 Okay so yesterday ended by stating this set of additive generators for the tautological brain. So I'll state that again and say a bit more about it. I think the set of generators was sort of common knowledge at the time but Graeber and Pandarapanda were the ones who first actually wrote down the details proving this is proving this theorem. So theorem is that r star mg n bar tautological brain is additively generated by classes iota gamma star psi kappa monomial. So remember that iota gamma is the gluing map associated to a dual graph gamma. So it's like a like the basic gluing maps originally were done by gluing over multiple pairs of points at the same time. And I'm pushing forward along that map some monomial and the psi and kappa classes. So example of what how do we think about such a class? Well we first draw the graph gamma so it has some vertices connected in some way maybe we have a loop. This graph gamma remember has vertices has edges also has legs half edges corresponding to the n marked points. Example one there, two there, three there. And finally each vertex we have to remember what genus that vertex is. This is g equals one, this is g equals zero as well. And the way to interpret this picture is each vertex is a smaller mg i n i bar. The genus is annotated directly on the graph and the number of marked points is the valence of the vertex, the number of half edges out of it counting legs. So the corresponding iota gamma here, example, it should be from a product of three mg i n i bars. Corresponding to three vertices we have like an m zero three bar from the top vertex. We have an m one two bar from the left vertex. And we have m zero six bar from the right vertex. And it's a gluing map where I have to glue together many pairs of points here. Happening here is that these three marked points correspond to these three half edges out of this vertex. And for each edge, edge has two half edges and you glue together the corresponding points. In this case you're going to end up with only three marked points left at the end which haven't been paired up. And the genus, you have to think about what the arithmetic genus is of a resulting curve. And it's going to be zero plus zero plus one plus you have two additional cycles. So the sense of being a map to m three three bar. Do you ask about ordering the half edges? Yeah, implicitly one, implicitly I'm doing that. I mean, that's just a matter of automorphisms of the, of the, of this product of m is given by permuting some of the indices here. So for instance, the image isn't going to matter. But yeah, there are some, if you want to be very careful about precisely what the map is and you have to say of these six points here, one, two, three, four, five, six, label one, two, three, four, five, six in some order there. So, okay. So then what, one of these classes are, these classes are sometimes called, say these generators are called basic classes, basic total logical classes. To get a basic class here, I also need to choose a psi cap of monomial. And by that I mean on m zero three bar, you have some size and capas and one two bar, you have some size and capas. So on on the product, you can take size and capas from any of these, you take some monomial and all of them and then push forward that monomial. So on the graph gamma, the, the psi classes and these factors here, again correspond to half edges in this graph. If I took, let's say, an m one two bar here, I have classes psi h psi, say psi h one psi h two is what I'll call them, really psi one psi two. But I'm calling psi h one psi h two where those are names for these half edges here. Then I have all these capa classes, which if I wanted to give them names, all my capa one at the vertex v. So again, this is a graph. It has vertices, half edges, edges, and so on. Giving some of them names, this vertex called v at the h one, that's h two. And then giving names to the capa classes and psi classes at one m one two bar. Bracket v indicating that it's a capa class in this factor of m rather than one of the other two factors and so on. So these, this, this I've made here, these are the psi and capa classes in the tautological ring of m one two bar where I've just renamed them using the graph terminology. And then what, what we could end up doing is take, say, iota gamma star of, I don't know, psi, psi h one times capa one at v. And just not, not put any other decorations at the other things. And then we would interpret this class as this graph here with additional psi and capa annotations. If we put a single psi along this half edge and put a single capa one at that vertex. So basic classes, so these generators correspond to table graphs, vertices, and half edges are decorated by psi, by capa and psi classes. Any questions about this? So these are sort of, these are the basic tautological classes. And for the course of this week I'll be talking about a number of formulas which are basically expressing tautological classes in terms of these basic classes, these additive generators. The