 I hope the technical problems are solved, so perhaps you can confirm to me that the microphone is indeed working, because it goes down during the lectures, so at some point it will disappear like it did yesterday. But anyway, yesterday I was a little bit late, so it was stopped when I was discussing some results, which appeared in the paper of Gabriel and Panna di Panna, and I think the important thing is that I described to you how to obtain linear generators for the tautology. This works in Chao, in tautology, they are given by decorated starter classes. So the idea is that for every time we have a graph, we have a gluing map which goes from some product of modular spaces to MGM, so in each one of them, on each one of them we can pick some product of psi and kappa classes, and if you want also lambda classes, so one can take simply one if I want to take the fundamental class of something, and then one takes the image under the push forward of this. So they arise at the image of, let's say, tens of products, NOMIAS, kappa, lambda and psi classes under the push forward. It doesn't matter whether, in general, when we pass from the Chao groups to the tautology, we lose a lot of information, but somehow in this case it does not matter, we can take the generators in what cases, we can take just the same, that's why the map, that's one of the reasons why the map is well-behaved. And then I wanted to reassure, so this is the gluing map associated to the gamma. And I don't think it's a, I expect this has been known before, but anyway, yes, yes, first in explicit form in the same paper where I've been about the repandent, which is devoted to the construction of non-tautological classes. So if you remember, we defined the tautological links by saying, well, it has, they have to form a system of links for our GNN that is closed under push forward by the standard forgetful and gluing maps that once associated to devices. So you may worry about what happens if you take the pull back instead. You get something that starts in the coromology of M bar GN and takes an image which is, which is a tensile product of coromology classes on each one of the factors. Well, actually if we take the pull back under the gluing map and we start with the tautological class, then we get an image here, which is a tensile product of tautological classes. So the system is automatically also closed under pull back. We can fix an arbitrary graph and take the associated gluing map where then it's a tensile product of tautological classes on the factor. Of course, you say tensile product of tautological classes. So actually the reason why the ribbon family panda were interested in this property is that it's a way to exclude, so to prove that there exist classes, which are algebraic in the sense of the image of some algebraic cycle, but which are non-tautological. If I simply say there exists subvarieties, substacks. Now perhaps I should have written the course, not the rest places, I wrote variety, but that's not exactly the any specific issue whose fundamental class is non-tautological. In particular, if we take the tautological meaning in coromology for M bar, this is of course contained in the even degree coromology of M bar, but in general they are different. Well, why is this interesting? Well, the point is that if the genus is sufficiently small, one can actually prove that these two things are the same. So if the genus is different one, by definition, I define it as a sub. So if you remember, I have two notations. This is the one in coromology. This is the one in ciao. So if I define it directly as a sub-algebra, no, the one in ciao is even worse. And the reason is very easy to state because of the fact that in general, I mean the modular spaces don't have any, they are not nice in the sense they are no longer rational at some point. So if you consider, you can easily construct modular space of curves that contain infinite dimensional ciao groups because not all points are linearly equivalent, sorry, not all points are algebraically equivalent and that kind of things. So one of the reasons one wants to restrict your tautological classes in coromology is simply to be sure that one is working with something which is finite dimensional. So in coromology it is automatic, but in ciao it is not. No, that's the other way around. So let me just simply write this formula and then I will make summary of what I stated yesterday about this. What I wanted to say is if the genus is zero or one, it is known what the coromology of m bar g n is. And in these cases, so this is follows from work of Keel in the 90s. This part has now been conjectured by Gessler, but the first proof to appear had been by Peterson. Let me see whether I have the year, unfortunately I can't find it, but roughly five years ago I guess. So what I wanted to say is well if the genus is small, actually the two things are the same. If there are no marked points, you can go only a little bit and prove that every class is tautological, but already for genus to graven and panda to panda were able to construct counter examples. And now let me comment on what I wrote here, that the fact that it doesn't matter whether we work in coromology or we work in ciao, but one can always construct generators for the tautological ring, which have a nice description, which I explained here, and these generators are exactly corresponding to each other in the two situations. So sort of, well I don't know whether it follows already by definition, but