Deformations, symmetries, operads & noncommutative geometry

0:00 Mr. Mardin said that once he was at the college and he was waiting for someone to give him a medal and he hoped that this time he wouldn't get a medal because he had already won one. He was accepted into the association this week so he still had the medals. But he doesn't know anything about that. He has one hour. I learned that he had a medal. Thank you very much for your time and I hope to see you again in the next lecture. I'm very sorry I won't speak French because my first courses are in English and this is a foreign language and it's a very, very foreign language in France and at that time I was approached to groups and at that time I made it between the big 12 and the small 12, meaning that perhaps it's just one group.

2:30 For example, there are three metacognitive systems, which are actually classified and are considered to be a general philosophy and practice. And for example, there is an empire of logic, which is explained in the second part of this lecture. So, only one of the eight most famous quotations are in science. All I want to write for now is to ask if you look... It's not about that, it's not about integers. It's about the order in which Kantor was despised, and his way to express this view was his respect to children and the body of creation of the world, and it also inspired him. Let me forget about this historical context, and by the way, I wrote elsewhere that Kant was not really... It was much more like an extremist study on the art than on mathematics, and I would give you a comparison between these days and the number of examples we've had even here on the opera, which is called The Carpenter, The Measurer of Genetics. Opera is a very good trick, to an extent. Okay, if we forget about this historical context, I will try to interpret it in my own language. I would say that there is some kind of mathematical principle based on mathematical objects, but I don't want you to accept it as an actual rule or a theory of observation or whatever, but rather some kind of intermediate statement, heuristic principle.

5:00 I tried to interpret it in this way. Let's look at the first example of what is or is not. There is a more recent example in the library of sciences. Again, we need some ideas and we need a general model and one of the illustrations of it is very simple. The algebras in particular, of course, they are the basic symmetry objects of the classical geometry. But we have knots. Knots is an interesting class of region structures, because knot as such is not a region in any sense, and the example of knots, as you can see, are quite old. The fact that we don't know where knots are coming from is usually somehow important to be clear that all the formations are there and the things are there. And we enter this way in the realm of homology theory, which provided a decryptable amount of increasingly complicated region structures, namely hematopoies. Get their answers and then manage them.

7:30 My studies speak of more examples. It's obvious, much more than modeling spaces or stacks of stable algebra and columns of geometric shapes. The algebra is not what we need. It's kind of what we're all thinking about. Again, it's just very, very not great. We don't know that much yet. The single cohomology class is actually a universal cohomological operation, which has cohomology spaces for all cohomological operations, which is slightly non-traditional in the sense that it has not one arm, but ten arms, so it's not quite like what you say, stick-and-crawl, but it's an operation of dividing one class into two classes. And also, there is one of these spaces, Everett Hall, of intermediate objects in algebraic geometry.

10:00 In theory, we say that intermediate objects are the basis of all continuous information in the world. In algebraic geometry, spaces of the combinations of various objects are essentially functions. So, the algebraic geometry for us is about 23,000 pieces. The x is deformable. Consider the total of x1 over x2. The x1 is still deformable. The goal of this device and the computation of this form is to make a conjecture. I've been speaking very informally about deformations. If you are taking some mathematical object from the study form, what is the information that is required to address this problem as well? Algebraic geometry did most of the job, most of the systematic job. Algebraic geometry did most of the systematic job. But okay, how, given an object in classical algebraic geometry, or at least given an object for which we want to find the moduli space in classical algebraic geometry, Well, classical approach is somehow five coordinates trying to manipulate this object such that you have some least functions of the space of the object. You say coordinates of this object are present in this function. They are probably, they are not free, most of the cases they are not free, so there are some relations, five relations of the object, very few relations.

12:30 And then finally calculate which values are relations of the size of one. And if you are very clever, then you can calculate the algebraic rules. Well, this is a night which, I think, almost ever really starts in the ocean. It's really a problem of course, almost ever really starts in the ocean. Well, this paradigm is much more simplistic. We find, a priori, two determinists of objects, characterized by universalist thoughts and universalist thoughts, as well as some of them, are understanding that determinists involve some variations which are flat. For example, if I just make a point in the play mentions, maybe it's because it is a kind of a point of justification, and then it's edited into another else's topic. If you impose too strong continuous restrictions, then you will not be able to go to the boundaries of your language despite the generations of your culture. So, the notion of continuity should subtly balance continuity and discontinuity. And, of course, it is one of the first discoveries of the notion of flatness in our generation. Usually, the most part of your logics are non-degenerate, and when you go to the boundary, the boundary is there, you get degenerations, but in a sense you get all possible degenerations of the boundary, you have extremely ample dormancy, so if you want to have a unique form, you have to have the notion of stability, and then say, well, I'm going to say, well, this is the game, this is the reason for us. We will study both possibilities and the possibilities of interagency and the ways of interagency in mathematics. Okay, now, my idea was that one of my subjects of this paper is that we want to perceive the formations which are connected with non-communicable.

