 It is a pleasure to introduce the first speaker of this afternoon session, Ivan Tomashich, from Queen Mary University of London. And I mean, he will present a topographic view of difference algebra. So, Thomas, Ivan, this is yours. Thank you very much. So first of all, I'd like to thank the organizers for their kind of mutation. It's a great pleasure to speak in this conference. So, let me start with a little introduction to difference algebra because I understand it may not be familiar, widely known. So, Ritt introduced difference algebra in the 1930s as the study of rings and modules endowed with distinguished endomorphisms. And this was meant to be an abstract framework in order to unify the previous studies of, say, recursions or difference equations in the calculus of finite differences, various functional equations, the study of algebraic functions on dynamical systems, etc., etc. So, I will now present a kind of Ritt style difference algebra in a more categorical way. So, if we start with an arbitrary category C, we want to define the difference category, sigma C, or maybe the better name would be the category of difference objects. So, objects are pairs X, sigma X, where X is just an object of C and sigma X is an endomorphism of X. Amorphism of a pair of different objects is simply a commutative diagram in C, like that. In other words, it's just an Morphism in C which commutes with sigma X and sigma Y. In the rest of the talk, I may refer to sigma X as the difference operator on X. Okay. So, typical examples that we'll use quite a lot in this talk are different sets, which are just pairs of a set together with a self map, a difference group, which is just a group with a group homomorphism, and so on, difference rings. And for a fixed difference ring, we can also consider the category of difference R modules. So, now, out of these categories, the most important and the most interesting one, one is the category of different sets, because as we observe here, it turns out to be a topos. So, if you denote by underline N, the category associated with the additive monoid of natural numbers, then different sets are nothing other than actions for the additive monoid N. But you can also view them as pre-shifts on underline N op. And therefore, it becomes a growth in the topos. People usually refer to this object, especially when it's denoted to be N, as the classic point topos of the monoid N. Let me tell you a little bit about the topos of different sets as a topos. So, being a growth in the topos, it has a canonical geometric morphism to set. And somehow, the global sections functor is this functor, which I denote by fix, which simply corresponds to taking fixed points of sigma on X. And we have the natural forgetful functor from sigma sets to sets, which simply forgets the difference operator. And that turns out to be an inverse image of an essential point into sigma sets. Basically, this slide is very well known and understood. The only point of having it is to actually introduce the notation fix and the notation for the forgetful functor, which I'll use quite a lot. So, as already mentioned in a number of previous talks, a good thing about the topos is that it can serve as a universe for developing mathematics. So, we kind of redefine difference algebra to be the study of algebraic objects internal in the topos of different sets. And we study ordinary algebra in a slightly unusual universe, which is a very nice feature for me, you know, working a lot in difference algebra, you become sort of self-aware that it's a little bit esoteric or specialist subject. But viewing it this way, it becomes just normal plain mathematics. So, indeed, viewed this way, the previous categories, for example, sigma groups become groups, group objects in different sets. Sigma rings become ring objects in different sets. And if we fix a difference ring, then the category of our modules becomes the category of modules on a ringed topos. And you might ask, okay, so this is a rather trivial observation, why is this any useful at all? Well, let me show you the immediate success story, which is, you know, you can suddenly develop cohomological methods in difference algebra, which is an area which remained cohomology free for decades. So, if you fix a difference ring, then the category of difference modules is monoidal closed when you equip it with the structure of internal home of difference modules, which comes from topos theory. Now, you know, since the beginning of difference algebra, nobody actually detected that internal homes might be useful. So, it's immediately kind of a new way of seeing things. Now, of course, another advantage is that in any ringed grotenic topos, the category of modules is automatically abelian with enough injectives and enough internal injectives. So, you can immediately start developing a homological algebra. And your cohomology is simply the topos theoretic cohomology. So, we view cohomology of difference modules as an instance of topos cohomology. We can immediately use the machinery of X groups between different modules. And we have a cohomology defined through the higher directing, higher direct images of the global sections function, which, by the way, later I will denote fix. So, to show you some concrete