something which, but if we restrict the natural map, there is always a map from the ciao groups to coromology and genus, but of course so far it will just work everything, the ciao groups are generated by classes of cell varieties, and so one would map this to the fundamental class. So of course inside here the ciao groups of the tautological ring, and then if we take the image here, one gets this, one could sort of prove it recursively using the definition, using the gluing maps, the fact that this map is subjective, or one could prove it as some kind of corollary of the fact that we have generators that map, so that each generator here can be lifted here by taking a generator, which is a decorator, a decorated starter class. So the generators correspond to each other, of course, expect that there should be more relations here, so what one expects, when in general one may well expect that there is a non-trivial kernel, but it's not known. In all known cases they are the same. So let me discuss this motivation, so I was saying for genus zero and one everything is known about this, and actually it does not matter whether we work with the tautological ring or we work with the even coromology, we have just the same situation. Actually for m bar zero and all coromology in even degree, so this is easier to check, actually one gets the whole coromology, for m bar one and there is coromology in odd degree, so this is something which we can never get because the fundamental class is always in even degree. In general, in the image here we will never get any class which is not algebraic, so which is not, we can't be lifted to an algebraic cycle on the variety, but even if we restrict to the good classes there are counter examples, so there are examples of classes which are of coromology classes which are coming from the fundamental classes of some variety, but which are not tautological. And as I said one of the ways to identify them is the following if we can find a class which under the pullback under some gluing map is not given by a tensor product of fundamental classes, so if I find a component somehow which lies in the wrong space then clearly what we were working with is not tautological. Again for genus two at the beginning everything seems to go nicely, so if n is smaller than 20, all or even coromology classes are actually tautological classes, but then from n equal 20 the things bake, so they made some kind of ingenious construction in which they were looking at the locus of smooth curves in n220 which have a bioliptic structure, so which have a double cover on the elliptical, let me denote this cover structure by pi, of course pi should do something with the marked points, otherwise we would not need to add them and we simply require them to be marked parallelized to have parallelized the same image in e, so if we have the point p1 then the other, then p1 and p2 are mapped to the same point on e and so on until we take p19 and p20, so if we take p2k minus 1 then it has the same image as p2k and we do this 10 times. There is a special locus of curves of genus 20 with marked points, this only makes sense as the definition of smooth curves, we don't care so much about what happens, so we will need to consider the generations as well, so we take the Zariski closure in n220 bar, so this is something we could have mentioned 11, but this class is not tautological, what was the issue with that? The issue is that if we take the intersection of Z with the locus of curves which have two components of genus 1, then one needs to subdivide the marked points on the true side, then if one takes the pullback under the associated gluing mark, one gets a class such that one of the components is not coming from a tautological class, so let me write a little bit more about this, so we need to take two components of genus 1 and we subdivide so that we have 10 points on one side and the points on the other side, so this will correspond to a curve which has two components of genus 1 and we can choose them, for instance, to be both the same elliptic curve and then there will be some kind of trivial structure as a double cover simply by fixing an isomorphism between E and E and mutting them to the same point, so that is that we sort of glue the curve to itself at one point and then extend the identification to an isomorphism and then one needs to fix 10 points here, and take the pre-images according to the labeling which holds, so this kind of curves belongs to the locus Z because this is some kind of the generation of this situation, this is of course something one needs to check explicitly and if we consider the associated gluing mark, then it goes from M1 with 11 marked to M2, 20 bar, the image is contained in Z, so if we consider the associated case, practically if we take the intersection of Z with the image of this map, we will get this kind of curves, so the restriction of Z is actually the diagonal embedding. The key point one wants to give this 11 point is that this is the first case in which M bar 1N has a homology class in odd degree, and somehow this fact forces that if one takes the pull back of the class of Z under this gluing map, one of the components is the tensile product of two copies of this odd degree class, so this guy is actually a two dimensional and the geometry of the situation, I'm expressing it, not very elegant, so I guess that I should some, if we take the pull back to Z, will model it is more or less the same thing as intersecting Z with the