 from Lehi, who will talk about Motivic Geometry of Two-Loop Fine Minutes. Okay, thanks, Matt. So first of all, I'd like to thank the organizers for inviting me to come here and speak at the store shop. It's been really nice so far. Thank you, everyone, for coming as well. Okay, so first, a couple comments about my title slide. So I'm going to talk about the Motivic Geometry of Two-Loop Fine Minutes rules. So first of all, I'm not going to say anything. I mean, of course, this comes from physics quantum field theory. I'm not going to say anything about that because I think the audience is divided in two groups, people who know that part better than me and people who don't, and the people who don't. I think you can just sort of think about this. Essentially, the way I do is like a really nice problem in podge theory or geometry coming from combinatorics. So that's the way I do this whole project. The second comment that I want to make about this is that I use the word Motivic. So I mean this in a very loose sense. I basically mean podge theory when I talk about Motivic. Okay, so, great. So with that out of the way, great, thank you. Okay, so I'll give some basic definitions first of all. So let's start with the graph, which I'll call gamma and some integer, some positive even integer called the lifetime dimension. So for me, a graph that's gonna consist of the basic data. So you have your edges, your vertices and your half edges. So I'll sometimes call these external edges. I mean, that's maybe the only modification you need to make here. And so to each half edge, I'm going to associate with a momentum. So this lives in a Minkowski phase dimension B. That's what the one D minus one means and these vectors will satisfy conservation. So if you add up all the momentum coming into this graph, you're gonna get zero. And so to each internal edge, we're going to associate a positive real number, positive. So I'm gonna simplify things a bit. So this is sort of the basic, starting point for this whole project. So the simplifications are the following. So first I'm going to view masses and momenta as complex numbers. So this is just to make dealing with them the outbreak geometry a bit easier. I'm going to restrict to the case where I have no one valent vertices on my graphs. This is just again a simplification that I can make. And I'm also going to look at only connected graphs today. And for each vertex, I'm going to associate a single half edge. So in fact, like we can sort of rough the half edge condition I introduced it and sort of got it immediately. And so I'm gonna write, I'm going to associate the momenta to half vertices instead of half. Okay, so these are justifiable simplifications. Okay, so once we have this data, we can associate a bunch of polynomials. So these are called the semantic polynomial. So they live in the ring of polynomials with variables corresponding to the edges of the graph gamma. So there's a U polynomial. So this is called the first semantic polynomial. So if I take a tree, a spanning tree of my graph gamma, so when I say spanning tree, I mean take a tree inside of the graph, tree is tree, spanning means that I contain each vertex of the graph in my sub tree, in my sub graph. And when I say this, I also mean that a spanning tree is simply connected. So a single component. So there's a unique component to this spanning tree. So to each spanning tree, I can associate a monomial. This is what I'll call x to the t. And that's just the product of all variables corresponding to edges not in my tree. I can take the sum of these things. And this is called the first semantic polynomial and I'll denote it U of gamma. So this is closely related to what's called the Kirchhoff polynomial of the graph. And that's obtained by taking the condition here and just taking the product that is inside of the tree. So you can get this, this other sort of commentarily interesting and important polynomial about it. There's another polynomial that I'll attach to this. So the first polynomial you'll notice has nothing to do with anything except the graph, right? There's just something about the graph that says nothing to do with the other data that I had these masses in the mental. The deep polynomial, this has something to do with the masses in the mental and it'll show up sort of in this question here. So we'll do the same thing. So I'll take a spanning two tree of my graph. So when I say spanning two tree, I mean a tree. So simply connected sub graph, which contains all vertices of the graph. So that's what spanning means here. And I want two distinct components. So two connected components and I'll call that a spanning tree tree. So again, we can associate the same types of monomials to a spanning tree tree. So take the product of all variables, not in that tree. And we can take this sum attached to this choice of three. So for each two tree, I construct this polynomial. Okay, so the coefficients that I introduced here, these have something to do with the momenta. These are the sum of all the momenta coming into each of the two