Research Working Session: Infrageometry

Okay. Hello, everyone. Welcome to a working session about foundational things, sort of physics. We are going to talk about infrageometry. And Jonathan and Carlos and UTI and others have been thinking about this, and they. They have interesting things to say. So, Jonathan, over to you.

Okay, sure. So I'm just here as the kind of Shakespearean prologue. So Carlos, I think, has prepared some expository stuff. And then I think the meat of what's currently going on at the moment is the work that Carlos and UTI have been doing on weighted derivations and their connections and their relationship to partial derivatives and covariant derivatives in this setup. But I just want to give the. I'll give the broad context, and then I'll hand over to Carlos to kind of fill in some of the details of what I'll say. But so the basic idea is if we have a graph. So, okay. Also another point of clarification, I'm going to talk about graphs in the first instance. But an important feature of what we're doing is that nothing I'm saying is specific to the Arity 2 case, right. All the formulas we're dealing with can in principle be extended to arbitrary arity hypergraphs. So you've got a graph. And one notable feature of graphs is that they are naturally metric measure spaces, right? So they have a metric on them because you can take any two. You can take a pair of vertices and you can calculate a shortest path distance between them, and they have a volume measure that is like a discretization of the Riemannian volume measure, because you can do the geodesic ball construction, right? You can. You can grow balls from a single vertex and you can calculate discrete volumes just using the counting measure. So we automatically have a metric space. We automatically have a measure space.

Hold on. When you say we have a measure space, I mean, you're saying for a given node, we can associate a geodesic ball of size N number to that node. Exact number of nodes within that distance.

Precisely. So for each N, there's a different measure that we can define on this that we know that we can define on the space.

Okay.

So if you impose a kind of a volume cutoff, then your graph canonically becomes a measure space.

Yep.

With effectively a Romanian. Something like a Romanian volume measure.

What is the definition of Riemannian?

So, as in generated by. As the determinant of a Romanian metric. So, you know, so typically in differential geometry, you take your manifolds, you would define a. You know, you define a metric on it. If that Metric is positive definite, then it's Romanian. And you know, normally the, the, the volume at a given point is proportional to the square root of the determinant of that metric.

Okay, so let's walk that through for this case. Okay, yeah. So the metric in this case obviously doesn't have standard kinds of, you know, canorial legs. It has these much finer grained legs which are all the possible connections to a given node.

Right, right. Although, I mean, it's, you know, one of the, one of our core claims is that it's sort of trivial to, to equip it with, as you put it, tensorial legs. Right. So, you know, one of the things that I, that I want to get to, and again, I think Carlos will probably fill in more details of this later is, you know, how, how to, from first principles, equipment, one of these graphs with, you know, the direct analog of a tensor field of arbitrary rank.

That's very easy to do for which in. Okay, so hold on, that tensor field, I'm trying to understand the, the, the standard Romanian claim that, you know, volume is determinant, square root of determinant of metric.

Yes.

And yet we say we have a metric here, but our metric is defined in a more fine grained way than a standard metric in Euclidean, you know, in a manifold like space.

Right. I forget what it's called. Well, I can just do reduced determinant, I guess.

Let me interject very briefly to say that actually making that statement precise even is one of the challenges. And I'll go over the discrete version of tangent space and tangent vector and so on to try to formally get at the statement that the metric is something like a tensor in this discrete setting. So you're absolutely right. In this setting it's kind of a discrete information object that is far from a tensor. But we're trying to work towards that, be made precise.

Right. I mean, you have to define equivalence classes of kind of directions in the graph. So think of them like coordinate directions.

Yeah, yeah, something like that. Yeah, exactly.

Exactly.

Yeah.

Right.

Okay, that's cool. I didn't know we had this reduced scalar volume expansion. That's cool.

So, so, so this is, you know, for something like the FLRW metric, this is the volume form which, which you can see, you know, depends on the, on the square root of the determinant, the absolute value of the square root of the determinant.

And what's that? Reduced volume expansion.

So this is showing the discrepancy between the volume form in this manifold and the corresponding volume form in this case in Minkowski space, in the flat manifold of the corresponding signature.

I see, okay.

And this is the thing that gives you the ricci curvature, the ricci scalar. But anyway, whatever, the point I'm making is that this is the analog of the measure structure in Romanian geometry.

Yep. Okay, Onward.

Right, okay, so the claim then is. So, so this is effectively, you know, the undirected graph structure is effectively our metric. Right. Amongst other things, but it's good.

But distance on the undirected graph is the metric, right?

Exactly. Then the claim is that if we construct a. Okay, so then the question is what is a connection? And so one of our basic claims has been that if we take, you know, if we construct a random graph now with directed edges. Oops, sorry. This is quotes like a connection. And I'll explain in a bit more detail what I mean by that. But so in this case, you know, this is something like. This is like a generic position, potentially non metric sort of affine connection. But if we took a metric like this, if we take an undirected graph and we just construct its directed graph analog, the claim would be that this is now a metric connection. So in other words, it's compatible. The fact that it was generated from this undirected graph is saying that this connection is now compatible with the underlying metric tensor.

I think I don't understand enough about, okay, define a connection in its simplest form. No fiber bundles, no whatever. What is a connection?

So, okay, I would say there are a few different ways to think about it, but at least the way I think about it is it's the rule that tells you how your coordinate basis changes as you move from point to point.

Yep.

So I have a local choice of coordinate basis and I want to kind of transport that coordinate basis to some other point in my manifold along some direction. And then, so the thing that tells me the new coordinate basis at that new point is essentially given by the structure coefficients of my connection, except that.

There are coordinate basis here. So I don't really know how that works.

Right, okay, but, so that's not the intuition. So the intuition instead here is to say, well, the connection is the thing that tells you at each point essentially what the tangent space is. Right?

Yes.

So the idea is. Okay, so, so now if, if we. Okay, so if I select a random vertex, let's take random choice. Let me call this direct as well. So I select a random vertex in that. And then now I do something like, let's highlight directed graph. I'll style that in green and let's do directed edge, vertex, hash, Graph, vertex 1. That. Okay. So that's a tangent space at some. Let me choose different vertex.

Yes.

Yeah. All right. So that's the analog of a tangent space at that point.

Right? Yep.

So the intuition is that each one of those outgoing edges is, you know, is a tangent vector. And so the sense in which this directed graph is like a connection is that it's the thing that allows us to construct tangent spaces at each point.

But I mean, it allows to construct a tangent space, but it doesn't. But the transport of a coordinate system is something that has several levels further of definition. Well, it's constructing a tangent space. Yeah.

Right. So one thing in particular that it doesn't allow you to do is to directly say, you know, this tangent vector maps to this other tangent vector when I transport it over to this point.

Yeah, right. But the point is there's no way, I mean, with different points.

Right.

But there's no correspondence between tangent vectors. Go ahead. Yeah.

You're not going to be able to do that. Precisely. Because there's no structured coordinate system here. Right.

Yeah.

Because the number of tangent vectors, because this is not a regular graph, the number of tangent vectors at each point is not fixed. You're not in general going to be able to construct a mapping of that kind. So in a sense, I mean, this is maybe a broader philosophical point to make about the infrageometry program is that what we're essentially asking is what structure can you retain? We obviously can't build a connection of the normal kind of what structure of a connection can you retain as you go to this more general setting.

Indeed.

Right.

Because there's equivalence classes. I mean, if this was a grid graph, there would be an obvious correspondence between, you know, labeling of tangent vectors at one point and another.

Right, exactly.

And there isn't in this case.

But the important point is that this directed graph contains all the information that you need to say, if I move to this point, what is the new tangent space?

Sure. Which is what new tangent space is, have nothing to do with the previous point. It's just any point. You can construct its tangent space by just looking at the out components.

Right, right. But if you only knew, you know, if you only knew a subgraph of this, you would be able to, you know, given local, you know, in other words, given only local information, you could work out the tangent space of the.

Exactly. Yes, Yep.

Right. So it's in this sense that we say that this is sort of like this is akin to this plays the same role as a connection would play, at least in context.

Well, at least a tangent space. I don't quite understand the connection part.

Yeah, actually I wanted to say the same. That's one of the current degenerations of the discrete case that I think the role of the tangent bundle, which to me is also more intuitive to spot there, as I was going to explain later, seems to be in terms of what object it represents or is represented by, seems to be the same as the connection at this point. But there's many aspects of it, like additivity, what it means to transport, and all these things that are still subtle, that they're not clear. But I think as a tangent bundle that that structure is much more straightforward.

Okay, maybe let's do that. Let's. So if I now say. Okay, so if we construct one of these tangent spaces for each point and I say directed gra.

Yeah, when you're saying, Carlos, tangent bundle, that's assuming some kind of fiber bundle structure, which I don't really see here. I just see a tangent space here.

You don't need to assume the tangent bundle is the fiber bundle.

Yeah. It's just the collection of all the tangent spaces.

Okay, fine.

So look, if I take that. Oh, sorry. What am I doing? Directed edge, not directed.

Okay, you're going to draw the tangent bundle.

What am I. Wait, what did I do?

You want to thread.

There we go.

There we go.

Sorry.

Yeah. Now you want a thread. You want a thread.

Yes.

No, map thread. Map thread onto the tangent.

Okay, let me.

Hang on, wait. Oh, you want them different colors.

Yeah, exactly. The point is it needs to be in a. Needs to be a disjoint union. It can't just be a union. Right. So hang on.

Yeah, let's see. This. This is going to be interesting to.

See if I do.

Something like. Yep.

That and highlight graph. Let's call that tangent bundle. Okay. So then that's our tangent bundle. So each.

Wait a second, wait a second, wait a second, wait a second.

So there's overlap. Obviously. You know, the same vector can appear in multiple tangent spaces.

Right. So I was thinking we would actually show the thing sort of off graph as an additional little tuft at every point.

I see. Okay. So. So in that case, we could literally just say. I guess if I took keys. No. If I took values of tangent bundle and I did a catenate of that and then graft that, maybe. No, wait, hang on.

No, no, you need to make. Because you need to make at every point.

Hang on. Why Shouldn't I get multi edges? Why am I not getting multi edges? There.

There's no overlap, right?

There is no overlap.

No, because.

Go back to your previous.

No, yeah, yeah, yeah, yeah. No, you're right, you're right, you're right.

Because, because everything. Okay, yeah. Okay, hold on. Let me understand this. So every node. If we colored the nodes in the same colors as. As we were coloring their outgoing.