15:00 If so, then we should talk also about how we should correct the definitions. The north of the paradigm certainly doesn't look good for several reasons per se. We start with a bit of amnesia, which is morphism. The non-communicative geometry of the situation between morphisms is extremely bad. Either morphisms are unknown in isotope, or they do not behave well, or else they behave more like, not maps, but correspondences. Or, and they can call geometric fibers because those points are rare or absent. So, I will demonstrate, okay, just a little bit, what are, let's see, what are, there are several of them. They mentioned they don't make results, but there's a combination of them. They are extremely interesting. There should also be counter-intuitive problems. I said 3D X here on the screen. Now it's 3D X on the screen. I'm using 3D spectra in some scheme.

17:30 It is in the 5G 3D X of the space spectrum. Then it does it with the geometry you can produce. It's very nice to watch. The space of Bausch is really funny. But in non-communicable geometry, you have another person. You have consumed the whole green room, so you can put the nations together. Multiplication will be twisted when the name of the group advises the green vertex. In the non-communicable geometry, this is a case of one of the various non-dark prescriptions, is that these two objects represent one and the same non-green. But from the viewpoint of the social degrees learned, you could imagine that modernism is not a component of the basis. Modernism is a component of the planes. You have a totally prevalent situation. In the first presentation, A maps to H. Sorry, H sits inside A. And there are four spectrums. A maps the perspective of H. Of course, the covering space maps under the space of H. But in the second situation, A... There is a lot of information inside the interest. Therefore, you are imagining that the space that is being recovered maps upon the space that is being recovered as well. From the viewpoint of the field of geometry, it is totally counter-intuitive. Whereas in the field of geometry, it leads to extremely few severance in the beautiful constructions of all the regions of the universe. So, you have the simplest idea of how to do it. Thank you for your attention and see you in the next lecture.

20:00 The point is that the second object is totally universal, it's more universal than the first one, and this is actually the right to the correct point, so the action is very comfortable, you don't have to be a little bit afraid of it, whereas in the other case, if you have to move this to the right, you see normally the first one, okay, you can say, I'm very sad, or something like that, but fine, I'm happy. Not a nice need to write objects in such a situation. So, I've been given the suggestion that read themselves or your objects are not good. This is okay, that's in the sense that I can't imagine to be true, but then, as I said, the intuition becomes wrong. There is another kind of idea. If you choose close from community geometry, you can try to declare that in non-community spaces, the categories, say, of right-wingers or left-backers of organisms are common, and there are two categories. The natural properties of such categories are the so-called marina morphemes, as we say in Poland, and by projecting such kind of morphemes, there are very applied classical morphemes, and there are more right-classical morphemes. And then, of course, amortism ceases to be imaginable as a series of close, fibrous, marked tries by simple or something, and then we cannot continue. And then, of course, why are we considering going into more and more difficult resources?

22:30 All of these are suggestions that we've received, and a lot of interesting work was done, but it's largely on the job of the class that we've created very interesting. I will finish the story by talking about the general moral when we want to study the new topics. To kind of coordinate, to try to find non-competitive coordinates, to look at paradigms, coordinates for the kind of algebra, algebra, algebra. Up to now, for example, just an iceberg. I have in mind a general metaphor. If you are imagining that the geometry should somehow be embedded in the planet's mind, it is embedded in the normative one approximately like the theory of singularities is embedded in the theory, say, of non-singular objects. If you deform the singularity, it generally ceases to exist, but such hiding information is called unfold. When we embed a cumulative object in the normative geometry and then start to report it, you should not expect that the fibers are similar to each other.

25:00 They lose a lot of the characteristics of the initial object because the initial object should be considered as a highly legitimate one. If it was the general expectation that something in the cumulative geometry was rigid, then it was a basic objective for the normative. It will stop the division in which three numbers satisfy one zero, which was suggested, which is just one parameter. If you define the integer n, one time, the left b is one that you can apply. It will just show that many of the formulas, objects, construction theories in which an integer counts as a parameter can be consistently and interestingly formed. By introduction, hypergeometry accomplishes this.

27:30 But that usually doesn't interfere with the calculus, it's a very elementary non-clinic theory, so it's quite... Imagine now, in a subject grid, a couple of elements, x, y, and x, y, and y, which is in the center. Then we get one of the first examples when, due to the binomial formula... Deforms according to Gauss definition. Then we deform binomially. Form binomially expressed by deform. It works in exactly the same way. The left hand side is essentially x and y. Now it is lost to the bottom. The bottom plate will be some imaginary spectacle. Now we need to predict and consider. What automartism is this one? The automartism should ask what kind of solution? Well, first of all, classical theories of a few languages.

