 He's going to talk about Adelic geometry via toposphere. Yes, hi everybody. So thank you very much to the organizers for inviting me. So what I'm going to be talking about in this talk is a joint with Steve Vickers, who is in the Zoom webinar. So he's here to be my conscience in case I say anything silly. But yeah, it's a great pleasure to be here. So thank you for the organizers for inviting me. And thank you for everyone for being here. I'm going to be slightly self-indulgent here because I think it's just nice to show some pictures. So this is the first conference I ever read, proper math conference I ever went to. And it was the 2018 version of Topos's online. And so here's a wonderful picture of all of us. I think this is Olivia and Laura. I think this is Engel and Ivan. And I'm over here. So I'm this adorable cluster of blue pixels, which you can't really tell. But yeah, it was very interesting times. And we were in Koma, which in case you're wondering is this place. If I had a bit more time, I really thought maybe I should tell the story about how Engel almost killed me in one of those mountains. But I think, well, we'll leave that to some other time. But basically we went hiking and I was convinced I was going to fall off the mountain because Engel wanted to go all the way to the top. And I was like, oh my God, I think I'm going to die. But as you can tell, I didn't die. So I'm here to share some mathematics with you guys. So yeah. Okay. So let me just reset a little bit. So here's on this slide. I'm going to give you a program for the talk. So what am I going to tell you about? I'm going to talk about two basic things that cross cut different areas of mathematics. So the first question is what kind of information can homotopical data encode? And the second question is when can we solve a problem by breaking it up into smaller pieces? And then, you know, one, the first slide I said, a deli geometry via topos theories, like the subject of this talk, right? A lot of buzzwords there. I'm going to try and introduce some of the main ideas and discuss how it serves as a very interesting test problem for understanding how these two themes sort of illuminate each other. All right. So, all right. So on the first theme about like homotopical data, let me just start by recalling a much beloved example from algebraic topology, which is lime bundles over circle S1. So, all right. Suppose I'm standing. Well, you can't see it. I'm not actually sitting, but suppose I am standing in a, on a circle and I have a pole tied around my back and I walk around in a circle, right? And suppose I reach my original position, I'm still upright and that then, you know, the pole would have traced out, you know, a cylinder as you can see over here. But suppose on the other hand, you know, if I walk around a circle and then, well, maybe I'm feeling a little bit tipsy because, you know, why not? And then I end up, you know, standing on my head. And then in which case the pole on my back has traced out, well, something that looks like a mogus strip. There's sort of a twist in this particular setup. And so here's a very basic question from algebraic topology. Are these the only one-dimensional vector bundles over the circle up to isomorphism? And the answer is yes. And well, there's sort of like a couple of different ways that you can sort of show this, but the typical ways to exploit the theory of clutching maps. And really there's like, okay, what I want to say here is that there's a very tight relationship between the homotopy classes of certain kinds of maps between certain spaces and homotopy classes of maps between certain types of spaces and the vector bundles. And so the general argument usually goes something like, well, there's a close relationship between these two things and through some additional orientation arguments, we see that there's essentially only two relevant homotopy classes for us to consider. And one of the homotopy classes of maps corresponds to the cylinder, the other one corresponds to the mogus strip. And this is very, very interesting because this gives us a first glimpse of this really tight relationship between homotopical data, as encoded through the maps, and geometric data as expressed through the vector bundles. And this actually gives us, well, a first glimpse of a much more general phenomenon. And well, we have this particular classification theory which says that, well, suppose X is some sufficiently nice space, like just think of the CW complex and consider the vector bundles to be this, well, vector and X to be the set of isomorphism classes of n-dimensional vector bundles over this particular space. And this theorem says that basically, if you look at the homotopy classes of maps from X into, well, it's class of the gross minus, but basically it's classifying space. You look at the homotopy classes of maps from X into its classifying space, they are in bijection with the vector bundles, the n-dimensional vector bundles over this particular space. And notice what this is first saying is something really, really interesting in that the homotopical data in some sense is sufficient to, well, it does determine in a very strong way the geometric data from the vector bundles. And from the point of view of an algebraic topologist is very exciting because it gives us some indication about how homotopical invariance might encode very deep and very strong geometric