 Okay. Thank you very much. So thank you for inviting me to give this talk. So I have to say that not only we are very grateful to Huawei for its very big support to VHS, but it is also a great pleasure for us and in particular for me to exchange with people working to Huawei for Huawei. So it is a great pleasure for me today to give this talk. So I hope you already see my slides. So the title of my talk is on the creative power of categories. So my talk is not meant to be technical of course, but maybe nevertheless it will be too technical for people who don't know what is a category, but even in that case I hope that my talk will make you dream a little. So it is not at all intended as a technical talk. And the theme of categories is as you will understand very natural because categories were developed a lot at VHS, especially by the most famous VHS professor, and categories as I shall also explain are of great interest to some mathematicians working for Huawei. And they were at the heart of all the exchanges between Huawei mathematicians, myself and other mathematicians big to EHS I will talk about. So first I want to give some elements of history. So categories were invented during the Second World War by two mathematicians Samuel Eilenberg who had escaped from Poland and Sanders McLean who was an American mathematician. So I don't want to of course to give a precise definition. I only want to say that category is some kind of combinatorial object which can be partly seen as a picture. I mean that this object consists firstly in some dots. So just imagine dots on the white page and these dots are related by arrows. And arrow always goes from one dot to another dot. And it is also possible that these two dots are the same. So in that case you have one arrow going from one dot to itself. And it has also to be mentioned that for any two dots there can be many arrows from the first dot to the second dot. And there can be also arrows in the two directions. So you see the two first data are dots and arrows. And the third data which defines the category is a composition law for arrows. So when you have two arrows relating three dots such that the target of the first arrow is a source of the second arrow, then you should be able to associate to them a composed row going from the first dot to the last dot. Okay. So on this of course has to verify some rules which I don't make precise. So the definition of a category is very simple. The next definition which was introduced by Eilenberg and McLean is the definition of a functor. A functor is a way to go from a category to another category, transforming dots of the first category into dots of the second category, arrows of the first category into arrows of the second category in a way which is compatible with sources, targets, and composition laws. And the last notion which was introduced by them was a notion of transformation from a functor to another functor. So here I don't explain at all what it is. I only say that these three notions in fact by mathematical standards are very simple. Samuel Eilenberg used to say that the most important paper he wrote in his last time was a paper without theorem, without result, and without proof, just with simple definitions. At the same time, as the same time categories were introduced, another very important notion for us was introduced completely independently by the French mathematician Jean Loret. This was also during the Second World War and in fact he was a war prisoner in prisoner camps in Austria. So Loret invented the notion of Schiff. So in fact his definition was slightly modified later by Carton, but essentially it was given by Loret. He also defined Schiff homology and already at that step he introduced Eilen non-trivial technique which he calls the technique of petrol sequences. So this is a computing technique. So you see that on the category called side what Eilenberg and McLean did was very simple, was basic. What Loret did already at that step was partly simple and partly technical. And here it has to be mentioned that later, in fact 15 years later, Schiff theory will become the most important part of category theory. This is why I mentioned these two subjects. The first thing which has to be said about categories is that in category theory the most important things are the most simple things. So I only want to quote a few remarks which have become extremely important for category theory. First there is a so-called Yoneda lemma which was introduced 10 years after the invention of categories. So this is just a remark with a one sentence proof by Japanese mathematician Yoneda. This remark was popularized by McLean who understood it was important and later it was transformed into a powerful tool by Grotendig. So this remark is that in any category when you consider a dot or object as mathematicians say, you can associate to it the network, the full network of relations to this dot. So this means the full network of sets of arrows whose target is the considered dot. And the remark by Yoneda is that if you know the network of relations then you know the object you are talking about. So once again the proof of this fact only requires one sentence. It is extremely elementary but it has proved the most important basic fact of category theory. The other notions I want to mention were introduced 15 years after the birth of category theory by the American mathematician Daniel Kan who I want to mention that is still alive and you can even talk with him. And so Daniel Kan introduced two extremely simple notions which I shall not define but it would just need a few lines. So first the notion of pairs of adjoint functors. So this is the definition of a relation between two functors going from one category to