Deformations, symmetries, operads & noncommutative geometry (partial)

0:00 There is a new group of stressors, a new group of stressors that we call the book of Lie, for those who at least admit, even if we know, without seeing it in the same way as Lie, there is a deep theorem that says that the set of hyperbolic mechanisms that establish an ordinary differential equation of order n constitutes a book of Lie of infinite dimensions if n equals 1. Why? Why is the book of Lie of infinite dimensions? Because a differential equation in order 1 is y' plus a0 of x equal to 2, okay? And so, when you fix the value of y in a point, there is only one integral curve, according to Cauchy's theorem. So if you take a point on the vertical, there is always only one integral curve at the bottom. This means that space is an effect that leaves the solution stable. And all the solutions are the same. It's just that, it's a pile of leaves. Moving like that in the leaves, or making transformations that look like the originality of the leaves, there are a lot of possibilities. Even if we can move the transformations of the leaf network, it depends on the arbitrary function of the variable. And that's the infinite dimension, what we call the finite dimension. I've seen you see things moving. In my opinion, that's how projective knowledge is.

2:30 In fact, this globalizes in the way I explained to you in the last session. So here, I also talked a lot about it because I wanted to characterize it with several X values, which are differentials, and here it's the same, it explodes in 30 minutes.

5:00 So now, I come back to the analogy with... Excuse me, but are you looking at, because it's a question about the dimension of the Galois group, when you look at the jury? And the dimension of the groups of different species, of the groups of life, because there is a relationship between them. I can't answer that, but I suspect that... because here it's the group... I don't know if we can talk about the dimension for the southern groups. It's the order, and the order here is really a factorial. It's a factorial of n. The maximum order of the group is a factorial of n. And then the maximum order is n to the square. And here it's true that there is no difference. There is no jump between 4 and 5. The jump is already located at dimension 2. For the group of lists, that is to say that from dimension 2 there are some that are not resolved by the quadrature. From there. The Riccati equation normally, in general, is not resolved by a simple quadrature. So the jump is located earlier, if you will. And there too, in the case of differential equations, there is really the case of the big objective which is exceptional. And he has more mathematical content than the equations in his study, we can see it. Even if it's because there is a knot here, I don't really know why. Because there is still a question of simplicity in group 5. There, there is still an analogy. That's right, it's this theorem that I have here, which is a profound analogy, it's one of Lie's theorems, which says that if we start from a differential equation, it's the ordinary, so order n. The analytical coefficient, where it's infinite, all of that is local. You'll notice that I got confused so that it's coefficient 1 here, because I used local, I didn't use the coefficient which is here. We're going to say that the resolution of the quadrature path is to have the only permissive operations between the algebraic operations and take the primitive function that we obtained from the previous operations. If and only if its group of linear symmetries, so the set of diffeomorphisms that stabilize all the graphs of solutions, This group is divided into subgroups, each of which is distinguished from the other, and the intermediate dimension is equal to 1. The dimension of the quotient groups is equal to 1.

7:30 So we say that the group is resolvable, because we demonstrate later that, in fact, when we take the hooks of Ranget with Ranget, so the commutators, and at the level of the algebra, it means that we take the hooks, we end up with G . And we come back to the subject of n-2, n-1, n, n-1, etc. up to the identity when we take the commutators. And so there, you can see very well that the analogy is very, very strong and that Kant added a primitive. We all know that a group of I has a group of I that can be seen as the whole of the vector fields that are invariant under the group of I. It means that if we take a vector under the tangent space in the group that is seen, it has the identity. We can, by pushing the whole group by translation, the differential of the translation to the left, for example, have a global lecture chamber which is I. And so the I hooks between these lecture chambers can be interpreted as abstract hooks of selectors on the tangent space, hooks of an abstract idea. And so we show that, locally, there is an exact correspondence between this third theoretical fundamental of I. If we have a string theory algebraic algebraic algebraic algebraic algebraic algebraic algebraic Would you like to answer them, yes? Algebra, Debye, Thames? Debye, Thames, Thames, Thames. Connex, simply Connex, Uniques. I've never really seen the demonstration, but the idea is that the way you push, is that, locally, it's just a matter of defining an exponential axis, which is in the middle of the course, or a... I think, often, if we go through the field of ADO, which says sub-algebra, and it's not unique, it's not quadratic, of an arithmetic algebra.

10:00 At Sotus-Lee, I don't remember how he does it, but it's his third fundamental theorem, which says that if we have constraints that satisfy what we can satisfy, the very inferior antisymmetry, and the sum, there is a unit that has to be equal to the sum of the terms. Which is not the case in the case of the Galois correspondence. Ah, it's not the case in the case of the Galois correspondence. You have the answer, but we demonstrate that there is a linear formula with the sum of the terms. I don't remember exactly how, but it's easy. So that any problem you solve can be solved in the same way. What do you mean by solving problems?

12:30 I don't know. I don't know the definition. It's a mistake. And all that, it's like watching a video. What is it? What does it do when it's not solved? When it's not solved, it's when you take the hooks from itself with its own hands. At the end of a certain number of steps, you don't use them. You stack them. You stack them. You stack them. Yes, that's the commutator sequence. Yes, that's the commutator sequence. So that's nilpotent. You get G1, first with half, then GG, that's G1. And then you do G1, G1, G2, and then G2, G2, that's G3. So nilpotent is a little stronger. There are fewer nilpotents than the other two. You take GG, that's G1, then 3G, 2G with G1, that's G2, that's G3. And in fact, it is in France that we had a specialist, who I have not yet met, but who classified Emotion 7, there are 21 pages, it may not be accessible, very explosive. Here, he said that everything is resolved. After, for the theoretical theories, the fundamental theoretical theory says that he classified all the groups that act on an event of Emotion 2. In fact, if there is a Roe, there is a need for an equation and it is a matter of interest. This year, in 1992, I think that Oliver, Cameron and Piquant-Salès-Lepes, in the journal London Mathematical Society, determined the classification in real time and in a test.

