Research Working Session: Infrageometry

Okay. Hi, everyone. This is a Wolfram Institute working session trying to figure out some science, or maybe it's mathematics, I'm not sure. We are working on what we are calling infra geometry, attempting to build up a kind of mathematics like structure around the kinds of things that show up in our physics project. In particular, building up something like geometry from the hypergraphs that represent space in our models. And the kind of usual way of building up kind of things like geometry and then calculus and so on, and differential geometry rely on the idea that the small enough scale space is just like the way Euclid said it should be. But in our hypergraphs, that's not true. When you go microscopically enough, you just, you see actual atoms of space and you see this actual hypergraph. And so the question is how one builds up kind of a mathematics like structure to describe what's happening in that kind of system and how one finds the kinds of constructs that have been discussed in sort of standard geometry and mathematics and makes a correspondence of those with what we see in our hypergraph systems. All right, now Jonathan is going to give a. His corrected version of my introduction, and then maybe Carlos will give a. A correction to the correction. Go ahead.

Right, okay. Yeah. So just to kind of contextualize this a bit, so you'll remember those of you who watched our previous livestream, I mean, I just brought up the notebook here that we worked through. We were kind of summarizing the initial insights we had about how to define things like tangent spaces, tangent bundles, you know, the beginnings of essentially defining what it means to sort of equip one of these graphs with a discrete analog of a connection or an affine connection, and by extension, a metric tensor and those kinds of things. And so the plan for today is to kind of talk about some stuff that we ended up thinking about in relation to trying to define what it means for functions to be smooth, continuous, differentiable, et cetera, over these discrete spaces. And in the process, one of the things we kind of found when discussing these things was that the answer to that question in turn gives better insights or at least a better contextualization of the definition of things like tangent bundles and connections.

Let me try and see if I remember how this works. So, I mean, this concept of a tangent bundle here is we've got, from a given node, we have edges that go in a certain direction and quotes direction to another particular node. And then, as I understood it, the sort of, there's the question of how do you continue from that node? And that's related to this kind of. That's related. So the connection, if I remember, I'm afraid it's been a month or two since we did this.

Yeah, exactly. So that was kind of the old idea, and that's mostly preserved in our new way of thinking about things with one key proviso, which is that these sort of things that we're describing as vectors are no longer pointing to nodes, they are pointing to. Well, in some sense, they're pointing to edges in a way that will hopefully become clear in a moment. And so I think. I don't know whose phrase this was. I'm are going to assume it was Carlos's this thing that we're calling the edges as places philosophy. So that was something that came out of trying to think about what it would mean for a function to be smooth or differentiable.

Can you recap the tangent bundle idea from last time? What is a tangent bundle?

In the last livestream we talked about? So you can take all of the outgoing edges from a vertex, and that gives us some analog of a tangent space, some discrete. Each one of these we can treat as being a tangent vector. Each one of these red edges. So then if we look at all the tangent spaces, one for each vertex, we can take a disjoint union of them, and that gives us a discrete tangent bundle. And then we can take a global section of that tangent bundle by just essentially applying a choice function by just selecting a single tangent vector for each element in the disjoint union for each space in the bundle. And that gives us a vector field. But then if we were to choose a pair of vectors for each tangent space in the bundle, we would then get a rank 2 tensorfield and so on.

Right.

So this is a global section through that tangent bundle, which therefore gives us a vector space.

Okay.

Right. But yes. So this idea partially came out of trying to resolve this confusion about what it means to have kind of path continuity. Right. If you've got. How do we define the analogues of things like parallel transport? If you set off in this particular direction in the direction indicated by the.

Doesn't mean to keep going.

Right, exactly, exactly. So the resolution to that, and I should clarify, I'm going to be talking about the understanding of this as it was maybe a week or two ago. I understand that Carlos and UTI and Nick and people have made a bit of progress since then. So I may say things that are a bit out of date. Please, you guys interrupt me if I do. So this is essentially the idea by.

The way was Nick going to join us? Is he here?

Nick was not here as far as I could see. He was sent.

Okay. But yeah.

Okay, Weird terms.

Afm. Okay, go ahead.

Let's start with the graph. And so the new idea is. Okay, so let me contextualize this by talking about the question we were initially trying to answer. So if you take this graph and we assign. Let's assign a scalar field to it. So if I take. Okay, so that's some. And so I say graph. I don't know vertex labels, scalar field. Okay, so that's some discrete scalar field that we've defined on the vertices of the graph. And so the initial motivating question was, what does it mean to say that this function is continuous or is not continuous if you're working in a fundamentally discrete geometry? And so the first thing that we kind of established or agreed upon was that if you just had the information at the vertices, that's not enough data to determine whether this is a continuous function or not. Because, of course, we know that if this were a continuous space and you knew the value of this function at 10 points, then you could always define a continuous function that would interpolate between those points.

Yes.

So just knowing the value of the function at the vertices is somehow not enough information. So then the idea was that, okay, when you define a function over this space, you are not just assigning values to the vertices, perhaps you're also assigning values to the edges. And then essentially, notions like continuity and smoothness become compatibility conditions between the assignment of values to the edges and the assignment of values to the vertices. So if we take a very simple example of that. So let's do. I don't really know the best way to do this.

So if I'm trying to assign a number to every edge.

Yeah, yeah. But in particular, what I want to do is let me do this. Let's make that an association. And so then if I just make this the mean. Scalar field first, scalar field last. Let's do that. Okay, so in this particular case, I've just assigned, I think, to each edge the average of the. Of the values of the. So the value of each edge is just the average of the values of the two instant vertices. So this would be an example. So within this. Within the prescription we were discussing, this would be a kind of. This would be like a maximally smooth function. And kind of the intuition is that you can think of the edges as being like limits. If we think about this as being a topological Space or the concept of an open set would be you've got a vertex and you've got some outgoing edges, but the outgoing edges are sort of dangling. They don't have. They're not closed with incident vertices. So the values of the edges tell you kind of things about the limit points of that open set. And so in this particular case, this is somehow maximally smooth. And then the idea is, okay, so suppose we didn't want to assign a value of 42 here, we want to assign some different value. The more different that value is from 2 and 82, somehow the less smooth the function. That's the intuition. And so as a very basic prescription, and no doubt there are more sophisticated ways of doing it than this, but you might say, okay, well, let's take a finite difference between 2 and 82, and if the value assigned to the edge lies within a multiple of, is sort of within one times that finite difference, then it's a maximally smooth function. If it's within two times that finite difference, then it's kind of the next, you know, the next most smooth case, and so on. So in other words, you characterize the smooth, the kind of the differentiability class of the function by the order of the finite derivative that you would need in order to interpolate between the values of the vertices and edges. And so that's the thing.

Hold on, hold on, hold on.

Sorry.

Okay, first point.

Yeah.

Okay, let me try to understand your definition of open sets. So on.

Yes.

So which we should be writing down some of these definitions. And I, I am. An LLM will conveniently summarize this, this meeting for us.

But open set claim, you know, a.

So collection of vertices with dangling edges.

Yeah. Okay. So, so, So, you know, a closed.

Set, Please, is.

There are no, you know, a collection of vertices with no dangling edges.

Is it a subgraph? Yes.

So each of these would be subgraphs. Yeah. In fact, let's say that subgraph containing dangling edges. Now, I don't know whether this is.

Where all edges are dangling.

No, I think, I think the sub graph, the closed set will be a subgraph, as you say, Steven, I think the open set will, Will. Will be a closed set together with, with a collection of vertices that are limit and then those have dangling edges on those.

All right.

Okay.

So closed, closed set.

I, I think it's, I think it's probably easiest to say, you know, open set is a subgraph that contains dangling edges. A non. Open set is a subgraph that contains no dangling edges.

Okay, but I was Thinking that the intuition was this, that a subgraph is just a collection of nodes with the vertices between them, and that is more like a closed set. And then that the open set, as Carlos has commented, there's kind of. Where's the outside? Well, the outside is nodes that are connected to nodes in the subgraph. But you don't see the nodes outside of the ones in the subgraph. Instead, all you see is that dangling edge.

Hang on, wait, wait. So let's be very clear. What do we mean by subgraph? Because. So I'm using the word subgraph to mean a subset of the vertex. A graph defined by a subset of the vertex set and a subset of the edge set.

Yes, I guess that's correct. That's the correct thing to say.

