Euler, Maxwell, Grothendieck & the Mathematical Representation of Space (contd.)

0:00 This move, I don't recall these points, however, to be honest with you or up front, we won't always get Cantor's answers, that depends on you, whether you're satisfied, or I'm rich enough to ask you, given E and now it's a pair of A, we might or might not satisfy each other. If A is a very natural choice, then S does not turn out to be a natural choice. Now do I say it instead. Still a very special choice, but not too long until the end of the text. And this makes a great difference in the simplicity of describing the conceptual relation. What are often mislead in the call to points is really the direct limit of Cantorius' set, sorry, of the Galois set. And this is the direct limit over a system that is not properly filtered, so it destroys even our key controllers. Thanks for all of the conceptual, some of it, but that's a fact we need to remove. So to express profit in this idea that we can model the general example of deep cohesion In space X, we bring object A into the subcategory and consider the basic mass, but now I'm going to consider the points of that. The E has given rise to an S value of the species, the speed or almost the speed of the species, so the adequacy of the talents paired is the same when you count them.

2:30 You may say, well I want two to be here, and that will generate this. You might say, you can tell them to be here. Typically, you get this pair, we're starting with just one or two very, very simple, intuitive examples. But whether that is adequate, what you need to know is whether this is a full and faithful function. So that is the crucial thing to know if we have made a correct analysis. Well, it means that if I, suppose I have two, two spaces, x and y, Now, I can relate them by saying that in each H-shaped figure is one that will be assigned by abstract mapping. I'll leave that for each of the A's eventually, which is five of these, but I'll require naturality, that it should preserve the inter-integrations, the inter-integrations should not be torn to be continuous in that sense. Then, that will come from an actual map, that is, a morphism of these. And that's the integral motion of the mathematical system. There are a number of different types of equations, and it should be either a theorem or an assumption or whatever, but it's probably not. It just doesn't work, and of course you'll have a subcategory where it doesn't work. Now what is the specific example? Oh, I'm sorry. Well, it will usually be the case...

5:00 In fact, the following discussion makes sense just when I have a string of adjoins like this. I'm usually thinking of the case where it has to literally be inside E in that sort of way, with the figure shapes and so on. And it's neatly done with the quality on an E. I'm thinking of the case where it has to literally be inside E in that sort of way, with the figure shapes and so on. And it's neatly done with the quality on an E. I'm thinking of the case where it has to literally be inside E in that sort of way, with the figure shapes and so on. And it's neatly done with the quality on an E. I'm thinking of the case where it has to literally be inside E in that sort of way, with the quality on an E. I'm thinking of the case where it has to literally be inside E in that sort of way, with the quality on an E. Categories, optimototals, you know, it equips with three fung terms, two of a chapter equal. And then a fung term, e to q, is half-compatible to this, so it's half-compatible. Well, turns out again, the same thing for the next section. Or at least, I feel that this is an appropriate word to apply to the contract, although I can't directly explain how this is related to the two kinds of terms and quantities. There's an extension of that. So F, a functor, either communicates as a functor to the basis of my study into a type K thought of as a measurement. So it's a measurement that should be compatible with the star points. And so if we could use an analogous name, we could take, we could call these Q.

7:30 In other words, F is compatible with points. So that's one that's an intensive quality. On the other hand, an extensive one is one that's compatible with components. Maybe I shouldn't use the same. The range is an extensive quality. It can be valued in two. There'll be various cubes that come out of two. It's fun because it's compatible with components. I'll tell you right away, there's an important extension of quality defined by H-quality. It's totally alternative after a few steps, but I get a new map. I define Q, I define the category of Q to have as maps from x to y, d lower trees, or q lower trees, of the function space y to the power of x. In this case, H tends to be a Hurray width with extensive quality to come as the homotopy type. Is it equality, the homogenism, does it satisfy this property that the two end-to-ends are equal? Well, yes, the traditional homogenism does. If you intuitively construct the sequence n equals 1, then the maps from 1, i.e. the points in the narrowest hands of y, is understood in the category of q, and if the component is 1, x is 1.