15:00 However, in what I am going to expose in January, on the problem of Riemann-Helmholtz, everything is done very rigorously. Key terms include invariants for pairs of points, i.e. if you want to act as a group on a three-dimensional space, you have a certain number of transformations, R for example, and you will ask that when you take two points, you make them move in all possible ways, by any group of elements, and you want a function of P1 and P2, which I will call distance between P1 and P2, generalized distance. The distance between G1 and G2 is equal to the distance between G1 and G2, between P1 and P2, for all G in the group. An invariant, what he calls a grand giant. And that's really what he was trying to do. In the case of complex...

17:30 He's the oldest on the planet. Really? There is a common philosophical principle between Gabriel and me, which is that we embrace all the solutions that we do not have access to, that we do not know, but we only authorize the gestures of calculation that focus on the concrete data to which we have access, which I may not do so much today, but maybe next time. I would like to make a little bit of a goal too. There are all kinds of philosophical terminologies, what you call the mixing, that is to say the fact of... We have to consider globally all the solutions and then virtually. There is a kind of individuation that occurs little by little as we get more precise. So there is something that is a virtual passage, a kind of actualization. We are in the middle, I agree with you, and it is permanently in the mathematical classifications. And I will come back to his method in the classification because precisely it is a way of fighting. There is also a problem in terms of efficiency. To go back to that, I would like to make general remarks. I think it is very important in mathematical statements to maintain the attention of questionnaires. There is a sense of openness to what I call the mystery, in quotation marks,

20:00 even in the most abstract techniques. There is a kind of questioning reflex and a reflex to see that the statement is unsatisfying. It's a researcher's reflex, a mathematician's reflex. It's that a statement says that an infinite part of the mathematical reality was very strong as it should be. Even if it's because we don't have an algorithm to calculate it. In Hilbert's theory of invariants, the example is quite the same. We have a base, it was right to make a shit about Hilbert, he broke everything. The truth is that the history of mathematical physics shows that most of the time the mysteries are not found in the past. It's Bolzano, it's Bolzano. But here, for once, what I like a lot about our original program is that the deepest questions are the core of mathematical creation. Especially for the problem of Hilbert and Molls, which will give itself an absolutely arbitrary application. It's an arbitrary function between two points. There is a rise in general. It's a bit like in Grothendieck, something like that. We try to get as close to the root of the content as possible to flood the point. And that, in my opinion, is one of the signs that we really see the extent of the questions. Then there is the idea of saying that a little better than what I said in German, of doing something written. I don't know if it's really a good term, but it's a kind of philosophical reflection on mathematics, but general, with the goal of classifying, perhaps in an inexhaustible way, the principle of the director. It's not well written, but it's about the structure of the theorem, and trying to see all the virtualities in it. It's not done yet, it's a project, I didn't have time to develop it, but I'm starting to develop it. It's just to say that the art, the art that really sees things, it has everything in it, it has not only, it has the perception of shapes, it has the mystery, but it also has the technique, and in every technical gesture, there is a perception of questions, it seems essential to me, it's like in every intuitive gesture, in every geometric gesture, there is an intuition, it's that the intuition, it automatizes a little too much, even if, for example, such, such, such, it is also necessary to speak, to feel drowned.

25:00 In the case of ancient things and ancient speculations, there is something that is non-technical, but a bit philosophical, but which is not limited only to the lovmanian scheme of genesis. This is also to be discussed by Lovman. There is a second part which says that the enunciation and the mathematical theorization always describe an implicit architecture of questions partially exposed. There is always the imperfect of the party. And what Luhmann said is very true from this point of view, we must try to reconstruct the questions that are at the level of the theory in order to understand it in depth, which is not the case with mathematical texts, be careful, they almost never do it. So, we stop, present, I told you that now we are going to show in conceptualized... Manifestation, Manifestation.

27:30 The fact of refusing to use the word variety in modern science, in some senses, is not the same as modern science. In OCR, for example, there is another reasoning that denies the technical term of variety, denies the term of what I just said earlier, which is complicated. So if you make a choice of translation like that, it's the same thing. I don't know who introduced it, but of course it's a very free term. I'm not interested in memorizing the memory on Helmholtz. I already have a number of pages and I know it's a very delicate term. But I would like to keep the numeric. Because for them, but especially there, there will be another question. The end of mathematical physics, it's not difficult. I don't know if they're going to be able to do it. They're not going to be able to do it. They're going to be able to do it, but there's no talent in front of them. Yes, exactly. I don't know, I don't know. They are numeric. They are not numeric, but they are not numeric. It's a question of the fundamental to know if we are talking about something. Kantor, before talking about Mergue, did he use this term together? Yes, after. After? Before? No, it's after. No, no, Kantor, before using the term Mergue, did he use this term together? So, that's it, we've discovered something. Cantor, Cantor, and then also Manish Pratishkait. Yes, but it depends on the context. But according to the context, it's still different. Cantor's theory, we call it, in 1999, the theory of Manish Pratishkait. It's a very specific theory. But it's already, it's already, there's already a... It's a very strong tradition. There are people who don't agree with each other. Yes, yes, yes. They know that this is not the same thing as the practical values of Einstein in 1955. You can talk about space and dimensionality, but it is not the same thing. We are talking about space. But here, he explicitly adjusts himself to Riemann and Engerich. So, it is not the same. Cantor.