But not edge subsets, right?

What?

No. So I think a subgraph could be any subset of.

Okay, so you're saying any subset of vertices and edges.

Yeah. So in particular, in order to make this claim work, you need to have the ability to have edges. To have a graph that contains an edge, but whose edge set contains a given edge, but whose vertex set does not contain the endpoints of that edge.

Indeed, absolutely.

So maybe we shouldn't call this a subgraph. Right. Maybe we should call it something more because it's, you know, that's.

That's a slightly closed. Look, I, I think there's a notion of a closed set, which is something where maybe. What is it? Graph? Theoretically, maybe it's something. Okay. It's basically something where you don't. That nothing is dangling. You don't miss. You don't.

No, but, but that's a. That's a non open set.

What's the difference in a closed set and a non open set?

A closed set is one whose complement is an open set.

Right. Okay. All right. So non open set, no dangling edges. So we deleted. So forget the subgraph claim. I, I think, I mean, you know, it's just a. I, I don't know.

We. We need. We need some kind of. So, you know, we need some kind.

Of word that is a subgraph.

What?

Wasn't that everybody? What's that?

Sorry, please.

No, I was saying it's like a monomorphism.

What is like.

Like the subgraph definition you're talking about. I think it coincides with the one that, that we use in category theory.

Right.

We map the vertices and edges.

Yeah, that's just an injection.

Yeah.

As would be any subset. But the whole point is that it's a potentially incompatible collection of vertices and edges. The Traditional definition of a subgraph would entail that if you have an edge, in other words, a subgraph still needs to be a valid graph. What we're talking about here is not necessarily a valid graph. Yeah.

It could have an isolated node, for example.

Well, isolated nodes are okay, but isolated edges are not exactly.

Yeah.

So this is not a graph. It's a.

Okay, could we have an edge which has neither node, neither of its nodes on it, so to speak?

Yeah, I don't think that would be necessary for anything.

It's not necessary, but we could in principle.

Right.

And what we're saying is we're defining a class of objects here that are something like subgraphs, where we are allowing the flexibility of selecting, sort of highlighting a set of vertices and highlighting a set of edges in an independent way. And we are saying it is not open when there are all the edges actually correspond to some vertices that have been selected. Right?

Yes, right. Okay, fine. Generalize subgraph. So, I mean, we can try a dangling edge. Let's define what a dangling edge is. A dangling edge is an edge that lacks either one of its nodes or both its nodes.

Yeah. An edge who's at least one of.

Whose nodes is missing.

And nodes is missing from the vertex set of generalized sub.

Right. Now, could both of them be missing?

So I should mention at this point, by the way, that there is a term for this in the graph theory string diagram literature, which is open graphs. So for instance, when you have things like circuit diagrams or you have string diagrams, you're allowed to have ports and sort of wires that don't connect up to nodes and things, and those are referred to as open graphs.

Okay. Make a note of that. That fact that a dangling edge is like a port with nothing plugged into it, so to speak.

Yeah. It's the right intuition. Absolutely.

Right. Okay. Okay. So.

So the. The idea would be, you know, we. So we can say. So, okay, if we were to select.

Sorry, can I understand a closed set now? So the complement of an open set. I don't understand what that looks like. Then the complement of the open set is surely something.

Yeah. So. So in this particular case, I think. I think in the way that we've defined it, open. I think you. I think you were perhaps I was maybe overly dogmatic. Non. Open sets, I think, are closed sets in this case, but they wouldn't be in general topology.

I think that's right in this context. Yeah.

And I think in topology these are called door spaces, spaces where every set is either open or closed, like a Door.

Okay, but hold on. Why is this the claim that the closed set is the complement of an open set? I don't understand. If the closed set has no dangling edges, why is that the complement of an open set? Because the complement, I would assume, is a graph complement, where you're taking the sub. Generalized subgraph that is the complement set.

Right, right. So if this is an open. If I. Okay, let me try and draw a picture of this. So let's take.

Wait, I think you only had.

Of course. Yeah, yeah, sorry. No, I forgot. There's a. There's a. Not vertex names. There's a.

Values.

Yeah, yeah. Okay, so. If I take. What's the best way to do this?

What do you want to do? What are you trying to do?

I'm just trying to. I just want to highlight. Hang on, wait. I want to highlight all the outgoing edges, but not the. Not the incident vertices. So if I just do undirected edge5 comma hash.

Yep.

And so if I take. Highlight, drop that. Okay, so then if I do. No, wait, I want to do. Join that with. There we go. Okay. So that would be an open set, I think. Right. Yep.

I agree.

Yeah.

Okay. But then. So. So if we remove that. If we. If we take the complement of that. This is closed. Right. So if. In other words, if I take the.

So, yeah, you're right. You're right, you're right. If you remove all the danglers, then nothing dangles.

Right, Right. Right. So if we call that. Let's call that open set. And I take graph complement of this comma open set. Can I do that? No, I need. This needs to be a graph. Oh, no, wait. No graph. Oh, no, it's graph difference. Wouldn't it be. There we go. Yeah. So that's a closed set with an isolated point in this case, because.

Cool. All right. Okay. So let me just internalize this for a second. So an open set has dangling pieces. Nice dangling piece picture there. And a closed set removes you. You remove. Like pulling out the spider, so to speak. I mean, you're pulling out this thing which has dangling legs. And since you pulled out all the dangling legs, there are no dangling legs left. Therefore, the thing is closed in the sense we originally expected.

Right, right.

Okay.

And so. Oh, sorry.

I was saying probably just to reiterate so that everyone's on the same page. The vertices that the red edges point to are like the limit points. And so the closer includes. Then the open set doesn't.

Right, exactly. Exactly.

Okay.

I'm sorry.

The incident points that are not contained in this case would be the limit points of this open set. Right. Because an alternative definition of an open set is one that doesn't contain its limit points.

Yes. Could you write that down? That's a good and important point. So, I mean, the limiting process here normally. Okay, let's talk about how you take a limit right in this set.

Well, yeah, so that's why the answer.

Is you just walk. The way you take a limit is you walk on an edge.

Right.

And that's a non trivial thing because normally when you take a limit in a, you know, in continuous spaces, topology, whatever, the process of taking a limit is something much more arbitrary. I'm making sense.

Yeah, yeah, that's, that's the whole point. Right.

So can we make a definition, please? A limit. A limit.

Well, I mean, the beauty of this is the definition of a limit is the same as it is in, in analysis. Right. It's, it's the, it's the open set definition.

I know that, but you can independently define a limit. To take a limit is to walk. You know, I'm a little. Okay, there's a definition here, right, because. Because normally it's a very floppy thing to take a limit. In this case, there's a very discreet way that you can take a limit.

Yeah, right. That's the reason for defining it.

I know, but what is the definition of taking a limit, of the process of taking a limit in these graphs? It is. You walk one step. Is that true?

Well, so you would take. I mean, the procedure is you look at what vertex bounds a dangling edge. That's the operation of taking a limit.

Yes, I understand.

The dangling edge defines a direction and therefore.

Exactly. Can we write down. I mean, the basic point is taking a limit in a certain direction. The only directions that you can take limits are, are the directions corresponding to edges. So the taking of a limit is just the walking down an edge to the other.

I just wrote this. I mean, in, in with more generality.

Well, okay, fine. I was, I was looking for a definition of the process of taking.

Well, evaluation of a limit is, you know, determination of the vertex that bounds.

A dangling edge on the other end of it. Yeah, okay, fair enough, fair enough, fair enough. All right.

Right.

But that's closer because dangling edges. So it's probably just a walk in a close set.

Yes. Limit in a closed set is just a walk and everything. And you always, you'll never, you'll, you'll always be in the closed set as you take these limits. This is very nice. I might Even understand open sets now?

No, and so as I say, this is obviously once you've got this apparatus, this definition of the limit is just the standard definition of a limit from analysis, right? Which means that we can things like metrics based completion just means doing exactly this process, right? Kind of putting in vertices at the ends of dangling edges.

Fine, you say it's just like analysis, but it's much simpler than the case of analysis.

What do you mean?

Well, I mean, you just got the explicit vertex right there. You just put it in. You're not having to worry about, you know, this limit set that some complicated infinite thing, et cetera, et cetera, et cetera.

But that's just like saying, you know, you've got an open set, you put in its limit point, that's evaluating the limit. It's exactly the same.

