Deformations, symmetries, operads & noncommutative geometry (contd.)

0:00 There is a space of four pieces. Each of the generators is represented by a specific component of the problem. This is a very general construction problem. It's time to make a break right now, and then I'll start applying the story that I've published. In two words, I will be explaining this last problem. We can say it's not a structural problem, it's a very specific problem. The category of progress is linear, and in some cases also too algebraic, but I will need to remind you of a stretch of a recitation of the work of some of the students. First of all, what is a collection? I will consider it linear or pluralistic. There are more technical elements still in the notion of connection. It's just a simple example of linear space, a statement. It can be symmetrical. It's an axis of nature.

2:30 Now, to get an accurate problem of connection, you need the law of multiplication rules. Whenever you choose some element in B or G, then the sequence of relations between B1 and B2, you should be able to compose them into one element of A, B1, B2, B3, B4, B5, B6, B7, B8, B9, B10, B11, B12, B12, B13, B14, B15, B16, B16, B17, B18, B19, B20, B21, B22, B23, B24, B25, B26, B27, B27, B28, B29, B30, B31, B32, B33, B34, B34, B35, B36, B36, B37, B37, B38, B39, B40, B41, B42, B44, B44, B44, B45, B46, B47, B47, B48, B49, B49, B49, B50, B50, B50, B50 But I will not be writing down that ordinary language is unreliable to the result of science. You are imagining some uncertain linear space. You are imagining that elements of a plane are pulling the linear functions of this linear space and energy space. So it's a very fancy word. There are a number of functions that can be used to solve the problem of S-F, that is, each poly-linear function can be used to solve the problem of S-F, that is, each poly-linear function can be used to solve the problem of S-F, that is, each poly-linear function can be used to solve the problem of S-F, Essentially, all axioms of absolute and absolute number include the more or less obvious and the more or less obvious problems of this general notation. And of course, this general notation can be illustrated horribly by just choosing a three-vector space and integrating them also.

5:00 It is important to bring back the space and consider, in fact, the quality of your creations on the details. So, the end part of the writing and the form is just this. And then you apply it here. Describe the toolkit and what your space defines as a program. What is it? Morbidism. Morbidism. In general, it should be more visible. It should be compatible. And symmetry. And in particular, even a general notion of an operative theory. And a complete notion of the right-hand organism, of the vector space, means a right-hand organism, an organism of the structure of the algebra of the universe. We have two batches of definitions. One which defines, sort of classifying the algebra, which classifies various types of algebra. And then we have another pageant in the year which tells you that all algebras will give you time, will give you class, and will direct you to more pieces of mathematics. In the list of analogies I will show you later analogies between associative algebras and algebras, one collection will be something similar to one linear sequence.

7:30 And an opera will be something similar to a social experiment. So, in particular, in the statement of main theory, I use the notion of operant representation here. The notion of an operant means that an existing group that any collection can run generates a free opera. And the string algebras generated by a collection corresponds to terms of algebra generated by a sequence. Yep. The despair and the operatic will end the action of the aspect of the space. Of course, of course, of course. And in the end, as I said, I did not try to explicit the axioms. But this relation, of course, follows from the intuition. If you take an operation with any of the many arguments, then you insert into each other multiple equations. So when you apply the action of this n, transforming the first... This group of arguments in this large function, I don't know if it's clear, but what is hidden there is some combinatorics of symmetric groups and problems between symmetric groups. It's very elementary and, as I said, in English you can reconstruct the necessary combinatorics as soon as you can use it. But when you write them down, you are gone. I am going to do this case again. Okay, so let me repeat these analogies. Let's see it here. It's not always easy, but here. Let's see it as a generalization of the sort of equations in which vector space is the response to the equation. It has the logic of the response to the equation.

10:00 In general, also, it's the logic of the response to the equation. And finally, it has the product of the product. We have to respond to a wide variety of products, a wide variety of products, a wide variety of products, a wide variety of products, a wide variety of products, a wide variety of products, a wide variety of products, a wide variety of products, a wide variety of products, a wide variety of products, a wide variety of products, a wide variety of products, And now what I want to say is that there are not only apologists, but actually we have social intellectuals who are embedded in the evidence of social intellectuals of the out there, that it is an operatic, in my opinion, that any social intellectual is just an operatic, which is not simply a degree of one. Is it more than an irony? I mean, is it true that operas are actually present in a certain degree of a sort of variable? This is also true, but that you don't get. Very good question. But then the point is that you should introduce the notion of weight, and the usual photographs are classified by the weight. And actually, if I have time, at the very end of this talk, I will very briefly explain that. The notion of Algorand itself is a big theory that is being taken on and a very, very vast generalization is also just arriving now on the exhibition, which includes a lot of what we would like, but much more what we need, and that is things like graphs and uncountable graphs. But take time when you wish to classify it in some kind of the same way as you would classify Algorand. Algorand, you should.

12:30 We just imitate a series of conclusions and statements about the social ecology of the software and its representations. We just let our students see how technology and the collection have three opportunities for the selection of the subjects. When you say, here, why do you slide that? I don't understand. No, I mean each component. And then we can define the category of A, W, B exactly in the same way as we prepare algebras by operands, rather than set up by whiteboards. Or at least begin with the same story, just equally repeat. And then the statement is exactly the same as in the project. Now it sort of comprises much more algebraic pieces. 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 operand number one and to operand number two, or whatever quantum matrices they choose. It describes the non-faulty space of more than one parameter, but this non-faulty space this time, as in my education example, is represented not by a social degree, but an opera, which is a much more general one. So I'm quite far away from algebraic geometry, but I still get some kind of point.