trees. And then I take the square. So when I say square here, I mean take the scalar product of these two vectors with one another. And so by conservation momentum, the sum of the momenta coming into one of these two trees and one of these components and the other component, these are going to be the same. The sum of things in here is the same as some, negative the sum of things in here, but when I square those. Right, so this is the basic setup. Right, so once I have that, I can find the polynomial that I'm really most interested in. So this is often called the second semantic polynomial. I'll call it F gamma associated to the graph. And so it's just the polynomial I get by taking U gamma multiplying by this linear term in terms of all edges. So here is where the maths appear in my polynomial and I add the gap. Okay, so a couple of comments about this polynomial. So the degree is equal to the loop order of the graph. When I say loop order, I mean first Betty number. So the degree of U is equal to the loop order of the graph and that's basically because if I wanna build a spanning tree out of the graph, I can just take a basis of boots and remove one edge for each of these elements in the base. Similarly, if I wanna get a spanning tree, I can just take a spanning tree and remove one edge. That'll be a spanning tree. So that will tell me that the degree of the F polynomial, which is the same as the degree of the B polynomial. The B polynomial times linear plus B is equal to the loop order plus one. Okay, so these are just homogeneous polynomials. So these are homogeneous polynomials. That may be the most important part. In this ring of polynomials corresponding to it. So a basic example. So this is the, I don't know what this is called. So this is called a bunch of different things in the literature. You can see this is called, I think we use sunset. I've seen sunrise, banana graph. I don't know, there's probably more. Banana is high, so this one is sunrise. If it's high loop, it's banana. Okay. All right, so in this case, you can write the expression out nice and very easily. So if I want a spanning tree in this graph, it's just any edge. So I have a vertex here, a vertex here, outgoing edges, outgoing edges, and edges connected. So a spanning tree is just any edge in this graph. And then you have the, so the complement is just the two edges that aren't in that spanning tree. So you get the product of those two variables. Okay, so that'll be my spanning two trees, product of all pairs of variables corresponding to edges, and a spanning, sorry, spanning tree. So a spanning two tree is just these two vertices. So in my formulation, a vertex is a tree. So the only spanning two tree is the two vertices. So if we compute this, we get this F polynomial. So this is, maybe it's good to observe that this is a nice cubic, homogeneous cubic. So if I look at the vanishing notes of this, of course, I get a family of elliptic curves living inside of, okay, so you should have been here. So Q is just my moment. Okay, so maybe another good remark to make here is that generically, I think for any, I think it's probably true that for any graph, we've ordered greater than one, except for this one, the vanishing notes of these polynomials, the F polynomial of the single. So this is important to note. I think it's something that took me a little while to really sort of realize that the sort of important thing to study here is the singularities of these graph polynomials. That's really sort of an important part. So in some sense, I chose a bad example to demonstrate this, but there are still bad things occurring in this example that we have to figure out in the test. Okay, so now, once I have this, I can express a Feynman integral, which, okay, there's a constant here that's missing, but I'm just gonna ignore that constant, some expression in terms of gamma function. So for me, this Feynman integral, it's just this integral here. So I have a rational differential form. I'm thinking about this as a differential form with, or a, yeah, a rational differential form on e to the e minus one, so this projective space. So I have my standard homogeneous differential, e minus one form on here. So this is a very standard thing that you think about any residues on hyperservices and projective spaces and things like this. And you have this cycle, well, not quite cycle, this is a relative cycle because it's just sort of the positive real values in this projective space. So this is not, this is not a closed cycle. This is something with boundary on the coordinate hyperclone. Okay, so this is an integral. This integral doesn't make sense for reasons that we'll see in a second. And maybe I'll explain this, but you can do formal manipulations or basically there's a way to get rid of the fact that this doesn't actually make sense because the denominator of this differential form can go up along this cycle. Okay, so I want to think about this as a period. I want to talk about hodge theory. I want to talk about mixed hodge structure. So first of all, I observed that this can be