image, and I guess that this should be done in homology, so in this way we are going to the locus of the intersection, which is more or less the same as M bar 111, and then this embeds diagonally into this thing, but then here we will find the class of odd degree which we described here, and so this, I guess I need to make some checks about this, because yeah, every time, yeah, but if we have, I mean, we need to have exactly, for the general point we need to have exactly one unique point and one unique point there, so we would need a degree, one map from one component to the other, so, and so for this reason we have to be isometric if I like it this way, yeah, this will not work, yes, yeah, exactly, I mean, the reason is exactly that, if I look at the proof, the reason is that the existence of odd homology plays a role in all these constructions, because as I said, one needs somehow to prove that the component by taking the pull back is not tautological, so an easy way to find a class which is coming, the only way somehow to find a class which comes from something as small as genus, which is not tautological is take an odd class, because as I said in genus 0 and genus 1, all even classes are going to be tautological, so the first count as example requires to work, so in genus 0 as I said, all classes have in and degree, so there is nothing to do, we need some count as example which is coming exactly for genus 1, but then we need to have a class which is not tautological, so we need to have a class in odd, in odd degree, so this is the first one we may use, because somehow if we look at the geometry of the elements, some components we have two lines inside here, so this is some kind of a tick, but the bottom line is that somehow there are more constructions associated, one could look to some kind of definitions of the tautological classes on fiber products over the curve, and then one there, one could study more naturally the properties of the diagonal, but somehow if one chooses the right geometry, then there is the possibility of creating geometric counter examples, and if one works in cohomology using the stratification with curves of, with rational type and of compact type, one can sort of control the situation more nicely and actually exclude the existence of any other non-tautological cohomology class with less than 20 points, so by now the knowledge of the geometry in genus 2 is sufficiently advanced that one can explicitly rule them out, so we want to understand the structure of tautological things, and to get a feeling for it, it looks like a good idea to start with the case which is sufficiently easy, so let's review a little bit what is known, so what is the state of the art for the cohomology of mg without mark points and not compactified, so which classes exist here, so the boundary stratoclases for sure don't exist, but we don't have any mark points, so the cyclases are no longer there, so a priority for what we know is generated by the kappa and lambda classes, so the question is how dependent are they, well a priority, I mean we have many of them and then we can of course, yes, well this is just by, so we can look for instance at the, at the description of the linear generators, so they have decorated starter classes, but here there are no, there is only just one starter, it's mg itself, so what we can do, what that theorem says by restriction to mg is we throw away everything which is not, which is coming from monomers of, yes, so that is telling us well if we work with smooth curves, then we can only take monomers in the psi, kappa and lambda classes, no mark points, no psi classes, so far so good, the first one to study this space also because he was the first one to give the general setup in which the theory of tautological classes work was Manfred in his 1983 paper, and his result is that we don't need the lambda classes at all, we don't need them not even on the compactification, so if we take the lambda classes on m bar G, so, but when we study lambda classes we don't need any mark points because if you remember they come from the Hodge-Bunder which is not related to the geometry of the markings, otherwise I mean we can always forget the markings and make computations on m G bar, and they can be expressed for monomers in the kappa classes and boundary starter classes, I'm just going to give you an overview of the steps in the proof, I mean the paper of Manfred is of course perfectly readable, but there are many technical steps which can be a little bit unknowing, so the idea is that we can use Glossendick-Liemann law for the universal curve of m G because that's where all classes we are considering originate from, so this would say that if we can take the chain character of the pointer, push forward on the universal curve, and study of the relative shift of the universal curve, then it's the same thing as taking the push forward of the chain character of the same thing, but then we have to multiply with some kind of code class, but of course we don't want to work on the universal curve, we want to work downstairs, so somehow this tells us if we push everything down that it's just equal to the chain character of the Hodgbender, I'm used to calling it E, but I think that I called it HG in the first lecture is just one plus, and then one needs to think about this, about what happens here, but the thing is that one needs to take the formal exponential of the first chain class, the relative bundle multiplied by this told class for the