idea is that if we, as I'll say in a second, we, we have a pretty good understanding of how to manipulate these basic classes, take push forwards, pull backs, multiply them together and so on. So we can really effectively do computations in the, in, inside the tautological ring once we've expressed our classes in terms of these basic classes. Any questions here? One thing I should say, by the way, is what's the tautological degree of this class? This should be in R something, the Rd of mgn bar. What is d here? Well the tautological degree certainly gets some degree from the psi capa monomial, the degree of that monomial, where again psi classes are degree one, capa, capa i is a degree i. The d is going to be the degree of the monomial, but then you actually, you have to add something for this push forward. I'm curious how the push forward changes the tautological degree, you have to think about what the dimensions are of the two sides here, but the end result that you get, you can check this if you want, is that you have to add the number of edges in the graph gamma. Edges meaning full edges. So for instance this class I wrote down here, I have four edges in this graph, I have degree two thing here, this will be in R6 of m33 bar. All right, so I explained yesterday why these classes are in the tautological ring. In the tautological ring because tautological ring is the smallest ring that is closed under push forwards by these gluing maps, and the forgetful maps, and slides and capas were defined using those operations, this generalized gluing map is a composition of gluing maps. So if you wanted, you could take this out of the gamma here and factor it as a composition of four gluing maps, one for each of these four edges, for the basic gluing maps that just glue a single pair of points together. Here we're gluing four pairs of points together. Okay, from the theorem, I mean in order to prove this theorem, Gerber and Ponderpunder had to show that the additive span of these basic classes is closed under multiplication and under push forward by the basic maps, the gluing and forgetful maps. There's actually a bit more that ends up being true. I'll state it as another theorem, is the thing to say there exists universal combinatorial formulas expressing the, they're take push forward, gluing or forgetful, so you can take the pullback by the same maps or you can take the product, take any of these operations, apply these operations to some number of basic classes. The result can be expressed by these universal formulas as linear combinations of basic classes. You give me a basic class and tell me the, give me a forgetful map or a gluing map, I can push forward or pull back that class along that map and write it as some of some other basic classes. We've actually already seen examples of that. For instance, we had the push forward by forgetful map of psi to the power i plus one. I remember that's how kappa was defined in terms of psi and that's an example of this formula for push forward, but there also exist formulas for pushing forward a kappa class by forgetting about some point or pulling back such a class and you, you remain inside the linear span of these basic classes. I'm not going to rate out these formulas because there are a lot of different cases to think about, but it's a, it's a fun exercise if you haven't done it before to think about the intersection theory of the situation a bit. You have to do things like use the, use the projection formula and try to compute some of these like the pullback of a psi class by a forgetful map. The, the most complicated of these formulas by far is take the product of two classes. You might wonder what that would look like since these basic classes, they depend on graphs. It's sort of clear that like if you glue together two basic classes by, if you take the push forward by a gluing map of two basic classes, that should more or less correspond to gluing together the associated graphs with their sine kappa decoration. That's basically all that happened. So like push forward by gluing map is very easy formula, but product is more interesting. I'm not going to write out the full formula, but the idea is that if I take iota gamma one star of some monomial alpha one, I want to compute the product of that with iota gamma two star of alpha two. What does the product formula look like for these basic classes? And it's going to look like a big sum over graphs gamma three, but I need to, some conditions on this new graph gamma three. Basically, what's going on is that gamma three is going to simultaneously refine both graphs gamma one, gamma two. So formalize this, I want a partition of the edges of gamma three into three disjoint sets, e1, e2, e3. And then I want isomorphisms between, for instance, gamma three with the edges in the set e2 contracted. I'm going to use this notation for contraction. That should be isomorphic to gamma one. Similarly, gamma three mod contracting the edges in e1 should be isomorphic to gamma two. So you pick all this