thought of as a, I'll mention a DRAM homology on the complement of whatever the boundary, whatever the denominator is, this is denominator. In fact, this is a good thing to notice that the denominator of this form depends on the dimension D. So the fact that I'm dividing by two here is why I choose even dimension D. Of course, the most important, I think dimension for physics is dimension four, but I mean, I'm fairly agnostic in terms of dimension. Okay, so I have even dimension. I have differential form. And depending on the dimension, the denominator can either be the vanishing locus of this F polynomial, or product of the F and U polynomials or the vanishing locus of the new polynomial. So for high dimension, I get new polynomial and low dimension, I get this F polynomial. So for me, what's going to be important is the vanishing locus of the F polynomial. And this is mostly, well, okay. So one of the reasons I do this is that if I think about the U polynomial, this depends on no parameters. So this is interesting in its own way, but I'd rather study something that varies. I mean, secondly is that the examples that I'm most interested in are graphs of what loop order two. So those are degree three hyper surfaces. If you look at the F polynomial, if you look at the U polynomial, those are just quadrants. So if you think about the poemology of a quadrant, it's very simple. I mean, there's a complete classification based on the rank of the quadrant. So in some sense, if I want to study the U polynomial or cases where the dimension is high for two loop graphs, I'm going to get a very simple answer, which is interesting in its own right, but I think I want to stick with the F polynomial. And so B here is, I guess I'm going to abuse notation a little bit, but this is just the torque boundary of projectives. Okay, so obviously when I write this out, the B is sort of extraneous. This is obviously an element in this gram poemology, but what I'm really interested in is integrating it over the sigma class. So this sigma class is almost an element in this relative poemology group. I would like to pair these two things and think about the intervals in terms of that pairing, but this doesn't work because if I take the intersection of the sigma class and the vanishing locus of gamma, I don't get zero for the interval below. There's a nice theorem of blocky node primer. So blocky node primer do this in the case of B equals four and primitive divergent graphs. And then Brown has later work that does it more generally that says that what you can do is you can basically take a quark wolf of projective space. So the underlying projective space here, I'm going to call that gamma and I can take a modification of sigma. So it's not quite the proper transform of sigma under this flow, but it's essentially the proper thing to do to transform sigma under this book. So that the Feynman integral becomes qualifier. So it becomes a relative period of this relative mixed-hodge structure, of this relative hodge. Okay, so the Feynman integral is a period of a mixed-hodge structure. This is really the point. Okay, so it's a period of the mixed-hodge structure, the canonical mixed-hodge structure on this relative poemology group after BOA. So I'm going to use the same notation for the boundary here before BOA and after BOA. But what I really mean is I take the preimage of the quark boundary after some quark BOA. And so maybe just an example of what exactly happens. In this case, to do this BOA, I need to blow up the points 1, 0, 0, 0, 1, 0, 0, 1. So the quark fixed points. So if I think about the vanishing notes of that, it'll contain all those quark fixed points. And after blowing that up, the integral becomes more than 1. Okay, so this is a setup. Everybody's doing okay, questions, comments, corrections. Okay, yeah, yes. Yes, right, exactly. Yeah, I don't know the physics part of it well enough, but this is true. So yeah, so that's what I'm going to focus on now. So from my perspective, what I want to do, so this is a relative mixed pod structure. But I mean, you can think about this, I mean, I'm not going to say exactly the words you're saying, but this is an iterated extension of mixed pod structure coming from, well, the intersection of this hyper surface. So this, well, if I take, what am I going to do? I'm going to take the intersection of this thing with the boundary, so the quark boundary. So if I take the intersection of that thing with the quark boundary, after doing the correct sort of manipulations, you see that the intersections look like, well, project this space minus a graph following a mill obtained by deleting some subgraph of gamma or things that sort of come from contracting subgraph of gamma. So this is something that Brown works out. So the point is basically that these, this is an iterated extension of mixed pod structures of this type. So if I want to understand this mixed pod structure and where it lives, it's enough to understand the mixed pod structure without the non-relative. That's totally enough. And so that's what I'm going to do. And