10:00 So you've made the components become points. But then there's a bit more to show that that's actually still the right kind of algebra. And that turns out to be the case with topology and physics, but not for superficial text. So this is one of the main contradictions in Gabriel Wiesmann's book on categories and fashions. They make exactly this construction, but then they have to do one more step in order to get to the classical or algebraic category or... From my point of view, one more step in order to get these two adjuvants to be really the same. To say that they're the same, I mean, you can write down a very important point. You have the category, I'm pretty sure, if it preserves products. On the other hand, the field of retreat always does preserve products if S is extracted from D by the diffusion that I mentioned. Because, well, it's clear that if I say S equals S to the A, well, that obviously implies S to the X. In particular, s to the x of the a of the x times a of the a to the x times x. So, the subcategory formula, when you define this discrete subcategory formula, the Cantorian type of formula, automatically gets closed under explanationation by arbitrary explanation, and it's a trivial exercise to see if that's equivalent to the fact that the left adjoint is a finite product. You need the preservation of finite products to make this into a category, because if you take all the ways to compose these sets of components, you have to interchange product terms.

12:30 There are always finite products if we're here, but the additional thing that's required in a Q quality type is that, basically, the preservation of infinite products. I should say the category Q is automatically Cartesian closed as well, and the country is preserved for differentiation under those. But this formula, which we're going to hear is not true, as we said, it necessitates the third category of fractions. And it's that fact which gave rise to the whole industry's weak equivalences, the gap between combinatorial approximations. There are a lot of other extensive qualities, most of which are derived from this one. Like, for example... It has an underlying structure in the category of ordinary computers. It picks off the zero-dimensional components of the greatest computer. Picking off that component can equally well be seen as taking the connected or the anti-connected part or

15:00 As opposed to by bonding out all the higher degree stuff, there's two descriptions of both the right and the right to ordinary community analysis and theory theory and so on and so forth. And then, and then, like the Tunis Theorem, many homology theories are going to fly to spaces and give others to infinite knowledge, and so forth, and then you have a community of total, a great community of total literature. The interesting part is the intensity of this equation, and just reading the quality of that, the obvious way to get that is to just take all the objects that think it's true, right? In other words, the situation, consider, let's use that zero, with all those spaces X, right, for which the canonical math comes from. Points to components in a graph. Now these are tractors in a category. So then the original piece can be factored through this by, so I'm encouraging all of us to include it, but then it turns out that it has this very excellent, excellent category. I think I should have said at some point that in order to get at least a simple computation of the properties of this. This is an intensive quality. This is an intensive quality. Now, in order to get good behavior, you need to be called a null cell. You need to be reminiscent of the Hilbert class. It makes sense.

17:30 I'm sorry, even if the field isn't that great, it's probably good. Now, no-skill math is just a requirement of this math. In each case, the math goes back to two components. The least epic. So epic is kind of an eternal existence assertion. So we're asserting that in each component, there is a point. No-skill math is just a point. So these special axes are... Now, what have we done here? I've ventured to give all sorts of words, mainly what in certain cases is known as homotopy type. I give it the name form in the sense that there are lots of extensive qualities. This is clearly a kind of canonical one. Actually, I hope it's really some kind of adjunct. It's certainly a canonical. Extensive qualities. On the other hand, this one is a canonical intensive one, so I'm calling that substance. The idea is that the Poincare conjecture, why should you even guess the Poincare conjecture, because there is a kind of credulous thinking that, well, at least in a simple case, if I know the substance and the form of the thing, then I know it up to an isomorphism.

20:00 This is usually not true, but in some ways it is, and this is a concrete idea, but this is the way of spelling out the philosophical precedents coming up tonight. We can always ask whether an object is eternal, not true, but it's immortal. Because the substance will actually mean things like I have a three-dimensional manifold, and made of... Maybe it's good to have some examples. Let's take the category of reflective graphs or truncated simplicial sets. In this case, truncated simplicial sets, directed graphs, but reflexive directed graphs. It works also for non-directed. We need the reflexive. He said that it should have been obvious, it should have been mentioned in the first five categories, or not necessarily, but it's both an HMU and an adequately separated thing, so it's doubly belonging to the other categories, as I've worked on, and it's a lovely thing that we couldn't have. We take these things, and we have some idea of what homotopy type is, but first, basically, all the homotopy types that you get from simplistic sets only appear in graphs. It's extremely complicated. But, so what is the set of growth? Just think about it. It turns out that the points of the graph are what are the points.