calculations, let's fix a difference ring and two difference modules with M et al. And that is a kind of technical condition, which means that M is sort of projective. And it has a kind of good comparison with its sigma R twist. So, then you get that, for example, X1 is calculated as a sigma co-invariance of the internal home MA. And it vanishes above degree one. In particular, the derived functions of fix are very easy to compute. So, in degree zero, as it should be, it's just sigma invariance of module. And in degree one, it's sigma co-invariance. And then it vanishes above one. So, encouraged by this success of homological algebra, we want to continue and redevelop difference algebraic geometry in a similar way. So, we proclaim that difference algebraic geometry is going to be algebraic geometry, relative algebraic geometry over the base topos of different sets. And the plan to develop it is, first of all, in kind of phase one, pursue the development of algebraic geometry over an arbitrary base topos, S, starting, of course, with Hakim's monograph, and reformulating when necessary in terms of more contemporary topos theory. So, somehow, Hakim basically almost started the concept of classifying topos, but the theory has advanced incredibly after that. And then, while Hakim, you know, went up to a certain degree, we went to extend the work to include relative etal cohomology and relative etal fundamental groups and group points. And then, as a particular case in kind of phase two of the work, we specialized what we obtained over an arbitrary base topos to different sets and interpret the results in the spirit of difference algebra. So, okay, as in any development of algebraic geometry, we start with affine schemes. And they're given in this case by, in the relative case, as Hakim spectra. So, Hakim proved that the two inclusion from locally-ringed toposes to ringed toposes admits a right adjoint, which she calls a spec czar, or the risky spectrum. And to just explain qualitatively what it is, so for ringed topos SA, the risky spectrum is a locally-ringed topos equipped with a structure morphism of ringed toposes, which we call by czar, to the original ringed topos. It's sort of universal localization of a ring, but that cannot be found in the same topos. You have to change the topos in order to do that. So, moreover, for any scheme topology tau, which for example may include czariski, et al, flat, et cetera, using Hakim's techniques, you can define the tau spectrum with denoted by spec tau, and it's going to be tau locally-ringed topos with a structure map to the original ringed topos. Now, it's really a kind of beautiful connection to Olivia's lectures on relative toposes from last week, and her forthcoming work with Ricardo on relative toposes. So, I will kind of give, I will attempt to give a sketch of construction in terms of this forthcoming work. So, if we choose the site for the base topos, Cj, and we choose the relative tau site over A, that is going to be a sort of indexed category, indexed site, which is indexed as follows. So, over U, the indexed component over U, is the usual tau petite site over the spectrum of A of U. And then the spec tau is constructed as the topos of sheaves on sort of a semi-direct product of these two sites, which is sort of understood as, you know, J providing the horizontal topology and tau providing the vertical topology. So, you know, people who heard Olivia's last lecture probably remember those keywords. Okay. So, before Olivia and Ricardo's work, the only way I could sort of fully understand Hakim's construction was by finding suitable internal sites and then picturing Hakim's construction as an externalization. So, let me give you an example. Say, in the case of Zeriski spectrum, Zerial and Tierney, a long time ago, had an idea that Zeriski spectrum can be constructed using the internal frame of radical ideals. So, you can do that. You construct kind of internal frame of radical ideas and gives you internal coverage. And then the Zeriski spectrum becomes the topos of internal sheaves on this internal site AJ, which of course, using this familiar externalization procedure, can be written as the topos of sheaves on this semi-direct product of the site with an internal site. And this construction here appears, for example, in Moerdijk's 86th paper. And it's also in the elephant. All right. Now to more general schemes obtained by gluing affine pieces. So, for base ring topos SK, we define this object kind of schemes over SK as the stack completion of the vibration given by this rule. So, to an object U of S, it assigns the category of schemes over K of U. And Hakim then considers the fiber of this stack completion as the category of S schemes over K. Now, given an S scheme X, this is just somehow gluing information. But for a given scheme to apology Tao, we can form a Tao realization of X, which we denote by X Tao, as a Tao locally ringed topos together with these structure maps, you know, to the topos S, but also with the global sections, the morphism to the category of sets. And we can now do some Tao cohomology and relative Tao cohomology. So if we choose an abelian group object M here in X Tao, we can, of course, take the higher direct images of the global sections factor and end up with