Fine, it's clearer for me. Okay, let's keep going.

Right, okay, so anyway, so the reason for talking about this is. All right, by the way, I should mention this was the original intuition that we started off with right at the beginning of the infra geometry program. So now we have a much more kind of robust reason for making this initial claim. But okay, so then the idea is, okay, so now we have values of function defined at the edges and we'll proceed to generalize that a little bit in a moment.

But so then you're describing different kinds of continuity for different. I mean, so you've got a scalar field.

So let's take the example of this edge, right? So right now I've just got this right in the. I've got the edge value right in the middle of the two vertex values. But okay, so the finite difference in this case would be 80. So one very crude definition, but which actually kind of can take you a reasonable distance of continuity classes or differentiability classes would be to say, okay, so if the value of the edge of this edge lies within distance, lies within 80 of either of the vertices or maybe, or perhaps between the midpoint, then it's kind of. Then it's somehow. That's the smoothest possible class. If it lies within distance 160, I.e 2 times 80 of that, then it's the next smoothest possible class, et cetera.

Sorry. The most naive thing is it lies between 2 and 82.

Right, right, right.

It's an undirected edge.

What you are essentially. Okay, so the idea behind this is. Okay, so now we have to. This is where the, I mean, this is one of the places where this edges as places. Intuition Comes from. But. So if you imagine. I'll make a picture of this in a moment. But if you imagine there being a point in the middle of this edge and we want to assign a value of the function at that point, the idea is, if we were to construct a polynomial interpolation between the function value at this point and the function value at this point that goes through the value at that point, what degree of polynomial would you need in order to make that work? And so the higher the degree of the polynomial. So in this case, you can do it just with linear interpolation. So the degree of the polynomial is 1. The higher the degree of the polynomial the claim is, the less smooth the function.

All right, let's make a picture with a manipulator. What we've got is we've got two anchor points, 2 and 82. We know keep. Don't mess up your beautiful graph. Scroll down.

Okay, the whole graph or.

I wouldn't think you would manipulate the whole graph. I was thinking you would just make a plot of function. I mean, a very simple thing here. Just where the end values of the function are 2 and 82. And you're simply taking. I mean, if you make a manipulate that just has an interpolation of a different interpolation order between. With the two value. Some value in the middle. 82.

So, okay, I don't. I don't know the syntax. For instance.

Okay, all right, so just say two and list inside there. Two comma, two, comma, X comma 82. Okay. All right, so now, now, now say I should call. Call that middle thing, I don't know, M or something. Okay, call. Call it instead of X, call it M. I see.

Okay, yeah, yeah.

And then of X. Okay, now say plot that from 0 to 1, and then you're going to manipulate that as a function of M. Okay, now, now put a. Manipulate to give a value for M. Yeah, And.

Let'S do M between 0 and 100 or something. It's not often.

I don't know why it's pink. Why is it pink? Oh, requested order has been reduced to two. Maybe it just can't interpolate. Hold on, let's just look at that interpolation at 2.

Yeah, I mean, it had that. It was complaining about this even when we constructed the insert. So If I do 2, 10, 82, what does that.

Oh, I think you need. I mean, you might need interpolating polynomial. There's another function in Tupling polynomial. Get rid of comma x there and get rid of the X after it.

Nice. Okay. Okay, cool. Is that the wrong way around? We want it 82 comma 2.

I don't think so. I think you want 2 comma 82. But I don't know what the. Okay, maybe you want the values. Maybe you want in that interpolation polynomial. We want 0 comma 2, you know, and then a half comma M and then 1 comma 82. See what I'm saying? You've got to have x and Y values.

I see. So 0 comma 2, a half comma M. Yep.

I think.

Yeah.

Yeah.

Okay. Yep, that makes sense. Cool. Okay.

Okay, so now we've got those things pinned at the end. So now there's a question of what's the degree of that interpolating polynomial? So that one was of degree three. But now what you're claiming is. Yeah, so what are you claiming here in this case?

Yeah, this is simply. This is. This is always. Yeah, this is always going to be quadratically.

It's always a. Right. So look up the documentation for interlagand function. Maybe it has a way of having a different order.

But crucially, I mean. So the.

No, I think it just makes the polynomial through that.

Yeah.

Okay, so maybe it isn't the right thing, but go ahead. What were you saying?

Okay, but. Okay, crucially. So in this case. Yes. So you. For a binary edge, you can always. It will be at worst. In this case, you can make do with a linear interpolation, but it will be at worst a quadratic.

Yes.

The point is, for hyper edges, that's not true. Right. For an arity N hyper edge in general, it will be at worst of order n. So that's the least smooth that it could be. But if you could make do with a polynomial of less than, of a degree less than N, then that function, that's somehow a more smooth function than the one where you needed degree n. If that makes sense.

Yep. Yeah. So I mean, the question is. Okay, hold on. In that picture you made. Okay, but back down to the function, the function. Okay, so here, for what values of M, is it a linear function? Answer. If M is really at the 0.5 point, there's only one function. There's only one linear function.

Right.

Is that true? It's a piecewise. No, wait a second. I'm. I'm. There's only one possible value of M for which that thing is linear, which is the average. Okay, so what does that tell us in terms of degrees of. Of continuity and things?

So somehow the more distant this value is from 42 I. E. From the value that would make this linear, the less smooth the function is. But this is somehow this assignment of values to edges corresponds to the most smooth possible function that you could define given the vertex values that we started from.

I agree, but you can put arbitrary values on those edges if you want to. It just won't be a very smooth function.

Right, right. And so one of the insights is that, I mean, one of the claims would be that in this infrageometric formalism, there's no such thing as a truly discontinuous function. There are only functions that have lower and lower degrees of smoothness.

All right, let me understand that claim. That's an interesting claim about what it means to be discontinuous here. I mean, let's use my favorite. Let's use a one dimensional grid graph. Okay. Maybe you can do it at the bottom. Put it at the bottom of your notebook. Otherwise we'll never keep track of what the heck was going on. Okay. Grid graph. Okay. So now we do the same jazz for this, because now we've got something where we've got. This literally is a function of position because there's a canonical labeling of the nodes by position.

Right.

So now what you're saying is there is no. Which is not, in a sense, totally surprising. There is no discontinuous function here because, I mean, you can have a function. We could actually just draw the function. Right. Whatever the values are here, we literally can draw the function on top of this graph.

Right.

So there will be functions that will be wiggly or not so wiggly. And so a smooth function might be one where the actual, you know, where the actual differences here are small, I guess. Okay. All right. That was just. Just to help me understand. I think that's a useful. I mean, good terms of this continuity thing, that grid graph is a case where one can already formulate all of what one needs to formulate.

Right, right.

Okay.

So that. Hang on. Okay, let me try and figure out the best way to do this. But.

Okay, what are you trying to do?

I want to now show a point on each edge because I think that's a useful visualization.

So I think. Jonathan, I think Nick made a function for this, and I could find basically where he sent it to me, so I can paste it on the Zoom Chat.

Okay, sure. If you have a function that does it, that would be good. Can I do graph? Is it graph plot, epilogue?

I don't know. I mean, you're trying to actually put those values at those Embedding positions.

Yeah. So hang on, wait.

If I do edge, I think you need next function. This is going to get messy. Well, you're going to just draw the thing if it is sort of within the, you know, if it is between. If it has the inequality between those. Those end values, you can draw it as an additional point on the graph. That's what you're going to do. Right, Right.

Hang on, wait. If I take that. So that's for each edge.

Yep. Looks promising.

Yeah. Yeah. Okay, so. So actually, I don't. I don't even need the edge names, do I? I can just take. So that's. Can I do epilogue point.

Why we do point mapped onto edge coordinates.

Yeah, I got something.

So you make the point bigger.

Yeah, so.

Ouch.

No.

Oops.

Wait.

Why don't you make point size?

Oh, is it, Is that an option?

No, no, no, no, no, no. It's not an option to point. It's just a. It's just a. It might be. You might be able to put it into the style directive or you can put it into go. No point size. Well, whatever. That. That's fine.