data, graph gamma three, stable graph gamma three, and pick basically two disjoint subsets of the edges such that if you contract some of the edges, you get one of the graphs, contract some of the other edges, you get the other graph. And then the thing that you put in here is then basic class iota gamma three star, and then I have to tell you what to insert here. And it will essentially be alpha one times alpha two times some self-intersection thing coming from these edges e3. Since edges e3 are sort of the overlap between two graphs. This formula is combinatorially quite cumbersome. I mean, this is a, you start with two graphs, then you can have a lot of different output graphs. So although there exists these universal formulas, they are hard to use directly. In general, multiplying topological classes, we can do it, but it's, it's combinatorially not very nice. And the reason for that is that the new graphs you end up with are sort of combining the two previous graphs in different ways. All right. There's some other details here that you might want to think about. Like what, when I say contract an edge, what should I do with the genus of the corresponding vertices? Well, if I merge two vertices together, I should add up, I should add their, their genera in the new graph. If I contract a loop, then I should add one to the corresponding genus. Things like that in this formula. Again, I, I, I don't want to get bogged down the details. The important thing is that there exists these formulas. And the theorem is, part of the theorem is basically how they proved, is basically the proof of that first theorem. Part of it also has a consequence that topological rings closed and their pullbacks as well as push forwards. They're, it's nicer to define the topological ring just using push forwards, but the corollary is r star mg n bar. Take the set of these for all g and n are closed. They're pullbacks, gluing or forgetful maps. Okay. That's, that's what I wanted to finish saying about this, this theorem, the set of generators. Again, we, we think of these generators as graphs decorated by Psy and Kappa classes. What did you say? So it's a corollary because this theorem is saying that, I'm saying that there exists some universal formulas, but in particular it's saying that the pullback of a basic class is a linear combination of basic classes. So if the ring is generated by basic classes, then it's closed under pulling back. Okay. So, so I want to, in a minute, I'm going to shift gears a bit and move from mg n bar to mg, the classical case first considered by Mumford. But before I do that, I just want to give an example of what a formula in terms of these basic classes might look like. This is a relatively simple formula for the churned characters of the Hodge bundle. So first, what's the Hodge bundle? E g, the Hodge bundle is a rank g vector bundle over mg bar. Pull it back to mg n bar if you want. And I'll just say what the fibers look like. So the fiber over a stable curve should be just sections of the dualizing shape. This is a very natural object that shows up a lot when doing, for instance, localization computations involving much less space of curves. And one of the basic questions you can ask is, okay, teleological ring is supposed to contain geometrically natural classes. It should contain the churned classes of the Hodge bundle then. And so Mumford computed this standard Grundy Griemann rock computation, the churned character of e g, in terms of what I called these basic classes. Grun character should start with the g because it's a rank g bundle. Sum over l greater than or equal to one. I have some coefficient here, which is renewably number b2l divided by 2l factorial, multiplied by, okay, so first we could just have a kappa class. Here's now maybe some place where I should say that, okay, my basic classes all had iota gamma lower star. But if you take a special case of gamma with no edges, just a vertex and n legs off of that. Gamma is a vertex, penis g, legs 1, 2, 3 through n. Then, well, what's iota get, so this is the dual graph of a smooth curve. Then, iota gamma is just the identity map from mgn bar to itself. That's the sense in which kappa 2l minus 1 is a basic class with dual graph, precisely this, and an annotation of a kappa on the single vertex. Okay, but then there are going to be more terms, the remaining terms, going to write as a sum over gamma, a stable graph, really, I mean a stable graph for mgn bar, exactly one edge. These graphs correspond to boundary divisors in modular space curves. And what I want to take then, take one over the cardinality of the automorphism group of gamma. Automorphism, think about it a bit, it's either one or two. If you have a loop or if you break the curve into two pieces of equal general, no marked points on either side, and one otherwise. Factor, and then I take iota gamma, lower star. So, in this case, there are two cases for what does gamma look like. It either looks like you have a single edge like this with general g1, g2, or