Last session of conference / conversations, incl. P Freyd, FW Lawvere, A Kock

1:00:00 No, no, that's my talk. Well, except I tried to... Or give some more general background. No, rather 25 minutes. Oh, no, of course, it was only 25 minutes. You don't want to give more background, but I just... Sorry, Mike, I can't see you. Yeah, sorry, I was just around the other side. I'm taking somebody else's seat. And the other thing I was going to ask, the stuff that you were going over yesterday about the no one was sitting at lunch, further to your very clear little explanation of... I wanted to ask a bit more about that, but it's okay, the next talk starts with the technical audience.

1:02:30 Um, I think it would have helped me probably to have heard Andy's talk about a little bit of strong support of a little bit of strong support of a little bit of strong support of a little bit of strong support of a little bit of strong support of a little bit of strong support of a little bit of strong support. Second order, but not the full second order, if we were to naturally define the space of, you know, it's in turn, at this level, the differential of an algebras is in distance, quadratic. Dominique Bouron, who will speak on the three by three level, and Potter on what he learned. Mark, do you have a problem? Objects are very strong in classifying properties. I imagine that we can classify the concept category, the natural concept category. We can classify the category quickly. When you have a body, but a body...

1:32:30 So you're simply identifying the distribution. You want to distribute it over the velocity of space, the velocity of space is distributed over the velocity of space, so instead of this e to the minus x, you're going to find this e to the minus x. You could write it that way, but the exponential series is different, because the power itself is open. So maybe there's a way of approaching this in a moment now that we've got to derive. The dimensionality of the space itself, the domain space, of which the quantities are actually varied, taking into account the richness of its qualitative nature. If I could get it another way, in terms of the Taylor series, the quarter Taylor series, if the variable is nilpotent, So even though in general you have arbitrary smooth functions on certain very small domains, all the smooth functions are actually equal to fourth degree polynomials. Thank you very much for your attention.

1:35:00 Thank you for watching. Again, I'm Dr. Stephen Wood, and I'm a professor at the University of Canada, and I'm going to talk to you about the mathematical evolution of quantum physics, and I'm going to talk to you about quantum mechanics, and I'm going to talk to you about the mathematical evolution of quantum physics, Thank you for your attention. I want to ask you to go through another part of what you've done, to explain to me a little bit more about algebra, mathematics, mathematics, and the significance of the law of the law of the law of the law of the law of the law of the law of the law of the law of the law of the law of the law of the law of the law of the law of the law of the law Right.

1:37:30 Thank you for watching. There are currently two topics on my mind right now. Well, we're... Not yet, no. Well, Anders! Anders! What is it that you're about to talk? Oh, good one. Well, Eduardo said he didn't understand too much. I understand a lot more than... Thank you for your attention and see you in the next video. I'm trying to figure out what is the property of covering a map that's defined over S by one that's defined over E, so any kind of general construction. Yeah, well, certainly, you know, I was just taking a look at one of the two ladies at the desk, which was very quantified, but they want to see what some paper wrote. Oh yes, no, no, no, we need to go back.

1:40:00 So I didn't get the thing about, you know, these infinitesimal objects defined by God, I just said something about removing a point, you see, a biography of a moving point, dividing out by a general ideal, gives you something that's not your understanding. I'm trying to get a picture of it, it looks sort of like the interior of it. The general way you describe the things that come out of it, say on the second-rate space, i.e. Schrodinger's geometry, when you divide out, you just get the first-order neighborhood, because it's n cross 1 that makes it possible to divide out. Thank you for your attention. And then there's an electron which is placed in no particular place. Oh, okay, like that. Okay, that would be reasonable. He's trying to define a description. I use the word, it's totally isotropic. Because it appears for my target, it's... There is no other second order property except the square distance, which is of course also expressed in the way you constructed it, that the only information up in the top dimension is the metric. Yeah, yeah. The energy, perhaps.

1:42:30 Well, energy is not a geometric concept, but a kinematic or dynamic, or can it be construed as a fluid? Actually, I don't think it was construed as a fluid. Probably not, really. Well, I mean, I have some chance of understanding it, for example. Yeah. Oh, well, it's... If you suppose that with every point in the body, there's also associated such a five-dimensional cosine algorithm as possible for it, so suppose you have given the probability distribution of that thought of as a function space. Probably this is a very funny thing now. It's just the first few moments. So that seems to be basically momentum and energy. But you need the ingredient of the average in order to get something that's distinct from square velocity, to get something that's linear. There'll now be just a function on the manifold. So it's a function, as a notion, averaging in this case would be a function from the L-or neighborhoods down to these M-sub-L. That would be the set of pairs of L-neighbors, where L denotes... Whatever comes out of the quadratic form. Back to M.