All right. That's roughly what I was trying to do. So, yeah. Okay, so then the idea. So now each one of these edges is now is an object to which you can assign a value of that function. And in a moment, I'll talk a little bit about how this, this is one of the cases where hypergraphs become inevitable because there's kind of. There's no reason to terminate this just at the RT2 case. But one of the insights that again, I think Carlos had was that actually once you start to think of edges as places, this clarifies a lot of the stuff about tangent spaces. Because now you can think of a tangent vector not as an arrow pointing from a vertex to a vertex, but as an arrow pointing from a vertex to an edge. And you have a natural. The thing you were talking about in relation to walking along an edge and finding its endpoint, that now becomes precisely the definition of what it means to continue moving in the direction of a tangent vector.

Yes, but. So your tangent bundle now, I mean, the tangent vectors at a given point are the dangling edges.

Yes, exactly, exactly.

And the tangent bundle is then all dangling edges from both directions. Because an edge can dangle in two ways.

For a torsion free space. Yeah.

Yes.

Could we clarify something here, please? Yeah. So just on Steven's point. So basically what happens is you take your graph and then each edge becomes a new vertex and you essentially generate you can imagine that you're generating a new graph on top, which is in principle undirected, but then to that graph that has some label of what is a vertex place and what is an edge place, so to speak, to that graph, you apply the logic that we applied earlier to the definition of tangent bundle. Right? So you basically have an undirected graph as space, and then you do the directed, the induced directed graph as the possible directions of that space. And so a tangent bundle becomes the collection of outwards edges, directed edges from any vertex place into all the nearby edge places, right?

Yep. But this graph has two types of nodes. I mean, it's a, one of a bipartite graph.

It does, it does. And, but so part of the sophistication that comes with the formalism, hopefully I'll be able to present, when I get my notebooks and I get everything in order, will be that you, you can forget about that. They are. So for, for example, for things like walks, you want to remember that they are different because you want to. So basically you can only walk between actual places, which are the vertex places. But if you're thinking about vectors as fields of directions, where you want to do vector fields or something like that, then you can actually think of the induced graph between vertices and edges in a very homogeneous way. And you basically just rewrite and do partial operations like in this rewriting hypergraph philosophy in a very homogeneous way. But you are only operating at the level of directed edges. And so this tangent graph level instead of the actual spatial graph, and I think by generating this tower. So that's, this is what, what is very analogous to actual differential geometries that you begin with a space. In the differential geometry case that's a manifold. In our case it's a graph hypergraph. And then you generate these bundles in a differential geometry case, tangent bundle, cotangent bundle, tensor bundles and whatnot. In our case, we, what you generate is this, is this more refined hypergraphs that effectively have these edge places. And so you can construct similar graph theoretic or hypergraph theoretic methods on top of those that will basically tell you things about the spatial hypergraph that you began with. But they are technically new hypergraphs that live on top. Right. So in that sense it's very beautiful because we actually have a very direct analogy between differential geometry and this sort of infrared geometry, hypergraph geometry kind of perspective. And it's a matter of really fleshing out the definitions and actually making all computational. Because in the end these are going to be functions that you operate rewrite rules that you partially apply. And so you have everything working partially and only in the regimes where they are defined and so on. But because it's a rewrite system, we're all good. And so, yeah, it's working quite nicely that way.

Well, I mean, but one point here is that presumably there are things you can formulate purely in terms of tangent bundles without. I mean, you could sort of, for some purposes you can throw away the space and just look at the tangent bundle.

In what sense?

Well, I don't know. I mean, in, in, in. Maybe I need to understand more about the traditional formalism of differential geometry, but I would think that there are things for which there are computations that you can do about, I don't know, some topological property of the space.

Right, right, yes, so, exactly. So, for example, a very good example of what you're describing, I think is the kind of algebra that you get on sections of the tiny model vector fields, right? So they have a calculus of their own. So the analogy here, I'm quite excited because the analogy here is actually you have to take seriously this approach of rewriting hypergraphs that, I mean, it's kind of very ongoing in this whole physics project. And you have to take this philosophy seriously and say, okay, in the differential geometry setting I had to, only I had to work hard so that I had well defined operations. So I have additional vector fields or composition of vector fields or library of vector fields. And these things, you have to kind of work hard for them to be well defined in our context, as long as the notion makes sense in a local sense. Meaning, you know, you compose vector fields, as we said last time, by concatenating them and basically doing composition of paths, but locally, then composition of vector fields reduces to a simple single rewriting rule for your graphs. And so vector fields are just assignments of directed graphs or in some sense directed subgraphs of that sort of place graph, edge, place graph. And then what you do is you apply this rewrite rule in typical rewriting fashion. So it's multi way and you have this sort of selfie, but it works because, I mean that's the. It's no globally defined. You don't have an actual algebra in the conventional sense. But in our formalism we work with it because it's based on a. It's captured by a rewrite rule. And whenever you have several objects that are vector fields defined on your graph, you can just apply the rewrite rule and it will act partially and in only the places that it's defined and it will give you a new object of the same kind because the rewritten graph will be a new vector field graph. And so, and so you have a sense of closeness, the algebraic closeness that is not based on the fact that you have actual operations, but still you are computationally sort of closed. Right.

So yeah, although I don't understand the multi wayness and how that comes in.

So this is what I hopefully will illustrate when I get hold of my notebook. But basically you are replacing here. So in order to claim that you have something like an algebra vector field, like we discussed last time with addition and composition, you are replacing the notion of composing two vector fields with the notion of composition is a rewrite rule that is going to pattern match for the occurrence of, for example, you know, vector field X having a directed edge that has a head at the tail of vector field Y. Right. And every time that you spot that pattern, you're going to rewrite for an edge Z. That I understand. Right. So that's why you turn into this rewriting philosophy. And the beauty of it is that because we have this computationally implemented, you are de facto, and everything is finite. You are de facto defining something like an operation algebraically, meaning if I give you a vector field as an assignment of directed edges and another vector field as an assignment of directed edges and you apply the rewrite rule, it's going to apply, you know, kinds of ways, but the finalized process. So you, it is a multi way in terms of the application. But if you keep on applying it and you just keep the rewrites, the actual outputs of the rewrites locally, the final result of that is a single well defined vector field. And so that's the, I mean, and there's the subtlety when that is actually always happening or not. And that's kind of a subtlety that you have to grapple with. But generally speaking, you can, you can define this as an operation without it being mathematically an operation because it's finite and it's computational.

Yeah, yeah, right. And it has some confluence property, presumably that is what leads it to be not to matter in which order you apply these rewrites somehow.

Yes, yes. So that's the subtle part. We have to be careful because you might need to define something like what I normally call the one pass rewrite, which is essentially taking an empty graph and filling it in with the outputs of the rewrites, which, which is how you, for example, recover matrix multiplication from a graph composition. But up to some subtleties, essentially, yes, you are relying on some confluence properties so that you actually have this well defined.

All right, let's see. Jonathan had. And by the way, I need to remind myself, I mean, the addition of vector fields is simply. You just add up the values. Right.

That's complicated. Addition is actually. I don't understand it yet. I don't think.

Well, I mean, given two identical graphs, if you've got two different fields, two scalar fields. Okay. Additional scale.

Yes. Scalar field addition is easy. Yes.

Right. Now, in the case of your places. Edges as places where you're sort of reifying the vector field. Yeah. It is a little bit unobvious how that would work.

Yeah.

Stephen, Sorry, the question you asked at the beginning about surely there are things you can do by just looking at the tangent space structure. That's kind of what Riemannian geometry is. Right. The whole idea of Riemannian geometry is that you have this inner product structure on your tangent space that very smoothly. And if you only know that, you kind of know everything there is to know about the manifold. Which is another way of saying if you only know about the tangent space structure and you know nothing else, there's nothing else you can sort of infer.

Walk me through what that. Okay, so, I mean, so the case.

That you're referring to in our context is, you know, that's precisely the case where one of these graphs is equipped with whatever the discrete analog of Riemannian structure is.

Okay, but you're saying that it is not true of every graph that you can throw away the underlying graph and just keep the tangent.

Well, we don't know. I think it's certainly not true of every space. Not every space is Romanian. It seems plausible to me that every graph has a Romanian. Just like graphs are all door spaces, it seems plausible that they're also all Romanian. But I don't think we have a proof of that.

Yeah.

Okay, but so hold on, hold on. First thing about graphs, and we should write down some of these things. There are no discontinuous functions on graphs is the claim.

I'll put that over here.

Yeah, Once. Once you take this edges as places approach.

Yeah. Of varying degrees of smoothness.