information for us. So this is, you know, from the view of algebraic topology, how homotopical data might equals geometric data. And I put to you like a similar attitude occurs in toposphere in regards to geometric logic. So I'm just going to cite a paper by Vickers, continuous in geometric logic where he says, you know, simply but very boldly, in this paper, I shall survey some of the special features of geometric logic and a body of established results that combine to support the manifesto, continuity is geometricity. And so for the next few slides, I just want to put a little bit of flesh on this particular thought and start by just recalling some of the aspects of geometric theory and the theory of classifying toposas, which were already sort of touched on during the school. So, okay, what is the geometric theory? Well, a theory is basically some sort of axiomatic description of mathematical structures. I'm going to paint a picture in very broad brush strokes here. And the model is just something that satisfies this axiom, right? And geometric theory is just basically something that, well, it's a theory that's built out of the required list of logical connectives, right? Things like finite conjunctions, arbitrary disjunctions and the existential quantifier. So these are familiar ideas from the theory of classifying toposas. And, okay, so this is a particular example, which we're going to return to at various different stages of this talk and through different perspectives. And this geometric theory is the geometric theory or the propositional theory of Dedekin reels. And so I'm not going to give you the axiomatic description. I just wanted to highlight that, well, you have a Dedekin reels. And what is it? It's basically, well, a Dedekin reels consists of two sets of rationales, right? One of it sort of corresponds to the left Dedekin section and the other one corresponds to the right Dedekin section of the real numbers. And the axiomatic description sort of describes what exactly these properties of these sets of rational sorts have and how they sort of interrelate with one another. And so, right, there was some discussion during the first day about points of toposas. And it's well known that there's some toposas that, well, don't have enough points or don't have any points at all. And I'm going to give you a slightly different perspective here. So, all right, we call that any kind of, we know that the notion, the appropriate notion of maths between toposas is the notion of a geometric morphism, right, which are adjoint fantasies that possess the desired properties. And so classically, well, or traditionally, we think of a point of a toposas as some sort of geometric morphism from set into, well, into your desired toposas. But actually, you know, much of topos theory is about trying to figure out ways to vary the set theory of the situation. And so from this perspective, you know, it also makes sense to think of any kind of geometric morphism from any arbitrary topos into our topos of interest as some example of a generalized point. And we'll see why in a little bit. So this is, I think in the first, I think this was during Ricardo's talk where he talked about how toposas can be viewed as some sort of generalized space, right, because, well, that's what the purpose of the growth and excitement does. It sort of categorifies this notion of opens, living over a particular space. And here's a different perspective from the point of view of geometric logic. So recall that we have this particular universal property that characterizes what a classifying topos is, right? It's one that we have this particular equivalence of categories where, an equivalence which is natural in E. And we also have these basic foundational theorems, which really just says that this sort of description gives us an exhaustive description of growth and topos. Every growth and topos is a classifying topos of some kind of geometric theory. And every geometric theory, he has a classifying topos, you know, done through the well, syntactic constructions. So here's the slogan is that like, well, using this particular equivalence relation, right, I said before in the previous slide that we can think of geometric morphisms as some notion of generalized points, right? And using this description of the classifying topos, we can view the models as being corresponding to points of a topos in a particular way. In other words, again, we start to see in this particular guys how continuity or homotopical data sort of equals geometrical, in this case, logical data. And in particular to suggest something quite interesting because as category theories, you know, we're often used in thinking about what are the objects of the category, but in this specific case, in the case of topos, because of the universal property of classifying topos, it suggests that we can also think about that we can reason about topos in terms of the structures that they classify. And this gives us, you know, interesting amounts of leverage and a different perspective to do things. And so to sort of just see this, well, this inaction, so this is just me recalling the previous slide and using the theory of data can risk a particular example. So I'm just plugging in, you know, are in the appropriate places, right? And just plugging and chugging. And okay, so suppose I have a generic data can relax. I can add X to itself, right? X plus X is still