus one and I raise a to the power the product tau x times tau y so that means that if tau of x equals tau of y Then I do nothing and if tau x is not equal to tau y then I invert That's the rule now because I have an action here I can define a semi direct product and W is going to be the semi direct product with the action defined here So that's W and of course this fits into a short exact sequence where W is going to be the semi direct product of dp I'm dp by Cura and just to give it a name lambda is going to be this map that Takes me from W to dp times dp. I just forget about the CQ part Okay so now we want to build the graph so first start with a graph on vertices a and with edge relation r and Form the product Where you take a copy of dp for every vertex and the copy of W for every relation r for every edge r then of course you can take any elements of this group should be called h as D bar times W bar where this is a vector in dp to the a and W bar is a vector in W to the r and We have the associated projections, so I'm just setting up notation here so pi of a is the projection under the coordinate a and Pi r is the projection under the coordinate r and Then we define a group g gamma as follows so I look at the elements in this product I Look at the elements of this product Yeah, so W Is here so it's the semi-direct product so so maybe I should just write this because I think it's worth remembering So W is going to be CQ semi-direct product Dp times dp Okay, so Yeah, yeah, sure sure sure so so we have tau Which goes from dp to C2 which we're thinking of is plus or minus one No C2 and then and then dp times dp acts on CQ by X Y acts on alpha by Alpha to the tau x tau y So again you take x and y you look at their images under tau if those are the same then you do nothing if they're different than you invert That's the rule Sorry Yeah, yeah, so I said that I had a typo so H is what I'm calling d to the A times W to the R Okay, but I won't use H ever again, so you don't need to worry. Okay, so G game is this group so Here's here's the way it works so So maybe one more piece of notation which is worth remembering is you have lambda which goes from W to Dp times Dp which is the thing that just forgets about the CQ coordinate, right? So so lambda here came from this oops it came from the sequence. So here's lambda which which I'll remember here So it's the projection from W to Dp times Dp that forgets about the CQ coordinates Okay, so so here's the definition of the group so I look at The vectors d bar w bar so that if I if I have an edge in my Graph which goes from a to b then I look at the elements in the R coordinate and I take its projection in Dp times Dp so W are it's it's something here So it's in a copy of W so I can apply lambda I can forget about the CQ part Then I want that to match the Dp parts so if I have an edge from a and b and then I look at the Like the coordinate that is associated with the vertex Then that's Then they're sorry if I look at the coordinate that they come from the edge And I apply lambda then I get the coordinates that come from the vertices vertices they match so So this is the definition and then just as a trivial remark As defined if the graph is empty then you get the empty set here Yeah Yeah, I mean This is a homomorphism. So it's it's fairly trivial to just check that it's close to their multiplication It's also profite because this condition is closed So here you have a big product of finite groups, which is Which is obviously profite because it's by T. Knobs theorem. It's compact separable I mean compact totally disconnected topological group and Then this is close to it's a close-up group of profite groups. It's profiting. So this is a profite group and This is just a sort of side comment if a is empty or sorry I mean if the graph is empty then in principle this definition gives you the empty set which we don't want So if it's an empty graph, then you just define the group to be the product of the vertices and this this coheres Okay, so that's a that's a group now Now I want to tell you an interpretation of graphs in profite groups so Given a profite group G which you know any one at all I mean this might give you nonsense but but for any profite group at all of you can define a graph gamma G AR so the vertices consist of open normal subgroups such that the quotient by them is DP okay, and You you say that there's an edge between in one and in two If they're distinct and there's an open normal subgroup G so that the quotient of G by M is W and M is a subset of the intersection of these two open normal subgroups so this is a fact that if you Take the graph you build the group and then you interpret the graph again you get the graph back So this doesn't do anything and there's a this is sort of nice picture of what this looks like in the inverse system So the inverse system you think of the sort Okay Okay, so in the inverse system you have the sorts let's say one P or sorry one two P and four P squared Q and I'm gonna have several Elements here or really equivalence classes which correspond to copies of DP so so these are like in two so these are elements of