the another category in the two opposite directions. And a basic fact of this definition is that if you know one of these functors then the other one is uniquely determined. And here it can happen, it is a basic general fact that one of these functors is extremely easy to define, is trivial and the other one is subtle. And so this in fact this notion of adjoint functors provides an extremely general device to define interesting functors in category theory and so in mathematics. The particular case of that are the notions of limits or sometimes people say projective limits and the notion of curly means people also say inductively means in category theory. So these notions were also introduced by Daniel Kan. Okay so at that point I already told you that categories mainly consist of simple facts. And so this means there are reasons not to believe in categories. And in fact in the first years or decades of category theory many people expressed doubts about category theory. So first the first feeling when they have is that the notion of category and the derived notions which are associated to the notion of category are too general. Secondly you can have the feelings that categories are too big. In general they are much bigger than all objects of study previously studied in mathematics. For instance people were used to study groups which mean to study one group or another group or possibly a morphism from one group to another group. But if you are working with categories then you will consider for instance all groups at the same time and all morphisms between groups. So this is a category. So this means it is huge. It is much much bigger than anything which was considered before in mathematics. And the last feeling you may have is that category is theory is too simple. It is too easy. In particular in category theory usually there are no tricks. Okay and then I have to talk about Grotendig. So Grotendig is the first and the most famous UHS professor and I have to talk about him because he was a mathematician who illustrated that category theory is extremely powerful. In fact not only it is powerful but it has a creative power. So this means that category theory as shown by Grotendig has the power to create new mathematics. So how did Grotendig do that on many mathematicians did also that following the example of Grotendig. So using the language of categories what Grotendig did is to illustrate the depth and the power of simplicity. So I already said that category theory is simple but in fact Grotendig showed that simplicity can be extremely deep and powerful. And it is a lesson not only for mathematics but also I think for other sciences and maybe also for life. So I said that Grotendig turned categories into an instrument of discovery and creation of new mathematics and he did that always in the same way. He tried to reformulate and to understand in terms of categories key classical theories or results. So by now in the next minutes I want to illustrate that by giving an idea of what Grotendig did. So the first example I want to give is a categorical definition by Grotendig of the notion of modular space. A modular space is a space which classifies some type of algebraic structures. Some type of structures not necessarily algebraic. So for instance you may want to classify lines in the plane and what you get is so-called projective line but you can also want to classify more complex things for example to classify algebraic curves. And modular problems, problems of classifications of mathematic objects of some types had been considered before in mathematics. In fact they began to be considered in the 19th century I think with Riemann. But for almost a century there was no precise definition of the notion of modular space. There was a loose definition. People more or less understood what the word meant but there was no mathematical definition of this notion. A mathematical definition has to be precise. And so Grotendig proposed the definition for the notion of modular space. And what he did for that purpose is just to use the trivial remark of Yoneda. The trivial remark of Yoneda is that if you consider an arbitrary category then it is embedded by considering the network of relations associated to objects of this category. So this category is embedded into the category of networks on that category. And so the idea of Grotendig is that any classification problem defines a network on an underlying category. And then the problem of the the moduli problem is to check whether an object of this category of networks comes from the underlying category which is embedded in it. Here you see you should see you should think of the network category C hat as some kind of completion of the category C. So what you do is to consider an object in the completion and to wonder whether it comes from the smaller more meaningful object. So it is exactly as if you consider a real number and you wonder whether this real number is rational or not. Then another example is the categorification of Galois theory by Grotendig. So Galois theory was a theory in algebra a theory about algebraic equations but Grotendig made it incredibly more general. For that purpose he considered a category C and he defined on the general category a notion of topology. And then so this means a notion of localization. And then this allowed him to define the notion of cover. A cover is an euro in the category. So here I have written the euro X to S. And you will call that euro a cover if locally for the topology you consider it is identical to the projection from a sum of copies of the base to the base itself. Which means it is a cover if locally for the topology you consider it is trivial. And then Grotendig proved under extremely general conditions that