30:00 No, rather Cantor, because... This is the site. This is how it seems. Cantor, in the history of mathematics, I could not quote him. What begins to emerge in Cantor is the difference between the two problems. Everyone saw that there was something we couldn't ignore. It's easy to understand, so everyone can understand. So, everything is local. Local means we don't specify the words. Everything is implicitly local. Well, it's not implicit. No, the generic point is not implicit. It's not implicit. So, you follow Hawking's interpretation. The dominance of the local world and the dominance of the local point of view. I would like to say that there are hypotheses that are not always expressed in the statements but in the denunciations. And when they must be, when a genericity is necessary, it is what it is. We think of two in the pursuit of the other, for a C that depends on whether to put itself on a small point of view to say that it coincides.

32:30 But this is a diagram where I make myself appear and I only have two movements with A and B chosen. And you have to see it for all A and for all B. The synthesis between the two figures. The first one says that there is a space of liberty that unfolds in several directions at the same time, and that this space of liberty that unfolds at the same time, when it is followed by one of the others, is captured by a third of the other. Already, to think about that, it's difficult. I don't know how to explain it. And so, the technical method in mathematics is to diagram to better excite conceptual appropriation by provoking interrelations of subreactors. So, we're almost at the end of each chapter, if you like. It's not a sentence I wrote at home, it was me who wrote it all by myself. But I insist on diagrammatizing science, which requires effort. The purpose of science is non-trivial, so it is to gain attention. When I say to excite, the term is chosen because it is visual, isn't it? But it is also to act on the speculative system that is in us, because we are a bit philosophical. And so it is by acting on the system that is in us, the rational being of our fate,

35:00 We are more or less the same in terms of conceptual appropriation. The diagram is not necessary. And what is very important is that the integrations come together. All technical development is a power entity that demultiplies the angles of the earth. This is a very important point. In a complex material. The data is picked up and the intuition allows it to pick up an intuition or at least the discussion. And so, for the time being, I'll put it here. It's filtered, it's filtered. Well, after, n equals x, which is equal in place of xa. For a C which depends on A and B and not on B, if you are in the elaboration of a text, of a writing and of a thought, do we have to do it? Can we do it?

37:30 The answer is no, in general, because it depends on the group. So that means that in the economic presentation of the text, we have to admit that here, we have to place a ball saying that here, in fact, we accept to say the irrefutable definition of the donation, There is an evolution of Engels, for example, to put it back into practice. So this principle of composition is what gives you the principle of composition. That's what gives you the principle of composition. It's the way it's played. And it depends on the system. The way we write is not the only way it can be transformed. And so, at the end of the day, it's the way the composition is played, but we don't really see it, because the system is generally not us.

40:00 It is in the development of the infinitesimal transformations of the eigenvalues of the vector fields that things will become clearer, and it is one of the reasons for which, almost never, what they give is the eigenvalues of the vector fields, which are less clear, but they give the list of six vector fields, in three variables, without hooks, and without recomposing x' equal to f of xa, which would consist of affixing the first vector field in a time a1, taking the point f of x a1, And then take the second vector field and integrate it into two, then into three, then into four, and obtain the same expression as the x-axis. But, very often, when I look at this in certain cases, it gives very bad results, which also depends on the order in which we take the vector fields, even if the vector fields are normalized. And it turns out that in the years 1900, Cartan-Fadeau cites the work of Calculating the Egos of all the Egos. The inflection of the donation has, by the way, what I just said here, I didn't think I was going to say, has, has, has, how to see it when it continues, so that it allows us, and, and, and, the essence, the constraints that define, that make a mathematical concept, how to understand the constraints, a, a set of all the technique that we know.

42:30 Question, is this abstract formulation of the notion of continuous movement an adequate proposition to reinforce the notion of mobility in space-time? Is this how we think we are? I don't know how to answer this question. There is an answer, I think, mixed of course, I don't know anything yet. By the way, it seems that in the first of Helmholtz's, he reads this problem, but I have not yet been able to read the paragraph, because I do not say the name and I cheat on one page in 3 or 4 hours. Well, the fact that we are in a movement, or in a space, you asked a good question, is it the mobility of the space, or the mobility of the space? It seems to me that this decision would have been made by Linus. But I don't know what he would have thought. For now, I'm dry.

45:00 Yes, I don't know what he would have said. He would have said something like that. Maybe, yes. But he would have answered something like that. He would have answered a question. They do math, they advance, and that's true. They are all magnetic texts. Not all of them. A large part of them are written in literature. I know that we speak of Gallois, we speak of Gauss, we speak of Leibniz, but they are all written in literature. If you take the EGA, it's not going to disappear. I think that in 100 years it will no longer exist. By the way, do we know who will be with archives on the web in 100 years? Yes, everything that goes into... I think that in November... I don't know. I have a question that I may have answered at the beginning of the conference, but when you talk about this analogy of the theory of Galois and the theory of Lie groups, I, because there are a lot of people, I know this group of Seyssens who, the others, you see, who know how to do a kind of Galois theory. I don't know, but it's a different kind of project, isn't it? I don't know, but it's a different kind of project, isn't it? I don't know, but it's a different kind of project, isn't it? I don't know, but it's a different kind of project, isn't it? I don't know, but it's a different kind of project, isn't it? I don't know, but it's a different kind of project, isn't it? I don't know, but it's a different kind of project, isn't it? I don't know, but it's a different kind of project, isn't it? I don't know, but it's a different kind of project, isn't it? I don't know, but it's a different kind of project, isn't it? I don't know, but it's a different kind of project, isn't it? The question, because in this book I read that, well, it's a remark like that, not precise at all, that in fact, Lee himself, he had this idea in his head to make, let's say, a theory of law in the case of the differential equation. Is that true or false?

