Embodied consciousness & quantum science / Category theory & living systems (& others) (contd.)

0:00 In this lecture you will learn how to solve the equation of the equation of the equation of the equation. So now, if you do this, I'm going to write down x, y, and x, and that is equal to x, y, x, y, z, I've done that bit, x, y, I'm going to replace by w, u, I've done that bit, and then I'm going to do the same, z, w, is the same as x, y, and u, x is also the same as x, y. What have we done? So now, I'm going to rewrite this. X, Y, X, Y, X, minus one, X to the minus one, X to the minus one, X to the minus one, X to the minus one, X to the minus one, X to the minus one, X to the minus one, X to the minus one, X to the minus one, X to the minus one, X to the minus one, X to the minus one, X to the minus one, X to the minus one, X to the minus one.

2:30 And that is where the symmetry between the topology and the penrose. There are some important lessons there. First of all, we use algebra, and also, if you look at this thing, we actually use some elementary humanities. Changing one formula into another, using certain rules, and that's a very basic aspect of anything that one does in mathematics. The higher-dimensional problem is to classify the ways of taking the string off the knot.

5:00 So if you say, OK, well, the fact that this is one means it comes off the knot. You say, but how many ways are there of actually doing that? That's a higher-dimensional problem. If this uses group theory, the question is what does the higher-dimensional problem use? In our e-person, if the one-dimensional problem uses group theory, then the two-dimensional problem uses two-dimensional groups. I'd like to ask you now a question that was involved in your work, which was actually inspired by the work of Henry Wagner, because what I'm going to talk about is an introduction, a very quick introduction, to the ideas of quantum physics. So I have the K-theory of K-1, where it came from. Now, Euler's work on combinatorial-anthropology theory was derived from combinatorial group theory. The basic problem with that, suppose you have a group given by a presentation X-R, and which is asymptomatically given by a presentation Y-X. That's why it's easier that you can move from one presentation to the other. If I have finite people, moves, which is right here, then XR is T-circuit equivalent, and this is not strictly speaking algorithmic, but the kind of thing you have to do is to say, well, if I put this and take this by equals, then this element is a consequence, well, if I have an element of a group, this element is a consequence of an element.

7:30 And you can actually say how an element of this element is a consequence of these relations, or moved over to that side, is actually quite difficult. It is not common for all these in general. So is this quantified proof. Now, Henry Weiser has an extraordinary idea of asking, does this generalise to find the answer? Generalisation is that it is both not a test, but a test for why. These are finite visual complexes. In fact, in homologous equivalence, how do you get an homologous with that in an elementary span, if in all values? So, Pepe had an extra-wide higher sequence of elementary rules, and the extraordinary answer he gave was that in general, no. I'm going to list this related to Goethe's equations theorem, maybe not, but if you look, it's an amazing thing, when you look at the White House papers, just in the early 1940s, He poses an entirely new problem and, totally solved it, he verifies that it's a totally new algebra and a new process. And it's very well worth having a look at the back of those tables to see what he actually achieved. Because his papers were written by lots of people after the war. So, what were the basic ideas in that? Let's start with something everybody knows.

10:00 So that's the invertible matrix over the ring R. Now the R in the field, this implies that M is a product of E1 and E2, of elementary matrices. And how to do this is the kind of thing we teach first-year students. Of course, it should be mentioned here that technically doing this, you really want to vary. An important point to mention now is that there are other areas of concentration, This is about division algorithms and many variable polynomials, where the key difference between, if we're having one variable, we don't need to call it a variable, in Gaussian elimination we need to call it a variable, in Merkle Bay case, we're doing a polynomial division, so we call it a polynomial variable, which is not possible, and the standard example, if this explanation is x, y, minus y, y, In the ring of polynomials x1, that went out by the energy emitted by x2xx-1, so that ensured that this equation was determined in 1, and so is invertible, but determined actually by the approach of error-making matrices over this grid. Now, let's move to GLNR, and then in GLN times 1R, GLR.

12:30 In this, if you look at ERROR, it's the substance generated, and that is called algebraic K-theory. This problem, the point is that these hierarchical K-theory groups have become really not easy to recover these elementary operations that you see here. So turning back, in the stabilization, there are several stages to this, and there's also an expression in quadratic theory, this is called FIS.

15:00 We are not going to get to a formal and significant category theory totally in the last one of these works. Looking into that, we do the attaching algebraic structures. When you attach algebraic structures, the structure which was mentioned, when you have A to the subgroup of G, then of course you get cosets. A operating on G. There are two subgroups. What's the algebraic structure you get there to describe it? And in some sense, the idea to regard this as a global-to-global problem, you know, locally, this would take H-actinology, or L-actinology, and then how do that needs to go? Another thing, if you take X, and the equivalent of the equation, we'll find that, but if you have X and a number of equations, we'll find that, We have all of these related forms of the equivalent relation and what we want here is that what we want to reflect of the other one, this is about an action, this is an equivalent relation, of course action and equivalent relation has special cases in a group of these, this is how we would like to reformulate it.

17:30 And so the idea of the global action, the groupoid atlas, is to incorporate G-alpha. You want the interactions. You may also have... In this situation, in the general case, there is some kind of order being generated. The context we sometimes go into. What we want is halves in this kind of structure. But the definition of a groupoid atlas in the kingdom of time... All you have to do is incorporate the stuff that happens in a GR, a GR graph. So the kind of picture is that you have two common equations. You have a diagram and a graph that each represents the equivalence classes. And you want to describe a graph as being able to do something like this.

20:00 And you may use a construction using an abstract structure to illustrate it and make sure the kind of structure you want to be ready from is in the current situation. I'm very self-taught, I'm a mathematician, I'm a chemist.