you have a loop, g minus 1 there. These are the two possibilities for gamma. In either case, you have two psi classes. Again, psi classes correspond to half edges, you have exactly two half edges here. They're possible locations. So, in terms of psi classes, I have two psi classes to work with, and I'm going to take psi to the power 2l minus 1 plus psi prime to the power 2l minus 1 divided by psi plus psi prime. That's the formula here. So, this is a relatively simple formula as formulas in MGN bar go. So, if you look at what's happening here, the only graphs that show up are you have the graph with no edges and all the graphs with one edge. The graphs appear precisely the ones with less than or equal to one edge, and then you're taking some very specific capo or psi classes and pushing forward. So, what is the churn character of this hodge bundle? Now, it's a churn character rather than like the total churn class. If I wanted to get the individual churn classes of the hodge bundle, usually called the lambda classes, then I would need to take this formula and basically take certain polynomials in the individual terms here. And I mean, one can do this. The result will be total logical, but it will be much more complicated to write down the actual total churn class of the hodge bundle using this formula because I'll need to use this product formula to multiply together a bunch of these. So, if I tried to write down a formula for the total churn class rather than churn character of EG, I would need to use basically arbitrary basic classes because in this product, everything is going to show up. Okay. So, this is just an example of what formulas in topological ring in terms of basic classes tend to look like. You have some, some of our graphs, in this case, some of our very specific graphs with at most one edge versus size and cap as in. Okay. I'm going to shift now to MG. So, as I said at the end, yesterday, we can define total logical ring of any intermediate modular space M contained in MGN bar, just as the restriction restricted to M or in MGN bar. I mean, this definition might not make sense for some, I'm saying this for like arbitrary sub varieties, but sub stacks, but in the case of MG, this coincides with the classical definition. So, what does this mean for our star of MG? So, the point is we still have the same additive basis. We just have to restrict things. Then is additively generated. I'll just write it down. We take Iota gamma lower star of some cap of psi monomial, direct to MG from MG bar. But the point is this is going to be zero unless gamma is the dual graph douce moves curve. If gamma is any edges, this is just zero because by definition, this push forward is supported on the singular curves. So, when we restrict MG, it just vanishes. Yeah. So, see that here. But these classes vanish if number of edges of gamma is greater than zero. But that's precisely what restricting to smooth curves does. So, the result is that our star of MG is, I should also note that we don't have any marked points here. So, we just have cap of classes. So, our star of MG is just polynomials in cap of one, cap of two, and so on. If we were taking MGN, smooth curves, genus G, with un-distinct marked points, tall logical ring there would be polynomials in the cap of classes along with n psi classes. So, really we're throwing away most of the complication. The complication of MGN bar comes from these boundary strata, these complicated dual graphs. I mean, at the moment, I haven't said, but I mean, a priori, you can say that cap of 3G minus 2 has to vanish because you're going to mention 3G minus 3. They actually vanish starting at cap of G minus 1. You don't need them after about cap of G over 3. At the moment, just using the general theorem about these additive generators for the tall logical ring of MGN bar and restricting that, we get that it's polynomials in cap of one, cap of two, and so on. Say that, take polynomials, cap of one, cap of two, one variable in each degree, formal polynomials there that surjects onto our star of MG. The much simpler case here, we're just interested in some ring generated by one element in each degree, subject to some relations. This is actually how Mumford originally defined the ring, just as the subring of the Chao ring generated by the cap of classes. One can say, it's somewhat arbitrary. Why are we choosing the cap of classes rather than, say, the churn classes of the Hodge bundle, the lambda classes? And by going about things in this way starting with MGN bar, I hope you get the idea that although the gluing maps and the, the gluing maps and the forgetful maps are not visible in MG, you can't, they don't exist in MG itself. You don't have marks points to forget, you don't have, you aren't allowed to glue curves together because things have to be smooth. But the reason why you're using just the cap of class in the tall logical ring can really be traced to these maps on MGN bar. All right. So studying our star of MG, one way of thinking about it is it's a