cohomology groups here in set, which are just abelian groups, or you could go this way and take higher direct images of along pi, and you end up with abelian groups in S. And somehow from the internal point of thinking, from the kind of relative point of thinking, these objects should be rather interesting. But occasionally, one actually wants to consider the total cohomology. Okay, so this is the setup. And let me now speak of some concrete cases. So over the Zarisky topos, Hakim already kind of dealt with the cohomology of quasi coherent sheets. And in particular, she has this result. So if you take a ring object in S, and then an A module, if you denote by X the affine scheme corresponding to A, and M tilde is the induced module by M, then Hakim shows that the relative cohomology of M tilde vanishes in degrees higher than zero. And, you know, this is a relative analog of a famous theorem of Sarah. So that's already kind of well understood. But we go a little bit further and develop et al relative et al cohomology. And somehow we claim that kuma theory and art and trial theory work just as anticipated. So working on the et al site of a relative scheme, you have the kuma exact sequence involving the multiplicative group. I think there's a typo. This should be GM, a multiplicative group. And then taking the end power of an object and end roots of unity, sheave, and it is exact in the et al topos. And similarly, the art and Shrier sequence gives an exact sequence on the et al topos. And this is actually not too difficult to see if you immerse yourself into Gavin race paper on generic Galois theory of local rings. Somehow it just boils down to showing that certain polynomials are separable. So in this case, the relevant polynomial is going to be x to the n minus a. And given that this is separable under the conditions, you somehow get that this is exact in the et al topos. And of course, from these short exact sequences, you can then obtain the long exact homology sequence and attempt to calculate some of the terms. Onto Galois theory now. Let me explain how to sort of produce a relative version of McGeed's Galois theory of separable ring extensions. And it's going to be done using Janelidze's categorical Galois theory. So we fixed the base topos s. And we consider an adjunction like so of two functors. Where the top category here is the opposite category of rings in s. And the bottom is the category of stone locales in s. And by that I mean just compact zero dimensional locales. And s is the pierce spectrum factor, which to a ring a associates the local with frames whose frame is that of ideals on the Boolean algebra of idempotence of a. So we prove that this functor has right adjoint and call it c. So Janelidze's Galois theory then tells you that if you take amorphism in a of relative Galois descent, we obtain a P internal group point with object of objects s of x and object of morphisms s x cross x over y. And this P group point induces an equivalence of categories between the category of split objects over y, which are split by f and the P actions, the category of P actions of the P group point Gal F. So this is sort of a relative version of Magid's separable Galois theory for commutative rings. Somehow a difficult part in this theory is to actually identify morphisms of relative Galois descent. In other words, normal objects, because this type of theory doesn't give you the construction of normal closure of an object like you have been say the classical Galois theory of field extensions. So that's why we would like to kind of elevate this to the underlying topocysts and use Boong's theory of v-determined topocysts. So if x is a relative scheme, then the proetal topos of x is a P-determined topos over s in the sense of Boong. And then this fundamental P pushout construction gives you everything you need, including the construction of normal objects. And in favorable conditions, in favorable circumstances, it agrees with Yanelidze's theory. And somehow, if you have this relative morphism between the top category and the bottom category, somehow the point is that Yanelidze's theory uses only the left adjunction and somehow Boong's theory uses the whole structure. Okay. So that's what I'm going to say about the kind of general theory. And now I'm kind of going on to the phase two where I specialize to the difference case. So to start with, I'd like to give you an explicit description of the spectra of a difference ring. So let A be a difference ring and choose a scheme topology. I will denote by S tau the classical petite tau site of the spectrum of the underlying rings. So this is just the ordinary scheme. And we're looking at the classical tau site for that. And also the classical tau topos, which we denote by kind of underlying X tau. Now, because the difference ring has the difference operator, that difference operator induces a base change or pullback function on the classical site. The pair, so this data, the pair consisting of the classical tau site together with the base change function actually defines an internal site in the category of different sets. And the tau spectrum is the topos of internal sheaves on that internal site. Now, okay, this, I promise to make it explicit, but I didn't really make it explicit. I