Yes, so.

Sorry.

Directors graph. So how could I do this? I would do.

Try just doing the. We could taking that graph and highlight it again. Wrap graph around that highlighted graph.

Yeah. Okay.

And then just do it for.

Maybe you can just extract a list of sub graphs and highlight it and it will automatically highlight the same color vertex and the outgoing edges.

No, you want to make a list of the. Sorry, you want to list of subgraphs. Vertex list.

Just vertex list for every vertex, extract that vertex out component and then check.

The subgraphs of those and highlight those. But I think you can do it right there. I like graph. Just save for the vertex. Just say vertex list. That second thing.

No, but, but this is the vertex list, so I. I don't understand why that's not working. Oh, sorry.

Are they highlighted? Just.

Yeah, they are highlighted. They just all highlighted the same. Okay. Anyway, we. We know what it's supposed to do. We can, we can make this work. Search.

Yeah, there we go.

Okay, fine. So each. That's kind of interesting. Okay, so it's making a tangent, but. Okay, fine. But then I understand fiber bundles better already. Okay.

And so the thing that I like is that it makes very clear which. This is a tool which I wish I had when teaching differential geometry is it makes clear the connection, the relationship between the space of smooth sections and the space of permissible vector fields. Because if I go through and I just say so a smooth section is going to be a random choice. Here. And then if I do a highlight graph style. That. Oh, wait, what did I do? Oh, sorry, the. Right, there we go. And so that's now a vector field.

Let me understand that claim.

Right, So a vector field is essentially, you know what is a global section. A global section is basically a choice function applied to this tangent bundle. Right. That you're just saying let me assemble a collection of edges by just selecting one edge from each tangent space from each element of the bundle.

I see.

And the space of all.

Okay, fine.

The space of all such selections gives me the space of permissible vector fields.

Indeed. Okay, all right, so let me just understand this. So you're saying a vector field and you're claiming a tensorfield, a rank 2 tensorfield will be select 2 edges out of each of those.

Yeah, precisely. So if we select. If we go through our tangent bundle and we select an N tuple, then that gives me a rank N tensorfield.

Of rank N. Okay. Okay, so that's a vector field. So now, now what's your point? So you had a vector field, and then what? I mean, it's kind of cool that we can generate, you know, one thing that's interesting is we can enumerate all possible vector fields, which is something you usually are totally unable to do, right?

I mean. Yeah, so. So typically the space of smooth sections over a manifold, you would, you would reason about abstractly, you certainly wouldn't enumerate.

Right.

Jonathan, should I take it. Take it from here because this is.

Yeah, yeah, go for it. Go for it. Yeah.

Where are you saving that file, by the way, Jonathan? Are you saving it in our working.

I. I can do.

Yeah, why don't you save it there?

Okay. Carlos, do you want to screenshare? Sorry, I. I may have unshared too early.

Okay. Okay, so, yes, a very brief recap where it's been said in a geometry project. We're trying to. I mean, the goal here is to make statements of the war from physics projects more precise and have a way to, you know, treat continuous manifolds, traditional smooth manifolds and discrete spaces under the same hood in some sense. So generally you like to correlate manifolds with hypergraph or hypergraph structures. So Jonathan has mentioned, this particular correspondence is at the top. So these easily, quickly.

Oh, no, wait a second, wait a second. You're going too fast. Stop. The last element there, volume form to a simplicity hypergraph. What do you mean by that?

Oh, yeah. Okay, so this is a sort of a preliminary correspondence. It's just sort of tentative. So that will be so. So we could go into that example. I wanted to focus on the simplicity.

Hypergraph basically means a hyperg where every distance is like in a simplex. Every distance is less than something or other. And you're looking at. It's like, like a judicial ball, but it's kind of a simplex made in a hypergraph. Is that what you mean?

Well, by simplicial hypergraph you mean a simplicity complex. So every phase has the sort of sub faces and all that stuff.

Yeah, but there's no faces in the graph yet. But there is a construct that I was just talking about that might actually be interesting, which is, you know, right now we've been Talking about judic balls, okay, where you start at a point and you, you know, go a certain distance away. What about a simplex in which, if I'm not mistaken, the definition of a simplex is there are a bunch of inequalities that say, you know, when you. That everything is, is. Well, in, in this discrete case, that everything. You're looking at the set of points that are each within some amount of other points. Is that right?

That's not the sense in which I'm using simplicial. I'm using it in the sense of simplicial complex. So just, just to say that you could have a hypergraph that is simplicial, meaning that for each edge you have edges of lower order, their faces.

Right, I understand, but I don't understand how to get those faces out of the hypergraph. But the thing I was just talking about, I think is an interesting construct in its own. Right, Jonathan, do you understand what I was saying? That. Or anybody understand what I'm saying? So I'm doing what I'm saying.

I don't understand how it differs from.

Our standard volume element because I'm imagining a set of. Well, maybe it doesn't, but I'm imagining a set of nodes in the hypergraph. Yeah, yeah. Which have the feature. Let's see, in a simplex, in a simplex embedded in n dimensional space, it's a set of inequalities, right. You can define the region of the simplex by a set of inequalities. And is it not the case that those inequalities, a little bit like a determinant, have the feature that each inequality involves some variables but not others? Each inequality is a choosing of variables. So in other words, by the sort of single point geodesic ball says everybody's measuring.

If you're just saying that we can construct a volume measure centered not at a point but at a subgraph, then I understand what you're saying.

I think I'm saying a little bit more than that, potentially. I'm saying imagine that. Well, I don't understand it yet either, but, but what I'm saying is we take a set of points and we find the set of points which have the property that all points in the set. It's a way of defining a subgraph, okay, where, okay, the geodesic ball definition of a subgraph is every point is less than graph distance N from a given point. Okay? So what I'm saying is imagine that you have a, a multi base point version of that in which you say that every point is that it can.

Lie within distance 4 of point 1 and within distance 5 of point 2. Whatever. Right. Which is, which is just an intersection of music balls.

Yeah, I think that's right. But, but I mean, hold on, hold on, hold on. But I, I'm saying something more than that. I'm saying imagine that you construct something where every point, you've got a set of end points. Okay. And every point is at most distance R from K of those points. So in other words, it's. It's just a way of defining a subgraph. But I think it might be an interesting way of defining a subgraph that might. Because I'm trying to understand like the determinant that shows up. I kind of think this might give you a determinant. Does it make any sense so far? The. Jonathan? Nick? Anybody? Carlos? Anybody?

I think I'm. I'm seeing where you're wanting to go there. I'm just not sure that. I mean, the determinant is going to be an operation of vectors. And my entire thing is this is not so straightforward to understand. There's no.

I agree, but it's just a construct in these graphs that is expandable. And I don't think we've thought about. I mean, that is, that it's the, you know, it has three parameters. The number of nodes, the number of choices to be taken at each. I don't know, in each, maybe it's. And the maximum radius. Okay. Anyway. All right.

But again, I don't understand why that's not just Boolean algebra applied to, you know, the existing volume measure.

I think it may be, but I think it is. It's. It's just going to limit differently because. Well, let me think about this for a second. I mean, I think it depends on the N and the K and how they limit. So I'm saying if, If K is small compared to. Sorry, the. The usual case. No, it's not quite the same thing because we. We're saying in the usual case, we have a ball that we're gradually, you know, creating around a given point and we have a parameter and that ball can grow arbitrarily large as we increase that radius parameter. I'm not sure what I'm saying. Okay, let's continue. We can.

Actually, that interjection might have been useful for me to emphasize something more philosophical at this point, which is that part of the motivation is to try have a framework in our kind of a theoretical understanding that allows you to not think in terms of limits. Because if you think about it, you know, you have a discrete manifold, something like a discrete hypergraph, finite hypergraph, but very large. And you're saying, okay, this is tending towards or is approximating this smooth object. Actually there's some kind of qualitative jump that is made at some point from a discrete to a continuous thing. And we wanted to somehow understand them in, in the same framework. That's kind of important to keep in mind in terms of motivation. So just as a very quick recap, what we're doing with Jonathan just mentioned a few minutes ago, is that we have a finite set of points, and I'm going to highlight, you know, the graphic element and the geometric element. So here's a finite set which is to be understood as the points of your space. Then some undirected graph structure in it. I would like to call it the connectivity structure. Jonathan said, correctly pointed out that it's also like a metric structure on that space, but more generically, let's consider it as a connectivity structure. And that's going to be my definition of a, of a working definition of a discrete manifold. Now, the collection of length one paths that you can define, and this is, as we just visualized very much more interestingly with Jonathan, is the tangent bundle, what I call the discrete tangent bundle here. And of course, the tangent space at the point, as we just so live, is this idea. You pick a vertex and, you know, all the outgoing directed edges there are that are induced from the undirected graph. So they are basically all the length one paths from that vertex is the tangent space at that point. Okay. Similarly, we also saw live with Jonathan what a vector field is in this context, which is a section of the tangent bundle. For those of you who are familiar with.

That's cool. That's cool. Yeah. Okay, go on.

And in this case. Yeah, so. Okay, so the next thing to do is to develop some kind of calculus, right? So what is a function? Well, obviously a function is going to be some kind of assignment of values. In this case, let's consider integers just for simplicity. And I mean, it might be interesting to also consider functions on edges. But for now, let's. Because we said vertices are points, let's consider just functions on vertices. So let's denote this usual commutative F algebra. This is just to say that you can add functions, multiply functions, and they are, you know.

Just a minute, hold on one second here. Okay, wait a minute. What do you mean by. So you can add functions by just sort of superimposing you're just adding the numbers at every corresponding point.

Yeah, exactly. So the usual natural, what I say, the natural community of algebra is you do point wise addition, you do point wise multiplication. And so if F is a field, then you get a ring, whatever commutative algebra of functions is. The usual thing is defined. It doesn't actually care. Importantly, it doesn't care what kind of graph you put on top. That's important. Yep, that's, that's something to be noted. And so here the goal is to try to find something like vector fields in differential geometry. So for anyone who's learned differential geometry knows that vector fields can be understood in two different ways. They can be understood as sort of infinitesimal assignments of smooth curves, like an infinitesimal flow locally at every point, or they can be understood as the derivations of the functions on that space. Obviously there's a very natural correspondence, as you can imagine, but those are in principle different entities or different objects that you have to prove they are equivalent. So in our context, we have a very natural generalization or contextualization, I guess, of that context of that concept, which is what we call the finite difference operator, which of course is nothing special to our context, but it's, you know, appears in here. So we say that a discrete vector field acts as a finite difference operator. Whenever you take a function F in blue here and you know, given a pair of, a pair of points which, which has a directed edge in the graph pq, you take the difference from Q to P. I mean, that's nothing. Fascinating.