the data can reel. In other words, it's still a model of this particular topos with respect to the theory of data can reels. And by the universal property of what classifying topos is, this corresponds to a geometric morphism between the appropriate and the relevant topos, right? Okay, so, okay, I'm going to, this is sort of like a bird's eye view of what we're doing right now. A bird's eye view of the kind of, well, the sense in which we're talking about generalized spaces right now. And really what we're doing is trying, we're starting to pull away from the set theory slightly. So classically in terms of, well, points at topology, we think of, well, what is a space where a space is basically a set, right? And points are the elements of the set. And then topology is sort of some sort of extra structure that we equip this thing with. Wasn't the point of view from point-free topology is that the points become the models of the theory and then the topos has become this sort of universe loosely speaking in which these points live inside. Okay, so, all right. I also just want to highlight this particular interesting notion of the generic model which we've also seen before. So every single, well, classifying topos has a notion of a generic model or also known as a universal model. I think Olivia calls it a universal model. And this has characterized by this property that in any kind of model of this particular theory can be obtained as some sort of, obtained by the inverse image filter for some unique geometric morphism. And one of the important consequences of this is that any geometric sequence, a sequence of geometric construction that holds for this generic model will also hold for all the models of this particular theory. And so on the next slide, I am going to show you a picture which I have shamefully appropriated from Olivia's book, but it's in the bibliography, so I think it's okay. But anyway, it shows a very nice visual representation of what's going on for those of us who haven't seen this stuff before, that you here is the generic or the universal model and all the other models here in these other different shapes. So it corresponds to models that can be understood as applying the appropriate inverse image filter to the generic model. Okay, so that was the first theme and I want to talk about the second theme. And this is sort of the thing that we do as mathematicians to a great deal, where it's like we have a problem. It's very hard to solve and so what we do is we break the problem into many different pieces. We solve the problem from each of the small pieces and we put everything back together again. And it's a thing that cross-cuts many different areas of mathematics, but I just want to talk about it in the context of number theory. So, okay, here's a polynomial on the slide. And one can, you know, it's a simple question to ask, what are the rational or the integer solutions to this particular polynomial? And in general, because, well, there isn't a lot of structure on the integers or the rationals. This is quite hard to deal with. But, you know, here are a couple of observations that have guided much the development of diaphragm geometry. So first thing first, if you have an integer solution, obviously this implies that we have a real solution and also more P solutions. In fact, pietic solutions, right? And if we look at the real and pietic solutions, these are easier to check than just the integer or rational solutions. So let me just say quickly for the real solutions, you know, well, you have a polynomial. If you can just check the signs, then you're pretty much done because, you know, if you plug in a number, you get a negative solution, you plug in the number, you get a positive solution, but the intermediate value theorem, you know there's something in between. And so, of course, this algorithm says that, well, you have a real solution, but it doesn't tell you the necessary integer, but it shows you in some sense how it's easier to check, well, to check for solutions over these richer geometric situations than the case of the integers or the rationals. And so this poses sort of a new question. So like given a polynomial with rational coefficients, like, does knowledge about its pietic real solutions give us some information about its rational solutions? And, well, this sort of suggests something which I'm going to frame in a very general sense, which is some kind of local global principle that basically what we're asking, right, one direction of this principle is obvious, right? That if something is true for the rationals, it ought to be true for its completions as well, but we're interested in the other direction as well. So when does it existence the solutions over the completions of the rationals and so this sort of motive explanation via del ring here. So like the local global principle says that, okay, it's this particular principle right here. And this thing doesn't always hold, like in certain kinds of polynomials it's true. Like for example, in the quadratic forms, but in other cases, like due to work by Selma, there are examples of polynomials where this is not true, but even then it's a useful thing to try and reason about what is true or what holds for all completions of q. And, you know, it's not true, it's not true, it's not true. So this is a simple principle that we call completions of q. And the adoring is well classically defined as