shift of relative differential parts, and if one works with it a little bit, this means that the chain character of the Hodgbender, the issue is with the rewrite in this thing, because one needs to express everything as a long sum, and then because of the fact that we have a told character here, and the told character is related to the expression of the series of T divided by E to the power T minus 1, this is what the told character would do in the first chain class, we will get some Bernoulli numbers here, so we take the sum for all E, which is at least one, and we will get as a coefficient a Bernoulli number divided by a factorial of true i, and then we get a kappa class here, and then we get the correction term which is supported to the boundary, so the idea is that the lambda classes are the chain classes of the Hodgbender, so on this side we find all the information about lambda classes just packaged in a different story, and then by looking at what this, at what this sheath is on MGM, but one can, one can realize this side purely in terms of the kappa classes and something from the boundary which keeps track of the fact that we are gluing the curves together, so it gives some kind of correction term that depends on side classes when we identify two devices, when we identify two curves of the lower genus to create a boundary device, so we know that when we level the boundary devices we have no marked points, we can always take something of genus h and something of genus g minus h, we should, we can stop at g minus, roughly at g half, but it's no harm on doing everything twice and then divided by two, and then for h equal to zero we want to create the graph with a loop, so they both give gluing maps, then my mh they depend on the choice of h we did and so here the push forward by mh and then here we simply take the sum of all possible monomials of the appropriate degree to i minus two of the side classes on both sides, so what I was trying to say is that of course there is some kind of correction term here supported on the boundary, but if one looks at the devices one can say that this is sort of keep, that it expresses the fact that we need to take a side class on each one of the edges which are identified and then going deeper when we find this expression, so it may look puzzling without a proof, but anyway it has a number of things, but mainly it has as well, this means that we can express all lambda classes with these strange coefficients, this is a multiplication, in terms of these of kappa classes if it is actually in theory otherwise we get something which is quite controllable anyway, so it could be implemented. If we look at the right-hand side we find only, yes let me write what the definition is to the Bernoulli-Lambert's in this case, so the idea is that one expands t divided by e to the power t minus one and this is a function that is related to the way in which the stored character is defined when working with line banders and then it's extended multiplicatively then the coefficients up to a factor which comes from the factorial of i are given by the Bernoulli-Lambert's, so that's why we will always find something like this by working with this kind of character and then let me tell you quickly an application of this formula, well everything we get here you see the subscript they are always, we're always taking something which is of odd degree on this side, so this means that if we take an even degree part of the chain character unless it's one, unless it's in degree zero, we will get something which is trivial, so if we take the chain character of the Hodge-Bandler and we restrict to a dv which is even, it's zero in our non-trivial cases, so this is already giving us a bunch of relations between the lambda classes, it's sort of telling us we don't need for our theory any of the even lambda classes and then for the other ones we can find an expression in terms of the other tautological classes using the other side, but then one can actually go further and exploit this expression a little bit further, it's actually easier, it works much better of course if we just work on the interior and we forget about all the boundary starter contributions and for instance using this, Manfred could prove that the tautological ring is generated by the classes kappa one up to kappa g minus two, so one stops at g minus two, so with respect to the open part and we have no marks points, this is generated by the first class, this is not necessarily a vector space of course, it's an algebra, so where does that hold? Well first we can rewrite the expression we had before, so on the interior the relations coming from gothenic Riemann-Roch can be re-phased in a more compact way and one wants to extract relations, it's actually easier to write this as a formal series with some dummy variable t which is just keeping track of which lambda class we are working with and this can be written as the exponential of the thing we had before, so with the Bernoulli numbers, now the constants have changed a little bit because we have packaged the information in a different way, now there is the exponential which is taking care of i factorial but we still get some coefficients and this is the side telling us well we are starting with something which is supported only in an odd degree and if you can check I mean this should for instance give that lambda 1 is equal to 1 over 12 kappa 1, to find this you need to know that v2 is equal to 1 over 6 and then for instance from the