22:30 And so to say that every component contains exactly one point is to say what? The graph is just entirely loose. It can have any number of edges, but they all have the same beginning and end. So the subcategory U0 is actually involved in these. And actually, concretely, I mean, abstractly or concretely, it is just this category, because a graph, if you have a graph of operations, then call that operation both source and target, and you have a graph that consists of both, and conversely, this category, abstractly, concretely, plus M, which is the bigger one, absolutely. So, what is the difference between the two things? The difference between the two things is, for the Dressler star and the Dressler tree, what is the difference between the Dressler star and the Dressler tree and the Dressler tree and the Dressler tree and the Dressler tree and the Dressler tree and the Dressler tree and the Dressler tree and the Dressler tree and the Dressler tree and the Dressler tree and the Dressler tree and the Dressler tree and the Dressler tree and the Dressler tree and the Dressler tree and the Dressler tree and the Dressler We have, as always, the three of these are just kind of retro, but we have this connection map. Yeah, we can follow along with this. We have about five minutes. Five minutes, yeah, okay. I'm just going to say this here. I shouldn't have said pH, I thought it was pH. Yeah. But I'm going to go back to it, and if you touch the graphs, I have all these, and I have two, S3 and S star, S star, these groups.

25:00 Now, why does this make sense in substance? If you look at the reversible graphs of more complicated structures, you can see that at each point, at only each point, you can count the number of loops. Or, in the more complicated case, you can count the self-converted loops and this kind of thing. When you say that the substance of this sample is H2O, or if you're just saying that you think that each molecule has a certain structure of that sort, you are completely ignoring how these molecules are related. That's not part of the statement. Whereas the actual sample X does, for example, have both interactions. And that interaction comes into play when we calculate the components. But you can't lower a string. There's more than just a component. It's another graph whose components and points are just as numerous, but it has the same components as the one that we began with in that subject. So basically, if you take the sample of space, and you have, on one hand, superheated things in gas, but then you can see how many interactions. On the other hand, which creates these giant, giant molecules, each original component of the graph becomes all of its dots that they coalesce, and then you find the same thing all of its trivial roots that become coalesced, so you have this super molecule.

27:30 So this map that comes in to represent the process of cooling gives you more information. Neither of these things alone tells you everything about you. And this map doesn't either, but it tells you more because it tells you which of those many quantum numbers or whatever you call them in the supercooler theory actually came from quantum numbers in the previous one and which ones are really virtual particles because they came from them. So again, I'm not saying this is a model of physics, maybe something like it, but it's a way of picturing. This is available in every academic situation that I've described. And various categories are built from them. They all follow a canonical analysis of this. Well, thank you very much, Lee, Bill, for a quote incredibly... We have about 10 minutes for discussion, so Gonzalo? I was wondering, you know, I heard several names, but I don't see it on camera. I'm thinking about Euler. So where did Euler enter into this picture? Please, tell me!

30:00 That was going to be my question too. There is a kind of natural choice of this A. It's called B. Now, people have jumped on into the puzzle an awful lot. There is basically something not built up by the human hierarchy, if they follow something, but something extracted from our thousands of years of experience in the real world. Mainly, if there is such a thing, it's motion. But what is motion? It's that you can't step on the same building twice, and then the... The conclusion under Praslis said, well, you can't even step in the same river once. But the actual situation is more like, in a given instance, you can step in the river and also not at the same time. So there's only one point involved in an instance, but the instance is more than a point. So that idea is represented by this object D, which has one point. And so when I'm talking about stepping in the river X, I mean, I'm applying the motion over this interval of time to whatever Z, and I'll be stepping in sort of in one place X0, but it's more than that. It isn't just X0. This is a way of adjusting the many traditional formulations of this dialectic theory to make sense for me and for the last 300 years of analysis. I want to define S by my same relation with D, but not only D, but D to the power of D. D to the power of D, this is basically homogeneity, the speeding up process that we apply to the motion of lambda, that X lambda speed up or slow down of the motion X to that point.