categories of covers are equivalent to categories of actions of some group. So when you have a group you can consider sets endowed with an action of the group. And this defines the category. And so Grotendig understood that categories of covers under extremely general conditions are equivalent to categories of this time. And in doing that Grotendig unified two very important to most important theories of mathematics which so far were not related. And they went on Galois theory and on the other hand the point theory of fundamental groups. So using this categorical language Grotendig proved that these two theories are only one. And of course by doing that he generalized incredibly the scope he extended incredibly the scope of Galois theory. The next thing I want to mention is the categorification of the Riemann-Rohr theorem. So this theorem was born in the 19th century. So first by Riemann, then Rohr who was a student of Riemann. It was a theorem for vector bundles over a geometric object. So for Riemann and Rohr it was a vector bundle over a curve. And it was extended to vector bundles over geometric objects of arbitrary dimension by Irse-Rohr in the late 50s. So this was a formula for vector bundles over a geometric object. And just after this formula was established in general by Irse-Rohr, Grotendig decided to try to understand it. And for him understanding meant to understanding in terms of category theory. And in fact what he did was to transform this formula into a drawing. And here on the page I have written this drawing. So you see it is a square. I don't want to explain the terms. But the important fact is that this is a drawing. And this drawing is associated no more to one vector bundle to over one geometric object. But it is associated to some new variant associated by Grotendig to this geometric object. This is what I have written k of x k of y. And it is associated to a morphism from a geometric object x to another geometric object y. So this means that Grotendig had transformed the formula for one geometric object into a diagram associated to morphism from one object to another object. So what he did was really to make the Riemann-Rohr theorem categorical. And in the process of formulating and proving this incredibly more general theorem, Grotendig introduced a new invariant of geometric objects which by now has become a whole new part of mathematics. It is so called category. And I want to say a few words about that. So what he did is to associate to a geometric object x the full category of vector bundles on x or the full category of so-called coherent linear shifts on x. So these two categories are instances of so-called additive categories. And here Grotendig introduced an extremely simple definition for any additive category with an induced notion of so-called short exact sequence. He was able to define a group so the so-called Grotendig group of this category. And here in fact on the page I have written the definition. So I don't expect you to understand it, but I want to stress the fact that this definition is very simple. You can write it in a few lines. So you just consider the free abelian group generated by all objects of the category divided out by the relations associated to short exact sequences. And you see this is very simple. And so this is a definition of just a few lines. So if I summarize, when you start with a geometric object x, you also share to these geometric objects two categories. The categories vex of vector bundles and the category mod x of linear coherent shifts. Then there are the associated Grotendig groups. And if your category, your geometric object x has no singularity, in fact these two Grotendig groups are proved to be isomorphic. And this is a category of x. On the other, you have to remark something. We already said that categories are too big. They are too general. They have too many objects. So you may have the feeling you cannot work with them. But here what Grotendig did is to start from a geometric, a concrete geometric object x to associate to it categories which are too big. This is true. These categories are too big. But then it defines invariant from these categories. And here what he has shown is that this invariance with definition is derived from the categories he considered. This invariance of the k group k of x are amenable to computation. In fact, they are so interesting for mathematics that by now this object has become a whole part of mathematics. Then I also want to mention the categorification of homology. So homology is the most important part of 20th century mathematics. It has helped to solve many, many problems in many parts of mathematics. So before the Grotendig, homology was already defined for linear shifts on topological spaces. And it was also defined for linear representations of groups. So what Grotendig did is to study the categories of linear shifts and the categories of linear representations of groups and to realize that they share some properties. So they are so-called Abelian categories. And in the process, Grotendig defines a notion of Abelian category. And they verify a few extra properties which are exactly what are needed to do homology. And so the consequence of this identification of the needed properties by Grotendig was that after his paper it was possible to develop homology in any Abelian category verifying these properties, which was a huge generalization of what existed before. But then you may ask, is it possible to construct or how to construct Abelian categories verifying these homology-friendly properties? More precisely, is it possible to imagine a construction process which generalizes at the same time the construction of linear shifts from the construction of the category of linear shifts