1:45:00 You're averaging over this little Gorenstein space down to the point in the center. It means you're doing this at every point. Ah, okay, so what you're averaging is a function... If you have a function defined on the L-neighborhood globally, out of that you derive a function defined on M, on the space. The body thought of as having this augmented thicker structure down to the more classical body. Ah, okay, yes. It looks like a sort of a Kahn extension diagram. Yes, yes. That's right, yes. Looks like a, sorry. Here's a version of the diagonal, here's the scalars, and here's a map of the sink. Just a projection from the sink table to the manifold itself. And so we have these two arrows, and then we just fill in here, which is an averaging. So it's just the shape of the diagram that I was describing? What would a similar thing be just for the first open neighborhood, or if the thing becomes void then? It could well be that there is no significant average of vectors, standard vectors. Well, of course, given any function on pairs of neighbor points, which is essentially a scalar plus a one-fourth, What can we do to throw away the one from the deeper scale of our field theory?

1:47:30 Well, let's see. If you have a function defined on pairs of labor points, since you can form an affine combination... ...of any two neighbor points, because they're one neighbors. You can, for instance, take the midpoint and then just take... Good point, there's so many kinds of averages. Define the, oh wait a moment, yes, you have the function defined in pairs of points. What would you like to give for your new function on a point? I suppose you'd just take a linear interpolation because... Well, now you have to, now you are just given one point and you're not given the two points yet. I'm very happy, yes. It's very good to have someone else in the room. Oh, well, now your function has a derivative. No, no, because the function you are given is not a function on the manifold, but on the second manifold. Oh, that was, that's the one, that's just the decomposition into a function in the wrong form, if you take the derivative. Oh, yes, yes. I was just telling Michael that one thing that might come out of this is sort of canonical short-circuiting of the moment in power. There's a problem in kinetic theory where you define various quantities like temperature and pressure and so forth as averages. So if you want a differential equation describing the time evolution of the first 13 moments, you have to know the 14th moment now. So there's sort of infinite regress, there's no closed off system. 13 is somehow a natural number, I mean reasonable. I think it's a good place to cut off because for other reasons you think well everything really should just depend on the momentum and energy so you add up things so what people do they just make sort of relatively arbitrary cut off you see but it's not as physically as well motivated as the idea of momentum itself but if you're averaging you see if the

1:50:00 In the case of which you're averaging an infinitesimal, then there won't even be any moment behind a certain point, and so on. Let's see, the Laplacian is defined in terms of an averaging, really, over pairs of... I mean, suppose you have a function that is defined for the L-thick neighborhood. Now, given any point, take any pair of... All of these are symmetrically located L-neighbors that will again be mutual L-neighbors because they're symmetrically located and take the value of your function on these two points. Oh, that may depend on, I mean, it's only if the function is harmonic. Harmonic functions, you have it, yes. So it may depend on, okay, so that's... I wanted to say, and that's one of the things I forgot to say, that in this sense harmonic means one and a half affine and little more than affine. Every function is one affine. That's a Freud concept. Two affine is a very strong concept because that virtually means affine at least for spaces. Affine. That means linear. Inhomogeneous linear. But this neighborhood in between affine means harmonic. Yes, yes. One affine, one-and-a-half affine, two affine. Two affine is completely affine. One affine is nothing. One-and-a-half affine is... Harmonic. That's right. Harmonic functions and morphisms and pairs as well, right?

1:52:30 You've got geodesic pairs, of course, somewhere here. Yes, they can be... There is a notion of a harmonic map between two after-remenions, and I haven't been... I put that in my paper that I... Conjecture that this may, if you have two Riemannian manifolds, there are two properties that are not the same. One is to preserve the L-neighbor relation. That I know always means conformal. If it's diffeomorphic, but it makes sense in any case. And the other one is to preserve affine combinations of L-neighbors. And these two properties are independent of each other. You may have one without the other and vice versa. Clearly, for instance, any function from a Bohemian manifold to a line certainly preserves L-neighbors, because in the line, L-neighbors are the same as 2-neighbors, so their preservation of L-neighborhood relations is void in concept. Thank you for your attention. If you put them together, it's at least correct both for geodesics, because then they have to preserve affine combinations of two neighbors on the line, because two neighbors are the same as L-neighbors in that case, so hence you get the geodesics, and in the other extreme you get harmonic. Oh, that's very nice, to actually see these as two extremes. You move to the R, but... Well, I mean, you have a map for either domain or codomain. It might happen that they all would use the same thing as the full neighborhood. So I make a suggestion in that direction, but I don't dare to care. I don't dare declare that this means harmonic map because there is a notion of harmonic map between two remaining manifolds. Which unfortunately is expressed in terms of integrals rather than infinitesimals.