matter of understanding polynomial relations between the cap of classes. We know the relations and it's just the tall logical ring structure of MG is just the quotient of this polynomial ring by the ideal for relations. Part of Mumford's motivation when originally defining the tall logical ring of MG is something, a conjecture known as Mumford's conjecture later proven by Madsen Weiss, which says that these capas in some stable limit of the co-emology, they freely generate the stable co-emology. So to state that precisely, Mumford's conjecture, Madsen Weiss, it states that, state it precisely again for any D greater than zero, the map that I wrote there from this formal polynomial ring in the capas, co-emology of MG. Not just the tall logical ring, but to the co-emology, this map is an isomorphism, degree D for G sufficiently large. If I was a topologist, I would work with MG1 rather than MG. I would talk about this actually being a stable limit in the appropriate topological sense, but if I'm just wearing with MG, this is the only real way I can cleanly state the conjecture. So those are the same two things. It's saying both that if you let the genus be much larger than the co-emological degree, then all your classes are tautological in that degree. They're all polynomials in the capa classes. Also says that for G large compared with D, there are no relations between the capa classes. Informal thing that you can say is limit G goes to infinity, star of MG equals polynomials in the capa classes. This was the original motivation for the capa classes, one for each injecturing. Although the tall logical ring for specific G is not the focal homology by any stretch, it's not even close. I forget exactly what they proved. It's either something like G equals, probably they proved about G equals 3D or something like that, something linear in D. It depends on whether or not you're using the real degree or the complex degree. If I'm using complex degree, I think it's G equals 3D or 3D minus one or something like that. Yeah, so there aren't maps between these rings, but if you added in a marked point, it worked with MG1 and there would be maps between them. This has actually proven a slightly stronger statement for MG1 is what is proven by the topologist. This is an interesting theorem because the only known proof is tautological. Time remaining, I want to briefly state the next major advance in understanding structure of MG, which is Faber's conjectures. The tautological ring of MG, it's certainly simpler than the tautological ring of MG and BAR. I mean, it's sort of a restriction of it. There's much less going on MG than MG and BAR. We also have less to work with, so somehow the structure of MG is a bit more mysterious what's going on because you just have these classes, you don't have good maps between different MGs, but it is just a matter, the first setting is a matter of understanding what the kernel is of this map. One is a polynomial and the capocloss is zero in R star of MG. Faber's conjectures, these are from late 90s, I guess. Tautological ring was defined by Mumford in the 80s. Conjectures give a full description of the ring. They'll answer some of the questions before like when did the capocloss vanish? So there can be three parts, four parts, depending on how you count it, there are multiple parts of this conjecture. The first part is that Rd of MG equals zero for d greater than g minus two. Vanishing at some point, that's why I know that cap of g minus one is zero. Faber also showed that cap of g minus two is non-zero. Oh no, by the way, this is much lower than it has to vanish by dimension, but because we have some open sub-variety of MGN bar, you should expect that the comology will vanish earlier. It's not the comology, this is tautological ring, so it might vanish even sooner than the comology. But g minus two is a lot lower than 3j minus 3. The second part here, Rg minus two of MG, be one dimensional. Remember, I'm taking rational coefficients everywhere. So really, there's like a 2a and a 2b here. 2b is have this isomorphism, but moreover, there's an explicit combinatorial formula for that isomorphism. Maybe you'll state that formula in a bit depending on how much time we have. But actually, given any polynomial in the cap of classes of appropriate degree, and see what its image should be under some normalization of this map, it's, of course, only defined up to a constant. Part three of the conjecture, maybe I should move up here. There's the heart of the matter, which is that Rd of MG cross Rg minus two minus d of MG, one tautological class of degree d, one tautological class of degree g minus two minus d, two polynomials in the cap of classes, and multiply them together, get something Rg minus two of MG. By the proceeding part of the conjecture is one dimensional. This is some bilinear pairing of q vector spaces. This is a perfect pairing, q vector spaces. Okay. So those are the three parts of the conjectures. And to think a bit about why do these conjectures fully