Hold on, hold on, hold on, hold on, hold on, hold on. So these are two adjacent points in the graph connected by a directed edge, and there is a directed edge from P to Q. Is that right?

So let me say it like this. So I would say something like this, right? So I would say PQ is in tm, which is all the possible directed edges that connect the points, right? So if you have such a directed edge, then you can compute the finite difference we say at P, right? I mean, I say at P because it's the outgoing point, right?

Fine.

And so if you have a discrete vector field which is given an assignment of such directive that just at each point, you can effectively compute a new function just by computing the finite differences at every point. We call this G in magenta here, and I just visualize it here. Unfortunately, this is just computed quickly by hand, but that's effectively what this becomes. This blue function becomes this magenta function by applying the X finite difference operator, right?

Okay, hold on, hold on, hold on. So in this graph, the result of this is. I mean, the coloring of the edges is irrelevant here. What you've eventually just got is this derivative. What is this thing usually called? This, this field that. The field that you get, you, you started off by one scale of. You had a scalar field originally, if I'm not mistaken, and you've computed another scalar field which is essentially the. The.

This is what is called the directional derivative of the. Of the function.

Right.

So. But it's not quite the directional derivative, it's the finite difference operator of that. So the delta X.

Except that you picked random outgoing edges here. You could have picked this, others.

Yes, exactly. Yeah.

You picked the directional derivative somehow with respect to some particular choice of vector field.

Exactly. Which is entirely analogous to what happens in usual geometry. Right. You pick your vector field and you compute the derivative with respect to that vector field of a given function.

Yeah, it's essentially the gradient in. Yeah, right. At those points with respect to that field. Okay, I get it. Yep, exactly. Yeah.

So in conventional calculus parlance, it would be like the gradient that every point in here is just to find a difference. But that's. That's exactly the point. So to make that as.

So the gradient. Okay, so the gradient is indexed by. It's. It's grad. And then the subscript of the graduation is effectively the vector field, which in our case is being chosen is this kind of randomly chosen subset of outgoing edges.

Yes, one choice. I don't think random here is too much of a.

Well, I mean, a particular choice. Yes, okay, one choice.

Yes, exactly.

Yeah, yeah, yeah, okay.

Exactly. So in order to make this as tangible as possible, let's look at these things, right? These discrete calculus of this finite difference operators. So something easy to check is that, okay, these are operators that send functions to functions.

Right.

Just saw. And they are. They act linearly also by construction, very trivially. I mean, constant annihilating, obviously, because we're checking differences in between points, so, you know, you don't change. It's zero.

In other words, if the scalar field is constant, then the derivative is zero. Yes, I agree.

Yeah, makes sense.

Right.

And then, but most interestingly what happens is if we did this with vector fields in the continuous case, we would get the liveness identity, AKA the product rule.

Right.

For. For your, for your, for your functions in here we get something that is called the weighted liveness identity, which is the liveness identity, these two terms. But then it has this kind of quadratic term which it's easy to understand because you can imagine the way you prove the product rule with the definition of derivation of derivatives. You don't have the, the term that is going to cancel out that it goes to zero when you take the limit of, you know.

Sure. Because there's two derivatives in there and there's only one in the other ones.

Exactly. Yeah. So. So this is what it is. So this is the most analog that we could find. The vector fields.

Okay, hold on, hold on, hold on. Let me understand this. Okay, so we're taking these two vector fields, I mean two scalar fields defined by F and G. And we are. And we can get the product vector field.

Yep.

And by point wise multiplication. Yeah, right. So this formula. I see. I mean I, I'd have to actually work this out, but I, I believe you that this is. Yeah. I mean, can you give a derivation of this? Isn't, Isn't this equal to. Can you explain why this is true?

Yeah, it's very simple. Again, each delta here is just a difference between P and Q for some P's and Q's.

Right.

So when you work this out, many of us probably did at some point when we were learning calculus, you compute the definition of a derivative of a product of functions. So basically you have to complete the terms so that you find the, this quotient of differences. In the case of the derivatives, this is entirely analogous. You're just basically completing. So when you do complete, you get this derivation like property. This is the actual, this equality here is the actual Leibniz rule. It's the actual product rule for derivatives.

Right.

So in the usual computation, if you had sort of one over H everywhere and you take H to zero as a limit in the end, like you do in derivatives, there will be an H here.

Yes, I, I get this. I'm just, I'm just not. I, I just have to get a piece of paper and just write down.

I mean, you can just think of this as the second order term vanishing because in, in differential ordinary calculus you just said epsilon squared to zero. That's exact.

Yes, no, I understand all of that. I'm just trying to see why this formula, something much more obvious than this, the formula for the product of these things, why is the formula independent of the fact that that second order term vanishes in this limit. Why is this thing.

Yeah, that's interesting to try to understand. Actually I don't have good ideas to give a good intuitive understanding of because normally you. The. The product rule stands as something like. It preserves dimensions in some Sense like you have a, you know, delta dimension, F dimensions dimension. And this is homogeneous in dimensions.

Okay, but we should, we should just, as a matter of arithmetic, be able to work this out. Right.

Take it, take Stephen. I, I would say that's obviously true. Yeah, exactly. Because of arithmetic. Because you're, you're just taking the product of two differences. You're expanding out the brackets, right?

Yes.

Those are the terms you get.

Okay, that was what I was looking for. So, I mean, let's see. We've got. So if we write this out, It's. It's like P1. I mean, if, if we're talking about. It's product of differences. No, we're taking the F times G. So help. Just write out the arithmetic. Okay. We've, we've got. Let's, let's just do it for a single edge. Okay. We've got these values P and Q, for example, on a single edge. And we've got. No, we've got. Let's say FP F sub P, G, sub F, sub P G, sub P F, sub Q, G, sub Q. Right. Okay. You can write it like that. Okay. F times G at Q. Okay.

Yeah. And you want to compute the difference of.

Yeah, minus F time. F times G at P. Right. It would make. It will help me if you put parentheses around the F times G there.

Yep.

Okay. And now this is equal to F of. What is it? It's F of Q. Just, just write out the arithmetic. It's F of Q. Times. Yeah, I know, I know, I know. You're going to write out the same formula. I want you to derive the formula. Right. F times G. So at each. Go ahead.

Well, I mean, I. It depends on exactly how you want to prove it. I don't know if UT has a calculation that is already visible that it's maybe faster for you to see.

I mean, so first you expand this. You get F Q GQ minus F pgp. And then you split the F, the first term. FQ gq.

That's right. That's the right thing to do. Okay, so can you please write this out? F of. It's F of Q, G of Q minus F of P times G of P. Right?

Yes.

And then you split each of them, and then you take common, and then you etc.

Yeah, I mean, I'm. I was sure it was going to be like this. Okay, so.

And now you're.

Okay, so write it out, Write it out. So, so then this is equal to.

The delta, the, the term that it's missing. Right.

Well, no, that's an exact. That's an identity. That. First thing is an identity. There's nothing. Then you can add. Then you can break that up. Nick is. Nick is sending something in the chat. Why don't we run the thing that Nick just sent?

Yep. Yeah. That's. What UDI is suggesting here, is that you add zero effectively. So you add and subtract the same term. Right. So F, G, E, and you subtract same thing.

Right. And then you rearrange it times G.

Of Q. Rearrange it. You get what you want. So whatever.

F Q minus F of P times G, Q. No, no, no, no, no. Delta of delta of F has got to be F of Q minus F of P times. Right. F of Q. So write it out. F of Q minus there. F of Q per N, F of Q minus F of P. Right. No, it's not that. It's not that. It's. It's. Look, it's that times G. Right? F of Q minus F of P.

Yeah.

Yes, times. Well, whatever we can write, this is clearly going to work. Okay. But I'd like to actually see this written out as a piece of arithmetic.

Okay, sure.

Yeah.

So.

Okay. All right, fine. And then.

And then you. You get here. The same thing with. With F. Sorry.

Yeah, no, I. I get it. And I can. I mean, I can see it's. Right. I just. I. I want to. You know, the goal here is to make everything like self. Evidently.

Yep. So that. And then what you do is obviously.

Sorry.

F U.

Right. That's delta, P, Delta, F minus. No, it's. It's F of Q minus F of P. Right. Times G. Yeah. And then we should have written this out in actual code, and then we could have just expanded it. But anyway, that. Okay, I get it. Plus plus the deltas, which is the product of those. That's the product plus. No, no, no, no, no, no, no. It's not that. It's. It's the. Take those parenthesis terms, plus plus. No, plus plus. I mean, and then just take out each of those parenthesized things, G of Q minus G of P, and just copy them. Copy them and write them down there. Okay. No, no. Copy the thing that you have there in parenthesis. Copy the thing in parenthesis. No, yeah, that one. Yeah, that one there. And copy that and put it in the plus. And now copy the G1 and put it in the plus.

Well, but that's. That's not technically correct yet because this is F of Q. So what you have to do is then. So this term here is, Is not present yet.

It is, it is. Just put that down there. Put that down there. This is going to be. Okay, isn't it an algebraic identity that, I mean, you know, if you untangle this. I, I can't quite do it in my head because I, that's why I developed computer systems to do this. Is. There's just an. Okay, that's the point. Yeah, I know. Right, right. Exactly. Exactly. So, okay, we'll write it out later. Okay, we'll write it out later. But there's going to be an algebraic identity here, which is what leads to that weighted Leibniz identity.

Right.

A trivially derivable algebraic identity.

Yeah, yeah, very easy. I mean, it's almost done. I mean, if I just make it equals here, you just substitute F of q here for this expression, then it's done. It's all you need.

Okay, all right. Okay, fine. Please, let's write it out. Okay? Please. Right. Not, not here, not now, but let's get this as, as you know, in the next iteration, let's just write this out so we know that algebraically, why it's, you know, with trivial algebra, why it's true.

Sure, yeah. This is written out in a different notebook. This is just.

Okay, all right, fine. Okay, all right. Okay, okay. I just, I was trying to get an intuition for why it's true and it's just a rearrangement of terms. And I get it, and I just like to see that. Okay, all right, so we, we assume it's. Let's go on from there.