1:55:00 Well, that we are able to decode. Well, I mean, if it's equivalent to some infinitesimal integral, might be put as a global... Yeah, of course. I mean, the average in probability... Classically, of course, can only be expressed in terms of an integral around a small geodesic sphere. Yes. Harmonic property. Harmonic map. Ah, yes. If it's an integral, it's hard to, if the value is not quantity, but... Right. ...but a more general space. On the other hand, since there are two extreme cases, one would certainly hope that there is an infinitesimal. Well, at least it's quite natural to consider these two things together, since in the extreme cases they give significant notions, and both are so intrinsically comfortable. The definition is something about minimizing energy, where energy is a global integral. Ah, yes. That's a conceptual thing I would like to understand, the G by which you encode the Riemannian method. The way you encode the key here is, think of it as a square distance between two points, and somehow that's just pragmatic, because why should you be interested in square distance? Really what you're interested in is distance, I have to admit, but it doesn't work. Yes, I know. We were just on Samos. Did I tell you this? No. The high school in the town of Pitagorio, in Samos, on the wall. I was going to say, you know, if I thought it was a theorem, I would imagine. No, no, alpha, beta, gamma. Anyway, there was some reason for considering squared distances. What did you say about the triangle? No, it's just the right triangle with alpha squared plus beta squared. Oh, but with a, b, z. Yes, but I was just thinking, well, that's probably the most appropriate place in the world to have such a display.

1:57:30 He was born in that town. Tradition has it. But also, in a way, think about these special quadratic Gorenstein algebras, which is, that's a flavor of taking the square root, by the fact, like the Clifford algebra. In a way, you're taking a square root because, well, there's all the vectors in the vector space. They have the square equal to the generator in the top dimension, the one dimension at the top, you see. So in some sense, they are all many square roots of that quantity. In fact, that quantity, you see, if you take the zero vector space, what you've got is just the dual numbers. So this quantity in the top dimension is really the epsilon whose square is zero. I'm introducing space. You've taken a whole number, a lot of square roots of epsilon. Ah, like that. Oh, say it once again. Epsilon cubed is zero. Of course, the cubes are zero, but in a way, it's a little more refined than just taking a cube and a zero. In fact, it has this sort of one and a half particular... Seems to. Well, why don't she count them? Oh, I think she certainly does. But what I do want to do is go to FedCrafty though before I go, so I'm going with that hand as a sign. What's that? Paperwork, yes. I have to go back and pay my bill. Right. Well, I have to go and pay my bill, so I'll come back and see. Yeah? Are you paying it back at all? Yeah, I've got to get back to the town and get my bill. So hopefully I'll have a chance to see. And you're going to have to go off on Monday, aren't you?

2:00:00 No, we're going off on March 1st. Oh yes, of course you should. Thank you for watching this video, please subscribe, like, share and leave a comment. Two half-intervals, in the case of the double covering of a trivial, non-topologically undue, the example of the double covering by Z2 of a space with a twist, the discussion that Anders was giving yesterday of the case of the Meebus strip, discussed that. Attention that you have to pay to the, as it were, the formation of the union of that double, in the case of that covering, the formation of the union of the structures of the domains of variation of the respective, what is in the case of a two topological domain of variation, that how it illuminates the remarks, see the point about having to pay attention to continuity conditions at the boundary of... Demands of variation, with respect to the unification of structural demands of variation in general, that remark of Bill's right back in the 1973 or possibly 1975 paper about having to pay attention to continuity conditions at the boundary of portions of domains of variation under unification of the structures of domains of variation in general, this is an example of where you have to pay attention to, for instance, the The information of the quotient, the information of the covering of the space, the double covering, as a quotient space, is the loss of the quotient decidability, character of the objects and local separability, see the point about local separability, and the case even of the double covering by, for instance, Z2 of the...

2:02:30 For the loss of the property of the question space as a locally separable space, so that you've got the example there of just the action even in one point, the kind of internal variation going on in the case of one point because of the action with which the space with which the structuring question is equipped, you've got... In the way that is reflected in conditions on the sections and fibers of the maps, for instance, particularly the loss of stability of the global sections functions and of extensionality conditions considered geometrically in respect of formation of coverings of spaces and the properties of, in the case of two topological domains, of variations of the quotient space, as illustrating Laubier's point of In the 1973 or 75 paper about having to pay attention to the point about there needing to be a continuity condition at the boundary of the domain of variation, having to pay attention to the continuity conditions. So the point is that in the case, for instance, of two topological domains of variation, this is illustrating the point about internal variation, the action of the group. Being such, because of the behaviour of the space with respect to covering spaces, the relationship between the local and global structure of the space and how this indeed might connect with the way that the spaces can be molted together in terms of, for instance, Posnikov towers, but that's of course a more abstract concept, possibly not directly relevant here, to the point about the way in which the structure of domains of... The structure of domains of variation with respect to the amalgamation of the structure of domains of variation with that of other domains of variation is not in general constrained by and is not in general intelligible in terms of the structure of analysis of those domains of variation in terms of the sets of points of the space. It's certainly the case.

2:05:00 The case where the increase in, or the way in which, with respect to the local and global properties, topological properties of the space, in the case of two topological spaces, it demands a variation. For instance,