30:00 But since that's widely known to you, we should allow ABCD. Then how do you calculate? You compare two H-I squared ET. In relation, you get two expressions, ET minus 3 minus 1 CD. You know both expressions. You have to go through more steps in order to get to the point you need to. First of all, you should add what is sometimes called basic predictions, where you want the matrix itself to act as a linear algorithm, but also the transpose matrix, not just the matrix itself. Then, of course, you might be interested in our inertical, and it turns out that exactly those, it turns out to be a central element, the inertical, which you can see multiplied by the term minus one.

32:30 So what do you get? I don't know. You get the following. You have three or four spaces, quantum space, quantum plane. You get the non-communicative spectrum of the nations, whatever you want. If you get the quantum proof, you add additional relations and you determine the situations, the necessity or the use of getting these equations and they spin also in various different ways. The second one is this. If you take only the first half of the equations and impose them on the ABCDE, you get a very large degree. It's obviously graded. If you add the missing relations, you get the ring, which is exactly of the same size as the ring of commutative polynomials in the KDC, so it's really much nicer to write. This is one of the reasons you add the missing relations. Another reason is that if you add them, you can construct a very nice theorem complex. What happens next? Now let's play a game, which, so far as I know, was not easy, but somehow we could go about it. So let's try to define what is a deformation of an outward space, a deformation of, say, a finite dimensional, or, sorry, or a division. The whole space of a finite dimensional, or, sorry, of algebra. Okay, you choose a basis. You choose one binary equation.

35:00 Of course, the operation itself is obviously defined by the terms that are out of structure, such as C, I, G, and so on. If you have some structure constants, if you have some symmetry properties, like relativity, and so on, you may need computation. If you have associativity, then it's cubic relations in terms of s, quadratic relations in terms of c. Why do we assume that C, I, J, K are computing? The algebra itself is not. Then we take the space generated by C, I, J, K and then impose these relations. But do not ask any more than C... You get some quadratic relations in a non-communicative algebra, and you reason that they can't be thought of as an algebra describing the non-communicative formation space. It's just an extant sequence. I assume that extant sequence. As I said, in non-communicative geometry, you always do something like that. I'm quite a protestant, I suppose. I am on the stage when I want to produce linguistic examples and worlds. So I, in the absence of you, don't do it. A is an algebra over a number.

37:30 I would like to remind you of that. And therefore, in principle, we might require, if we are classifying now algebras or not algebras, we might require this geometry of mathematics to form an algebra of one more, which one we do not know as yet, but we do, and I, for some reason, think that we should choose another algebra which is somehow related to this one. Any more assume the multiplication. No, we now treat the multiplication itself as a variable element in our operations. Some kind of binary operation. And the relations we get, none of them take the same form. Without the binary operation itself, and the origin of the coordinates would become extremely non-classical. Coordinates are elements of something. And the duration is now all set. What I will explain now is a kind of interpretation of the construction of the continent. It's kind of an example again, but it just shows you that the construction of the continent would be more rough, as vast as you'd like. There would be no formal knowledge of trying to be more complicated.

40:00 It's more of a little showing of what we can do to clean our construction of the continent. So let's consider, for the time being, just associated algebras with one additional algebrism. This essentially means that I want to generate a linear space of G-rate. So here we have a subspace G-rate, and later on it will just depend on the tensor algebra. And now let's introduce two such variables. Now let's define a category, an object. There could be an F, not exactly from K or D, but from A to F times a B, where F is the algebra's linear in terms of A1 and B1, and a morphism of two such objects, one doing up and one doing down, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, and morphism of two such objects, This category has an initial option, and some might go, and this initial option is required uniquely, I think, in mathematics, and I guess it's required in physics. What I'm saying is that this is a natural generalization, where we need to start a generalization of our construction of quantum matters. We may choose to work in the quantum way.

42:30 There are some subspecies that you can read, but I think that's exactly the initial object that we need to write the written one. Here we stop at the stage, we don't have this information, we just write the written one. It's actually very, very simple, but at that time I was surprised to see that you didn't have it. First of all, it's very, very simple. We need to construct a model that works in this way. All these have been refined into spaces created to check on their possibility. In terms of the space, I don't consider monoidal category, algebras as representations, and sometimes that's a problem, a fighting problem. So object is almost the same as it used to be. In high algebra, we have a subjective matter. Delta-algebra was a statement. They only are your models, so you don't know what they are. We presented algebra A, washing algebra, or free algebra. And order should be linear. And while the product, essentially, is only a natural thing, like the delta-product of the subspace structure, There is a pretty nice categorical statement in this form because instead of imagining that between two objects

45:00 You are imagining now that instead of the sets you have objects in some amount of time. The arrows are inverted. This is also important. You see how whenever I try to explain these short lectures, I feel simultaneously amazed by the ways in which... All of this makes us think and requires a certain reluctance to do it because everything becomes so compact and so obvious that whenever you state it, you sort of lose the whole intuition you got when you started.