of course, the obvious statement is that, if I want to understand the mixed pod structure but complement the hyper surface in productive states, that's up to mixed state factors or up to the state factors, just understanding the co-mology and the hyper surface itself. Okay. So I want to study the mixed pod structure on the co-mology groups of these hybrids. So, I mean, the whole philosophy here is, I mean, I could study the Feynman integral itself. I could think about that as a theory. But what I want to do is I want to think about these as periods of this mixed pod structure and understand that space of two. That's essentially the ideal. And secondly, as I mentioned a little bit before, I'm going to focus on the case where the polynomial z-diamond d is just the vanishing locus of the atmosphere. Okay. I'm going to break that. I'm not going to do that all the time, but for most of the time, I'll tell you when I don't. Okay. So what is note? I'm going to say stuff about the mathematical literature because I don't know the physics literature long enough for other people to explain. But I'm just going to say what appears in mathematics. So here's a bunch of exam. So for the case of the wheel, so this is called the wheel with end spoke. So this is studied by Blockino and Primer. They show that the mixed pod structure that you get is mixed tape. There's the Sunset family of graphs, which is also called the nanograph. So these are studied in, I guess lots of different works. I think Lot, Kerr and Van Ho have specific indicators where they study these things. I mean, other people have studied these things as well, but I think in the mathematical literature. So in this case, maybe a good thing to point out is that if you take the graph hyper-services of actually, these are degree N plus one hyper-services in the end. So they turn out to be clavial. So that's a very interesting fact, but those are the only clavial graph hyper-services vanishing modes I have at polynomials that you'll find. So you want to think about clavial directly. Those are your friends. There's also an example in the literature corresponding to the double box. So this was worked out by Block. So in this case, this is, so it's a degree, free hyper-surface. If I take the vanishing milk, it's the F polynomial. So what's gonna happen? So it's a degree hyper-surface. It's a five-fold. So it's a cubic five-fold. And Block shows that the fifth weight-graded piece of page five is more connected to the homology of the milk with the curve, one that I mentioned is the physical. Okay, so yeah, so it's sort of a hidden clavial in that case, right? And that's sort of the theme here. The theme is, except for the first case, we're looking for hidden clavial. So we see obvious clavials here and a hidden one. So maybe just to mention a bunch of other results. So a general result of Beltel and Brosnan, which studies the motives of the new polynomial, says that there's, in some sense, general. I'll mention also that Doran, not Doran, Doran, basically applied the same method as Block and Owen Primer to a family of graphs called the zig-zag, family of graphs. Get the nice results there. Similar results. I'll mention because in the audience, so Clometyle and lots of collaborators have studied the Sun-Tech graph and the Edgman graph. So, I mean, this is maybe breaking my rule about being in the mathematical literature, but it's closed. And I'll also break the other rule about being in the literature by mentioning that Matt has a bunch of unpublished results about various two-loop graphs. Okay, so soon to be, okay. So close enough, close enough. Okay, okay. Both rules are broken because of audience. Okay, right, so how do I study these? So this is my background portion of the talk. I'm gonna talk a little bit about some of the results that we have. So this is work with Chuck and Piravino. Okay, so here's like the basic motivating idea behind all of this. It's a very simple idea. I don't, I mean, I think, I don't know if I want to attribute this to me or even us. But, you know, I learned this from Pierre, so. So if I have a graph, a Feynman graph, and I have the quarter greater than one, and I have a chain of edges. So when I say chain of edges, I mean something like from the double box, I have an edge connected to another edge, connected to another edge, and these are basically biodeal invertecines connected. So if I ignore external edges, these are connected by biodeal invertecines. So if I have a chain of edges, then on the vanishing locus of the graph polynomial, I get a quadratic vibration over some projective space, and that's the projective space attached to the edges that are not in the chain. So I get a projection. So in this case, this'll be a, this'll correspond to a cubic bi-fold with a quadratic projection, production, sorry, a quadratic vibration onto P3. So you get like this nice geometric spectrum in Sancton. This is not like, not every cubic bi-fold has that structure. So this is sort of a special structure, which can also be seen as something that comes from the structure