from topological space, and the construction of the category of linear representations from a group, or more generally a monoid? And here Grotendig proposed the following answer. He considered a category endowed with the topology. In fact, we have already considered that when talking about Grotendig's categorification of Galois theory, the notion of topology on a category. So then he considered a pair, which he called a site, and he showed that any time we have a topology on a category, there is an associated category of linear shifts. And this process generalizes both the definition of linear shifts on a topological space, and the definition of representations of a group or a monoid. And in fact, the categories of linear shifts always verify the homology-friendly properties which are needed to do homology. And at that point Grotendig did another step, which he called a child-like or child-minded step. He just remarked, but why do we consider only linear shifts? We could consider shifts just of sets. It makes sense. And then he decided to consider the categories of shifts of sets on an arbitrary site. And the categories define in that way, he called toposis. So here I denoted, I denote them in this way, the category C with an hat and with an index J, because it can be understood as some type of complication of the category C. In fact, category C has many complations. It has a complication for any choice of a topology of C. And if the topology is discrete, is trivial, what you get is just the category C hat of networks of C, which we have already talked about. Okay, so here it has also to be said that there is a double characterization of toposis, either a constructive definition through the shift process, but also an axiomatic characterization. It was put by a grotendic student, Giro, that a topos is a category which verifies the same properties, which has just as good properties as the categories of sets. Okay, so I think I will try to go to wake up. So another thing grotendic did is to categorify Poincaré duality. So Poincaré duality is the most important general property of homology. So for Poincaré, it was a duality between homology spaces. On the other hand, it has to be mentioned that grotendic did that categorical in categorical terms. He understood Pancaré duality in categorical terms. So I don't want to, I don't have the time to explain it, but it is really a miracle of mathematics here. It is incredible that grotendic was able to imagine such a reformulation of Pancaré duality. So this is the so-called six-operation factors. And in the process of proving that, he defined a new class of categories, which he called derived categories. So you see there were these new definitions. So here on the page, you can see this notation F with an explanation point either up or down on this notation D of A. So these were new definitions by grotendic. And this new definition of made category theory, especially shift theory, extremely powerful. So here, I just want to let you dream on the drawings you see. So this is the homological translation of the geometric notion of smoothness, no singularity, and of compactness. And here it is another miracle of category theory that it is possible to translate these notions in categorical terms at the level of homology. So here you see these notions can be translated in the way I have drawn. The left diagram should imply the so-called commutative square on the right. So you see this is an implication between diagrams. And another miracle of this formulation is that the local property of smoothness and the global property of compactness are expressed almost exactly in the same way. They are expressed by the same implication between diagrams. For compactness, F is fixed and G is arbitrary. For smoothness, G is fixed and F is arbitrary. So you just exchange the role of what is fixed and what is arbitrary. So then I have to mention also the categorification of homotopy theory. So here, the most important names are Daniel Coulin and also Grotendick himself. And so this categorification has given birth to a new part of mathematics, which is called higher category theory by now infinity category theory. On the other hand, I have to mention the most important name, which is Yuri, Jacob Yuri. So this theory has become very important in mathematics. It is studied by many mathematicians, not only mathematicians. It has become important even for some physicists. But we have to mention it was not done in a systematic way. So the development of higher category theory is really driven by applications. And it is really wonderful that something so abstract, because it is extremely abstract, in fact, is driven by applications. Okay, but I will not talk any more about higher categories, which in fact is another subject. I want to come back to toposis. So here, I come back to toposis. And to what Grotendick proved about toposis. So I already mentioned that there is a topos category of sheets of sets associated to any category C, and odd with a topology J. And then a miracle of category theory is that it is possible to define, in purely categorical terms, the notion of sub-toposis of a topos or the notion of open or closed sub-toposis, the notion of intersection or union for the topos, the notion of point of a topos. So you see, all the words I am introducing here, all the words I am using here are words borrowed from geometry. But it is possible to define this notion in purely categorical terms. So this means in purely combinatorial terms. And you do it in such a way, or Grotendick did it in such a way, that the notion of the topology geometric or topological side and the notion of the categorical side to inside. So these things that for instance, the points of a topological space