some sort of restricted product of all the completions, but morally it's a way of trying to reason about all these completions simultaneously. And so just recall what we said from the previous slides about the generic model. So the idea, the starting point of our research project was sort of thinking well, how about instead of thinking about reasoning about properties that hold simultaneously for all the completions, we reason about what holds for the generic completion. And then we go from there. And so, it's just sort of a suggestive quote. I'm just going to, well, this is a quote from an accident survey by Barry Mazer. And he says that, well, a major theme in the development of number theory has been to try and bring somewhat more in line with the periodic fields. And a major mystery is why resist this attempts is strenuously. And we saw echoes of this during Schultz's talk where he was talking about how, you know, the liquid tensor experiment is not locally profite, which causes a lot of issues and a lot of technical adjustments that needed to be made in order to bring, you know, some of the condensed analysis to bear on, in the case of R. All right. Exploration begins now. So let me just start talking about what we're doing. So I said that we want to think about the completions of Q, right? We want to think about what these things are and what the generic completions might look like. But what exactly is a completion? Well, we have a notion of distance, right? That's a metric, if you like, that's defined on the rationals. And we define, and these are, well, these notions of distance, these notions of metrics have to satisfy these basic conditions. So, right, it has to be non-negative. It has to be multiplicative. And it has to respect some notion of the triangle inequality. So these are fairly standard notions from undergraduate real analysis. And what's interesting is that we're not interested in just every single kind of completions, but up to some notion of equivalence. So we define a place as an equivalence class of absolute values, where two absolute values are equivalent if there exists some alpha from this interval, such that if you exponentiate, they're basically equal with respect to this particular relationship. And as it turns out, if two absolute values belong to the same place, then it turns out that they, when you complete them, they yield topologically isomorphic completions. So this is why this notion of places is an interesting one. And so before we start talking about what the generic completion is, we want to understand what the generic places. And so intuitively, what does this topos look like? So, right, the first few slides, we're trying to tell you about how we should think about the points of the topos or the models of this topos. We want to basically be equivalence classes in some sense, right? That any two absolute values belonging to the same equivalence class ought to obey this particular relationship. And so we're interested in this particular, well, as a starting point, it's interesting to think about this diagram as a starting point. So Pi in this case is a projection map, which, okay, AV is basically the classifying topos of the geometric theory of absolute values. And if you're wondering whether the theory of absolute values can be cast in geometric terms, the answer is yes, so let's just go with this. And so you have an absolute value here and you have some exponent here. And the projection map just sort of forgets what's happening in the second tuple right here. But the exponential map is saying that, well, I actually want to exponentiate this. And what we would like to do is basically quotient by this sort of equivalence relation, right? But before we even do that, we need a notion that this is actually, this exponentiation map actually is a geometric morphism and that this is a well-defined diagram of toposes. And so let me just quote a quick result that we've done recently that's on the archive. So this is where a geometrical point-free construction of real exponentiation is helpful. So if we can define, do a geometric construction of exponentiation that here's the geometric logic and sort of by the reasoning that we showed in one of the previous slides, this also massively because continuity equals geometricity by the universal property of the classifying toposes. This corresponds to a map that we can use to build our diagram of toposes. All right. So let me just sort of return to this picture. So we said that we want the points of this topos to be equivalence classes of a particular kind, right? So actually we also, right, there's sort of an algebraic action here, right? If I exponentiate this absolute value with the specs of one, I want to get back the same thing and I want them to compose in this obvious way. So it seems that we want to quotient, right, just to, well, we want to quotient this topos by an algebraic action. The question is what is the appropriate quotient in construction here? And the answer is that, well, if the action is reversible, then we're going to use what I'm going to call the standard descent construction. And if it's not reversible, then I'm going to use the lax descent construction. And so, all right, I've been painting fairly broad brush strokes, but if you don't absorb all the details of the next two slides, don't worry, but let me just try and say something a little bit more formal. So suppose I