sort two P in the in the inverse system, so this is S of G and I have an edge between these two things if I can find some in here So this picture tells me how to interpret this Infer system so this interpretation although I've described it in this language is really an interpretation in the in the inverse system Okay, so this tells us how to build a group but the the real question is can I possibly have this group as an absolute Galois group of a PAC field if not then it's not so helpful so in order to talk about that I'll have to tell you what a profite group is Or say a projective group so a profite group G is called projective if it satisfies this following is this property so I take an epimorphism from G to a and from B to a and and What I want is I want to be able to complete the diagram with a homomorphism Yeah, it's the same thing as being a projective object in the category of profite groups Oh Sorry, sorry. No, I really meant to say that a and B are finite. I mean they could also be profite But yeah, so yeah, sorry, so a and B are funny. Thank you Yeah, so It's known that you can't always complete this diagram with an epimorphism that would be the natural The natural guests for this and notice that that would be dual to the extension property that you're familiar with for say for say classes So when you have the stronger property that you can complete the diagram with an epimorphism that that's called super projectivity or sometimes the Iwasawa property and Zoe showed in a thesis that the inverse system of a group with the stronger property is Omega stable And so these are very very nice groups and the PAC fields with those as their absolute Galois groups are NSP one Which which is a consequence of the theorem I mentioned earlier, but it was known much before this is something that you tie and I worked out Okay, so Oh, then I oh shoot. No, I just I just use the wrong arrow. So Okay, many typos here. So a and B are finite and this The text is correct. Although the diagram is wrong. So there's a continuous homomorphism Here Yeah Okay, is that good cool, so We need to know the answer to the following question. What are the absolute Galois groups of PAC fields? so the Is this sort of striking theorem of van der Dries and the Botsky which shows that the class of absolute Galois groups of PAC fields is Exactly the class of projective Prophetic groups so secretly here. There's there's one direction which was known already to ax which is that if you have a PC field Then it's Galois group is projective that occurs already in the 68 paper so the the thing that Lobotsky and van der Dries do is they go the other direction they show that if you have a Projective profanity group you can build a PAC field that has that as its Absolute Galois group. So what we want now is a construction not to just take a graph and build a Profanity group, but we want a construction that takes a graph and builds a projective profanity group such that I can recover the graph from that group so How do we build projective groups so if you have a profanity group, there's a certain subgroup called the Frittini subgroup which is Denoted capital phi of H and this is generated by the maximal open proper subgroups of H so From the definition you can tell automatically that it's a characteristic subgroup. So it's normal and a Frittini cover is an epimorphism Such that the kernel is that should say phi of G. Yeah, so the kernels contain in phi of G Now every profanity group has a universal Frittini cover, which is the same thing as the smallest projective group that surjects onto H So so this is defined. This is unique up to isomorphism and Here's the second key fact if you take the graph You build the group Then you pass the universal Frittini cover and then you try to interpret the graph again using the interpretation I described before you get the graph back So actually this process of replacing G with its universal Frittini cover does not affect the graph that we interpret in the inverse system Okay, so this is wonderful because what this means is that if we take this graph We we build the universal Frittini cover then we can realize that at the absolute gaoual group of a PC field and then we get what we want No, no, I clarified subgroup of G. Yeah, it's just a typo is subgroup of G Of 5g yeah, well in particular the subgroup of G and it's also a subgroup of 5g that one actually has content Okay, thank you Okay, so now I want to talk about how the analysis of these inverse systems of Universal Frittini covers of G gammas go so we want to understand certain important quotients Of this group so notice for example if I take in copies of DP So I look at DP to the end there's a copy of Z mod PZ to the end sitting inside This is generated by the betas in each copy so Now if I take the quotient of that then I get a big copy of Z mod twos so I can quotient by this group and I get Z mod twos each the end Now sitting inside of Z mod twos each the end as as a normal subgroup. I have the diagonal subgroup So these are the things which are either zeros and all coordinates or one and all coordinates Wait is this what I want? Yeah, I guess