32:30 So you have the space of all these lambdas, and it's a two-point space, at least, as we have learned. It has zeros, but also has their identities, and the name of the zero, and the identity of the identity. Those two are actually contained in the part, I call R, of zero preserving, the kernel of the evaluation of zero. So, the claim is that this is the real R. These are ratios. Ratios are, you know, this is one thing you have to teach children. Division is not like addition and multiplication. It's something you have to think about in each case. Classical cases. Localizing rings shows that. Difficulty. Difference quotients and calculus is easily eliminated if you just face the fact that division is not an operation. There's a problem of can A be multiplied by something to get it to be? And that's called the ratio. So, the ratio is in some sense, well the space is the two of the things that you're ratioing, but the choice of the individual elements comes after you choose the ratio. The ratio of lambda might say a is the meaning of a times b. So b to the b, or r, r more exactly, because that would be one of the ratios, or zero. This is the space of all ratios in the assessment. And being automatically a monoid, a commutative monoid, in fact, no self-function space could ever be commutative, basically because two constants couldn't possibly, but no, no, the thing is, d to the d, like any monoid, has a commutative reflection.

35:00 And R is contained here. So one actually says this composite is either. So that implies that R itself is either, even though it's either B or not. It also says we can retract either B onto R, and that retraction is essentially what people call dots, in various operations. So the thing is that R is actually some kind of intrinsic multiplication. Now, we know from the claims of other people that... Often times the category of multiplication determines the condition. So there are several, actually three different formulations. Common, contrival, integral, trapezoid, two questions, trapezoid. Ways of formulating certain conditions on these conditions imply that this R has unique conditions as well. So this R is the real numbers on the point of view of Vatican. Further analysis should map into the Vatican view, so they can go for a map into the Vatican view. It's not injected because it's precise to know that it's intesimal if it lays out. We can map to, then we can say, now this is a statement that will be a different story that will have to be down to a counter. The auto-morphism is a theme. So it has its components. The picture that you have of the line is that if you take the invertible out, it's intesimal.

37:30 If you do the other way, you get two pieces. It's a component of the identity. And that's the positive, so you get the ordering of it. It's a strictly positive, but there's a trick for deriving the common notion of strictly positive, the notion of less than or equal without, as they would call, boolean logic. So you get a less than or equal in this form, and you made it now. The India right hunch distinguished really actually the theorem. These are the kind of properties that are expressible. So we've got Euler in there as well, isn't it? I think we have time for one more question. And yes, there is this very deep theory, the theory of biology and theory, which is of general, great general counsel, as David Sutherland. This is at this point a categorical approach or conceptual approach to the proof. The proof is very, very involved and documented. So, it kind of bothers me because the seasons over here is so much like mid-seasons, you know, you can't join the spots and that sort of thing. You know, it's clearly a very categorical nature because these are minors. But one problem is graph theory is never quite totally adaptive, right? You see it one time and another time and that's it. As far as I understand minors, the basic idea is, damn, you're the island.

40:00 You have between two graphs, you have spans where you have arbitrary workings in the second part, but in the first part you have something that's five years out of par with a disconnection. Yeah, so it's all about the evolution and the progress. Yeah, so these moves, these movements are those connective stages for calculating, in this case, a certain kind of thing. We never say exactly what this kind of thing is, but we count a lot of what we did. We think we've figured it out. In this case, whatever says it's some kind of span, these spans can be composed. So you have another category. In other words, one in which maps are looser than the rest of the work. So, we don't have enough solutions. We have tried. Shandoran and I have thought about it. I'm a little confused about this, and what is the sense of decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal, decadal There's a lot to do with the sub-objects, so to speak. We would love to talk for a long time, but we're out here still a lot. We know a lot, too, about the section. This is going to be a place where the concrete channel will be. Right, at that point, I think this is a very interesting topic, but if at this point we could draw this discussion to a close, bear in mind we do have two hours of general discussion with all the speakers tomorrow afternoon.

42:30 We're going to start again in ten minutes sharp with Jean-Pierre's talk, if you'd just like to take a couple of minutes of coffee break until then. We don't need to do it now, we can do it in a little while. Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Are you sure you want to do it first? Thank you for your attention. There are a number of different fields of study in the field of mathematics, such as physics, geometry, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra. Basically all the problems of mathematics, mathematics, chemistry, mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, and mathematics, Well, it's, it's, it's, it's, it's, it's, it's, it's, it's, it's, it's, it's, it's, it's, it's, it's, Thank you for your attention.

45:00 Thank you for your attention. Thank you for watching this video. Well, because it goes here, you present it here. Now, where did this guy come from? I mean, just look if you want to do this. Well, then, what are you going to do with it? You're going to suggest, right? There wasn't a lot of information. Unfortunately not, no.