Yes. Okay, so now what would it mean in a trivial case like that grid graph, my favorite kind of grid graph thing. What would it mean for that to have a quote stream alien structure when it's labeled.

Yes. Well, okay, you wouldn't label it. Right. I mean, Romanian structure is something that belongs to a manifold, not to a function on the manifold. So what you would do is you Would say, okay, so if I only know the information about the tangent spaces and how they kind of plug together, which is really what the metric tensor is, all of that information is somehow summed up in the Romanian metric. If that somehow tells you everything there is to know about the geometry of this graph, then it's Romanian.

How do you construct this, the construction that you previously described for the tangent?

Let's see, in other words, that's saying, if you like, if I only know the outgoing dangling edges from each vertex, is that information enough to construct everything I might care about in relation to this graph? I think the answer is almost certainly yes. Right.

When it dangles, does it know where it connects?

Yes. So it would know its limit points.

Okay, fine. So if we labeled every node here just, you know, trivially, you know, just vertex label our automatic in that thing. I don't think you don't need it. You just put it inside the grid graph. Okay. All right, so we've labeled it. Okay, so now every dangler, you know, there's a six, five dangler, there's a six, seven dangler, and if you, all you have is a bag of danglers, you can clearly reconstruct this graph from a bag of danglers.

Right, so, so, but again, there's, there's an important subtlety here. Right? So if, Hang on, let me reuse my code from up here. No, where was it down here?

Coordinates.

I know, but okay, if I just take this example, right? So you have a bunch of arrows that are pointing to these interstitial points between the vertices. The connection is then the rule that tells you precisely what is the endpoint. In other words, the connection is the thing that tells you, if I follow this tangent vector, where do I go? So all this is saying is if I define. So the statement we're making up here is almost a trivial statement, which is that if I introduce a metric compatible connection that's metric compatible, then I get a Romanian manifold, which is a topological statement.

Okay, hold on. So I'd like to write down the statement that the connection is the establishment of dangling edges.

So yeah, claim connection is the rule that sort of tells you the end. The terminating vertex of a dangling edge.

Okay, but so normally. Okay, so in, in, in the normal world of connections and so on. Let's see, what, what is the. Okay, let's look at covariant derivative for one second. Okay, so we can define a derivative. I mean, we can define of the scalar field. There's perfectly reasonable derivatives to be gotten. I agree. In this, they're just, you know, how the field changes from one node to another. Yep, the. So now walk me through how the covariant directive works in the presence of this connection.

Well, so what you need is. And you know. Okay, so I don't have an answer to this question. I don't think Carlos does either.

Can I jump in for a second?

Yeah, yeah, please. But.

So actually the notion of a metric is not entirely obvious yet. I don't think so. Our initial guess back when we were not thinking of edges as places, but it applies here, I think very directly, is that you want to take the infinitesimal notion of metric, meaning the tangent sort of quadratic form notion of metric. So you want to somehow. So basically a notion of angle. So if you take two vectors defined in this way and you want to assign some kind of scalar value to them, the only information that you have going to get going is the incident structure of your graph. Right. So you can define some kind of discrete or boolean valued or maybe finite field valued angle simply by doing some incidence condition. For example, last time we said if the vectors are pointing to edges that are incident, then that's it. Sorry. If they're pointing to edges that have adjacent vertex elements, then you say that the angle is zero and otherwise you say it's one or something like that. I think, yes, we have that somewhere in that notebook.

Yeah. So this is our rank 2 angle tensor. Yeah, in that case.

So that definition still applies in this context. In fact, it applies in a more tidy way because you have the edge that is sort of highlighted. They are a different kind. And so defining this metric structure is not entirely obvious at this point. And so the other angle is the distance function. Right. It's having some kind of distance function that is, that is assigned. But now you have to think that it's assigned to paths to some notion of a curve in your space. Right. And so because the norm of a vector doesn't really make a lot of sense in our context. So in the sense of distance of paths we do have an.

Sorry, what is a vector?

A vector? A vector is. Definition, is a directed edge of the edge place induced graph. So the graph that you used with the red dots, so it's a blue.

To red, that's like a tangent vector. But I mean there's also a whole path made of many of those things.

Yes, exactly. So that's where the sort of subtlety comes in that you have paths on the tangent vector on the. Sorry, let me call this dotted red Dotted graph, the tangent graph for short, that graph. So you have walks on that graph and they may or may not correspond to actual paths on your spatial manifold. So that's the special graph that's the notion of integration, right? In the sense of vector fields, that you have some distribution of tangent directions in space. And the question is, is there a curve such that its tangent matches that field of directions? Right?

Yes.

In our context that is expressed as I give you a walk on this tangent graph and the question is, does it correspond to some path on the, what I'm calling a path which is a non overlapping walk on the spatial graph. And obviously there are many walks on this tangent graph that do not correspond to those. In particular, those that correspond connect red dots, right? The red nodes cannot be connected because they are. You always have to walk through an actual point of space, an actual vers. Right. So, but also not all the paths that connect the vertices are actual spatial paths because they might not be connected by some edge, right? So this notion of integration is going to be closely connected to the notion of connection. And then if you have a way of saying how your tangent paths or your tangent walks on this graph correspond to actual paths on the spatial graph, then you might be implementing, you might have the notion of, or the possibility to implement the notion of a metric as the relation between the scalar pairwise scalar relations of vectors and the scalar assignment to the distance of a path, right? As in usual Romanian geometry where you have, you know, a metric tensor allows you to define a geodesic distance measure. So but at the moment it is not entirely clear how to give the sort of like full, full stop definition of a metric tensor because there seems, there seem to be several options here. My canonical choice is that you look at the incidence structure of where the arrows are pointing and you see whether there are intermediary or the number of maybe the number of maybe the shortest path between intermediary that could be an integer valued angle to basically count the number of edges that you have to go through in the shortest path between the edge places that you are pointing at.

Right.

But it's not clear yet entirely how to make the definition because there seems to be choices and it's not obvious which one to go. I think that if I were to. So what I'm focusing at the moment on is that it's trying to recover this lie algebra story that we mentioned at some point because we were running into issues and I think that this edges places is going to effectively give us a very Analogous story to the differential geometry one, because it effectively recovers all the algebra for free. Because we now have actual tangent spaces as separate to the underlying intrinsic space. And we have the notion of, for example, the directed edges ingoing into a vertex node. That's the negative vector. Basically. The positive vectors are the ones outwards and the negative ones are the ones inwards. And you can think that the actual tangent space is the collection of all of those. Right. Both inwards and outwards. And so you have. You basically have a. I mean, you mentioned in the last live stream that you said, oh, it would be great if differential geometry is explained like that. This is exactly what we're doing. We're basically having a discrete analog of the differential geometry. And what's great is that we are actually developing it by necessity and not by pictorial analogies. So it's. Nice confluence there.

Yes. What's Jonathan up to here?

I'm plotting it back to feel. Sorry, ignore me.

Can I jump in for one second? So, Carlos, you said you'll recover the algebra. So if I'm interpreting it correctly.

I think he means Lee brackets and things. Not. Go ahead. What, what were you. I mean, I don't even know what is the relationship. Okay, yes, go ahead.

So it's like. I mean in differential geometry.

Like you.

Can take the leave bracket, it forms V algebra. So in here, like we first, when we viewed. And we didn't think of edges as places and we just thought of the endpoint as just. Sorry, we just. We just thought of as a vector from point A from point one node to the other. We do not get a L structure. And the weighted derivation story that we introduced, that was like we got an extra term that caused it to. Not cause it to have.

Remember you were talking about that last time? Yes, yes.

So in here, since we are introducing like these midpoints as, like these extra places which are in between the edges. If I have a function at a node to the function over there would be like something plus epsilon, like a small perturbation. And since epsilon squared is zero, you would recover from the algorithm. Is that what you were going for, Carlos, or.

What I'm. What I'm suggesting is. Yes, morally that, that, that you are going that now we have a notion of infinitesimal display displacement. Before we only had. We only had whole displacement that. And that's why we talked about discrete difference operators and so on, or finite difference operators. Now there is a. There is a way. It's still discrete, still graph based, but there's a way to tell apart a whole displacement, which is vertex to vertex, or potentially an infinitesimal displacement, which is vertex 2 edge place. So that's exactly right. That's that we now have a. And a discrete handle, a graph theoretic handle on what it means to be near something but not quite there, which is.