of the singularities of that. Okay, so this is just a general fact. You can sit down and prove it in probably about five minutes or so, but it has sort of nice consequences. So I mean, basically the consequence of this is that, well, here's this proposition in quotes because it's not quite precise, but this is essentially the idea. It's that if I have a quadratic vibration, and so when I say quadratic vibration, I mean a math whose generic fiber is a quadric. I don't need smooth quadric fibers. I don't need anything like this. I just want them to have a generic quadric as a generic fiber being quadric. And I want to think, I basically want to think about where these quadrics become more degenerate. Okay, so if my generic co-rank is R and I let D be the divisor along which the co-rank of the quadric's increases, then if the generic co-rank is greater than zero, the middle-dimensional co-amology is mixed up. So I don't have anything, well, interesting from my perspective at least there. If my goal is to hunt down cloud yows, I'm not going to find something new. So if the co-rank is zero, so that is I have generically smooth fibers and the relative dimension is odd, then the co-amology, the middle co-amology is basically determined by a double cover of D, the base. So that's in this case, yeah, and I guess I wrote this three different ways. So this should say PN here, ramified along the discriminant side. And if the co-rank is generically zero and the relative dimension is, I shouldn't say even, then the middle-dimensional co-amology is determined by the co-amology of the double cover of the discriminant visor D. So the point, I mean, we're getting all of this is basically that the place at which this discriminant sort of exists, this the place where the quadrics regenerate, this is going to determine a lot about the co-amology. So if I have a quadratic vibration, I just want to look at the discriminant mostly. This is a relatively easy thing to compute if you have a equation, so it's very nice. I should mention that, so I gave an expression for the finite interval before. And so, I mean, this is not, maybe the most general expression for a finite interval, like there's a lot of them in the literature, but the way that this is obtained is basically applying essentially, maybe a slightly different version, but a version of this trick to the most general sort of expression in the finite interval. This is the schwing. Okay, so here's a couple of examples. So based on these like little observations, essentially we can get a bunch of things. So we can first of all see that if I take a graph like this, I'm going to call this an A11 graph. Then this is going to admit a quadratic vibration over P1. So I've drawn here, I guess this is a 411 graph. So four edges on the top, then I have one, then I have one. So this admits a quadratic vibration over P1. So there's a bunch of different cases. So first of all, if I make the dimension big enough, so that is the space time dimension becomes big enough, then I can force all the fibers to become singular. They all degenerate, which means that if I want to understand that the middle dimensional co-amology of this thing, I just get something that's big. If my dimension is small, and I mean, I can make this precise if you want, but if I'll just say small and A is even, then again, the mixed hot structure is big, because this is a quadratic vibration. So maybe I'll say this in more, it'll come up here. If dimension's also small and A is odd, then what I get is a vibration, a quadratic vibration over P1, gonna have five singular fibers. One of these fibers is two copies of A minus something, meeting in a point. So that's a degenerate quadric, a very degenerate quadric. And then four of these fibers are just a nodal fund. So in this case, the co-amology of the hypersurface itself is essentially determined by the double cover of P1 ramified in those four points. So the four points corresponding to the nodal fund. And that just sort of falls on it. So up to maybe some, yeah, I don't do anything with that. No, no, yeah. There's no regular technique, so fixed dimension, right? So let me see if I have examples. Yeah, this is my one example. I think I have a lot of data. I'm here, so I'll continue with my example. So we have an ice cream cone with N scoops. So if I take this graph, I have a chain of edges here. So I get a conic vibration, so a vibration by one-dimensional projects over PN minus one. And you can compute this and the really interesting thing that comes out when you do this computation is that the discriminant locus is a union of two distinct sunsets, flabby out N minus one. There's a little bit extra that I'm not saying, but it's essentially just flabby out N minus one. So if I wanna think about the co-amology of this thing, the co-amology basically comes from those two flabby out N minus two, okay? Yeah, I should say N minus two, sorry, N minus two, four. So N is the number of them. Okay, so this is something that's also appeared in the literature and work of Dur, Flam, Negan, and Kredi. So they