determine the structure of the ring? So first two parts clearly fully determine the structure of the ring in high degree. Intermediate degree, what's going on is just that if you have a polynomial in the cap of classes, and you want to know whether or not it vanishes, well, you can compute its pairing against all cap of polynomials of complementary degree. If that pairing is always zero, then because this should be a perfect pairing, that means you should have started with a relation that cap of polynomials should vanish. If you ever pair with something non-zero, then you know that it had to be non-zero. So in other words, assuming these conjectures have a straightforward algorithm for computing whether or not a cap of polynomial is zero, you just compute its pairing with everything else, see if its pairing is ever zero. And this pairing is fully explicit because we can multiply polynomials in the cap of classes, and then the isomorphism to the one-dimensional space is explicit by this formula I haven't told you. Fully determine the structure R star of mg. They're very appealing statements. I mean, you have this, maybe you could write it that way, but the, I mean, the usual way of writing it is as some, some of product of like double factorials and such. Maybe, yeah, I have enough time, but I'll write that down explicitly. So this formula is easiest to state. Over here, this is now a natural statement of 2b. So, for any, so first I need some notation. For any cap of monomial, capa a1, capa a2, capa am, I want to let curly braces over that monomial. There's going to be some polynomial in the cap of classes. Really, this is a change in basis for the cap of polynomials between the standard basis of monomials and some other basis. So here I want to take a sum over permutations in the symmetric group on M elements. For each such permutation, I want to take product over cycles of sigma, cycle decomposition, and then for each cycle, I write down kac, where capa ac is equal to, I'll write it as, ac is equal to the sum over i in the cycle of a sub i. This is a lot of words, but it's just saying that I sum over all the permutations, the each permutation is a way of grouping together terms and combine the indices in those, those capa classes. So for instance, sorry, what was that? If sigma is identity, then you get the original product. An example, we have like capa a1, curly braces equal to capa a1. You have two capas, then you have two terms. One of them is the thing you started with, the other is capa a1 plus a2, and the next one will have six terms, although some of them will be equal and so on, which I guess two of them will be equal. So the reason why I want this notation is that then the map in 2b, 2b is that if I take one of these capa polynomials, you can track that this gives a different basis for capa polynomials, signifying things linearly just using the spaces. Capa a1, capa am, you could send this, sending this to an element of q. Here I have it right, 2g minus 3 plus m factorial, 2g minus 1, double factorial, double factorial, of course, meaning you multiply together all the odd numbers, less, odd positive integers less than or equal to 2g minus 1, divide by minus 1 factorial, product i equals 1 to m to ai plus 1, double factorial. It's some combinatorial expression here, and I mean, if you want, you can invert this change of basis and write this as a function of polynomials as sum over, sum over like set partitions of things that look like this. But it's fully explicit, 2g minus 3 plus m factorial, plus m, m being the number of capas. I mean, I'm always assuming that my modular spaces are in the stable range of 2g minus 2 plus n, greater than 0, which means that g should be greater than or equal to 2. For g equals 2, of course, this is not saying that the saying that the topological ring is not very interesting because all the capa classes vanish equals 2. Okay, so that's the actual statement of Fauper's conjectures here, and now with this explicit formula, you can actually, completely effectively, assuming this conjecture, determine whether or not any capa polynomial you want to track is a relation. So what's the status of this conjecture? So the first part is true. Parts are also true. So I think that I should be attributing these to like Lujanga and Fauper, and then this third part, this explicit conjectural formula, was actually first proven by Getzer and Pantara-Ponda by showing that this formula is equivalent to, or mostly equivalent to the verisoral constraints on the grovelin theory of P2, just proven by Given-Tall. So there's a fair amount of work that went into proving, especially 2B, but 1 and 2A also. But those were established fairly soon. 3 is still open, and I think it's sort of the heart of the matter. I mean, it's good to know that Sol-Logical-Ringfem-G, it becomes one-dimensional at some point, above that it vanishes. We have this nice sort of integration formula in that one-dimensional space. But part 3 is what's determining most of the interesting structure of