just gave you a recipe. However, we can make it more explicit. So explicitly, X tau is actually the category of sigma equivalent sheaves on the classical tau topos. In other words, those classical tau sheaves, which are equipped with the morphism from F to F composed with the base change. And you see this sort of supports our premise that kind of M-equivariant geometry corresponds to geometry over the classifying topos of M. So this is where we see the kind of equivariance. If you did this for any other monoid, you would get some kind of equivariance conditions like that as well. Okay, so a good thing is that our construction allows you to recover the original difference ring as the ring of relative global sections of the structure sheet. Now, there have been lots of earlier attempts in difference algebra to define the spectrum of a difference ring, but those constructions attempted to create a different spectrum as a ringed topological space where the topological space would be, say, the space of different primes. But the problem is that you may not have any sigma prime ideals, so you won't be able to occasionally, you don't have sufficiently many points, and you cannot recover the original ring. So the point why this construction is successful is that you abandoned the idea that the spectrum has to be a ringed space and we treated it as a ringed topos. Of course, you could also think of it as a ringed locale. Good. So I promised an explicit calculation of ethylcohomology in a different case, and usually the first result that everyone proves is ethylcohomology of a field. And in this case, we're going to show, we're going to calculate the ethylcohomology of a difference field. So kind of the relative version of Hilbert 90 tells us that H1 of the multiplicative group is the Picard group, the difference Picard group, which in this case turns out to be the sigma covariance of the multiplicative group of the difference field. And H2 of GM is somehow the difference Broward group, which has a very easy comparison to the Broward group of the underlying field. And it's just the sigma invariance of the classical Broward group. And using Kuhmer theory with this information gives you now that the object we want to calculate, the cohomology of the field with kind of torsion coefficients in the roots of unity, is sandwiched between some rather interesting looking objects. So the notation here means when sigma is upstairs as a superscript, this means sigma invariance. When it's like this, it means sigma covariance. And then when n is to the right, this means co-kernel of multiplication by n. And when it's to the left, it means kernel of multiplication by n. So you can do some explicit calculation. All right. Now, what about Galois theory in the difference context? Well, if we attempted to use Grotten de Galois theory for this purpose, we encounter some rather difficult obstacles. For example, a different scheme can be typologically totally disconnected space, something like a contour set. And yet, it can be indecomposable as a difference object because the difference operator may kind of permute these components in a sort of wild way. And so X may not have any geometric points. In the difference context, geometric points correspond to fixed points. And also there's a problem with the fact that the base topos of different sets is not Boolean. And somehow Grotten de Galois theory relies very heavily on the fact that the notion of connectedness, connected objects, is well behaved. And kind of this relies on the fact that you have a Boolean-based topos. So basically, that kind of immediately tells us that we cannot go along that route, but we have to look to more general topo-theoretic Galois theories and that perhaps we should anticipate to get a kind of locale group point as an object rather than a group. So, all right, let me first do the kind of Janelidze style or Magid style difference Galois theory. So we let K bar be the separable closure of a difference ring K. And that simply means that the underlying ring of K bar is the separable closure of the underlying ring of K in the sense of Magid's monograph. And in a recent archive paper with Michael Wibmer, we show that Janelidze's Galois theory applies to this context. And when you apply it to the discover K bar to K, which is of relative Galois descent, you obtain a group point in difference profaned sets. So you get a difference profaned group point, which we call pi 1 K K bar. The separable closure is not unique up to isomorphism. So for rings, it is the case, but when you have this difference operator in the picture, then you could have non isomorphic separable closures. And that's why we include it as a part of the notation. So, so this difference profaned group point classifies the category of difference locally at all K algebras. This is a technical notion, which maybe is not the best time to talk about, but essentially somehow component wise, so you can have totally disconnected space of components. But over each component, you sort of have an end etal object. All right, now in forthcoming work, we want to do, so this is more like a topological version. And in the forthcoming work, we want to complete the picture and do a localic version in the determined context of Boge that I