approach this, I think, from the perspective of periods more than geometry, but I think the idea is the same. You can go up in dimension, why not? I made up this name, I don't think anyone else is using this. So in our paper, I don't know if that's the general thing. We call it like the observatory, if it's just like no. Okay, yeah, okay, so anyway, whatever. So you get a two-dimensional quadratic vibration over PN minus one. You do the same computation, and it turns out to be the same. The discriminant locus is again, a union of two distinct sun type flabby out N minus one. And I got my N minus one folds. And in fact, in this case, okay, but in this case, it's a two-dimensional quadratic vibration. So you're getting something that comes from the double cover of PN, gram-fied in these two collabials. And what you can sort of see, if you do the computation, is that what you get is a collabial N fold entering into the column of the minus two fold. And so the reason for that is that if I took the sunset graph here, I would get a double cover coming from the chain of one edge. So anytime you have an edge, you just get a double cover onto a projected slope. So from that double cover, you can present the sunset collabial at the double cover of PN, gram-fied in union of two collabial N minus two fold. All right, so you get sort of this periodic iteration where you can just add any number of edges here. So I put two, I did one and two, but you can just add N here. And for odd N, you get a motivic collabial and for even N, you get two motivic collabial. So that's sort of a nice general easy application. And the last thing that I wanna say is I wanna talk about the cartograde graph. So if I take the graph that I have here, so I have two edges, two edges, two edges. So this gives me a family of cubic four-fold inside of Q5 and taking one of these chains of edges, I get a chronic vibration over P3. So the discriminant locus is up to little bit of extra. This is a nodal portic surface along with a hyperplane. So the motiv basically comes from a cakely surface. Okay, so in fact, in this case, this is maybe, that's probably not an interesting statement in this case because this is a cubic four-fold and the four-folds basically, if you look at the chronology, it looks like a cakely surface. So in some sense, this is generic. So what maybe it's the more interesting thing is if you take any other new graph with the same number of edges, so six edges, you don't get a cake. So this is probably the best case. In the other case, you got to do things that are worse. So we did some computations. You can see that this is generically a card rank 11. And you can do this more generally as well if you have an NNN hyper surface. So it's three chains of N edges along these things, you'll get sort of motivic clavial NN. So sort of fun applications of a little lemma. Now I want to spend maybe the rest of my time here talking about theorem, a maybe more general statement. And I want to focus on what I'm going to call a time improvement. I think this comes from that as well. So this is the case where my leap order is two. And then my graph is of the form gamma ABC, where A is the number of edges on the top, B is the number of edges here. So the degree of this F polynomial is three. So these are mixed and the degree of the U polynomial is two. So as I said before, in this case, it's not really very interesting to study the vanishing of the U polynomial. What's more interesting is the F polynomial. Okay, great. So this is the statement here. So what I want to understand is the vanishing of the C. So if you dig into the physics literature, there's lots of computations around these things. And lots of them sort of suggest that what you're gonna get out of the five minutes those attached to these graphs come from either elliptic curves or mixed statement type. So you have lots of elliptic functions, elliptic dialogs and stuff going around when you do this. Maybe a sensible conductor that came from block paper about the double box is that if you take a graph like this, at least when one of A, B or C is equal to one, you shouldn't basically come up with elliptic curves. That was intended to be expected. So since this isn't a mystery novel, the answer is to this question, like is this actually gonna happen? The answer is sort of, right? So if you restrict your face time dimension to four, you actually do. This is actually good. If you let your space time dimension get big, even in the double box tape, you get curves higher G. But you still get curves. That's maybe the interesting part. So I wanna explain the following result. So let's suppose that B is equal to zero or one. Why did I choose B to be fixed? I don't know, looks nicer. So I mean, this is basically, if you think about the graph theory, this is the same thing as saying like the whole graph, what's happened to edges is planar. No, no, yeah, yeah, right. Yeah, this is not like a well drawn, but if I add zero edges in the middle, this is a very planar graph. So this is the case I wanna think about. I don't know how