vanishing of the second of the degree part of the degree 2 part of the Chen character one finds that lambda 2 has to be equal to 1 half lambda 1 square and then going on for the other classes one finds something which involves some Bernoulli number and then the vanishing gives additional relations, so the point is the lambda i's can be expressed in terms of the odd kappa i's and then what is the next step, the second step is to give some kind of dimension of bound on the rank because at least on smooth curves the relative dualizing shift is generated by global sections, implies if we take the pullback of the hodge-bunder to the universal curve we get the subjective map of the relative shift to the relative, the shift of relative differences, so this is a subjective map and so here the key point is that we need some kind of geometric input and this is the global generation of the relative dualizing shift, so if this is subjective we can look at the kernel and the kernel will be some locally free shift of rank g minus 1, yes so on each curve, so somehow we need to, yeah not on the whole and that's why we get the hodge-bunder here and not the river-bunder, so the kernel is a locally free shift of rank g minus 1, so in particular its chain classes will vanish in a degree which is higher so formally we need something there, chain classes of the kernel which at least the growth in the group is something which we will view as the difference of the two shifts we are taking but you can think of this some kind of notation for the kernel, the multiplicativity one can equivalently look pull back of the sum of all lambda classes which is what we called as this side and divide it by 1 plus psi 1 because it's coming from the other side and this has to vanish if j is larger than g minus 1 so if j is at least g, we can see this but the point is well if we have a bound on the dimension of something then this gives us a bunch of vanishing so once we have all this vanishing of the universal curve we can push down the vanishing relations to find relations between lambda and kappa classes in every degree which is at least g and now what Manfred did was to prove that these new relations are independent from the ones that express the lambda classes as polynomials in the old kappa classes so they can be used you sort of solve out, you eliminate the lambdas from the expressions and you remain just with the first g minus 2 kappa classes and we find many of them because we can find it for every possible degree which is so we start as there is no, there is a lower bound but there is no upper bound and now the key point is to prove that what Manfred did was to establish there are sufficiently many relations that everything that starts with kappa g minus 1 need not be considered, we can take the previous group of relations this now given liminter, some more vanishing reasons and we can use them to eliminate the higher kappa classes so this is a psi, this is a construction which is particularly easy but the key point of this proof is something which we are which people working on pathological things are still using because they, one needs some kind of geometrical input one ideally one wants to find some kind of map between vector banders, locally free sheeps over the modular space of hers and then there is some kind of geometrical information that gives bounds on the rank and from this one can produce some kind of, one produce a vanishing of some classes on the universal curve or any other auxiliary space which is what we have over mg n bar and one push them down to produce even more relations so this was sort of the baby case and this had been used hundreds of time and exactly as in the case of Manfred part of the problem is to find a way to control the relation to patch a kitchen together to understand because of course if one creates infinitely many relations not all of them are going to be significant someone needs to also to have some kind of combinatorial skill to be able to select the minimal number of relations that can be used to reproduce larger objects by x because it can be anything but it should map flat way of mg which is pretty loosely we can construct in some intrinsic way vector banders so the idea is that in Manfred's book once it was simply taking the universal curve and then as part of the definition of the universal curve we have omega which we were using in the computations but one can add even more structure so like fix a device on each curve and take a space of global sections that vanish there from each curve and try to glue it together to a vector banders something like this then one would have a larger I mean one can insert more data about the curve here to construct automatically some kind of vector band and then one should use the geometry of the situation to find some form of vanishing usually produced by the fact there are some bounds on the rank we are working on the auxiliary space the one in which we have chosen more data and then we take the push forward to mg and we find a huge number of relations so for instance father obtained more relations with some construction which is related to the Brill-Neuter theory of curves so that will give the geometric input for the vanishing and he was simply taking fiber products of the universal curve with the device of a certain degree which we are ordering the points which we consider and that gave incredibly many relations and then this kind of approach had been generalized by Fander-le-Pander together with with