Okay, so anyway, so the, the, the point here is that this is the closest we can get to something like the vector fields of our manifold and something that looks like calculus with these things. I mean, calculus at the end of the day is going to be some form of taking directional derivatives of other objects, not just functions like what we call lie derivatives or covariant derivatives. We are doing it with respect to a connection or a metric or something like that. So before we can go to those heights, in some sense, there's some interesting questions that need to be answered. So here's where uti's work was particularly illuminating. So, okay, so given. Arrived at this point where you find this, this expression, and as you said, just trivially check it, it's true. There's no escaping because there's no limits to take. That's. We have to take at face value. We have to sort of ask ourselves, what are these weighted derivations doing? What are, you know, what Kind of.

Hold on, hold on, hold on. What, what you've defined here is the derivative, directional derivative. And the next obvious thing is a covariant derivative.

And well, depends if, if we don't have a metric structure.

I mean, but we do have a metric structure and the covariant derivative is surely something very straightforward, given that this is. What's that? Go ahead.

Can I interject here? So.

Yeah, please.

A notable thing about the, about this definition of the derivative is that it doesn't use anything about the structure of the graph. Right. It just uses the fact that there's an edge. So what is the essential difference between a directional derivative and a covariant derivative? Well, a covariant derivative has these additional terms that relate to the connection coefficients. So the obvious. Right. So morally the obvious definition of a covariant derivative is something that actually takes into account some aspects of the underlying directed graph structure.

Yes.

And I, once Carlos is done, I mean, we'll see how this, we'll see what the overlap is. But I can give some indication of how that would work.

But yeah, yeah, because I would think that what you're doing here is you're taking a. Just as you've defined an arbitrary vector. Yeah, I think I can see how this is going to work. Ok, go ahead, go ahead, go ahead.

So basically, I mean, that's basically my landing point and we should see UDI's.

But I want to see the rest of this here. So.

Yeah, yeah, so I'll finish, I'll finish this with. The point of all this is that you get to understanding, okay, we have something like directional derivatives in our context, this fine difference operators. And there's something that is a little bit uncanny already because we cannot treat them as. So, okay, so you can't compose them like derivations. Right? You can effectively say, okay, I'm going to take the finite difference of this function and then I'm going to take the finite, this other finite difference of the result. So you can effectively compose. This is what I mean by this circle here. You can compose these things. But if you think kind of geometrically discrete, geometrically about these things, what it means to compose is not so obvious. So the obvious thing to think is that, okay, if I have a length one path that. Or two length one paths that could compose to a length two paths.

Right.

And so, but obviously that's not going to be true in general. So my, my statement here is that in, we know in, in the, from the smooth case, from the continuous geometry case we know that you take two derivations, you compose them. So in this form, which is unfortunately here, sort of in this line, so in this, this form here is not a derivation, right? You take two derivations, you compose them. That's not a derivation. In general, we know that in order to make that a derivation, we have to take the commutator. That's kind of the hallmark of all these result of differential geometry. So what is the analog of that in the discrete setting? That's a kind of the mystery, right? That's what we're trying to solve here. What is interesting, and this is the observation I wanted to make, is that we can certainly think of discrete vector fields being assignments of directed edges to each vertex of the graph. We can certainly think of some kind of transitive construction out of or transitive completion of this Length 1 paths into Length 2 paths, Etc.

Right?

So it might be that, you know, you have two vector fields that compose. I mean, you can. Okay, so if you think of just the highlighted vector fields, imagine I have two such choices, this orange one and a yet and a green one. You can simply compose them. The problem will be that you will define length two paths that might not correspond to a length one path by. By transitivity.

Okay, hold on, hold on, hold on, hold on, hold on, hold on, hold on, hold on. All right, so this vector field picks an outgoing edge from every node.

Correct?

Correct. Okay, so let's get another vector field that also picks an outgoing edge from every node.

Exactly.

Then, then what you're saying is the composition of those vector fields, which would normally be kind of like if this was like a flow, you're applying the thing, you know, one thing, then the other thing. So you're getting, you get to a certain place with this composition of vector fields. So now this derivative operator acting on the composition. So in other words, you've got a directional derivative grad. Sub, you know, direction and sub vector field, basically. And now you are, I mean, the concept of composing vector fields is not really a thing of the continuous case. In the continuous case, you would be composing two, you know, operators.

Yes.

And those operators would be, would be, for example, flows that go in the direction of a certain vector field. And they are, you know, they're derivatives defined by a vector field. And you're integrating that vector field and you're, you're doing integrate one vector field a certain distance, then the other vector field a certain distance.

That's right. So the analog. So that's right, the analog here is what I'm trying to Bring to this discrete context is some kind of discrete version of that. That seems to be quite interesting to me because as you described, you have two vector fields. You can effectively drop yourself in any of the vertices. You flow with the first one, you land on the next vertex and then you flow with the second one. So that's what we call composing flows, right. In the discrete context. So the net flow, it's something that takes you from one vertex to another. The question. So in general that's not going to be a. That's not going to define a connectivity of those vertices. What I mean by that is that's it's not necessarily the case that if you have a tangent bundle like this one.

Right, no, I understand that there's a tangent bundle that is also that the composed the composition of those two vector fields. Where we now have this notion of a composition of vector fields, which we don't really have in the continuous case. There's no guarantee that a composition of vector fields is also a vector field.

In that tangent bundle. But there might be a graph on that sensor vertices. Obviously that includes it.

You can make another graph. But I mean we're assuming. So we're saying within this graph there's no guarantee that a composition of these things. We're calling vector fields, which we shouldn't call vector fields because that's going to be really confusing. They are really. What the heck are they?

Well, these three vector fields.

But the notion of composition does not exist for vector fields.

It actually does. I mean if you think of vector fields as infinitesimal flows, you can just compose them.

No, I know once they're flows. But can you think of the vector field as. I mean, I guess you can say that there's a correspondence between vector fields and flows. So you can think of composing those.

Yeah, so the problem is that they don't. So that the flows themselves don't compose. And that's precisely this thing with the Lee bracket. You need to sort of get rid of the redundancy for the non commutative video of the flow, so to speak, you have to account for that. And so you can only compose them in this sort of anti symmetric way. But effectively the commutator is the composition of the flows. Right? So, so flows and vector fields are effect infinitesimal flow and vector fields are effectively the same. And by being infinite decimal, you are getting rid of that, of that, of that problem that we have in the description.

Maybe we should call this thing a discrete flow rather than a vector field. I mean we can Say it corresponds to a vector field.

But.

But I think it's more, more naturally thought of as a discrete flow.

Can I, can I, can I interject for a moment?

Yeah, yeah.

So that. Steven, they are both vector fields. The, the distinction is that one of them lives in the exponen and the other doesn't.

Right? Yeah.

They just live in different places which.

Live in different places.

The. So the, the, the. The vector fields, pre and post composite, you, you, you take two vector fields, you compose them together, you get another vector field, but that new vector field does not live in the same space as the old ones.

Well, wait a minute, there's an infinitesimal. Remind me. I mean, so there's the infinitesimal.

No, I'm precisely trying to formulate this in a way that doesn't involve infinitesimal flows. All I'm pointing out is that's a completely well defined operation in the continuous case. It's just that the result lives in the exponential, you know, in the. Whatever in the image of the exponential map.

But the exponential map is what you get by going from the infinitesimal, you know, thing to the. Or am I confused here?

Yes. Yeah, it's what takes you from the. Yeah. From the sort of the linearized space to the original space. From the tangent space to the original.

Okay, so which one are we talking about here? The exponentialized one or the. We're talking about something in the exponentialized case, are we not? As we look at these discrete edges.

I would say in the discrete case is both and same thing happened with the connection and tangent bundle. And I mean the discrete case kind of collapses several of the notions that in the smooth case are separate. So they are in terms of the data is actually both, I would say.

So look, the important point is that. Okay, so you have a. Let's take the trivial case of this, which I can show some images of once callus is done. But so you take the tangent space at some point, which for us is just. That's the vertex out component truncated at distance 1. But if you took the vertex out component truncated at distance 2, those paths are also valid vectors. They just don't live in the tangent space. Right.

Sorry, say that again. I was distracted because I was reading Nick's comment. Composition of vector fields is the same as multiplying corresponding subgraphs. Adjacency matrices.

Yeah, that's what we described as a two step flow, basically.

Yes. Okay, that's. That's correct. That's what we're talking about. Okay, now Jonathan, what were you Saying, Sorry, I was distracted there.

Right, so if you take the, you know, the distance, the length, two paths, those, those are also. That also forms a vector field.

Yes.

It's just not, you know, that vector field doesn't live in the tangent space.

Exactly. Yes.

Right. And so the map that takes you from, say that vector field, sorry, that takes you from the tangent space to a space that contains those vector fields, is the exponential map. So when you compose the two vector fields that live in the tangent spaces, the infinitesimal ones as you're describing them, then the image of that is a vector field. It just lives in the image of the exponential map. It doesn't live in the original space.

Makes some sense. Okay, all right, fine. Okay, so let me understand the Lee Bracket business. Can we go down to the Lee Bracket business?

Yeah, but Jonathan, do you want to say that these are operators on, so linear operators, but not necessarily derivations? Does that matter?

Yeah, I would rather call them linear. Yeah, exactly. Linear operators with derivation like properties.

Yeah. What's the them in this case?

So, yeah, so something like the composition of two derivations, right?

Yeah.

So as an algebraic story, it's quite straightforward and something that UDI probably will remind us. But basically if you have two derivations, they, they are linear maps that behave like derivations with, you know, the limits property.

Hold on, hold on, hold on, let's slow down, slow down, slow down. So you're saying I've got. Okay, I'm, I'm not understanding something here. We, we, this, we know how to take the directional derivative and get a new scale of field that is the result of. Oh, I see. We've got two different vector fields here and we're taking derivatives, We're taking just the product of the results. We got two scalar fields. One of them is the result of the directional derivative in one vector field, one of them direction derivative and the other vector field. And that's. And we just get a bunch of numbers, we assign numbers to every point.

Yeah, those are maps that send functions to functions. Yeah.

Well, each one D, sub X, D, sub Y sends one scalar field to another scalar field. Correct? Correct.

That's right. That's exactly right.

So now the result, the product of those two scalar fields, products. You're saying that one question you might ask is, is that product itself any kind of derivation? Is that, is that what you're asking?