That makes sense.

Yeah, yeah, exactly. Yeah.

Okay, so this is a vector field, except, I don't know, the red dot. Oh, I see the confusion here. You should make different colors the vector field. There's got to be a color of vector fields purple or something.

So Jonathan, in fact, you need also red dots to have outgoing vectors.

Ah, okay.

Purple is a bad. Let's make it. Let's make it, I don't know, blue or something. Purple is bad because we could barely see it. Okay.

Is that any better?

Not really, but it's. It's not terrible.

Okay, well, no. So in that case, then we can just.

The only constraint is that you don't connect red to red, but otherwise it's a general choice of edges.

Well, in that case we can just do that, I think.

Yep.

Yeah. So I think the important point is that a path that terminates, so you know, we have vertex points and we have vertex places and we have edge places, so to speak. And a graph path that terminates at vertex places is the analog of a curve and one that terminates that has one end at a vertex place and one end at an edge place is something akin to an infinitesimal vector. Right, like an edge vector.

Yeah, I'm calling them traces to distinguish.

Right.

And what about a red to red? What about.

I don't think those are allowed, are they?

Those are not allowed. In principle, I think we might have interpretations for those down the line, but at the moment, because those are. Could be something like higher order displacement or higher order derivatives, we are thinking of this as tangents, but at the moment I think they're just not allowed for the direct, the first order interpretation of tangent space. Yeah.

I mean, one thing that I think I brought up in the last Institute livestream that we did, which I think would be kind of cool to look at, is to see whether this interpretation of these edge places as sort of the analog of infinitesimals, whether we can make that algebraically precise, in other words, whether we can recover the axioms of something like non standard analysis by looking at essentially the algebra of how these infinitesimal distances compose.

I recommend doing that for this one dimensional grid graph. I think that will be a straightforward case.

Right.

Because that's like the One dimensional at the very least.

It's easy to see how fairly simple axioms like epsilon squared equals zero should be true. But the question of whether the full algebra of infinitesimals can be recovered.

Why is epsilon squared equals zero true? Because you can't go from. There's only one red dot between two.

Yeah, exactly, exactly. You can subdivide this distance, but you can't subdivide this distance. Right. So you can see that there's the beginnings of an algebra of infinitesimals. But I think we know what it is.

That sounds like an OOTY project to me to look at the one dimensional case and try and see what the algebraic version of that is.

Yeah, sure. Jonathan, I think just as a comment, are we actually expecting to get an algebra infinitesimals? I think we probably can only content ourselves with the discrete analog of that, which is going to have something like that. A finite. Finite element extended algebra.

Right, right, right. I mean, it would be very surprising if we got exactly non standard analysis, but understanding where they differ would be kind of cool.

No, this is like, this is discrete non standard analysis somehow. Yeah, and what, what the heck is that? Because, you know, the general topic here is discrete analysis, whatever that means.

Yes, yes, that's right.

Which is. Which is unfortunately already a term that does not mean this.

What does it mean?

Infra analysis.

Right.

What's that? Yeah, it's got to be infra analysis. But what does discrete analysis mean?

Discrete analysis, normally like there's a journal called Discrete Analysis which mostly, for instance, publishes results in like extremal combinatorics, which is related.

Oh my God. But it's not even. It's not like finite. It's not like recurrence relations and so on.

No, no, no, but it's more like, for instance, Banach geometry, right? You know, cases where, you know, you're doing something, you know, you're sort of investigating a functional analytic problem, but you're doing so using some kind of combinatorial structure, like, you know.

Yeah, fair enough.

Which is not quite the same, but so one thing that's important to just mention before we forget it is, you know, so obviously we've been talking a lot about the Arity 2 case, but so in. So here, wait, no, actually, let me show the. Where was the one where I had an interpolation. There we go. So here, you know, where we say, okay, we're assigning values of the function, you know, not just to the vertices, but also to the edges. There's no particular reason to just privilege rrt2 edges. And so in particular, if you take, say, this path here from this vertex to this vertex to this vertex, it also makes sense to ask, okay, given these three points on the boundary of this polygon, more generally of the boundary of this polytope, what possible value could be assigned to the interior? And so in general, for any length path, we want to be able to assign a value to that path, not just to the RT2 edges. And so that led to us, I don't know if we came up, or if Carlos or anyone came up with a more sensible name, but in the last livestream, started talking about what we provisionally called hyper hypergraphs, a kind of generalized transitive closure operation where you take, in this case an underlying Arity 2 graph and you look at all of its paths and you introduce all of its length n paths and you introduce an arityn hyper edge for each such path.

Right.

So in this case, okay, if we take this graph, so I guess we could do graph, say text list.

So these would just be basically like the path through point of the.

By the way, I mean, you know, an interesting point about graphs is if you're just given the graph, you don't have the topology of, you know, you don't have which edges bound a face. But what you're doing here is you're saying actually for any set of edges, they can bound a hyper edge. Right. So in the traditional way of saying, how do we construct ordinary topology from a graph? It's not obvious because you have to, you know, if you've got a triangulation of something, for example.

Yeah, yeah.

Then it's pretty clear how you construct, you know, set up topology, construct all kinds of things. So what you're doing here, which is rather interesting, is you're saying instead of asking to get faces which have immediate geometry, get hyper edges which are somehow combinatorial kinds of things.

Yeah.

Make any sense?

So. So yes, and I can maybe expand on that intuition. So the reason to include hyper edges, I think can be quite intuitively argued basically in this edges as places philosophy, you want to think that if you have two vertices in the binary case you have, you say there's two places, two landmarks in my landscape that I can look at. And then to say that there's an edge between them is to say that there is some place between them. There's something you have to traverse in order to travel from one to the other. So, you know, if you consider three landmarks, you know, abc, whatever, to say that there is a that there is a ternary edge between them is basically to say that, you know, there's some space between the three of them. And so that information is logically different from saying that, you know that there is some space between one of any of the pairs.

Right, sure.

So that's, that's why you got kind of self justify the. Okay, yeah, okay.

But I'm trying to understand the analog of a vector field as you, you know. Well, the, this tangent. The concept of a tangent vector.

Yes.

Becomes something more general. A tangent. I don't know what the heck it is.

Aha, yes. So. Yeah, exactly. So, so you. So, so this is, this is exactly what people in, in. In the case of the, in the directed case where you make everything directed, you will be defining what people would call blades or multivectors in geometric algebra language. So these are the sections of the higher anti symmetric powers of the tangent model. So these are very well known in differential geometry. And in this context you will be thinking as exactly as what you say, which are hyper edges which are made up of one vertex point and several edge points.

Yes, yeah, yes. I mean we're saying one vertex. Oh, I see. You're one of the vertices.

And so maybe it's easiest to think about it in terms of the limit construction from before. Right. So when you just had binary edges, what you're saying is okay, I've got a vertex place and I have an edge place and that determines the limit point, which is the terminating vertex place. But in the Arity 3 case, you've got a vertex place and an edge place, but that determines two further vertex places. And so if you constructed a bundle here and you take a global of these objects and you take a global section, you're not going to get a vector field, you're going to get something that's more akin to a rank two tensorfield.

Yeah.

And how does that relate. I mean I have never understood this geometric algebra business and you know, there are whole bunch of people who have kind of militated. I suppose there are geometric algebra activists who've said, you know, we should pay attention to geometric algebra for purposes of what we implement in W from language and so on. And I've never understood what was being said there.

I think just as a comment, it's a bit tangential. I mean, pun not intended, is that the geometric algebra business is mostly relevant for actual cleaning geometry or Riemann geometry in this discrete context where we are trying to first emulate general differential geometry. The geometry. You got the perspective. I don't Think it's necessary, necessarily relevant immediately, because I think what geometry algebra does is it, it condenses in a very nice algebraic package the notion that you have that the fact that you have an inner product in your vector space induces all, all forms of area forms as well the determinant and all the sub determinants and so on.

I see.

So it's a nice package, a nice algebraic package to, to have a geometry encoded in algebra kind of thing. But in this context I don't think it's particularly irrelevant. It's just that we are recovering the objects that geometric algebra speaks of, which is this set of vectors into a higher order, higher rank object called a multivector applied. And in this context it's just very clear that the arity of the edge is maybe minus one, the dimension of, or the order or rank of the multivector or tensor that you're representing.