like, in the cases that I'm considering no. Yeah. Okay, so the theorem is that if I take the vanishing locus, so the south polynomial, then this is contained inside of what I'm gonna call mhs view of height. So this is the extension closed subcategory of the category of mixtile structures generated by the chromology of the hyperlipid curves. So we have mixtape stuff in here and you have the chromology of hyperlipid curves and that's it. And their extension. The next part of the result is that if I replace A or C, if I assume that they're less than or equal to two, then I can replace hyperlipid with liptid. In fact, in the case where A or C is equal to one, we saw the easy proof of this already. That was the A11. And if the dimension is large enough so that this quadruped vibration that I'm gonna construct if that degenerates, then I just get mixtape. So maybe more generally, you see that this finding in motor based on the reductions we talked about between the motor and the clock. The finding in motor, so that is the relative commons of the group in which sort of the finding in flow rates. So that's contained in this category as well. So I mean, if you wanna think about this in sort of a down to earth way, this is just saying that if I compute the finding integral attached to one of these A1C graphs, then what I'm going to get is something that's built up from periods of hyperlipid curves and now the very, you know, the building up is the complicated process but the built up is, this is what we did. Okay, so I'll explain quickly a little bit about the truth of this. So the first proposition is that if I have a cubic hyper surface and I contain a co-dimension one in your subspace, then the claim that I made is true. That's a fairly obvious thing to do. Basically what you do is you take this linear co-dimension one in your subspace and take the pencil of hyperplane section passing through it. That'll give you a quadric vibration over P1 after blowing up along this linear subspace. You have a quadric vibration over P1 and you apply stuff that we had. So the observation to make, which is actually stronger than this, is that if you have a cubic hyper equation of the type A0C, so that is, so take that graph that I had and make a bubble out of it in the top and the bottom, then that vanishes along of linear subspace of co-dimension one. So that's great, we got that case easy. But you can't really do the same thing when you have A1C graphs in general. So they don't, well, I said they don't. They don't obviously vanish along with subspace of co-dimension. I wasn't able to find anything. But you can play a trick to sort of do essentially what you want with this. So you take the birational map, you construct this new cubic freefold or cubic arc, sorry, called XA1C and you build a birational map from the vanishing of that thing. And this is an isomorphism along two different open subsets, U and V, U and W. And this has the following properties. So this new thing here, this contains a co-dimension one linear subspace. And the things that are removed are complements of hyper plane section. And these are themselves qubits which contain co-dimension one linear subspace. So I take out, I put back things. I basically build this vanishing levels from qubits containing co-dimension one linear subspace. And so by standard arguments relating to co-mology, U and W to X and V of F gamma, we can patch together the co-mology of the vanishing levels of F to get what we want. All right, so I mean, this is just fairly straightforward stuff. Obvious stuff we can find in my theory. Okay, so that's basically the proof. It's not a particularly hard thing to do, but I mean, this step here required a little pep in it. There's a lot of other ways one can go about trying to use, but you can't sort of obviously, you're like stupidly apply this quadric vibration number that I had with it. That won't get you there in any way. So we had to take sort of this more circuitry. Okay, I think I have about five more minutes. Yeah. So I want to say a little bit about another aspect of the motivation for studying, which I think is maybe related to the, I mean, this was Pierre's sort of motivation, I think in ours, because of that. So Pierre and Pierre Loret, Loret is sorry, were studying Feynman intervals or they were studying periods to attach the pencils of graph hyper-surfaces of this one. So you just stick a T in front of the polynomial and you have this family of differential forms and you can try to compute the Ducard-Tukes equation of that term. Okay, so Pierre Loret has, he's got this really nice algorithm that allows one to compute Ducard-Tukes operators attached to singular half-surfaces. So they did that. They computed a bunch of things. And once, I mean, so the reason for doing this is basically that if you can find a differential operator that this form satisfies, then you know that the Feynman interval up to some small lie satisfies an inhomogeneous differential equation of this one. So this thing that annihilates the differential form here, this is going to not annihilate