so so what I want is the quotient to be Z mod 2z Yeah, so so I'm not sure what I mean Yeah, so there's a quotient of the Z mod 2z which which is okay, so fix whatever normal subgroup makes this true You have a quotient of this diagonal which Exactly, yeah Yeah, okay, so I said nonsense here Yeah, yeah, I'm not sure what I had in mind when I wrote this okay, so so anyway that Let's ignore this for a slight moment so Here's one thing that is certainly true so if So let me let me rewrite the fact in the following way so let me let me write it on the board very quickly So if you have a DP and a DP sitting at level 2p and you take a 4p squared q then there's a unique w sitting on top if there's one at all Okay, so this follows from the fact that in the graph interpretation if that the the the copies of DP you see are always associated with with vertices so every every DP is really the the kernel of a coordinate projection to a DP and And every W that you see is is really coming from the kernel of a pi r So all of the W's and the DPs that you see in the inverse system come from this configuration So that means that from from these DPs. There's a unique W that's sitting on top from the definition of the group now I want to define a system a definition called the graph closure and Here's the way it's defined so you define the graph closure a sub gr to be the smallest Subsystem containing a so that if you take an element and you get a Z mod 2z Then you include all of the all of the vertices that you that you see so So Here at level 2 you might have a DP above Or it might be finitely many DPs above So so if you if you have a Z mod 2z Which is a quotient of a DP then you put the DP in that's what this says so if if I see a element of this structure and it gives me a Z mod 2z then I put in the the the kernel of the vertices that are associated to the C meaning that that Surject onto the Z mod 2z non-trivial now Additionally if I have two things here which look like vertices so which give me Structures isomorphic to DP so if I see two DPs then if there's an edge between them in The subsystem then I put that in the closure so the idea here is The I take the small substructure that sees every piece of the graph so if if I take a Z mod 2z and I I see vertices above it Finally many then I put those inside the graph so they're all algebraic over the Z mod 2z And if I see an edge between the vertices that I put in then I make sure that I put those edges inside this structure So the the whole point of this definition is that the interpretation of the graph Applied to this substructure is an induced subgraph of the gamma that I've defined Okay, so forget about this nonsense that was written before this is really the definition so this is this is just a definition so it's It's a way of getting an induced subgraph from a certain set Which one Yes, sorry so in the in the subsystem I have this partial order right and then you have an equivalence relation and on the on the equivalence class of a particular element Because I have this this relation P which is like the graph of multiplication here This is isomorphic to a finite group So so each element in the inverse system if I take its equivalence class that is a finite group and I can ask What group is it isomorphic to okay? So so here when I say see is isomorphic to Z mod 2z I mean I take an element I look at the Isomorphism type of the finite quotient associated to this class Which is which is given to me by P in the language of the inverse system and if it's Z mod 2z then I look at which DPS are above Yes, okay, and then if there's an edge I make sure that I put the edge in Okay, so here's here's the theorem which sort of tells us how to analyze types in the inverse system so suppose I take two finite tuples from the inverse system and I look at this the graph closed Substructors that they generate so this is a subsystem which is graph closed so it means that the the interpretation of the graph here it's going to be an element you know an induced sub graph of gamma and the The thing here will also be an induced sub graph of gamma And I look at the induced map Which goes from gamma a to gamma b So I first take these substructures I form the interpretation of the graphs and I look at the induced map between those graphs and I ask the question Is that partial elementary? So if that's partial elementary in the theory of the graph Then actually the map that sends a to b is partial elementary in the theory of the inverse system So this tells you exactly how to analyze types in the inverse system all you have to do is understand types in the graph and the quantifier free type in the in the inverse system and Using that you can Well because the quantifier free type is encoded by the isomorphism type of the substructure generated by the tuple Right it's sort of like in for say limits right like two things have the same type if there's an isomorphism of the structures that they generate Okay, but okay, so quantifier free type is not strictly speaking true because the language is not correct right, I mean like the the the Substructures don't correspond to subsystems, but but you