Marianne and Opria to the concept of stable quotients in this way one finds even more relations and there is no reason actually to distinguish the mg one could also do it with the larger spaces I am stating it in this case because this situation is more controllable and with this kind of approach I am taking as an auxiliary space the fiber product of the universal curve of mg this kind of approach is actually u to father Eline-Leonard was actually able to prove that one does not actually need the first g minus two classes one can stop at g divided by three Opfer was able to construct many relations but he was just using one the geometry that was coming from the choice of one point if you take up to d points and general geometrical considerations then what you find is well you actually need much less if you want a set of generators as an algebra I am writing it wrong so if I want to have your next theorem that the one could read the algebraic geometric methods then of course she is proving it for Chao because actually the result in cohomology is also true but it had been proved by them in the same period but with actually more like topological methods by Morita I think this appeared yeah, sure I mean, yes I mean it's not so I don't think there is anything asymptotic about this theorem I mean the first one the first cases are only easier than the other ones and anyway there is no restriction you could always be able to use less of them but actually I think this is the actually and this is good because I mean this had been proved in 2005 but this kind of property had been conjectured by Karl Faber already at the beginning of the 90s in 1993 when he started to study the structure of topological links so this had been a long expected case but it took at least 10 years to produce a proof of that on the other hand here we are close to the case in which we know independence so this would not work if we are working we would not work directly if we are working with charolings but if we work with cohomology we can evaluate ourselves of the results about the stabilization of cohomology the idea is that the cohomology of mg and that of let's say plus one of mg for any g which is larger than the original one taken is not independent if one takes cohomology in a sufficiently small degree one will get always the same thing this is a phenomenon which is called cohomological stability so if one is working with a small degree there is some information about the independence of kappa classes coming from the fact that if the degree is small then it's the part of cohomology that has become independent of the genus so now I will try to summarize together the work of many people so the fact that the cohomology stabilizes due to hair in the 80s then the bounds have been refined in many ways many times first by Bolson first at Ivano and now quite recently by Bolson but then the characterization of stable cohomology has been known has been open for a long time and known under the name of Manfred's conjecture and it started here by Bolson and Rice you have a long timeline it starts in 1885 perhaps this may be 2012 but the really difficult theorem is this one and what they want to say well the point is if we take cohomology in a degree which is sufficiently small with respect to the genus then it becomes independent of genes this is a phenomenon it's now under the name of hair stability larger than 3 divided by 2 k plus 1 this is good because approximately it's saying that k should mean smaller than 2 than twice g divided by 3 so this is the degree of the last generator we have here and in this range cohomology is freely generated by the kappa classes so this means that the first group of kappa classes we have here is going to be always independent and this word was known under the name of Manfred's conjecture and now Malzen-Weiss theorem and this I mean this is something I guess algebraic geometers we algebraic geometers are very happy about the fact that we know this but at the same time I guess that when the announcement was made there was a lot of unhappiness because the proof of this theorem uses a stable homotopy theory and has nothing to do with the geometry of the modular space of curves so it's a great result but nobody knows whether it would be possible in the future to give an algebraic geometry proof of it certainly not starting directly by the way it's proof now so it's a good news but one feels a little bit indebted to the colleagues on the other hand this is clearly something that requires some kind of topological input because how is the stability map so the map inducing the isomorphism how is that defined as I said if an algebraic geometer you don't want to study Riemann surfaces with boundary because there is no way to express the boundary in an algebraic geometric way but this is just our situation not everybody if you are a topologist you may love surfaces with boundary and there are some advantages to them because if we take something of genus G with a boundary so this will be an element of something that stays in the modular space of genus G Riemann surfaces with a parameterized boundary then you can construct a map that associates to it something of genus G plus 1 again with a boundary simply by attaching a Riemann surface of genus 1 with two parameterized boundaries so one fixes this guy and one can attach it by increasing the genus of the Riemann surface in a controllable way so ok it's very clear that this map exists and it is well defined but of course it