47:30 That's what I said. This theory is what I said. Nothing else. But I missed the beginning. No, that is to say, the second theorem that says that it is solved by the literature, if it is the only one that is solved, is that of the boxed groups that are distinguished by one another. And it corresponds in the example of the theorem of Galois in the subject of geometric equations by radicaux. That is to say that the group of Galois is the result of the groups that are distinguished by one another. That is to say, he made the Ego-Lowex reference. No, he really wanted to have some details. Apparently, he never really understood the theory. He understood the statements, but he did not really understand the details. And then there is another very important thing that I have not said, which is the differential equations. All those that we have solved, it is because there are symmetries that are hidden. These are the things that we call, that we learn in preparatory classes to solve differential equations. Homogeneous, the constant method. All of this makes it seem that there is a symmetry of order 1, a group of symmetry curves at least of dimension 1, and yes, it would be an honor to show that everything that was given at that time, it would just lead to say that there is an infinitesimal symmetry of the differential equation, and then that's it. But it's quite scandalous when we do this report. Because there is a unifying principle that we do not understand. We can perhaps do a group work session with a little more technology. Today, I have a lot of work to do on mathematics, maybe more maths on the axioms. And also, as I said, it is the theory of dimension 2 classifications. This is very important and I wanted to highlight that there is a distinction, but I have to repeat it because I do not have the time, between groups called primitives, which do not establish me perfectly. And those that are imprimitive and that stabilize the approach. And Sopisky has classified primitive groups quite quickly in his memory.

50:00 There is the projective group, the affine group, plus translations, the combustion of the directs. And there is the special affine group, the intros, in fact, that are primitive, that do not stabilize any family of courses. Stabilizing a family of courses means that... When you make a curve act on the space of a point, a curve is sent on a curve of the other, not necessarily the same, not necessarily point by point. It can slide one on top of the other. And then, you remember a bit, but it's an impugnation, it's an impugnation of original algebra that I had already used. So it's to say that when you defend an algebraic structure, you change a bit the constraints of the structure, it's the same in fact. And so, in the case, I say it, in the case of Young-2-D, it's simple, it's all equivalent, it's the same, it's well understood, but for me it's something that I don't have the time to work too hard on. And all this is in the development of the 80s, of course. The differential equation, it's a little later, but it's Young-2-D, complete classification, it's in, what is it, 6 or 7 years after the discovery of the second group. You see, it's very, very fast. In 6 or 7 years, to have the theorem that gives the 30 components of the second plan, it's very, very fast. No one worked next to him, there were no curves if you will. Not often there are people who work, who are competitors at the curves, who progress together even if we are not. And there, what I find wonderful too, and I would have liked to talk about it today, but I did not have time to prepare that, is that in the case of primitive algebra, it is to establish a type of curve, there is an induced action on the space of the curves, the space of the curves is the transverse. So there is an induced action on the transverse which becomes an immiscible action.

52:30 And when we look at it in detail, to classify in dimension 2 those that are additive, it consists of taking those that are non-negative and looking at how they can be elongated in this way. And in the first theory of transformation, there is a whole theory of elongation, but which in my opinion gives a general framework to this idea that when you look at a non-negative action on a sheet of paper, you can reconstruct the group with no problem. And it's quite wonderful because there is a very quick thing that is immediately put in place. We think that if we have dimension 2 and if we have primitive reactions in dimension 3, for the primitive reactions, we will have deflections and we will be able to see how it will turn out. And Samy has spent the rest of his time to do that. What we are going to do, perhaps, is already well understand the volume 3, the 456 or 457 volumes of the volume 3. In my opinion, this is the principle. And so, maybe I'm just right. And then, in fact, to go and work a little bit on what he didn't do. I don't think it's a complete question. But we can have... There is also something that I defend about this, is that when we are in the concrete of the calculations, ideas come faster than when we have an abstract language. Except if an abstract language is so well worked, that it fully corresponds to the intuitions that we had in the concrete things. That is to say, there, for once, I think we are working on the same line. It's algebraic, it's combinatorial, it's algorithmic. To put TGF in the space, for example, in the TGF, you have to classify local actions, because any global action will be located in the main part.

55:00 It was done a little bit in 1905, but not for the TGF, but for today, if there is a seminar, it will be done. So, I would like to make a reference to the ideal mode for the specificity of the attack. At the same time, it is important to make a translation in English of Hemox by the question by Brede, since we have the reference. In the last lecture, I talked to a lot of people who have studied a lot of things, such as Westmark and Forman, or Mann. I think the code is really easier to understand. What's wrong is that I don't want to react to my intuition. Yes, I think I could do it later, but we won't do it later. I'll explain, there's a note that says that, and it's in the second part. Ah yes, there's a note in the second part. This is a little more beautiful, x prime equal to x, y prime equal to y, plus a constant, and z prime equal to z. When you linearize, it means that all the terms in between disappear, so it becomes Z equal to Z, when you look at the action induced on the tangent space.

57:30 So here, Helmholtz said that we are going to infinitesimalize, we are going to suppose that it is transitive. If we look at a transitive material, we see that there is no tangent space. This is a great way to learn interesting territory about geometries and various images of a non-linear space and morphisms and how to do that. So, I will finish this philosophical part of the story by deploying several morals and general morals when we want to get interesting examples. We will be looking, we will receive two kinds of coordinates. We will try to find non-communicative coordinates with basic information or objects, but we will try to link to the Grotendieck paradigm as close as possible. Coordinates will form a kind of non-communicative link or an operand or algebra or something like that. Plus, we also want to try the general principle that non-productive geometry or basic constructions must be non-productive species as well. Up to now, for example, there are just a nice and very interesting theory of the formations of the surface rings, but the parameters of the formations are too limited. So you get questions about homology and things like that, which are very interesting, but it's very nice, but the parameters are still in use. I will also have in mind a general metaphor. Namely, if you are imagining the universe of geometry, then the universe of geometry should somehow be embedded in it.