the ring. So what's known about 3? Part 3, the only part still opens. It's usually so many people just say Fauper's conjecture for the third part now. It's also called Gorenstein conjecture. It's one way of interpreting this as it's saying that Sol-Logical-Ringfem-Gorenstein-Ring, basically it satisfies this Poincare duality type statement. In this conjecture, it's saying that the tautological ring of MG looks like the comology of a compact manifold of dimension G minus 2. Unfortunately, we have no idea what that manifold should be. So although clearly the form of this makes it look like there should be some compact manifold working somewhere that this is the actual comology of, there isn't actually any motivation to believe the conjecture based on that since we have no clue what that manifold could be. But this third part of this one criteria, it's Gorenstein conjecture and it is Fauper's conjecture for genus less than 24. Conjecture is open for all G-graders and are equal to 24. So I should say that this is a very attractive theorem. For a while I think everybody sort of believed it was true for genus up through 23. I mean originally checked it like it was 13 and 19 and up to 23. But I think at the moment, those people including Fauper himself think this is more likely to be false than true. And tomorrow I'll get into some of the specific reasons why this conjecture is maybe less believable than it once was, although I mean still open. I should say some words about what's going on with it being checked up through genus 23 but not further. This isn't a matter of Fauper's computer isn't powerful enough to check genus 24. I mean he got to 23 quite a while ago now. Basically the way Fauper's checking this, he had a method, very classical method involving maps between factor bundles of constructing lots and lots of relations between the Kappa classes. Just had a machine which produced sort of somewhat random looking relations between the Kappa classes. And the way this conjecture works is that if you prove that all the relations that predicts are true are true then you're done. Since this conjecture, one way of interpreting this perfect pairing is saying that the maximum possible number of relations are true. Because you can't have any more relations because if something pairs with something else, something non-zero then it can't be a relation. So this is really Fauper's conjecture saying the maximum possible number of relations are true. So I mean if you have some way of proving all those relations are true then you're done. You know the conjecture is true. You know the tall logical rings exactly as it's described. His method of producing relations for genus up through 23, it sort of immediately produced all the relations. Like some specific location, there's 20 relations. He ran his program to produce sort of 20 relations. They would be linearly independent and then he would get that the conjecture is true. For genus 24 certain missing started to miss relations. So the way this works is that if you prove lots of relations between the Kappa classes but you're missing one and you don't know whether or not it's true or not. It's very hard to give a polynomial in the Kappa classes determine is it zero or not unless you have something like this conjecture already. And we mainly just have machines for producing lots of relations, not machines for proving something isn't a relation. So the point of view of Fauper's computer calculation to happen to G24 is that suddenly his method of producing relations isn't good enough to verify the conjecture. It starts getting more and more off as G increases. Yeah, so it is possible to write down an explicit polynomial in the Kappa classes in R12 of M24 such that if this specific polynomial in the Kappa classes is zero then the conjecture is true in genus 24. If it's non-zero then the conjecture is false. 24 is exactly one missing relation R12 of M24. All the other degrees are fine but I believe that the Delta Fauper's computer searches Fauper's computations and goods that the dimension of R10 of M24 equal to 36. 36 is less than or equal to the dimension of R12 of M24 is less than or equal to 37. And I should say that this one missing relation of course it's only able to find up to adding in the other relations that exist. So you shouldn't expect that if you write it down it will necessarily look nice. It will just look like some random rational linear combination of all the partitions of 12 many monomials in the Kappa classes not to gray. Maybe there's some really nice element in that coset of possibilities but we don't we don't know if there is one. That's the status of Fauper's conjectures. I should say for G equals 25 we're again missing exactly one relation and then for G equals 26 we're missing I think two and start missing more and more relations always near the middle of the brain. All right. I should stop here. The plan next time will be to talk about some of the more recent evidence for disbelieving this third conjecture. Nice though it does.