correspond to the points of the associated topos of sheets on this topological space. And the same goes for open sub-objects, for closed sub-objects, for intersections, for unions, they all correspond. But the advantage of that is that the notion of topos is much more general than the notion of topological space. So in this way, you can bring the vocabulary and the intuitions of geometry to a much more general setting than the setting of classical topology or geometry. So here, there is also a notion of morphism between topos. And in fact, if your two opposites are associated to topological spaces, a morphism on the topos side correspond to a continuous map on the topological side. But here we have to remember that the topos is defined in general by a site. And here it has to be mentioned that in fact, any single topos can be defined by infinitely many different sites, which means couples consisting in a category endowed with topology. Okay, so this means there is incredibly more flexibility in the notion of site than in the notion of topological space. I also have to mention that toposites are not only related to geometry, but also to logic. So here I have mentioned the names of the few people who have played a role in that respect, beginning with Rotendik himself. So it all stems from the fact that a topos is just as good as just as good properties at the category of sets. So this means that it is possible to do mathematics in the context of a topos, of an arbitrary topos, exactly as we are used to do mathematics in the context of sets. So in particular, if you have any so-called first order theory, for instance an algebraic theory, the theory of groups, the theory of things, the theory of vector spaces, and so on, then it is possible to consider models, which means realizations of this theory, not only in sets, but in an arbitrary topos. And so for any theory T on any topos E, there is an associated category of realizations of this theory in the topos. And here comes a very important and wonderful theorem, which is the following fact. Start with such a first order theory. Then for any topos, you can consider the word, which is a category, in fact, of realizations of this theory in that topos. And if we have a morphism from a topos to another topos, there is an associated functor in the reverse direction, transforming realizations of the theory T in the second topos into realizations of the theory T in the first topos. And here, the most important theorem is that this functor is representable by a particular topos. So this thing, there is a particular topos E T such that for any topos E, the realization of the theory T in E corresponds to the morphism from E to E T. In particular, the set theory realization of T, for instance, let's say the set theory T groups, correspond to the category of points of the topos. So this thing, for example, for the theory of groups, there is an associated topos such that groups in the usual sense correspond to points of that geometric object, the classifying topos of the theory of groups. And here is a very important remark that for any topos, there are infinitely many theories which define the same topos. So here, this remark is extremely important. In some sense, it is some kind of duality between language and meaning. What can also say a duality between freedom and truth? The topos incarnate truth, but there is freedom. And the freedom is in the way to speak about truth. There are infinitely many languages which allow to talk about the same truth. Okay, so here I want to mention the development of category theory, which was invented and developed by Olivia Caramello in the last, so she began in 2008, 12 years ago during her PhD work. And Olivia Caramello is one of the Gelfand Chair Holders at THS, for example. So her idea is the following. Her idea is to take advantage of the fact that any topos can be associated to many different theories or to many different sites. So here I have written these possible equivalences between toposes defined in different ways. And then her idea is to consider invariance of toposes, which means operatives or constructions associated to toposes and stable under equivalence, and to express them in terms of the defining sites of theories. And in this way, you can get correspondences between sites or between theories or between the site and the theory, which are completely different. And in this way, you get correspondences which are most of the time completely unexpected. So this I think is very important because it opens the possibility to develop a general mathematical theory of relations between the contents of different theories. So in other words, it is a possibility to develop a mathematical theory of translations. And it is extremely important because in mathematics, most of the time, a result, a formula, for instance, is a relation and an equivalence between two ways of computing, for instance, or between two theories. An equation is an equality between two things defined in two different ways, which means these are the result of applying two theories. So this means that mathematics consists in building relations between different theories. On the other hand, you have the possibility to study this thing in a mathematical way. You see, the reality of sites and theories on the one hand and toposes on the other hand, opens the possibility to develop a mathematical theory of relations between the contents of theories. Okay, so Caramello has applied that to many different contexts. And I want to mention the application she had made to Galois's theory because by this very simple idea, she was able to generalize, and for me it was really a surprise. I didn't expect that at all. She was able to generalize the Galois formalism of