Yeah, that's, that's one way to put it. Exactly. Does this composition satisfy this sort of formula? And, and the answer is in general, no. But when you com. When you combine it in the commutator, then it does.

Okay, so hold on, hold on, hold.

On, hold on, hold on.

Why? I mean, can you show that to me on these graphs? We should be able to just show that on these graphs.

Well, no, this, this is the continuous, this is the continuous case.

I understand that, but I'm asking what, what's the analogous thing Discreetly or do we not know yet?

Oh, so this is the thing that, what I was trying to build towards is that the analogous thing doesn't seem to be. I mean, that's, that's an algebraic. The way I understand that is algebraically. Right? You write it out, you have extra terms and you can rearrange those extra terms to look like minus the composition the other way. In the other way. But in this context, my, my, my intuition is telling me about this, this comment Nick made where you understand composition of, of this local flows or this sort of finite flows as the. Basically, you know, going into a step flow one after the other and asking whether that is a, a possible vector or a possible directive edge of your tangent.

Right?

So. Which is very reminiscent of exactly what the Lee bracket is telling you, which is, you know, effectively measuring the non. Commutativity of, of infinite decimal flows of, of the vector fields.

Okay, hold on. There's an obvious thing we could do here. You've got a, A flow. Let's call it a discrete flow. You know, let's call it, you know, I don't know. You know, I don't know what you want. What, what, what do you, what are you calling these flows? The. It's. It's sub X. Yeah, right, right. So you've got a flow, an X flow, and you got a Y flow. Okay, so what you can clearly do is you've got the X flow followed by the Y flow, or you've got the Y flow followed by the X flow. Yeah, right. Does that commute? No, it doesn't.

In general. No, it doesn't.

Yeah, right. On a graph. So let's just understand what it does. Right. So we've got, we apply one sort of flow on this graph, you get to a bunch of new nodes for every node, you get to a bunch of new nodes.

Yeah.

Now in general, since every node has an outgoing edge in each of these flows, in the end it maps graph, you know, the set of graph vertices to the set of graph vertices. Right. But the permutation that it generates will not in general be the same. The permutation associated with X compose Y is different from Y compose X. Am I correct?

Yep.

Okay.

It might be that it doesn't match to a possible vector field in this sense, a possible flow. Because it might not.

No, I understand. So we need to write that out. So we've got X compose Y, Y compose X. Each of those leads to a permutation of edges, which may not be a one step permutation. There may be no Z that is a one step flow, so to speak, that corresponds to X compose Y or Y compose X. So now the claim you're making about these Lee brackets is a claim that even though the, You know, even though this, even though that those z's may not live, there may be no, you know, elements of. There may be no vector field Z that can be defined here, it is still the case that for some algebraic type reason that this quantity, that is, you know, this quantity you're writing out as a Lee bracket, somehow, like the weighted Leibniz identity, that that thing together has to be something you could generate from a vector field. Does that make sense? I mean, I, I can't see why that's true. And my guess is that there will be an analog of that light weighted Leibniz identity where it will be true up to higher order terms. And I think, yeah, right. So you should just work that out. I mean, the thing I just described, namely. Jonathan, do you, do you agree? Does that make sense?

I think so, yeah.

No, no, that's, that's, that's what I would try to capture in this language here. So that sounds perfectly right.

Okay, So I mean, we can just work it out. We just, it's just like an algebraic thing, like that Leibniz thing. And let me, let me just. Okay, I'm trying to computerize, computationalize what we're doing here. Okay. That chunk of text shouldn't be. That chunk of text. We should literally be able to just write out, you know, here's this thing, we can derive the following thing from it, et cetera, et cetera, et cetera, it should be a computable statement, so to speak.

Sure. So that's. Yeah, I agree. So this was more of a summary sketch to present. But one thing that I, that I want to point out in relation to that comment is particularly when you think about addition, for example, which we haven't really discussed so far. My guess, and again, this is if we take the original motivation seriously of this infrageometry project, we're not just trying to do discrete analogs of manifolds mathematically, but we are actually Trying to do something more computational, more similar to the physics project on all this hypergraphic business. My, my feeling currently is that the way to understand these operations, I mean, we can use this algebra to guide ourselves. This is mostly my sort of background in mathematical physics that is helping me to guide myself. But eventually what I would like to have is something that doesn't really care about algebras or brackets or anything, but actually uses hypergraph rewriting to, to implement this kind of operation.

Sure.

And that's, that's why I was trying to point out that this composition of vector fields seem to be very directly connected to basically taking, you know, flow with one, flow with the other, and then check if there's the sort of the transitive completion of that in your graph. I mean, that's effectively like saying use the rewrite rule. You know, arrow one, arrow two goes into arrow three between the endpoints.

Right.

So that's, that was kind of my current understanding. So it makes perfect sense where you say it. Right.

Okay, so look, what I think is going to be true is some things will be have a natural limit and be naturally definable in the hypergraph, you know, infrast setting, and some will not. Like, I think X plus Y will not be naturally definable. X compose Y is naturally definable.

Yep.

X +Y is something where we've got kind of two graphs laid on top of each other and it's all rather confusing.

Yeah, I think it's interesting to think about because plus, if you think where plus comes from, which comes from the local chart. The reason you can add vectors is because you're modeling on a fine space or a vector space locally in your manifold. So it might be that your analog of linearity here is some kind of analog of isometry or isomorphism or graph isomorphism for the each of the tangent spaces. And there might be some permutational operations that can be done with those discrete elements that will run into addition.

Let me offer a suggestion. Okay, so you've got a graph and you've got certain edges that join nodes in the graph. Imagine that you double the graph just like going from the, directed. From the undirected graph to this directed graph. Right.

Okay.

Okay, so the, the. Just like this. Right. You're doubling it like that. Okay, so now I'm going to claim that the notion of adding two vector fields is the following thing that you take the. Okay, so scroll down until you've got a vector field. Just, just. Okay, there you go. There's one nice vector field okay, so now I'm going to claim that the addition of two vector fields is. You double this graph again. Okay. So that every edge has. There's a doubled version of every directed edge. Okay. And then the addition of vector fields I'm going to claim is you pick. I'm not sure if this is right, but then you pick of those for the, for the double directed edge from node A to node B in one of the vector fields that that edge is lit up and the other one it is not. Yeah. And. Well, so, so Nick, who seems to be unable to actually say anything but is commenting, isn't X plus Y just a graph union? But, so what I'm suggesting. Yeah, but, but Nick, the, the. The graph is the graph. What I'm suggesting is that the addition operation is this labeled graph. Because the addition, the vector field is a labeled graph. Right? It's a graph where certain edges have been identified. See what I'm saying? It's, it's more than just a graph. A graph just has those edges. This vector field concept is a, Is an annotated graph. Right. So what I'm suggesting is that the addition of two vector fields is. You're essentially doubling. Yes. Which is essentially like what Nick is saying. It's like the graph union, except that it's. Yeah, it's putting. Right. It's putting that graph and then another vector field and it's then making a graph where there are, you know, in this particular case, where there are two directed edges going between every two vertices. Does this make sense? I think this makes sense. Jonathan, do you have any comments on this?

I mean, what you're describing is a. Is a graph tensor product with some contraction.

Yeah, exactly.

I was going to say. Yeah, that's the tensor product. More like, I mean, having graph union where you have choices of couple vectors instead of a single vector. That is more like a tensor product. But I think, additional, we should remind ourselves that addition is a remnant of the fact that you're modeling on something like an affine space or vector space. So it's actually composition of the flows that are generated by the vectors themselves. Right. But because these work infinitesimally, you actually have a closed operation at each point.

Right.

So that's why the closest analog I can think of is if you go on a path on your manifold, on your discrete whatever space, the shortest path you can take is length one. So addition and difference in this context is going to be something to do with. You look at two vectors that point out of a given vertex and Then you try to make some statement about vectors that are relating, possibly connecting the endpoints, the two out vertices that we are pointing two. Right?

Yeah.

Stephen, there is a way that you can make the graph tensor product into a sensible notion of vector addition, which I think is close to, but if I understand correctly, not quite what you were saying. Okay, so, all right, let's ignore the vector field case. Let's just go straight to the vector case. Right. So let's start directly from the vector case. So if you've got, if you have two directed edges, and obviously the end point of one is the same as the start point of the other, you can trivially add them by doing what Carlos is describing as this closure operation. Right.

In other words, if you go from A to B and you've got another vector going from B to C, then.

The sum, I can just introduce a new directed edge that goes from A to C and that's my sum.

Right? Yeah.

So the complexity comes when the endpoint of one doesn't match the start point of the other. Because then what you need is some analog of parallel transport. Now you've just got two arbitrary directed edges and you want to be able to say, well, if I moved that one so that it was. So that it was, I don't know, coterminous or whatever with the first one, what would that look like? And that's what your connection gives you in differential geometry. So one way that you can make that work, which I think is close to what you were describing, is you just say, okay, I create a larger graph that effectively says for each edge in the graph, what would that edge look like if I transported it over here? And so I have one edge emanating from that vertex, one for each other edge in the graph, if that makes sense.

Well, I mean, this is.

And then I convert the second case of addition into an instance of the first, which by the way is strongly is closely related to the, you know, to the definition of covariant derivatives in this, in this setting.

Hey, hold on. What? How big do you have to make? Maybe you should actually show these graphs. I mean, how. Why don't you share and show the graphs?

Sure. So, all right, sorry. Actually, let's keep the same notebook. Okay, So I have the same thing. Okay, so yeah, obviously if I picked this edge and I picked, you know, this edge or something, you can add them. But one thing I could do is say, okay, so let's imagine, okay, let's, let's pick.

Let's take two edges, we highlight two Edges that we're.

Let me start by highlighting one edge.

Okay, Let's just go off comma, edge.

Yeah, let's style this in our usual way. Okay, So I have an edge there. So now, yeah, obviously if I. If I take any edge that's emanating from that vertex. From vertex. Yeah.

It's easy to compose.

It's easy to compose. So now what I want to be able to do is say, let me take some random edge like this one over here and turn it into one that's effectively emanating from Vertex 6. So one way I could do that is by constructing a larger graph that takes every directed edge in this graph.

Exactly.

And puts it over here.

Yep.

Right. So we can just do that trivially, I guess it'd be graph union, directive graph. And then I suppose the out component.

Of that node, isn't it?

No, no, because what we want is. I think what we want is. Is. So I think what we want to do is we want to take edge list of directed graph and then I want this underneath.