the Feynman interval, it's going to be the homogeneous part of an inhomogeneous differential equation. Oh, right, so, okay. So this was not done in general. This was done for a specific amount. Yes. Okay. And so what they observed is that in these cases, they found the card-to-to-operators that came from families of lifted curves. And they found, I mean, these operators, they can factor them. They break up into different pieces. And so the pieces that they found came from the lifted curves, so hyper geometric stuff. And there's stuff that came from a million ODEs. So this means that the differential Galois group is solvable. And I take this sort of as saying roughly that the monitoring group, well, I mean, okay, that's the same thing as saying that the monitoring group is solvable. It says something about the monitoring group being very simple. So they did this because these are the things that one can access through computational methods. Like this, the paper that they wrote is very computational. So these are the things that one can check in like Navel. And so thinking about this more generally in terms of mixed hot structures, the solution sheet of this differential equation that they get, this is a sub-portion of the variation of mixed hot structure attached to this family of hyper-zone. And so if I have a weight filtration on this thing, that corresponds to a factorization on this ODEs. And the weight-rated pieces are basically local systems corresponding to the factors. So the theorem that we get as a consequence of the bigger theorem that we talked about earlier is that the differential equations that Pierre and Pierre got, they must factor as things of the following type. So if I have an A1C graph, then the annihilating differential operator factors with finite monodrome. So it factors into pieces with finite monodrome. So that's essentially not exactly, but roughly saying new billion. And it factors into pieces which are the card-to-box operators of hyper-eliptic curves. Or, I mean, I should be careful and say, does it necessarily need to be a hyper-eliptic curve? I mean, it could be a factor of the OD annihilating periods of hyper-eliptic curves. But in the elliptic curve case, it is exactly the same. Oh, yeah, we don't know if it's a homework or if it's a French. Like, I'm not, yeah, yeah. Yeah, if this is your bar for collabianus. No, no, so yeah, yeah, yeah. No, no, yeah, that's one of the points here. I don't know. We don't know anything about this differential. It's just there is something and we have an annihilating OD. Okay, so I have about zero minutes left. So, do you mind if I take another piece? Okay. This is my last one. Okay, so this is a comment that I made earlier, that if these people before, then we only get elliptic curves for all A, B, A and C and B equal. The second comment that I wanna make is something related to this paper of blocks that I mentioned earlier. So, if I let my dimension be six and I look at the double box example, if I take B less than or equal to two, then I only get rational curves. That makes the cubic degenerate enough that the hard structure degenerates. If I take D equals four, then I get a family of elliptic curves. But if I take D to be six, then I get a family of D to be six. So, in this case, if I let D be six, in the first case where I actually happen, I do get genus two curves appearing. So, this is sort of a counter example of the expectation that we get elliptic curves or elliptic curves in general. And the last thing I wanna mention is that this theorem that I've proved earlier, essentially for arbitrary A, B and C, I think the same truth will apply and basically we'll get a bound on the hard structure both way through there. So, for instance, if the minimum of A, B and C is equal to two, then the hard structure on this hyper surface should basically come from surfaces or surfaces and curves. If I let the minimum be three, I should get three fold surfaces and curves, et cetera, et cetera. So, the lowest of A, B and C should give the bound on complexity for the motor. Okay, so that's all I have to say. No. Okay, questions for Andrew? I think, there, yeah. So, do you know the nature of this inovogeneity in some generality? No, no, no. I mean, there's obviously these death terms for a pond where it goes on the boundary. That's right. But we haven't done that in computation. In the physics literature, that's quite interesting. This is basically corresponds to the subgraphs. Yes, exactly. And so you can actually, in most cases, Yeah. Yeah. All right. Yes, yeah. And so to say for the, in this case, the sunset graph is super special because if you look at the subgraph, it's all tadpoles. Right. And therefore there's no momentum integral anymore. That's right. That's why they are basically constant. But in the other cases, it's like it should be very complicated, Yeah. I mean, so more generally, I think what you'd probably want to do is just get a bigger operator but annihilate the entire. Right. Yeah, exactly. Questions? Any questions online? Not? Let's thank Andrew again.