1:00:00 And the metaphor I have in mind is that the universe of geometry is embedded in the non-universe of one. Approximately, the life theory of singularities is embedded into the theory, say, of non-singular objects in cumulative geometry. When you deform a singularity, it generally ceases to exist, but such kind of deformation is called unfolding non-deformation. And when we embed the commutative object in the non-commutative geometry and then start to deform it, you should not expect that the fibers are similar to your initial object. They lose a lot of characteristics of your initial object because the initial object should be considered as a highly degenerate particle. Okay, and then the general expectation, which is more or less logically connected with this one, is that if something in community geometry was rigid, so it was a basic object according to the common error, it will stop to be rigid in non-community. This is the general expectation. And let me now illustrate all these principles and philosophies by examples. Here is an example of Gauss. Example number zero will tell you that even integers stop to be rigid, they are not immutable, only at first you don't see the deformation is somehow related to non-immutable geometry. Gauss suggested the following series of integers or thousand integers. And you define the integer n, which is quantized with the parameter v, by this explicit form.

1:02:30 So when v is 1, then you get just the pure old integers. Now, we need a general justification of these things. And the general distribution given in Gauss' time and many decades afterwards. All these terms run approximately like this. You've just shown that many, many other formulas, objects, instructions, theorems in which an integer counts as a parameter can be consistently and interestingly deformed by introduction of this theory of cubic form, hypergeometric functions and so on. And usually... Non-permanent relativity doesn't appear there, but of course it's a very imaginary non-permanent relativity, so let's introduce it right away. Imagine now an associative ring, a couple of elements x, y on the screen, and an element b, which is in the center of the screen. This is very important. And then we have one of the first beautiful examples when Newton's binomial formula deforms accordingly to Gerrard's definition of deformation of features. So this is the formula in which, if you disregard hours of day, then the deformed binomial features. And the deformed binomial of the equations is expressed by a deformed vector, in exactly the same way as for the classical A to D equations. So, in fact, you have Newton's binomial for the deformed, and the left-hand side is essentially Newton's plan, and the decommutator X and Y now do not compute the equations, the decommutator is the same. And then included by mathematical coefficients are the gaps. So, let's consider the example of one group, which was essentially the subject of the lecture 17.

1:05:00 Symmetries lost and found for the morphisms of one group. The quantum plane will be some imaginary spectrum of non-convenient degree. We take two free elements x and y, whose pure connotation vanishes. Here I use slightly different connotation. The whole thing is square. And let us consider what kind of automartisms has this quantum plane. So we write this formula x'y' equals to xy to which we apply a matrix ABCD and we ask what kind of solutions. Well, first of all, classical symmetry is if one easily sees that the vector square is not 1 and ABCD parallelized to mu, then just D and C equals to 0 and you have essentially a variable of things. But since x, y did not commute, we should allow a, b, c, d not to be commuted. And then how do you calculate the necessary relations? You write down some stars, and then you compare the additions. x squared, you get relation a, c equals 2 minus 1, c, d. Then at y squared, a, d equals 2 minus 1, c, d. And that is the relation when you compare the additions of x, y. You get two expressions at-q-1cd which should be equal to ga-qbc. You denote both expressions. You declare this quantum determinant of the natural state of the object's polarity. And you should work a little bit more in order to get the quantum group ga-q2. You should do two more steps in order to get what is called quantum 2 and 2. First of all, you should add what is sometimes called missing relations.

1:07:30 Declare that you want not only the matrix itself to act as a kind of linear endomorphism, but also the transposed matrix, not just the matrix itself, but also the transposed matrix. Then you get additional relations. And then of course you might be interested to see which matrices are inductible and it turns out that exactly those are for which we want to determine. It turns out to be a central element and this central element can be made invertible by the algebraic which is a new variable T multiplied by the determinant minus one. So what you get, you get the following, you have three or four spaces, the quantum space, the quantum plane, which is space related to physics, you get the non-communicative spectrum of relations between A, B, C, D. The first kind of relations along the matrixes and you get the common group you add additional relations and you may determine missing relations the necessity or the use of getting missing relations can be explained also in various different ways the simplest one is this if you take only the first half of relations and you close them on ABCD you get a very large ring it's obviously graded. Degrees of ABCD are one, the relations are all quadratic, and then the dimension of the gradient path grows exponentially. But if you add the missing relations, you get the ring, which is exactly of the same size as the ring of commutative polynomials between ABCD. So it's really much nicer to have. This is one of the reasons you add such relations. Another reason is that if you add them, you can construct very nicely the wrong complex, not using the wrong complex.

1:10:00 So what happens next? Now let's play the game which, so far as I know, was not explicitly played in the literature. It's very simple and it is mitigated by what we did now, but somehow people were reluctant to do it. So, let's try to define what is the deformation of outward spaces, the deformation of, say, finite-dimensional associative algebra, or if you wish, the common space of finite-dimensional associative algebra with a given support, a given linear space. Okay, you choose a basis, you choose one binary operation associated with that function. Then, of course, the relation itself is defined by the tensor of structure constants, C, I, G, X, Z. Then you get equations on structure constants. If you have some symmetry properties like relativity or linear, you can mean no equations. Then it's cubic relations in terms of X, the quadratic relations in terms of C, this area is quadratic relations. Now in the classical approach, this I, J, K are of course naive coordinates on the space of all structures of physics or whatever algebra we have in our space, just our coordinates, and then we have relations between them and everything is infinite. Now let's try to play the game that we play for one group. Why do we assume that C, I, J, K are commuting if the algebra itself is not commuting? That would be non-commutable. Then we take the space generated by C, I, J, K form of variables. We have some quadratic relations in a non-communicative algebra, which really can be thought of as an algebra, describing the non-communicative deformation space of the initials.