Gotendic. So let's recall that Gotendic had characterized the categories which are equivalent to categories of actions of a group. And so the simple idea of Caramello is to characterize the sites, which means the category endowed with a topology, such that on the associated topos is equivalent to the category of actions of a group. And this is much more than the rules as the previous result. And then when you do that, you realize that the only possible topology on the topology is the so-called atomic topology. You can describe it. And so this is a property of the category itself. And you can explicitly write what are the needed properties for a category endowed with the atomic topology to be Galois in this sense. And here you discover, or Olivia Caramello has discovered, that in fact, in mathematical practice, there are incredibly many categories which verify these properties. She has also discovered that in fact, through this formalism, there is a possibility to unify Galois theory and an important part of so-called model theory in logic, which is the so-called Fricé construction of theorem. So this means this is yet another unification. You remember, Gotendic had unified Galois theory and the Poincare theory of fundamental group. And the other is yet another unification with the so-called Fricé construction in model theory, in particular with the notion of homogeneous model, which is imported in some part of mathematics. And this also is completely new. And it is just an easy consequence of the basic idea of bridge, of toposteritic bridges. So by now, of course, I want to mention the relationship we built with Huawei. So in fact, I first heard about Huawei by Jean-Pierre Bourguignon, the former director of USRS. But my first meeting with people from Huawei was just three years ago, in July 2017. I was invited to give a talk in a conference on the Matt Vision Forum organized by Huawei in Lyberon, South France. And it was my first contact with these people, Benjamin Coulomere, Mehrouane Débat, Jean-Claude Belfure. And in fact, I didn't imagine that I could talk about something which could be interesting for them. So this means that first I refused and then they insisted and I refused again, but they insisted again. And in the end, I say, okay, I come. And I gave a talk about opposites because I had been convinced by Caramello that opposites were extremely important, were some of the most important objects in mathematics, exactly as Crotendic had said repeatedly. But I didn't think that my talk could have consequences. So I gave a very informal and relaxed talk. And I also had informal conversation with the people there. So it was very nice. And it was especially nice because I thought that nothing was at stake. Once again, I couldn't imagine that this very abstract stuff I was talking about could be really interesting for these people working for an industrial company. But in fact, I was completely wrong. In fact, I discovered later, Caramello also discovered that these people from Huawei, especially Jean-Claude Belfiore, Mehrouane, Benjamin Coulomere, and others were really interested. But what we could say about opposites and categories, we were listened to. And so the consequence of that, of course, for us, was first the support for a top-class program at IHS, and also the support from Huawei for a PhD program for the PhD program of Olivia Caramello in Italy. Certainly it also played a role for IHS. So the support from Huawei to IHS was already very important. It became even more important. And the Huawei people know what was the role of this meeting. And the most like another really incredible for me consequence is the fact that this talk, this informal talk I gave also had consequences for some mathematicians working for Huawei. So they became really interested into toposites and categories. Some Huawei mathematicians attended summer school about opposites. Caramello, Alan Cohn and myself organized in promo two years ago. And Jean-Claude Belfiore began to work on toposites and related subjects as I will immediately mention. So stacks, derivatives together with Daniel Belka. So Daniel Belka is, of course, an academic mathematician, who had just officially retired from university. And he was recruited by Huawei as a consultant. And in fact, I will immediately say he has done wonderful, already done wonderful work with Jean-Claude Belfiore related to categories and toposites. And in fact, Daniel Belka had become interested into toposites before, independently from Caramello, from myself, from Alan Cohn, from all these people. But probably there was a contact between Huawei and Daniel Belka because of the new interest of Huawei for toposites. Or anyway, I suppose that. But anyway, Ben Kahn began to work with Jean-Claude Belfiore in particular. And this work for me began to become concrete in a workshop, a deep dive in topos theory, which was organized at the Huawei mad center in Paris at January. And so there was talk with several people. So by Caramello, by myself, by Daniel Belka, by Jean-Claude Belfiore, also by Juan Pablo Vigno. And in fact, I already was very impressed by how fast these people on the Huawei side had gone. In fact, even today, I am very, very surprised. Very impressed. And yeah, if I need wonderful. And in fact, Jean-Claude Belfiore and Daniel Belka recently sent to me some papers, they have written, so they are still drafts, but the drafts are already quite consistent. And so the name of the title of the first draft is Topos and Stacks of Hypno-Roll Network. Hypno-Roll Network is the basic architecture for artificial intelligence. And here you see topos and stack. And stack is another notion invented by Grotendic under the French name Jean. And it is closely related to