Sorry, just as a notebook organization. Why don't you do this underneath the picture you just made.

Yeah, sorry, sorry, sorry. But I think what I want to do is take that and somehow move that for each. I don't actually care what these are. Right. What I actually want to do is for each one of these, create a new edge emanating from Vertex 6. So actually. So I could literally just do this, I guess, and then I don't want to let me add 50 to that or something. And then directed edge6hash. So that. And then do graph union of those. So I get something like that, right?

No, I know some way that's useful because those things are often outer space now.

Right, right. But the point is that the larger graph that I'd be building, I mean, this is only one part of it. Right. But the point is, the larger graph I would be building would effectively transport all vectors that are in this graph to this one here so that they can be readily composed with the edge I started from.

Yeah, but don't you have to go back instead of those 60 things? Don't. Isn't it, isn't what you really want that. Can you make the following thing? That instead of that graph, just. Just copy down the line. You just have there. Copy down the line you just made.

Okay.

Okay, now just get rid of the plus. Specifically just do the same thing without the plus 50. No, that's not. Because you want the vertex list. Sorry. Just do that directed edge instead of that Range map it over vertex list, directed graph.

Yeah, so that will just connect that vertex to every point.

To every vertex.

Right, but that doesn't give me a way of adding edges.

Okay, explain.

Well, no, well, actually I think you need to explain. Right, like if I take this edge and I want to add this edge, how does this graph help me do that? Apart from the fact that I could just say.

Well, it allows you to get to that point. It allows you to go that.

Yeah, but there are multiple ways I could get to that point.

Whereas in this is arbitrarily adding in edges so that you've got a transport layer.

No, no, but under this scheme, the addition of this edge and any edge that leads to this vertex will be the same.

Yes, that's right.

This edge add this will be the same as this edge add this will be the same.

Yeah, no, I know it's not a great. It's not a very good definition, but.

These are not the same vector, so that can't be right. Right.

Yeah, it's not the right definition.

You need to have one for each edge, not one for each vertex. That's why I did this.

One Watt.

For each edge, you need to have a transported edge starting here for each edge in the graph. Because you need to be able to add this vector to any vector, not to. You don't add a vector to a point.

I get it. I think we need to see some pictures of this.

Yeah, sure. No, I mean, this is constructible. It's just. It will get pretty incomprehensible quite quickly. But like I said, this idea of building a larger tagged graph is also the thing that occurs when one tries to define a covariant derivative, for instance. So in what Carlos has been describing so far, I mean, you're dealing with. Initially you were dealing with scalar fields. So there's no difference between a directional derivative and a covariant derivative. But of course, more generally. So each one of these tangent spaces defines a coordinate basis. So if you had a vector field defined over this graph, that vector field is potentially different for each possible choice of coordinate basis. Right. So if I want to differentiate that vector field. So I have an outgoing vector at this point, I have an outgoing vector at this point. But the coordinate basis here is not necessarily the same as the coordinate basis there. So I can calculate the derivation assuming the coordinate basis here. But then if I want to get the covariant derivative, I have to apply a correction that makes use of what the vector field looks like. If I know the coordinate basis there.

So this question about coordinate bases and so on, this requires a construction that is some kind of sort of stacked thing above this graph. And you're saying, and this seems like the right place for us to go next is certain things. And again, we need to very clearly understand what things can be defined without kind of going to. What you're saying is there's some outer product. Ish graph that is your, your.

Sorry, keep talking. I'm just gonna. I'm right.

No, no, no, fine. It's. It's an outer product somehow of the original graph together with some edge, some dual graph that's associated with edges. I think.

Jonathan, when you say this coordinate dependence is that. I mean, that's analogous, right? That you're using some kind of analogous language to.

Yes, yeah, exactly.

Yeah.

So. So. So. Well, it's analogous in the sense that, you know, each one of these tangent spaces defines a local, you know, defines our analog of a local coordinate frame.

Right. Can you expand on that? Actually, that statement that doesn't connect immediately with me. Can you expand on, on. On what you mean by that?

So, okay, hang on, let me show this in one sec. What I'm going to try to do is. Okay, what I'm going to write hopefully in a second is something that shows. Okay, so these are vector fields, but these are all of tangent vectors, right? But if we now constructed a vector field where we had the paths of length two. So if you like, those live in a graph that is like the level two transitive closure of this one, if you see what I mean, the structure of that changes depending on which tangent vectors we picked. So in other words, there's that graph as seen from the perspective of this vertex, but there's also that graph as seen from the perspective of this vertex. Now, so that vector field is therefore sort of its representation as a graph is coordinate dependent in the sense that it depends on which tangent space we started out in.

But is that real? I mean, I don't see. Why would you. You would call that coordinate dependence when it's actually locus dependence, which is in some sense the analogous of intrinsic location in a manifest.

Right.

I suppose because it's like saying that.

Obviously you, I mean, okay, I see the point of calling independence, but the fact that your fields, I mean, that's, I think that's what you're saying is that the fields are in homogeneous, right? They have features that are different in different places, and so they're going to feature the manifold differently in some sense.

Right, right.

Yeah, that's what you want, comma, vertex.

There we go. Okay, so this is the analog of the. So like the tangent space construction, this is now the, you know, this is the kind of length to tangent space, so to speak. And if we wanted to, I could, you know, I could effectively compose all the vectors, you know, or add all the vectors using the procedure we were just describing. And I can build some larger graph that has all of these, as, you know, in which all of these vectors are tangent vectors. Yes, but if I picked a diff. But if I had a different tangent space, then the same, you know, morally the same vector field, so to speak, the vector field with the same geometrical properties would have a different representation. The resulting directed graph would be different. And so that's why one needs to keep track of the connection. Because if I now wanted to do calculus on that vector field, I then need to know if I'm differentiate, if I'm doing a directional derivative in this direction. I need to know what that vector field looks like from this vertex's point of view and from this vertex's point of view, which from my vantage point is. That's the thing that gives one the intuitive explanation for why the connection coefficients appear in the definition of the covariant derivative.

Yes, I think here, I mean. Right. This needs more working out. But yeah, I think that, I mean, the question really is, as we get to covariant derivatives and so on, are we going to have to erect a more elaborate graph or can we live in the same graph with just different annotations? And you're implying that we can live in the same graph with different annotations.

I think, and I think that the power, I think the power is going to come from using something like rewriting. So in some sense we don't stay in the same graph, but in a sense we do. Right, because it's kind of our initial state for rewriting.

Maybe, maybe. I mean, that's a claim which I'm not sure. It could be the case that we have to erect a new graph by rewriting to get the thing that we can meaningfully talk about as covariant derivatives. Or it may be that it is merely an annotation of the existing graph that I think has to be.

But I'm a bit confused about what you think exactly is unclear here. I mean, so from my perspective, what you're describing there is precisely the difference between covariance and non covariant definitions. If you can define.

Walk it through again, sorry, could you make the second level edges here, a different color that would help me.

There are no second level. I'm doing everything with the original graph. But what we could do.

No, no, I understand. But when you go from the green vertex. Okay. It's first level edges.

Oh, I see. So you mean the ones that come from here, so to speak, differently. We could do. I guess. Okay, so if we did, what's the best way to do that?

I think we want a second style thing.

Yeah. So hang on, if I just do that and then I do so style here B.

Okay, let's understand that picture.

So. So red is the tangent space and blue is not.

I mean by not blue is the extension.

Yes, but yeah, blue is not. Blue is the stuff that's not.

Blue is not in the green nodes tangent space.

So the point I'm making is any construction that you can do just on the directed graph is I would say by definition covariate. Right. It's by definition non coordinate dependent. The only case where you need to construct a larger graph is if some edge that you're dealing with depends on which node you started from. In other words, you have a construction that depends on which tangent space you're in.

Yeah. Okay, so walk that through how that relates to the covariant derivative.

Well, so the point with the covariant derivative is you're taking a non covariant concept, which is the directional derivative of a vector field or a tensor field, and you're making it covariant by essentially applying a post hoc correction.

Okay, but walk me through that in this case. So I mean, normally you're saying the directional derivative we get by going from the green node, the value set on the green node, and going to all the red destination nodes. I'm a little confused that there isn't an outgoing edge, that the whole out component of the green node, that there isn't a. You're picking ic. You're. You're picking only particular outgoing edges from.

I'm picking all the things that are in the out component that you know that go to distance too. So, so this edge for instance, is not highlighted. Right. Because it doesn't.

You know, that's confusing to me, but okay, because I thought that the original vector field was things that are in the out component independent of where they will eventually end up. Right to field.

So, so yeah, so. So that's that. That's the thing that gives you a section of the, you know, a smooth section of the tension bundle.

Yes, yes, but so this restriction to say well, I'm not going to include those pieces of the outgoing out component that land in the wrong place. Seems like a different thing.

I mean, this is not really a restriction. That's just that, that's a statement that if I compose these two things, I just get, I get the zero vector. So it does not. It's not.

Yeah, okay, well, but I, I still think that, that I'm, I'm not yet understanding the, the distinction between a derivative and a covariant derivative. So a directional derivative is just at a point. We pick the out components. We, we pick, we pick various.

Okay, so, so, so, so let's episode. We have a. Say we have a vector field. Okay, Right. So we have something like this.

There's a vector field. Fine.

And I want to get the directional derivative, say along this edge, right?

Yeah. Along which edge? I mean, there's an edge going along.

The blue edge here.

Okay, Right.

So I have this vector is at this point, and I have this vector at this point.

Yeah.

So I can just subtract these two and that gives me a directional derivative or it gives me a directional derivation. Right.

I don't understand. Wait a minute. We've labeled the things with some scalar field, we put values.

But you want to know about the covariant derivative. So we can't do a scalar field. Right. Oops, sorry, sorry, one second. I'm in a room with a dog and the dog is upset, but we're okay.

Okay, woof, woof.

Yes, apologies for that, but no. So obviously we can't use a scalar field. Right. So we have to do a vector field.

I'm a little confused, I think. Okay, just for the sake of structuring this meeting, I mean there's clearly.

But wait, hang on, I don't understand the question. You wanted to know about the covariant derivative, right?

I do.

So we have to start from something that's at least a vector field, right?

Yes, this is a vector field. The thing you've drawn on the screen, there was red arrow.

So what we're going to do is the analog of Carlos's scalar fields construction. But for a vector field or some higher rank tensorfield.

Okay.

Yep. So we take the value at this point, which is this vector, and the value of this point, which is this vector, and we subtract them just like.