1:12:30 I assume that X and Z go together. I assume that X and Z go together. As I said, in non-communicative geometry, you always do something like that. Otherwise, the process never stops. Yeah, it can be considered also not commuting. You commuting, for example, or just for you or whatever. There is a lot of extra possibilities, but I'm on the stage when I want to produce interesting examples and more of them. So I, in the absence of definitions, I'm adopting arbitrary examples. And therefore, in principle, we might require, if we are applying now algebras over another algebra, we might require an extra algebra on one more algebra. Which one we do not know as yet, but maybe we will be guided by some principle telling us that we should choose another author which is somehow related to this one. One of the simplest examples we have been considering is that we do not anymore assume the multiplication between two series was an associative multiplication. There's a variable element in our answer. It's some kind of binary alteration and the relations we get now take the same form of form, but now the binary alteration itself isn't so, and the only relations you know is so our model of coordination now is we allow our

1:15:00 Classical coordinates to become extremely non-classical coordinates are elements of something and the durations to compose these coordinates is now also new. What I will explain now is a kind of generalization of the construction of quantum proof. It's kind of an example again, but it just shows you that the construction of the quantum proof. So let's consider, for the time being, just a summary project for us, but there's one additional element to the actual presentation. This essentially means that I want to fix this linear space of generating. So here will be subspace generating A, and later on it will be just a method in terms of algebra. And now let's introduce, let's consider two such pairs A and B, and now let's define a category. An object of such a category will be a map, not exactly from A to B, but from A to F times B, where F is a multi-extension of constants and so on and so forth. The algebra and the movement of these new algebras, it should be linear in terms of A1 and B1, so U of B1 goes to F and so on, and F equals G, yes. And a morphism of two such objects, if we had one pair of a Q and another pair of prime Q5, and a morphism is a homomorphism of a social equality, which is compatible in the obvious sense.

1:17:30 So we get a text word and a theorem, which was broken. The first time in my PhD lecture, the moment I took job degree in the building, is the discussion of it as an initial object consisting of some algebra e and some math built from a to b, so a and b are fixed together with spaces of three-dimension radius. And this initial object is defined uniquely after unique isomorphism, the element is a five-dimensional substrate which generates... And what I'm saying is that this is a natural generalization of our construction of quantum matrices. If we choose for A and B the quantum plane and the linear substance, it's going to be generated by x and y, then exactly this initial object will be given by the ring of quantum matrices. It's actually very, very simple and at that time I was surprised that the scenery was very, very simple, a way to construct quantum and the workings of all these arbitrary quantum spaces you reach and you prove it in the same way as you construct it. To reinterpret the game we can be inspired with delta as the so-called internal cohomologies. This time I will consider monoidal category, algebraic representations and some kind of denser problem, light problem. So Alget is almost the same as it used to be, a chi-algebra A together with a subjective map. Delta algebra always stays A1 with A, only I do not see the generative of A1 maps into A, which I think is just a gesture. So we represent algebra A as a quotient algebra of a three-algebra theory. And Morpheism should be linear on A1 and A1, so any linear matrix descends.

1:20:00 And white product, essentially, is the only natural thing that you can produce by taking tens of products of two subspecies of two variables, A1 and B1, and then inside you look at what the relations should be. And the restatement is then simple but very clear. The restatement of previous theorem is that you have a functorial isomorphism between two spaces of markings. If A maps to F, y to the left of E is the same as to map your initial probability of E to F. And this is the general notion of non-forms and objects in a category which is kind of similar to that of a product. And the previous theorem tells you that such an object exists and more or less tells you that it doesn't exist. Now, it is a pretty nice categorical statement in this form, because it shows you that you can enrich your category of associative algebra in presentations. You can enrich, instead of imagining that there are two objects, which are just set in the multiplication, the position of the morphism. You are imagining now that instead of the sets you have objects in some another category. Objects in some another category. So you have a kind of internal homomorphism that the errors are inertial. Again, this is also important in the formalism. But you see how, whenever I'm trying to explain this initial question to Maria, of course, I feel simultaneously amazed by the ways in which

1:22:30 So it makes us think of things and quite certain reluctance to do it because everything becomes so compact and in a sense so obvious that whenever you state it, you sort of lose the whole intuition you got when you started it. And the way we are thinking it is the most classic style. So, moreover, the non-communicative space of morphisms in associative algebras, the linear of generators, is represented by an associative non-communicative algebras as well. This is a very general construction of algebraic functions. Probably it's time to make a break right now, then I'll start reminding you about what the rest is going to be. So, let's make it ending. Please, there's not too much ending. One last paper of David Sparks. We were to explain it in two words. I will be extending this last construction from the category of circuits and algebras to the category of operands and, in some cases, also to algebras. First of all, what is a collection? So, we will consider linear operas, linear semantic operas more technically, and that will state the notion of collection. It's just a sequence of linear spaces, a ring, and the semantic group, and then x for each of them.

1:25:00 Now, to get an operant from a collection, You need a lot of multiplication rules. Whenever you choose some elements with an a of g, then a sequence of elements with an a of k1, so an a of kg, you should be able to compose them into one element of a of k1 plus 7 plus h, rather than just axioms. Flores, Heinz, and so on. But I will not be writing down them formally. I want to convey the intuition, which is underlying to the result of this lecture. You are imagining some uncertain linear space, such as Earth, and you are imagining that elements of a limb are poorly delineated functions. On this linear space of n variables, with radius in the space itself, so instead of C or E, so just polylinear functions. How SN acts in this model? Okay, each polylinear function you can permute out until you still get a polylinear function, so this essential permutations. And what does this composition do? It takes a polylinear function of j variables and then, instead of variable number a, it inserts another polylinear function of kj. So you get a composition of greater duration. And this composition of duration is... Essentially, all examples, sorry, all accents of abstract and abstract culture impute more or less obvious and only universal properties of this general intuition.