topos. And yeah, I have just lifted from the introduction of the paper, the content of the paper. And this paper is interesting, I guess, for deep neural network, but it is also very interesting on the mathematical side. And you see all these notions, sites, topos, stacks, fiber categories over a site, the relationship with logic, the relationship with semantic, which I already talked about, the true category of stacks. So here you see the model, the coolant theory of models. You see homology and the homotopy invariants, which here are really invariants of topos. So all of these appear in the paper, in these preprints. And this is for concrete application to artificial intelligence. So this is extremely impressive for me. The paper is very nice to read. It is a pleasure for a pure mathematician. But I also understand that it is important for introduction. And in fact, Jean Claude Ben-Fior told a recently wrote to me that they already have very interesting experimental results. I only quote what he told me with some unexpected logical behavior. Okay. And by now, the last thing I want to mention is that, in fact, Caramello, ourselves, completely independently from these people, is working in particular on general questions, which are very close to what Ben-Fior and others also work. So for Ben-Fior, this is because of the interest for deep neural networks. And for Caramello, it is for general interest of the development of theory of category. So for Caramello, it is a theme of relative toposis. So the study of morphism from a topos to another topos. You see, not to consider only one topos, but the structure consisting of a topos over a base topos. And in fact, she's, so on the EHS website, you can find an EHS preprint of 100 and 70 pages on the general study of morphism and of toposis characterization in terms of sites. There is also a secret paper of Olivia, which I hope will soon become public about the full expression of morphism of toposis in terms of defining sites. And there is presently the development of Caramello together with PhD students Ricardo Zanfa of the general study of relative toposis, both with the language of geometry, which is largely the language of stacks. You see exactly as in the Ben-Fior, Ben-Camp paper. And also the study of this situation from the point of view of logic. On the earth, this is very interesting, because when you study that, you see higher order of logic coming in. You see logic of second order. And you can imagine that if you want to get logic of order n, you have to consider not true toposis, but a chain of n-toposis related by morphism. So this is presently a work in progress by Caramello and her student Ricardo Zanfa. And I think this is really very, very important. So I want to conclude with some lessons for the development of mathematics. So the first lesson for me is that applied mathematicians or pure mathematicians who care about applications may be very open-minded and have very good judgment. So this, I really learned a lesson from them. It is a pleasure for me to talk with them. I am very impressed by also the speed. You see for instance, Grotendi complained in the 80s that he complained in the 80s that the mathematical academic community was not taking seriously enough the notion of toposis he had introduced. But here you see, I talked informally to people from Huawei about toposis just three years ago. And three years after that, they are already not only working on toposis, but working on it in a really interesting way, both for applications and for theoretical mathematics. So this means that here there is a lesson for the academic world of pure mathematics. We really have to learn something. Yeah, there is also the lesson that, yeah, I want to say that when Grotendi insists that something is important, it is a good idea to take him seriously. And you see, this is true for the subject of toposis, and it is also true for the subject of derivatives, which was a notion invented and developed by Grotendi, completely alone in the 80s. This notion was not considered by the academic community, almost not considered, but by now it is used by Daniel Benka and Jean-Claude Belfure on their work on deep neural networks. So for me, it is very striking for me that this notion which Grotendi introduced and he wrote, I think, hundreds and hundreds of pages on the subject. This notion is finally studied seriously by people interested in applications. Here, I want to make a suggestion which would be, so maybe a suggestion for Huawei. We would be to create a new institute devoted to toposis on related subjects on two applications, for example, subject of derivatives also. I really think that toposis is a huge subject, and it deserves to be developed in a systematic way with important means. This is very important for pure mathematics, and then we can see that it also can have applications. And the last thing I want to say is the price of EHRS as an independent institute. So this is something I experienced by talking with mathematicians. So EHRS professors, professors Kahn, Gromov, Konsevich, these people are able to think and evaluate for themselves. They can have personal opinions. Whereas a problem of the mathematical communities of the academic world is that too often it is submitted to group thinking. You try to talk to somebody, but you realize that somebody is not ready. He doesn't want to formulate or to conceive a personal opinion. He's just waiting for the opinion of other people, for the opinion of the group. But EHRS is a small institute, and it is independent. It has professors who are strong personalities, and they are able to make personal judgment. And I think this is also very important for mathematics. So thank you very much for your attention.