already mentioned. Okay, so one thing to note is that if you have an object X Sigma X in Sigma prof, so when you have a difference profaned space, it actually lives in the realm of topological dynamics. It's kind of topological space with a shift and somehow you can view it as a dynamical system. So using this kind of observation, we provide a connection of our difference Galois theory to symbolic dynamics in the following way. So, you know, in a very special case, when K is a difference field, then the fundamental groupoid G is actually a difference profaned group. So it's just a profaned group with a continuous self map. And the Galois correspondence restricts to an equivalence between these interesting sub categories. So on the left, we have a finitely Sigma presented in detail K alfabris. And on the right, we have a sub shifts of finite type with continuous action of G. And these objects belong to, you know, the area of symbolic dynamics. And we observe that the algebraic difference algebraic notion of limit degree corresponds to entropy of these sub shifts. And we kind of wonder, you know, when you actually take into account the kind of full thermodynamic formalism of topological dynamics by well, you know, what are the translations of other very interesting kind of thermodynamic notions into the world of difference algebra. But in any case, this shows that difference algebra, you know, when you give up connectedness assumptions, and you work completely arbitrarily like this, it is going to be extremely rich and very difficult because it includes the whole symbolic dynamics. Okay, so in the slide, I'd like to talk about, I'd like to go back to RIT. RIT not only envisaged kind of the abstract formulation of difference algebra, but also of differential algebra, where the ring is given with derivation. Of course, you can also mix the two structures, and you can have difference differential rings and modules, where, you know, they're equipped with a difference operator and a differential operator. So with my PhD student, Antonino, we kind of describe the spectrum of differential ring, starting with describing rather explicitly the classifying topos for the theory of local differential rings. So local differential rings are differential rings which are local and have a unique maximum ideal, and the unique maximum ideal is also a differential idea. So we show that the construction of the differential spectrum, if you use the Hakim Kohl approach, which invariably uses this classifying topos, this construction actually agrees with the construction using the locale of radical differential ideas. And this is very interesting because these ideas appeared in a Carafero paper from 1990, and so it's a very interesting connection. This classifying topos has been mentioned by Marta Bungae a long time ago, but she hasn't made it explicit. Now, I have to emphasize that our construction of a differential spectrum differs from previous constructions of Keiger and Kovacic, which attempt to do it again as a sort of ringed space, and they use the space of differential primes. But again, the same problem, you might not have any differential primes, and so you may not be able to recover the original differential ring, but in our context, this is certainly rectified. And okay, so that's the spectrum of differential ring. Now, what can we do about difference differential algebra? Well, so Keiger showed us that you can consider the theory of differential rings in an arbitrary topos is pretty well behaved. And so we view difference differential rings as differential rings in the topos of different sets. And of course, we have the program to basically repeat everything I said for the difference case in the difference differential case. Okay, so to conclude, why pursue this program? And you know, what's interesting about these ideas? Well, the first point is that it sort of provides you with an Ariadnes thread principle in the sense that it gives you a template to adhere to. Usually, when you're a researcher in difference algebra, you often wonder what is a suitable generalization, a suitable sigma analog of some classical concept or classical notion. And you may have you may be presented with several options, and it's it's not clear which which is the correct one. Now, our approach provides kind of definite template through topos theory. And all you need to do is perform the, you know, operations using topos theory and categorical logic, and then interpret what that actually means in the spirit of difference algebra. Now, of course, that's far from easy making certain topos theoretic constructions and calculations explicit. And kind of also meaningful in a different area is not not so easy. And requires actually, you know, very sophisticated technology from topos theory, which you might not think that you would need. And then there's a great capacity for generalizations. Because of our motivation, we, you know, work over the base topos of bn, but we can replace n by an arbitrary monoid or group or category to obtain the corresponding equivalent geometry. And this, this may provide like a nice unifying machinery to kind of formalize equivalent theories. Okay, that's it from me. Thank you very much for listening. Thanks a lot. Thanks a lot for this talk.