Oh, I see what you're saying.

Just like we did in the scalar field case.

Okay. What is the result of subtracting one from the other?

Well, we know what the Addition is. So, I mean, well, we have a provisional definition of an addition operation here. So you would take. I can't, you know, the subtraction lives in some other space. Right. So I can't immediately visualize it, but it clearly is well defined.

Yep.

But then the point is, if this is a generic vector field, then this is only the representation of that vector field with respect to one tangent space. In this particular case, it's not because these are all tangent vectors, but in general there will be a different representation of this vector field for each tangent space in the manifold in the bundle.

Yes.

So I would need to know not just what is the. So if I'm at this vertex here, this is the vector at this vertex as seen from the perspective of that vertex. But I need to. I need to add a correction that's related to the vector field as seen from the vertex. And that's why I get the connection coefficient appearing.

Okay, we should work that out in more detail. This is a Carlos UTI Etc exercise to really write this out in more detail. That makes sense.

Yeah, actually. So I don't think I will have time to expand on it on this today. But the mystery. My idea for this kind of implementation, so addition, for example, or composition and so on, is really to use the. The rewriting mentality. So I think that something that, I mean, we're glossing over the subtlety of what it means to add vector fields because we are, you know, we made the entire analogy like complete. We said, oh, there's vector fields in the continuous case, there's vector fields in the discrete case. And this was mostly like cartoon sort of comparison to guide ourselves. But actually in practice, I think what's happening is that you are going to be pattern matching specific constructions. And just to be clear, in the case of addition, as we said earlier, I think that the guiding diagram that we should have in mind is the parallelogram rule, right. So you have a parallelogram that defines when vectors add. Right. So in our case, a parallelogram is actually more sophisticated than just a basic composition, which is just a V shape where you flow one and then flow the other. A parallelogram is much more constrictive because saying that you have two vector fields in such a way that when they happen to be, you know, forming a parallelogram, then you can compose them and you only compose them locally where those things happen. Right. When those matter patterns actually occur. So I think there's a. I mean, I will work in this for next time and I'll have a much more. And we will use the hypergraph packet for this as well. So it will be a nice connection there. So I think it will be a nice, nice thing to do.

Right. And by the way, I wouldn't be 100% surprised if some of these constructions blow out of being graphs and end up having to be hypergraphs.

Absolutely. Yeah. Absolutely. Yeah.

Well, I mean, it's arguable that the correct representation of say, these length two paths is an arity three hyper edge. Right?

Exactly.

Yeah. Yeah. Well, that's interesting because I think the hypergraph thing is the much more universal structure.

Right, right.

We are restricted to graphs just for simplicity here. There's nothing I don't think, fundamental about it.

So what may end up being the case is that you have a, that arity 2 edges are useful for representing the tangent spaces, but arbitrary arity edges are useful for representing vector fields in general. So starting from a directed graph structure, there is a sort of unique hypergraph that you can build using directed paths. And that's really the thing that's relevant to constructing the full tangent bundle. No, not the full tangent bundle for constructing arbitrary vectors.

Arbitrary vector version fields.

Yeah.

And, and, and to, and to circle back to your question about what I meant by simplicial hypergraph. So if you wanted to build the tower that usually is built in differential geometry of general forms and so multivactor fields and differential forms, that would naturally imply using sort of three edges, four edges, etc, that are kind of the dimension of those of those subjects. So these are clearly the, the one, the, the one edges are the ones that, that represent vectors. The two edges would be the ones that represent.

Right. So the P form in general is formed, is created by. Instead of. These are one forms and you know, by having a single edge coming out of each node that's making a one form having sort of a whole collection of edges. A P of P of those edges coming out will make a P form. And the claim would be that, for example, you know, this question of a, you know, like the Hodge dual operation on P forms, for example, Hodge star operation dual dual of P forms. Right. Yep. In, in this case, because we don't really have that. The, that would be sort of the complement in the gdesic ball. If you take the geodesic ball and you complement it with a particular P form, you will get the, the dual of that P form. I claim.

That sounds plausible. I also think that the, so the P forms would. You could induce P forms by collections of edges, but they will also be sort of more natively P edges. Right. So they will just be hyper edges of order P. That would be the.

Yeah, but I think it's more useful. Well, okay, whatever. It doesn't. I think there's a different concept, which is, I think there will be a more granular version of P forms where one is a P form, where P is an integer. And you're doing that by picking, you know, edges or hyper edges. And another thing will be you get a more granular P form where you say, I'm going to pick sort of one arity two edge and one arity three edge and so on coming from each node. I claim that's a more granular version of a P form.

Sure, yeah.

I mean, yeah, I was just going to show. I mean, so from one of our earlier working notebooks, if you wanted an example of a. A tensorfield or I guess a P form of higher rank, this is something Carlos and I constructed, which was the angle tensor on a directed graph. So for any pair we're looking, this is at vertex one in this case. And so for any pair of the outgoing vertices, it will tell us the angle between the corresponding tangent vectors, which. Well, some discretized version of the angle, either 0 or 1.

By basically asking, is there a way to get from here to there? Type thing.

And that's here to there in one step. So, yeah, we're effectively, we're saying, you know, the angle between these tangent vectors is zero because there's a directed edge between them, but the angle between these is. Or the angle between these, for instance, is one.

Yeah, it's really zero infinity or something. It's zero. I mean, it's zero. Yeah, they're either parallel or they're not.

Right. But then the point is that. So that's how it goes. So it goes for tangent vectors, but then if we then were to look at the length two paths or something, or the length three paths, this tensor construction then allows you to have more or less arbitrary angles as you take the limits.

Yeah, I get it. That's interesting. All right, let's see. We should wrap up very soon, but did UTI have some things he was going to present?

I mainly have algebraic results, so I'm not sure whether. I mean, I haven't figured out the connections to. On weighted derivations like, like the, the modified Leibniz rule. Okay, what, what properties do they obey?

Okay, but somebody. Carlos, you're in charge here. So should UTI show something or not?

Yeah, I think. Well, UDI did very Very valuable exploration of the algebraic behavior of these weighted elevations.

Let's see it for a second.

I think he prepared a minute.

I see some comments on our live stream here. Comments from glitch about simplices for and dimensional simplex. You'd have n plus one inequalities. Yes, that was what I was kind of looking for. Given the constraints of rnk. It's reminiscent of density based clustering techniques and data analysis. Yes, that's a good observation as well. That's a good observation that, that in, let's see in, in this, these, these subsets of the graph. I, I do think it's worth looking at this, these other. Can you share please? Ruti, you share your screen please. Thank you.

Oh, can you see my screen?

Yes, yes.

Okay.

Okay. But I just have another comment here. Yeah. As another piece of homework, guys. Can we please look at not a sphere, Right. We've got, you know, we can look at an, the generalization of a sphere in a hypergraph. I would look like to look at the generalization of various kinds of polytopes like a simplex in the hypergraph. Does that make sense? We understand the generalization of a sphere to the hypergraph case. Can we please try to understand the generalization of a simplex to that case? Simplex in the sense of a simplex and continuous space. Is that making sense?

Yeah, yeah. I, I think we all understood Carlos.

Since I, I'm considering Carlos is in charge here. That, that, yeah, that makes sense.

I'm just wondering, so you want the sort of equational definition of a simplex in the, the context of hypergraphs?

Well, no, I, I, I think just as it's a very straightforward definition to just define a sphere of in the hypergraph case, there is similarly going to be a very straightforward definition of a simplex in the hypergraph case. I just want to know what it is.

That's, that's not, that's not true.

What's not true?

The inference you just made.

No, I know it's not obviously true.

I'm, I don't, I don't think it's even non obviously true. I mean, so the reason we're able to define a generalization of, you know, n balls and therefore N spheres in graphs is precisely because of their metric structure. Yes, but just having a metric structure does not enable you to define a politic.

I get it. It's got faces and things as well. I get it. Right. So what I'm asking for is to think through what that. And I'm making This claim that there is this thing that is essentially a combinatorial picker that says things like every. That we are finding the set of nodes that are. For which every node is less than this distance from some subset of other.

I agree. You can certainly go a great distance by building by essentially. Yeah. By allowing for arbitrary unions and intersections of balls and N spheres, which is how higher dimensional polytopes are defined when you have a CW complex or something.

Okay. So I mean, following essentially a discrete definition of these CW complex things, which I vaguely remember seems like a good way to go.

Yeah.

And that's a good piece of muscle flexing of whether we know what we're doing is, you know, can we get these polytopes? Define a polytope. Okay. Because I think that that will perhaps back us into defining faces and things.

I think it depends on, on the kind of geometric structure that you want. So if you want something like oriented volume and oriented areas and these kinds of, these kinds of geometric abstractions. That's what I mean by simplicial hypergraphs. I think that's fairly doable. I mean it shouldn't be too mysterious. It's just you have to consider hypergraphs with specific edge structure, symmetry structure to do that.

I don't think that's going to be the case. I think there's going to be a general definition of what it means to have to put a simplex where the simplex is going to be parameterized by various kinds of combinatorial quantities in this graph. And I think what's going to happen is my guess is that. Yeah, I mean, just, just like the ball is characterized by two things. It's characterized by its base point and its radius. Okay. This is going to be characterized by more quant, possibly by k base points. What's that?

Yeah. So effectively more form. So the ball, you need one metric structure. So that's one form, one symmetric form to compute. When, when you do these polytopes, you might want several higher forms to do faces of different dimensions and so on.

So you might, so you might have construct it. That's my, my. Okay, uti, what were you going to show here?

Okay, so I have some results. So as Carlos mentioned this. The weighted Leibniz rule is something that is obeyed by the. By the construction that takes one scalar field to another. So if I have any vector field, then I can construct this delta X thing which takes in the. Takes in a vertex of the graph and a function and tells and evaluates the function at the outgoing at the Node that is outgoing. And in, in Carlos's example, the weight W was assumed to be one. So in a sense you can, you can. The way to think about this is that the label that you label the edge with with one. But technically you can label the edge with any value W where W is an element of the field.

Okay?

So, blah, blah. Now I have some algebraic results. So the main algebraic result I have is not the main, but like the most basic one is if you have A as K algebra.

And what is a K algebra? What is a K algebra? What do you mean?

An algebra is just a set with. It's like a ring, but over a field. So over here the field is K. Okay. And the way W is an element of K and home alge A is the set of all algebra homomorphisms. So it's additive and multiplicative and it respects all the linear algebra properties, etc.