Can we talk about vectors versus one forms and things? I mean, is there a relationship here? Go ahead.

That's interesting. Yes. So, so taking the definition of one forms as the dual of vectors, then you are essentially looking at functions over the graph of, of the. This thing I'm calling tangent graph. And essentially we should probably prove at some point, I didn't have time to go over this cotangent construction, but we should prove at one time that the information of what we should encode as a linear mapping of tangent vectors is actually also a directed edge of some other graph and prove that they are isometric, actually isomorphic. Actually that might be one of the cool exercises to do in a finite case, in an actual computational example, to show that we are on the right track, that we can recover the fact that the cotangent bundle and the tangent bundle are perhaps non canonically but isomorphic, that they are the same dimension. This is something that might be actually interesting.

So this is one of the reasons why in the original plotting of this picture, why I didn't include directed edges coming from the edge places. Because. So this may be completely wrong, maybe overly naive. My intuition behind all of this has been that the edges that go from an edge place back to a vertex. Those are the analogues of your covectors or your one form. Because then if I traverse from here to here and then back again, I get back to where I started from. In other words, I get a scalar, which is another way of saying that if I multiply a vector by a covector, I get a scalar.

Why do you get a scalar?

Because. So get a value. I just Get a, I get a. I get the value at that point. Right.

But why, where is the value coming from? You only have the data of two of two directed edges. Right. But I'm imagining a boolean function, for example, that.

Okay.

If you specify at a given, at a given vertex point, right. If you specify one incoming direction, you can just match against all the outgoing directions and say, are you closing the path or not? And that's a boolean function, right?

Right.

So that, that might. That actually, I like it. That's a natural possibility. But it's actually a function because that's the thing. Dual vector is a linear function on all the tangent vectors.

Right?

Right.

So that's a possibility.

Right, yeah. So our underlying field is always F2 somehow, for example.

Yeah, yeah. I mean, I guess. Yeah. That's interesting. Actually, I was thinking of the incoming vectors as negative vectors because that's kind of what negative means also in geometry. Right.

I would say that. Okay, I don't know. This is. We're resorting to battle of the intuitions here, which is not justify any of this.

Right.

But I would have thought that a negative vector would. So, you know, if you have a vector going from this point to this point, the negative vector would be the vector going from that point to that point. The covector be the vector going from that point to that point.

That's what I would think too.

Yeah, that's. No, that's. That's all too. That's right.

That's right.

Can we write down that. That guess. Okay, and then, then I want to understand.

Vectors take you from vertexes to edge places. Covectors take you from edge places to vertex places.

Okay, so can you explain to me what a P form?

Yeah, so I was just going to comment that, you know, I don't want to seem too down on geometric algebra, but so my view is that it's just. Carlos, correct me if I'm doing a grave disservice, but I've always kind of viewed geometric algebra as just a way to get geometric intuition for some kinds of tensor products. So the idea is, okay, if I have these two. So let's say I take these two vectors, I can compute their exterior products, which gives me a two blade, which in this case is going to have the interpretation of this triangle, which I can represent as an arity 3 hyper edge, as Carlos said. And that just gives me. And then there's a vector space of those arity three hyper edges of those two blades. And then I can take products of those and so on So I get this whole infinite hierarchy. And so it's just a way of geometrically visualizing and reasoning about what it means to take tensor products of tangent vectors or tensor products of higher rank objects.

Well, that together with a notion of dot product. Right, right. So you have those notions and then the moment you throw in a dot product at the sort of vector level, then everything, you know, the Clifford algebra gymnastics sort of closes everything into a nice fold. And you have the geometric algebra. But yes, it's a good characterization. I never saw it as anything particularly revolutionary or, you know, breakfast. More of a nice package of known structure.

Okay, can we talk about exterior calculus? And, and you know how we get you. We said a one form is associated with. With, you know, going from an edge place to a vertex place.

What's so in that? Modulo this guess. Right. Then a two form. Okay, so if you take this is like our two blades. I can equivalently think of it as being the tensor product of two tangent vectors.

Sorry, the dots there are vertices. Is that right?

The dots here are vertices. Exactly. So this is. Yeah.

Vertex places.

Vertex places. And then this is a single edge place. But it's a. It's a three edge place.

Yep.

And so I can consider a path going from any. You know, so. So if this were embedded in some larger hypergle.

Well, you think a 2 form is basically just the going out from that. Whatever is in the middle of that. Whatever is the red dot.

It's the thing that's going from the. From this hyper edge to the vertices. Whereas the, Whereas the rank. Yeah, exactly.

Tensor. Right. Is the ordinary rank 2 tensor, as opposed to the 2 form is going the other way somehow.

Yeah. So, yeah, the contravariant. Is that right? Yeah, the contravariant rank 2 tensor is the thing going from the vertices to that edge. The covariant rank 2 tensor, ie, the 2 form, is the thing going from the edge to the vertices.

Okay, so now let's do the next exercise. Can we write down this point? I mean, I like this idea that you, you erect to get a P form. You. A rank P tensor is just a bunch of. It's just P dangling edges going out from a given vertex, is it not?

Yeah. So I'm going to be a little bit. So I'm not sure if we should settle exactly on that. I do like the definition because it's concise, but.

Let'S take this as a hypothesis. Carlos. I'm not sure it's right either, but. Okay.

The contravariant case is I'm not sure it's right either. But.

I mean, I do have a comment that maybe. I'm not saying we shouldn't try this definition, but the comment is if we are thinking of a spatial hypergraph, the data of a spatial hypergraph which is going to have a mixture of order of edges. If we're thinking of that as this is our primary spatial data, right? And we are thinking that only. So we're basically saying only RET2 is going to be the tangent vector, right? So the tangent vector sort of data. If I want to traverse an edge place that happens to live in one of these ReT3 or higher ReT edges, then I will never pick that up in the sense of paths because I'm not considering them for tangents. So that's why I like the conciseness that just RD of the edge determines the order of the form or the order of the tensor. But I think we might be going a little bit against the base philosophy of the hypergraph is the spatial data. And then it's our parsing of the hypergraph, how we get all these higher forms of algebra and sort of tensors and so on.

Okay, but, but I, I still want to in this theory that the higher hyper edges, the, the what we can say the.

Because so, so can I. So I can defend what, what I'm saying by, by. By the following. So imagine so you have a random hypergraph like the graph that we drew, but with higher order just in between. You draw the red dots in between edges, regardless of their. Then you can generate the same way that we generated a directed graph out of those to define tangent vectors and so on. You can generate higher order hypergraphs in the same way where you're basing your vertex points as different as the edge points and you're generating hyper edges, this sort of tangent hyper edges and higher tangent hyper edges, this multi tangent hyper edges and then you generate these higher hypergraphs, right? So it's not that you generate all possible hypergraphs that way that you are also constrained, but you are somehow mixing the orders of the edges and you're just counting the number of them. So basically, and this is actually what we want because we want to have the number. So the order of the hyper edge that you can create out of out of a vertex place is the maximum number of the maximum arity of that edge is the degree of that of that vertex place. So that's exactly matching the intuition that the dimension of your space is directly related to the degree of the hypergraph.

Right.

Well, okay, but hold on. This is exactly what I was going to ask. I was going to ask about Hodge star operations and so on, and I was going to, you know, dual operations because, you know, there's a point at which you've kind of, I mean, the local saturation of edges is different from the global saturation of edges. The global saturation of edges is what gives a global sense of dimension, the local saturation of edges. I mean, you can have a, A, you know, five dimensional object where every local vertex has just, you know, I don't know, three, you know, three nearest neighbors, so to speak.

Yeah.

So I'm trying to understand, I mean, you know, in these interpretations it should be the case that you can start seeing what you can. These membranes. You can start seeing sort of as you try and globally construct these membranes, you should start seeing something that reveals the effective dimension of the space.

Yeah, and what I'm saying is that. I agree. And what I'm saying is that it is not yet obvious to me that we should be thinking of the order of a given edge as an effective token for dimension, which is what we would do if we do this particular choice of edge placed in, into vertex place, directed edge to be the placeholder for a, you know, higher order tensor, the rank, higher rank tensor.

Well, wait a second. There's two different things. There's the rank of a tensor and there's the dimension of a space, which are very different things.

Yes, yes, but, but yes, they are. There are different things, but the, the dimension of a, the, the rank of a tensor, the rank of a form, of a P form or a multivector is, cannot be, cannot be higher. I mean, higher than.