have a truncated simplisher topos, right? Where all the maps here are, well, between these toposes sort of commute up to some sort of coherent isomorphism. And really what these simplificial identities sort of encode is basically the algebraic properties that I listed as desired algebraic properties on the previous slide. And so in the case of standard descent, you know, well, in the case you have a topological group or you have a space of objects in your space of arrows, right? And in the case, because it's a group, we know that all the arrows are invertible. And so, all right, we can define by some sort of nerve construction, some kind of simplisher topos. And we're just going to, when I say two truncated, there should be, you know, E0, E0, E2, E3, E4, but I'm just concentrating at E0, E1, E2. So that's why I mean by two truncated. And through work of people like Murdike, we can construct a descent topos. And explicitly, what are the objects of this particular category? Well, it's a sheaf of G0 plus some sort of action map, right? And the action map exists in G1. And this is necessarily invertible because, well, as we said, the group what has all the arrows are invertible because we're dealing with a groupoid. And it satisfies some special conditions known as the unit in the co-cycle conditions. In the case of lax descent, where we don't necessarily have invertible arrows, then what we have is, well, we can do a similar construction, right? And the objects, again, is some sort of sheaf. The lax descent category has as objects some pairs, which are sheaves of E0 and also some sort of action map. But the thing is that we don't require that data, the action map, be invertible. And so this sort of interesting distinction sort of gives us a very, well, an interesting perspective on the different notions of quotienting that might exist in the context of topos theory as generalized spaces. All right, so just sort of plugging in the various relevant details here. So this suggests this particular picture. So I'm going to, this is a word of warning. I'm going to suppress the sheaf notation, the S, this is what you see here. I'm going to suppress it for the ease of readability, but they really should be there in the subsequent slides. But this is just to put something on the slide so that everybody can look at, that we are applying the same recipe, the recipe suggested in the previous slides to our context. And what we're interested in is getting and obtaining a good explicit description of this universal code kind of this diagram of topos is. But this is the global picture. And it might be really, it looks kind of tricky to deal with. So maybe one thing we can do is just sort of break it down into like local pieces and see what we can do with this. So this, the way to do this is suggested by Ostrowski's theorem, which says that every single absolute value of the rational is equivalent to some notion, some kind of non-archimedean Piati absolute value for some prime P, or the Archimedean or the Euclidean absolute value. And you might wonder if you're constructively minded, is this a, we know that this is true classically, is this true constructively and geometrically as well and the answer is yes. So this forms a valid organizing principle for how we're going to proceed with our plan of attack. So let's start with the non-archimedean place. So already I've made some simplifications, right? So in this case, instead of, so for any non-archimedean absolute value is determined, and this comes out of the, well, the analysis from Ostrowski's theorem that is uniquely determined by where it stands the prime P and this lives in this particular open interval zero to one. So that's why I've replaced this here. And for any non-archimedean absolute value, because of the ultrametric inequality, the exponentiating to alpha for any alpha from zero to infinity still use a non-archimedean absolute value. It was only in the case of the Archimedean case where exponentiating to something that's greater than one use something that violates the triangle inequality. So in this case, we have a greater level of flexibility and we do in fact get some sort of group action. All right, which is established like what is D? Well, D trivializes. So basically everything is set. And all right, I'm not going to go through the technical details of the proof. I'm just going to show you some pictures and hopefully this works. But basically, all right, the crux move for this is basically just note that the connected components of the sheaves, well, we already know that these things are locally homomorphic to zero one, right? And sort of hinted at by this sort of different perspectives we can understand as sheaves as, you know, a local homomorphism but actually the send data actually imposes very strong conditions and this basically forces everything to be homomorphic. And so once you've realized this, then this, well, you can use this for leverage to prove that actually the category of descent data really does, you know, trivialize and give and is equivalent to set. So what about the Archimedean place? So in the case of Archimedean place, the setup is slightly different. So any Archimedean absolute value is equivalent to well, the Euclidean absolute value raised to some power of alpha from zero to one. But in this case, notice that it's stuck between zero and to one and anything greater than one is not well defined. So