Etc. So wait a minute, is that, Is that for us something about graph automorphisms or something? Is that the analog of the homomorphisms of these algebras I haven't figured out?

Yeah, sorry. Those induced algebra homorphous.

Right. So the. Any automorphism of the graph is a homomorphism of the algebra on the graph, is that right?

Well, it uses one, yes.

Okay, what's the algebra on a graph? Like what algebra are we talking about?

The algebra functions.

Oh, okay.

The algebra functions over. Over the same field. That's. That's what you have.

Okay, Right. Where. Okay, got it. So then if you have any algebra homophism, then this turns out to be a W derivation where ID is the identity map. And you can check that easily.

What is a W derivation? What does that mean?

This is a W derivation. It's a linear map from the algebra to itself, which satisfies this, which was what was satisfied in the graph case.

Right.

Where you have the W over here and it turns out that this is a W derivation. If you take.

Well, but. Right, but I mean, insofar as the claim is that the homomorphism is, as you say, induced by automorphism on the graph. This is just saying that you can relabel the nodes of the graph and the thing remains keeps that derotive structure. Is that what you're saying?

It's a bit more than that, because it's. In fact any graph endomorphism will induce such a derivation or just automorphism.

What is the. Sorry, what's the difference for a graph?

So the difference is that you don't assume injectivity.

It's any map I see, I see, I see, I see, I see. Any map, including a contractive map of the graph to itself.

Exactly.

Yeah. Okay. Oh, wait a minute. Is that obvious? Maybe, somewhat. Okay, well, but, but okay, but let's walk through. Where are we? Okay, so I think we. Okay, go on. So you're going to. We should probably wrap soon. But where is this, I mean, algebra. Where's it going?

Okay, so I'll just present the main result, all of the algebra. So you can show easily that all of the W derivations are of this.

Form.

Where phi is an algebra homophism. And from this you can get that you can, you can find like a operation which, which is closed on like you can. Which. So that you can combine W derivations in a way so that the resultant is a W derivation, which is a non trivial question.

That was another that, by the way, that was an open mathematical question that, that was not known before our work.

I mean it's, it's kind of very easy to drive. I don't know why someone else did not see that. But it turns out that this is a, is a weighted derivation where you multiply the weights together and so the, so the. So when you ask where this is going, I can skip over all these. I mean, I think these are relevant results and pretty new results, I think. But the geometric part comes in is that let's say you have two vector fields on a graph. Then we know that a vector field induces a W derivation on the set of functions from the. And so the question is that if I combine like the Delta X1 and the Delta X2 in the way that I had, does that correspond to some other vector field? Or if it like doesn't, then what does it.

What do you mean by combine like?

So if I have. Yeah, so I said that, that if you have 2W derivations, there's a way to combine them so that you get another one right.

Show the formula. Because it's a composition.

Yeah.

So you compose the functions together and then you add this. So just for the sake of clarity, let's assume that both of the weights are like, are one. And so this is a one duration. If these are one dimensions. And so the question is that, the question is that if I have two vector fields, then I can construct like, analogous, you know, I can construct the derivations corresponding to those vector fields, each of which have weight one. Then the question is that like, if I compose them together. Oops, yeah, if I compose them together and then I add them also, then this is an Element of this.

Right.

I, I showed that.

Yep.

The question is, does that correspond to something in here? And that that would, I think, depend on the structure of the graph.

Okay.

In, in ordinary, in ordinary differential geometry, the operation on the set of derivations where weight is zero is the leap, is the leap bracket that Carlos talked about and from. And, and since in ordinary differential geometry this is not only injective but also surjective, it yields an another vector field. But on discrete drafts, I'm pretty sure that this is not subjective. You can. Because we don't have any structure on the set of functions also. So I mean, the question to ask is like what kind of graphs would, would be so that this operation is closed, essentially, that's the geometry question.

Or, or more to the point. Okay, so what is the physical interpretation?

I have not thought about that only.

Well, but is it, is it basically a flat space or something that doesn't have curture or some such other thing?

We wouldn't, I don't think we would necessarily have to be with curvature of the space. I think it has to do with how you parallel transport around it. I mean, you could think of that as curvature in a generalized sense, but in terms of the connectivity, yes, it could be about curvature.

I think it would mean about like, it would depend on the connection, like how connected the graph is. The more connected the graph is, the more subjective this map becomes.

I see, exactly. Yeah. So maybe what that is instead is a measure of something which we don't usually talk about in manifolds and so on, which is, you know, the thing might just fall apart.

Well, so to, to, to really characterize this quantitatively, you need to have an, a generalized notion of curvature that is not a measure of how much your tangent space gets distorted, which is the normal notion of curvature, but how much the cardinality of your tangent space gets distorted, which is not normally talked about.

Yes.

Right.

Let me think about that for a second.

So you're saying which is the very, very micro scale version of dimension change.

I guess that's interesting. Yes.

Because what you're basically saying is, so if your graph is perfectly regular, so the number of outgoing directed edges and therefore the cardinality of the tangent space is the same everywhere, then there's no problem. Right. The cardinality of the, of the.

But Jonathan, I think what you just invented, right, is, is, you know, what we've been looking for in dynamical, you know, dimension is a dynamical variable. We have had trouble understanding what the continuum limit of that is.

Right.

That is, whereas we Know what the continuum limit of curvature change, you know, continuous curvature change is.

Right.

So you're saying the continuum limit of or somewhat continuum limit is this changing cardinality as opposed to change in. In direction? Effectively.

Right.

I mean, in other words, curvature is a change in direction of the coordinate.

We used, we're used in differential geometry to the idea that your tangent space changes shape as you move around. We're not used to the idea that it changes dimension, but this one does as soon as you have a non regular graph.

So that's what we have to figure out is that's our dimension field. Basically that's going to give. Just like there's essentially a curvature field, we're going to get a dimension field from this, from cardinality change. Lovely.

And so the interesting thing here is that the measure of how non, I guess how non subjective that map is gives you exactly the analog of curvature, but for this distortion of cardinality.

Yeah, right. So it's the. Instead of.

So from my perspective, the next obvious steps are. Okay, I still want to check that all the obvious things. So for instance, if we have a lie bracket of vector fields, check the obvious consistency conditions that the Lee bracket of vector fields vanishes if and only if one has a torsion free space, which in our case just means that the directed graph generated from the undirected graph edge weights the same in both directions. So there are basic algebraic consistency conditions. But then the next obvious thing to do is to say, well what, what can we do that's new? And you know, an obvious thing to do that's new is yeah, describe distortion of the cardinality of the tangent space.

Which is by the way, torsion freeness. You know, the fact that we are assuming we can go from that we have our base object is an undirected graph that we lift to this sort of doubled directed graph and therefore it's torsion free. It's not obvious. That's what's actually. I mean we don't think that's what's actually happening in physical space.

Well, that's what's so beautiful about. I mean I skipped over all this in my introduction. I can talk more about this, but I know we want to wrap soon. I have other pictures I can show of this. But. So we've just been dealing with metric connections so far, right. Effectively with Levi Civita, with the Levi Civita connection. But in principle you could generate. Well, first of all, you could generate an arbitrary directed graph that doesn't have an undirected graph structure. That would be an Affine connection that is not necessarily metric compatible. But you could also generate a metric compatible connection, but where the weighting is not the same in both directions. So that would be a non levi civita connection. And then so something like the fundamental theorem of Romanian geometry becomes a more or less trivial statement that there is a unique directed graph compatible with the undirected graph that has this.

What is the fundamental theorem of Remahan's?

The Levisabita connection is the unique affine connection that is metric compatible and torsion free.

Okay.

Which for us is sort of obvious, right?

Yes.

That I would say is the kind of the landmark, the landmark reference result that it's kind of a trivial statement in the graphic side and it's a non trivial statement.

So can we please write that out?

I mean, things like we have. I mean, I can show you. Wait, if you let me share just for a moment. I mean, yeah, there's much more we can show. But here in the formalization stuff, where do we have this? Yeah, so you know, this is formalization of affine connections, formalization of metric connections. Yeah. This is what it means for a connection to be torsion free. This is an example of a connection that would have torsion. Yeah, this is the fundamental theorem of Riemannian infra geometry, I suppose, which is kind of a almost tautological statement in our case. But as Carlos says, it's useful as an anchor point to make sure that what we're doing actually makes sense.

Mm, I'm sorry, these. Can somebody organize in a nice way the names of these files and stick them in the working files directory?

Sure, sure.

I mean, maybe we need a subdirectory for this stuff if we're going to have lots of different names so we want. So they won't be sprayed out in different places and we won't be able to find them. All right, I think we should wrap up here. But this is, this is very lovely and I mean, I think we're, we're chugging along. I really like this idea of the dimension field thing and I mean, presumably we're going to be able to formulate in this setting, we're going to be able to formulate the Einstein equations and all this kind of thing eventually.

Right, right. And it'd be good to check that we meet the formalism that we already have somewhere in the middle.

No, I mean this is a much more step by step version of what we think we already got in bigger steps. But that's the goal is to have this be so self evident. I mean, the goal is can we in 10 pages of almost self evident material derive the Einstein equations?

Right.

And I think we're well on the way here. This is good.

And by the way, I mean this kind of that construction that we were just talking about in some sense belies the notion that there is any sharp distinction between dimension and curvature. Because the curvature field in the discrete case that we've been dealing with so far involves growing out paths of, of length N and looking at cardinality. Is there. The dimension field involves growing out parts of length one and looking at the cardinalities there. So somehow there's some kind of smooth transition from the dimension field to the curvature field, which is quite interesting.

So the other thing we've been looking for is can we formulate Einstein equations in terms of dimension change?

Which it's now pretty obvious that we can because standard continuum curvature in the, in this setting emerges from lots and lots of microscale, you know, from lots and lots of discrete dimension jumps at the microscope.

Yep, yep. Well, that would be interesting. Okay, and then we have to make the jump to dark matter. We'll see how that works. Let's see. I just, I'm just looking at the comments on that live stream here. Yeah, right. Okay. Lots of interesting comments there. Okay, we should wrap. To be continued. Let's continue in another week or two. And you know, I think we've got a. Yeah, this is, this is going well. Thanks. Thanks guys. And thanks to folks on the live stream. And we're slowly, you know, at the end of this, I might actually understand differential geometry. Who knows The. Okay, see you guys soon.