Exactly. But that's not true of a, of a, I mean, the. Right, so in these P forms what should happen is as you try and construct, I mean, the point is that constructing it very locally, it's going to depend on the arity of, of edges and stuff like this.

Yeah. So I guess. Okay, I can, I can boil down the two options that we have as follows. We can either look at a vertex and then basically parse the edges by arity. And then that's going to give us a token for a dimension. Or we could look at a vertex and look at its degree and then take that as a token for a dimension. That's basically the two options I'm seeing here.

Okay, but I think we already have a, you know, a definition dimension that we like quite a bit, which involves, you know, creating bigger and bigger geodesic Balls.

Yes, yes, yes, that definition is. Yeah, no, but I. But that's kind of the estimate definition. That's why I said the placeholder for a token for dimension, which is like the infinitesimal dimension, the analog of the dimension of the tangent space at a point, which in differential geometry happens to match the global definition of the manifold. But in the discrete case, it might be different.

Right, so what would happen is there's a. A local dimension that is 2, 3, 4, 1 2, et cetera, et cetera, et cetera, but which somehow averages out to the.

Or somehow. Yes, I agree. Yes, yeah, yeah, exactly. But what I'm saying is that the local dimension. I'm seeing two choices. I don't have a particular preference for either, but I don't see a good reason to choose one over the other, which is that you either take the arities of the edges that are incident in that vertex as the. As the token 4 dimension, or you look at the degree of. Of that vertex as a token for dimension. I don't have a good sense of what. Which one should be.

I understand.

Yeah.

Cool. Jonathan.

Well, okay, so the claim would be that that's. That that's a two form, right? A two form at vertex. A vertex place a.

Yes, yes.

Wouldn't that be a two vector?

No, it's a two form.

Wait, wait, wait, wait. No, no, you're right.

Vector.

Hang on.

Oh, you are right. Yes. A2 form would come out from that node.

Yeah, yeah. So that would be. Hang on, let's get this right. Yeah. Okay. No, so it would literally just be the reverse of this, right? Yeah, you're right. Okay, so then that's a. Yeah, that's a two form. That's a two vector. And then you can even have a mixed index tensor, rank 2 tensor, which would be. That case.

Yes. All right, listen, are we at a reasonable stopping point? I'm supposed to go and do something else. Did we. I mean, this seems to me it's progressing and there are obvious questions to ask here.

Carlos, uti, Were there other things that were figured out that I. Or that we neglected to mention?

I don't think so. At least not without having the time to expand the next time we can do.

So.

I think it's a good stopping point here.

Okay. I mean, progress is being made here. It's. When you build things up from such a low level, it takes a while to get. I mean, it feels like the roadmap. There's a. Okay, just. Just give me one. A little bit of a sense of the roadmap we've got, you know, we've got the notion of open sets and things. We've got just what is the, what's the roadmap? The notion of, you know, p forms and things seems to be coming.

Yes. And to me, the most obvious thing is to try and have a computationally operational notion of tensor algebra of a manifold which is going to replace operations with rewrite rules and complete global operations with partially defined, partially matching patterns. And so everything is going to live in this rewrite. So at each. So now that we have the special hypergraph that we are not going to rewrite, at least in this context of tensor algebra, but then you have all this tower of hypergraphs on top, this P form, you know, tangent hypergraphs, those we are going to rewrite, we are going to be applying rewrite rules to.

And okay, so basically what we're trying to make is a compiler that goes from the operations of differential geometry to a bunch of graph rewrites, essentially.

Yes, that's, that's, to me, that's one, one big goal, to have a nice correspondence, you know, one, one, one column differential geometry, one column hypergraph rewriting and be able to.

No, but I mean, we can then operationalize differential geometry. That is, you know, this is a way, if we were doing numerical analysis, for example, we want to take differential geometry and we want to find some discrete approximation to it. Sure. This is a way in which we would say we are literally compiling this symbolically represented piece of differential geometry into this collection of hypergraph of graph rewritings.

Yeah, exactly. So I would say it's not hypergraph rewriting, it's graph rewriting.

Right, That I think too. So.

Because as Carlos mentioned, you've got, you know, those rewritings are not occurring on the underlying hypergraph, they're occurring on this kind of. I think we were calling, Carlos was calling it the tangent graph or whatever we're constructing on top of it. And so then, I don't know, it's pretty clear that things like index contractions, you can just operationalize as if I contract along this index, that's adding another edge that takes me back here. So I reduce the rank of the tensor and that's just the graph rewrite rule. Right?

Yeah, right. So what I'm saying is every one of those things that can be described symbolically, and in many cases we already have symbolic descriptions in Wolfram language. We could write a compiler that turns that into.

Yeah, operations.

Yeah, yeah, but so horizon for sure. Yeah.

Okay, so where do we get next. When do we get to metric tensors? When do we get to curvatures? When do we get to. I mean we already think we know what those are, but it would be nice to be able to see that more rigorously.

Feel like there's a lot of scope for topology. The, the diamond edge construction was pretty nice and the complement of open sets, like it was like it really matched it nicely. So I'm kind of excited to explore like what do H source spaces look like, compact spaces, etc.

What, what the. Sorry, what, what do we say? So, so, so for example, things like.

What the, what the analog of the hierarchy of separation axioms would be.

Yeah, what is that?

You know, things like Hausdorf spaces, ticking off spaces, et cetera. You know, essentially asking like, you know, if you've got, you know, if you've got two open balls that are, you know, that are non intersecting, you know, do they always, you know, you know, construct open sets that are, that are, you know, that don't share limit points and things like that.

But what about all the results about sort of, you know, all the things analogous to, you know, Euler characteristics and Gauss's theorem and all this kind of thing. The relationship between vector fields and.

Right, but I mean of course that's, that, you know, that's we want to get there, but the problem is to make. So with, with graphs, understanding what things like topological indices are is fairly straightforward, but you need to, you need to have a bunch of the underlying, you know, analysis machinery operational before you can really start making those kinds of claims.

Right, No, I understand, but, so getting the relationship between whatever the analog of the topological indices is for graphs, which I'm not sure is totally straightforward, but we can imagine it. I mean, I don't know, there's probably chromatic numberish type things that may be related to that. I'm not sure.

I mean there's a fairly well developed theory of graph homology that's quite sensible. Right? You know, general graph homology.

Okay, but I think, I think, let me emphasize because I think it's relevant here that I do think that we're doing something that is slightly different from that. Because if we are taking our base graphs as sort of, you know, the token for space or this is what space should be in a discrete sense, then we're going to be analyzing graphs that are emergent from that in some sense that as we described this tangent graphs. So it might be that obviously we can pull techniques from graph homology and all these existing theories, but it Might also be that we need to rely more on this sort of rewriting perspective, where our algebraic notions are now more graph rewriting notions, and we have to work more sort of native to this graph land just than applying existing homology to. To that or existing formalisms to that. It might be that there are interesting techniques there, but it feels to me. I mean, maybe I'm just doing cultural now to thinking in terms of hypergraphy writing, but it seems to me that if we're going to do something that is really proper to the name infra geometry or infra calculus, something that goes below, then we should do something that doesn't actually use existing formalism in an explicit sense and maybe recover somehow. But there has this sort of infra. Infra engine, right. That is running in this sort of rewriting way, and we are just clever enough to spot the right patterns, and then we can say it's actually doing what we want, which is very similar to, I guess, what the physics project does with the rules.

I guess the argument there would be that what you want is something that just as we have the objective is to build something that operationalizes differential geometry and tensor algebra, we would be to have something that operationalizes effectively simplicial homology defined over one of these hypergraphs.

Yeah, you know, exactly. In a way that. That is purely graphical in some sense.

Yep, we're gonna get there. This is. This is cool. I mean, it's. It's. And it will be very computational and operationalized and, you know, some results will be very straightforward, others will be complicated. Limits of what we're talking about.

Yep, very likely.

Cool. All right, well, thanks. And we had some comments on live stream here, which I'm.

Let's see, before we go, I think we. I mean, Jonathan and Carlos, when do we want to discuss the Causality Project? I have some interesting.

We should do that as a. As a. As an institute live stream. Maybe next week.

Yep. Okay. All right. I got to go to something completely different, guys. Thanks a lot.

All right, thanks, everyone.

Okay, bye. Bye.