I've made the appropriate adjustments here. And also the space of this is where monoidal stuff comes in. So like the space of Archimedean absolute values is acted upon by a monoid action as opposed to a group action. So in particular, it's not invertible. And so, well, we had an interesting, I showed you a picture in the previous slide about what's it happens in the non-Archimedean case. Can we play the same game as we did in the Archimedean case? And here's the big surprise. The answer is no. And so, okay, I'm going to show you some pictures and we'll see if this works. So we said that shoes can be understood as local homomorphisms. And here is a particular type of local homomorphism that does, that exists over zero one. And it turns out that this is compatible with the lax descent data. And so what's in previously that the standard descent data is, you know, it basically flattens everything. You don't, you just basically get a homomorphic homomorphism if you have a connected component of the shoes in the case of the lax descent data is much more forgiving and forking phenomena is allowed to persist. And so, all right. This is just a quick exercise, but I think I'm running out of time on it. Okay, so I'm going to skip this, but basically this is the idea of like how crazy can the forking phenomena get. And, well, this type of forking phenomena is eliminated because of how she's understood as fiber-wise discrete spaces, but let me just skip that for now. So the insight comes in is that we have, I've said before about how we have a geometric theory of Dedekin reels, but we can also forget the left Dedekin section. So this gives us the theory of upper reels and we're going to denote zero one as the space of upper reels living between zero and one. And, okay, I'm running out of time, so I don't really have time to go through some of the technical details, but basically the purpose of this particular slide is to say that we have a particular lifting lemma that says that there's a canonical way of extending sheaves that are selected by the lax descent category to the sheaves living over zero one of the upper reels. And this is essentially being proved by Steve and myself, but there's still some constructive details that need finessing. So, okay. Right. The purpose of this is also just to say that this forking phenomena actually gives us a very interesting insight into what exactly this lax descent data is doing. So in the case of the descent data, standard descent, all this potentially unruly forking behavior is completely flattened. Whereas in the case of lax descent, globally forking phenomena still might persist, but as you know, through verifying this claim locally, well, local forking phenomena is actually eliminated. So this gives an interesting sense of contrast between the standard descent story and the lax descent story that mirrors some notion of, well, global stability versus local stability. And so here's, I said it was a bit of a surprise as to what this candidate, so the candidate picture is this, and I haven't proved, we haven't proved this yet, but I've given you some indications about how the Archimedean place is not trivial, right? Through this language of forking phenomena, why is this surprising? Well, out of Keelow geometry sort of guided by this fundamental analogy between, you know, the point and affinity living on the projective line and the Archimedean place of the rationales. And this sort of suggests that, you know, really, really should be thinking about the situation as some sort of one point compactification. But that's not what this lax descent data seems to be indicating here. So our candidate picture sort of suggests that there's some kind of blurring going on at infinity. And the infinity is not just some sort of classical point with no intrinsic structure. Okay. I don't have time. The point of this slide is just basically to say that I'm thinking about some seriously and about some of the potential connections between my work and homotopy theory. And, well, here's a particular pullback square and it's just interesting to think about. It's called the Chasa square, which is well suggested, but misleading for pretty much the same reason that there should be the Archimedean place there, the Archimedean completion there. But it's kind of difficult to figure out how to incorporate this in. And ideas from our previous calculations might give us some interesting insights about how to do this. Okay. So let me just very quickly conclude here. So what have I shown you? I've told you about two different themes, right? Continuity equals geometricity. I also said something about local global issues and how generic reasoning sort of allows us to have these two themes engage in dialogue with one another. And that through our analysis, we can see that the two themes are very similar in terms of the relationship between the two themes. And the other themes are very similar in terms of the relationship between the two themes. So I will just say that pulling away from the set, they reveal some very interesting insights between the relationship between topology and algebra. And that descent and lack of descent gives us an interesting understanding of what it means to quotient these poplices as generalized spaces. And that lack of descent sort of indicates some kind of blurring going on in our picture. And so much more careful thought is needed and about how to