1:27:30 Of course, this general intuition can be illustrated formally by just choosing a concrete vector space and considering the impact for linear operations on D with vectors in D. The nth part of the orthoradic antinormals of the inner space V is just the space for the inner maps of the inner space. And then, here, apply the composition rules and exam actions. I've described a little bit of them, but any vector of space defines the orthoradic antinormals. What is a morphism of operands? Well, in general case it should be obvious. So you have two collections, A and B. So a morphism is a set of linear maps, A of n to D of n, or A of n. And it should be compatible with the musicians. Can you understand? And symmetry. And symmetry, yes. And graph. And in particular, you had a general notion of an operant A and the complete notion of operatic endomorphisms of the vector space B, so you can see that in principle I meant from A to operatic endomorphisms of B and morphism of O. And by definition, endomorphism is a structure of an A algebra on the universe. So, you have two batches of definitions. One, which defines, sort of classifying the object, which classifies various types of algebra, various composition laws and identities between them and so on. And then, you have another batch of definitions which tells you that all algebras are a given part or a given class, are actually morphisms of operands. In the list of analogies I will show you later, analogies soon, analogies between associative algorithms and operands, one collection will be something similar to one linear space, and an operand will be something similar to an associative analogy.

1:30:00 So, in particular, in the statement of main theorem, I use the notion of algebraic representation. Here I look at the notion of an operatic distribution. It means an existent one. You should be able to prove that any collection A1 generates a free opera. And these three operators generated by a correction correspond to the terms of the algebra generated by the linear series. Do you think it is fair and realistic that the operation of the distribution laws and the action of SN on the space... Of course, of course, of course. And again, as I said, I didn't write these reactions. But this relation, of course, follows from the intuition. If you take an iteration, depending on many arguments, and then you insert into it other functions, so when you apply the action of SM, transforming this group of arguments into this large function, then, of course, it's clear that those are also functions. What is hidden there is some combinatorics of symmetric groups and products within symmetric groups. It's very elementary and, as I said, any of you can reconstruct the necessary combinatorics as soon as you can. When you write them down, you're going to lose time. Okay, so let me repeat these analogies. The linear apparatus will be considered here, it's not always in this way.

1:32:30 Here we consider the generalization of associative algebra in which vector space corresponds to the vector, tensor algebra corresponds to the vector, the general associative algebra corresponds to the general algebra, and finally tensor product, tensor product between associative algebra corresponds to white product between operands, white product. The collections just componentize the sort of product. The component of like product number n is the denser product of V1n, V1n, and everything else is the denser product of the data given on the new characters. SN representation is obtained by diagonalizing SN representation V1, V1, and V2. Composition, everything is just. And now what I want to say is that there are not only analogies but actually associative algebra are embedded. If you have an associative algebra, well, without any, then it is an opera, very structured. I think it might be very structured. But any associative algebra is just an opera which is concentrated in degree one. There is only a periodic multiplication that can be non-trivial in terms of the usual multiplication of a specific function. So, is it more than an anarchy? I mean, is it true that algebras are very specific in terms of variables? Yes, this is also true, but then you will get to a very good question. But then the point is that you should introduce the notion of paint. And the usual operas are classified, I think. And actually, I have found at the very end of this talk, I will very briefly explain that the notion of opera itself, the main theorem that I will state later on, has a very, very vast generalization. It's also described in our paper, which includes the world of opera like a much more complicated object like a cross.

1:35:00 But any time when you wish to classify some kind of things in the same way as you classify algebras, you should change some variable parameters, these are what you should classify. These series of definitions and statements about such a class are also operative because the presentations are just better than a system graph of an A and a collection and a pre-operation rate of its collection in the subject. When you say A1 is my language, by the way, it's only my language. No, I mean each component is my language. And then we can define the category R, A, double arrow, B exactly the same way as we do now, with the gesticulates everywhere. Algebras by operas and product that is there by white product. And morphism again with the same story, just literally repeat the same step for replacing them. The theorem is exactly the same as in the case of certain problems. But now, it sort of comprises much more objects than it used to be. Now, what I'm saying is that for any pair of operands, they can construct a new operand, which is kind of quantum matrices connected to the operand number one, to the operand number two, or the operand of quantum matrices. I am quite far away from algebra and geometry, but I still have some kind of modular spaces in this non-community degree.

1:37:30 There is a distinction in terms of internal pronouns. Authorized is a game, as I said. It's used to be just a place for verbs referring to a series of algebras to the respective verbs referring to the verb. This is the last mistake. But here you have an internal pronoun object in the category of authorizes. Classical authorizes. A link to Romney back in time, in general. Explaining how classical operands can be extended blocks. Again, the same theory will fade in this much time. So, is there any kind of second vision in this process? No, I don't think so. At least if you don't do it by brute force, like you do it in the Algebras, for example. I do believe that there is an interesting limit, but I couldn't give you that. So the moral is that we have been able, by imagining non-communicative spaces represented by algebras and set of allogic parts, we have invented the procedure of constructing new non-communicative spaces of this kind, morphisms from the ancient media, which are really non-engineerable, so we get new algorithms of mathematics. And so we have two particular cases which are interesting to teach. First of all, we are considering co-and-demographic studies in a big way.