think about it really being secretly quantifier free I mean you can you can you can formulate an appropriate language in which it's quantifier free and it's not too bad But okay, really I mean the isomorphism type of the substructure which which we think of as model theorists is coding the quantifier free type But up to an appropriate choice of language Okay, and then using using this analysis of types you can prove that for all positive integers and here I was careful to write it that the theory of the Inverse system of the universal for Chini copper of the group that you build from the graph is Inesopian if and only if the theory of the graph is So this is enough to tell you I mean not really the theorem But really the proof of the theorem tells you how to analyze types and you can you can code the relevant amalgamation problems in terms The graph and then solves it Okay, so now let's just put it all together and Yeah, this is right on time. So for ingrid and they're equal to three there are graphs. There, you know, they're known examples of graphs These loop-free graphs of a certain kind which are SOP in and in a sopian plus one So you just build the the right graph now There's a general fact which is which is easy to show although. I don't know if it's written anywhere Well, I mean I wrote it down, but I don't know if it had been written before anywhere else Is that if SOP 1 and SOP 2 or so be two or not equal to so be three this will be witnessed by a graph gamma and you know to prove this it's very simple you just open up your copy of Sue gows and variant descriptive site 3 you look at the proof that graph isomorphisms broth pleat and what you do is you just code arbitrary structures into graphs you it's very simple to show that SOP n for example is preserved under this operation because essentially definability is not changed What that's the idea yeah, so you Yeah, yeah, but okay. You there's a small check. I mean I I was clear that it's trivial. I mean it's not a deep fact Okay, so then you build g gamma from that graph and then you pass the universal for a teeny cover and we know that that operation preserves the graph Then you use banded Dries-Libotsky the inverse gawa problem for PC fields to realize G tell the gamma as the absolute gawa group of a PC field and then Gamma is interpretable in the field. So this is something which Well, this is the universal for teeny cover of the group g gamma then gamma is interpretable in the field so this is a somewhat delicate point in the sense that sg of gamma is not actually interpretable in the field Gamma because it's the the the gawa groups are really only interpreted up to conjugacy So sg of gamma is is is interpretable in the pair the field together with its algebraic closure the field together with separable closure But not necessarily in the field itself However, the graph is interpretable because you can write polynomials that say I'm separable and my gawa group is DP and I'm separable And my gawa group is W and using that you can code the graph even if you can't get your hands on the gawa group itself and So this gamma is interpretable in the field and so f is SOP in But then also it's in a sop in by this theorem of Zoe Which shows you that if the if the group itself is in a sop and then that passes to the field Oh shoot, yeah, in this op in plus one. Yeah, this is a remarkable theorem, right? So Which which one Zoe's there? Here I Mean so for this one this this is me and then this one is Zoe So so so look so there's there's two steps, right? So one is that you have a graph then you get a group and then you have the group and you get a field and then You want to prove that in a sop in is preserved at each step of the way So I start with a graph and I build a group I show that if you start with an in a sop in graph Then the in the in a sop then you get an in a sop in group So it shows that if you start with an in a sop in group and you build an in a field you get an in a sop in field Yeah, yeah, so so that so so Yeah, so so this theorem is a corollary of this there Because you code sop in as a as an amalgamation problem They use it for all three So so okay, so you have to use strongly indiscernible trees or came independence to do the one one in two case It's not such a direct translation, but but you use but you translate everything into an amalgamation problem And you cite this theorem Okay, so Enrique standing up so I should stop thank you. I
construction will be germane Pure fields in the an absolutely reducible variety to find over the algebraically closed by the Nostalon thoughts, recall that a field K is called bounded if K has the work of Trillian van der Dries Macintyre characterization of types that Trillian van der about the Galois group. They introduced this question of amalgamation in the Galois group fields With prescribed model theoretic properties question about the absolute Galois group So to the r and We have the associated projections, as an absolute Galois group of a PAC field if not groups and the PAC fields with those as their theorem of van der Dries and the Botsky which this is a somewhat delicate point in the sense