only makes sense if one considers an auxiliary map in which one is filling the gap by attaching to it some kind of cover and the same map allows us to compactify the other side so there is no direct way to go from here to there but if one can prove that the maps going down are isomorphisms in a certain range the horizontal map are isomorphisms in a certain range one gets the mapping homology which gives stability and then one can try to be clever to find some kind of meaning of this also from an algebraic geometric point of view but the fact remains that you can't map mg to mg plus 1 in a natural way the question is well using this kind of so using stability we know that now the result one has to further generators is sort of optimal but of course there is no guarantee about how to find all possible relations between the polynomials in the given kappa classes so what is the evidence up to now about that first of course we don't expect a homology sorry a tautological groups to be non-trivial in arbitrary high degree because they are of course dimensional constraints but when do they exactly become trivial this is a problem that has been studied by Luyecha and it would conclude that anyway I guess I can state it in child because it implies the other one if the degree is at least g minus 1 then homology vanishes and then this is something Luyecha found by giving a very precise geometric description of generators so he was not working directly with the kappa classes but working with classes of multiple covers of p1 such that the mark points are lying over 0 and infinity in the verification locus so the question is what happens for g minus 2 this is generated by the class of the hyper elliptic locus at the time it was not even known that it was non-trivial so it was either one dimensional or trivial itself but then Faber was able to prove that there is a class in degree g minus 2 which is non-trivial one can take for instance the kappa class and so this is a generator so this means that the tautological links start in degree 0 where they are one dimensional and they end in degree g minus 2 when they are one dimensional again but actually there is more evidence of a symmetry between a low degree and degree close to g minus 2 and even the shape of the normal relations between the kappa classes was actually particularly nice and this is part of the conjectural description for the tautological link of mg that Faber gave during the 90s so first there are explicit formulas expressing to express the intersection numbers of the kappa classes so to express all monomials of g minus 2 in the kappa classes as molecules of k g minus 2 this was known by the name of the intersection number conjectural this has been proved but this was really tough indeed the first proof consisted in saying that this expression could be interpreted so would follow from a special case of the conjectural this is something that the guest letter of course the conjectural was not known in that case but then it was just known later by work of and then after that more geometric proofs as always so by now many different approaches to prove that the intersection numbers between the kappa classes are what one expects so once one has a nice formula for the intersection of any two kappa classes now the next question is is it enough to know the intersection numbers between kappa classes to find all relations between kappa classes so the idea is that we have a paving between tautological classes in degree G one can be stated in an incormology from previous the conjectural exists for both spaces and then one takes something of co-dimension I because of the fact that we want to end up in degree G-2 this is the top degree we take so the intersection will give something in degree G-2 but that isomorphic to and this gives a perfect parameter is to or if this were to this will simply mean that we have all information to determine the ring structure of tautological ring because we have all generators and we can calculate all relations from the intersection numbers so of course before conjecture is something like this father check that this property held in all possible cases he could control so when he stated the conjecture the first time I think it was known up to genus 15 by now we know that the property holds up to genus 23 so anyway if you remember the variational geometry we are now out of the range in which mg is not of general type or at least in which mg is that is this what happens in genus 24 so when it takes the same construction using auxiliary bundles of fiber products of the universal curve or spaces of a moderate space of stable quotients as a pandery panda does well what does one get exactly in this case how far can one get some kind of unfinished work here because if we take the homology of 24 in degree 10 it's known the dimension is 36 and I think it's even known that the 36 generators that are left out are actually independent if we take all known relations what is known is the homology of L12 which should pair with L10 is most 37 so somehow one would need just one additional relation to prove that the property holds also in this case but I mean by now incredibly many such relations have been existent but not a single one had cut something new here and the expectation is indeed that this is going to lead to the first counter example so that one should need to prove the intersection of these 37 things are different from something of complementary dimension which is not ontological so thank you