1:40:00 Then, of course, you get not just opera, which is an analogous creative algebra, but you get an analogous biology, such an opera, because it meets a diagonal map from itself to its wide square. So you do have... it's not so often when an opera is... It means that on the category of the respective algebras, you have a denser product. Now, for example, it's well known that there is a denser product of associated algebras, but there is no denser product of the algebras. So the category operand that classifies the algebras does not. In this way, we get a universal construction group which produces a lot of operands with interesting properties. Another application, and here we return to the very beginning of my talk, we can now produce a construction of a periodic modulus space for the structure of the A algebra on the B. So, you know, she was an arbitrary author of classifying whatever you wish. Now, we know that modular space, classical modular space, was captured with such an algebra. But now, instead of this set of maps, instead of this set of morphisms, we consider the internal, the opera, which is the internal homomorph. Again, this is a Grotendieckian way of thinking about these objects. It's extremely compact, but it hides a lot. You should really start to unravel everything in order to see what happens. So what has happened is we have produced a non-commutative space of deformations of any structure, any given type, in any linear space. This non-commutative space of deformations don't allow it to be offered. But you didn't even have to guess what kind of effort. It's just a universal thing.

1:42:30 The only non-universal thing that we met here was the necessity to choose a linear space of children. Well, that's usual. When you do it, this is the remnant of our classical non-rhythmician way of thinking. You want coordinates on your space, you choose a basis, and you write down the structure. This is the remnant of this part of our work. Before passing to the navigation of this story, let me show you several complete useful constructions. This one is old. We are returning to Algebras first. This is an old construction. At the end of the web pages, I suggested it. So we returned to what Algebras we want now to see as quite a more basic object in a more complete way. Consider an associated typology for this presentation, which is quadratic, meaning that the kernel of such a presentation is generated by some sort of space quadratic, which is the delta of this typology. Now consider only quadratic. There are two interesting products on this category, which are called white and black products. You consider the space of generators, which is the tensor product of the first space and the second space, and the relations are essentially either, all relations on the left-hand side and the relations on the right-hand side. And to define the left product, you consider the same space of generators, V1 and V1, but as relations, you get tensor products. It's not separate V1 and V2, but tensor products. And there is a very fine dual operation, the dual duality. The dual algebra is defined by the linear dual space relations and the orthogonal space, sorry, the linear dual space generators and orthogonal space relations.

1:45:00 And it turns out that it looks like a very nice generalization, non-commutative generalization of the category of linear spaces. It turns out that this generalization or interchanges white products and black products, whereas for unilinear spaces, the duality, I think, the duality just makes a tensor protocol a tensor protocol. Here there are two kinds of products. And if you consider a quadratic algebra with a black product, then it has a monomorphism object, which is the generalization of the classical formula that if you have two linear spaces A1 and B1, then the linear space monomorphism is A1 dual, that's the right way. Here, the structure of the monorheval product is black, but in this formula, the monorheval derivative is white. And, respectively, the cohomology for the category of algebra is dual, i.e. the white product is given by the same formula where the white product becomes the white product. So this is the internal cohomomorphism object in the category of quadratic algebra, and the theory, the claim is that actually if you embed this category into the category of algebraic representations, then the formula remains true when both arguments are correct. This is a very explicit formula which tells you in a very, in a rather expressive way in what sense this internal problem of algebra is the space of quantum matrices. On January, this is exactly a matrix of many calculations which are very explicit. It turns out that this story can be generalized to occurrences, not quite, while at least I do not know the protocol.

1:47:30 I just briefly say that one can define some monoidal structure on the category of collections in such a way that another article is just a monoid on the collection and the reality is expressed by the formula which is very similar to the fourth or fourth-tenths of algebra. The only thing is that this square in terms of product is not commuted. It's not even bilinear. It's linear only if you respect the first argument. And then, the operand representation is called binary property if its representation consists of only binary operations. The kernel of the presentation model is misgenerated by its base in global applications. It turns out, and it was the result of the important experiment 10 years ago, that on this category, the logic of the runs. Binary algebras also derive two monolithic products, white and black, and the duality complex. And then the theory about the structure of the objects becomes exactly the same as in the case of quadratic algebras. That's quite good for the story. And quite good for example that all three examples of classical algebra, They include three graces, so we have three graces, three classical operas, and the associativity opera is self-dual, whereas stability is dual to the E and B is dual to the E. But I'm not sure that people really find the operas that are really obtained from three graces by applying to them white and black products. But anyway, this is a direct generalization of the story.

1:50:00 Let me show you once more after I told you at the very beginning of my lecture about an exotic example of a very rigid object whose homology classes serve as universal operations. I will define now the generalization of new algebra, which consists of the following data. So everything is the origin of 3d0. And I will say that the structure of secret security algebra is just given by many composition rules. One is multiplication of fill arguments, another is multiplication of fill arguments, and so on. Whatever number of arguments you have, you have a specific multiplication rule, which is written. And it shows that the science reacts. One is stimulativity, which is in this case just a sense image, which will keep you oriented the same way it keeps you in the science field. The other is secrecy. If you multiply n elements, then take the scalar product. And finally, there are infinitely many identities. You choose any number m, you choose any m plus 3 elements in our space, and you kind of put brackets. In all possible ways, with how many brackets? Four brackets, with the first sum, four brackets, etc.

1:52:30 And you require that the result wins. There is also an optional axiom about the existence of identity in the ORC, the algebra, and we show you how, especially to you, to use more than one. So if all your multiplications are zeros, then they are just the provenance of a really lovely broken environment, so otherwise you get this. Turns out is that any cohomology space Any projective smooth algebraic method has a canonical structure of such a scheme in its biology course, not just the usual one. This was not known until this is explained. Any cohomology algebraic, any smooth projective system, is clear.