15:00 It's again just the same. It used to be just the same. The theory you have in terms of the object in the category of a plastic model later on you may have time to explain you have a plastic model that's been fixed and it works again the same theory Is there any kind of stabilization to these processes? 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 interest in it, but I don't think so. The problem is that we can be able, by imagining non-verbal spaces, to present five operands in a large class. So we have two particular cases in which our industry functions as a whole, and that's a kind of co-and-partism that is a big, big way inside. There, of course, you get not just operator, which is an algebra, but you get an algebra, my algebra, which is actually an operator, because it's a type of math from itself to its widespread. So you do have, in this, it's not so low.

17:30 When an operator is such a powerful life structure, then it would be on the category with respect to algebra, as you have an answer about how, for example, it is well known that there is an answer about the possibility of algebra, but there is no answer about the idea of algebra. So, the category of an operator with respect to algebra does not. This way, we get a universal construction, which produces the law of operands. Another application, and here we are at the beginning of my talk, we can now produce a construction of what the rapid modulus means of the structures of an A algebra model. Modernization, whatever, CRMG, ACMG, whatever. So, you know, choose an arbitrary, but we are classifying whatever you wish to associate with it. Also, we should be whatever all of us want. Now, we know that modulism is classical modulism. It is structured in such an algebraic way. In linear space, E is just a set of names of A and 2. But now, instead of this set of maps, instead of this set of logarithms, we consider the integral, the automatic, which is the integral number of objects. Again, this is a great idea in the 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 one wants to get. So what has happened here is we have reduced the non-communicative space deformations in the structure we see here on the slide. This non-communicative space deformations don't help with the algorithm. But you don't even have to guess what kind of algorithm. It's just a universal thing. The only non-universal thing right here was the necessity to choose a linear space.

20:00 Well, that's usual. When you do it with the remnant of classical, you want coordinates of your space, you choose a basis, and you write down the structure. This is the remnant of this part of our class. This story that we showed you is a real complete construction. This one is gold, buried under the object of stress. This is a gold construction. When it is suggested that the quadratic algebra we want now to see the square root form is an object in the form of a quadratic algebra, consider the sub-relative algebraic expression, which is quadratic, meaning that the kernel of the sub-relative expression is generated by some subspace of the quadratic expression, which is the algebraic expression. Consider only quadratic. There are two interesting products in this category, the white and the black one. In order to produce the black product, you consider the space of generators, which is the number of products in the space of the same. And the relations are essentially even. All relations, the left hand side and the right hand side. And to define the black cloud, I cannot say that the same space generates E1, E2, E1, but there's relations. You get tens of products, not separate, but tens of products. There is a very funny, you know, variation, which is a little algebraic. It is defined by the linear dual space generators and orthogonal space generators, and it turns out that it looks like a very nice generalization of the category of linear spaces.

22:30 It turns out that this dualization, the way to interchange white product and black product, whereas for linear spaces, it's a duality, it's a linear equation, it's a duality, it's just a way in terms of quantum mechanics, but here there are two kinds of products. And if you consider a quadratic algebra as a black product, then it has all of its functions. This is a generalization of the classical formula of that. If you have two linear spaces C1 and C1, then the linear space of all the morphisms is C1, dual, that's why we have here the structure of the monomical product is where, but in this formula, the level of the product is Y. And, respectively, we go over... More than a category of algebras is dual. It is a white product. It is given by the same formula as a white product. So, this is the integral of the category of quadratic algebras, and then here the claim is that actually you embed this category into the category of algebras. Then, this problem remains true when both are present. This is a very explicit formula, which tells you in a very rather expressive way in what sense this integral goal of algebra is the space from which it increases. In theory, this is exactly the matrix of the calculations which are very explicit. It turns out that this scoring can be generalized to operands not quite well-adjusted. I'd like to briefly say that one can define some kind of structure for an academic lecture in such a way that an operator can just acknowledge the lecture.

25:00 The algebra is expressed by the formula which is very similar to the formula for the tensor algebra. The only thing is that this square tensor product is not commuted. It's not even binding, it's linear. And then, the photographic presentation is called binary photographic, meaning that its presentation consists of only binary operations, so A1. And the kernel, for instance, of the brain, that is, it turns out that on this category, the photographic photographs... And then the theory of the structure of objects becomes exactly the same as in the case of quantum mechanics. There are three increases in classical mathematics, and the associativity of them is self-dual, whereas the derivative is dual, the median of B is dual. I'm not sure that people really understand the operand that we have today in quantum mathematics. Increases are minor, minor, minor, minor, minor, minor. More on gravity. I told you at the very beginning of my lecture about an exotic example where each object was in topology classes, so it's universal. Topology classes are universal operations.

27:30 Right, so let me show you the structure of our universe. I will do it in an abstract way. I will define now a generalization of the meaning of the word. Which means it's sort of the following two words. So, everything is a working word. So, that's what's considered by the future. Space, linear space, and singular space. Everything is centered on linear space. And I would say that the structure of the cyclic scene in the algebra problem, the structure is just given by many propositional one-isms, two-arguments, non-isms, three-arguments and so on, whatever number of arguments you have, you don't have to see it, you just develop it. The number is cyclicity. If you multiply m with a scalar product of plus 1, the result is symmetric with respect to s of plus 1. And finally, we have associatives. There is the remaining identities in general. For example, m, you choose any m plus 3. Speakers include mathematics, geometry, algebra, analysis, quantum mechanics, physics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics.