First of all, yeah, thanks for the great organization of this conference. So I'm going to talk about a project that of course wouldn't be possible without Dirk. So probably, I hope you can see it, but this is a picture of the group a couple of years ago of Dirk's group. So I was in the middle of my PhD back then, and I mean, I cannot really point at them at the moment, but you probably can recognize Dirk there on the right and a couple of other faces. And yeah, it was a great time. And Dirk was a great supervisor. And he, I mean, already like Bob de Borro wrote in his letter, he really implemented these ideas of letting everybody develop their own scientific style and develop themselves individually in the scope of the school in Berlin. And so it made a very colorful picture, just like this picture there in the summer of 1950, I think, of 2015. And a couple of months afterwards, Dirk introduced me to Karen Vogtmann. And there she first introduced me to this problem that I'm going to talk about today. But it turned out to take quite a while until I realized then that the tools that I'm going to or that I was going to develop in my PhD were actually capable of solving this problem. And yes, so this is the story that I'm going to talk about connected to this Euler characteristic of RFM. And naturally, this wouldn't be possible without Dirk, because he made this connection, but also because it's connected to the work of Dirk, via this hope for graph that plays an important part in this path to this proof of this problem. All right, so I'm going to introduce this problem or try to introduce this in a very like pedagogical way. And yeah, so if you don't know anything about RFM or outer space, you should also sort of at least understand something about it. So I will be very slow. And if you don't understand something, please just unmute yourself and ask a question or put it in the chat. I don't know if I can see it here with the chat, but probably someone else can just can just ask the question then instead. So this is about groups. And especially it's about automorphisms of groups. So there's a second order group, so to say. So we take all the automorphisms of a given group. An automorphism of a group is just a map from the group to itself, an isomorphism, which respects the group property. So it's pretty basic. And for every group, this group of automorphisms has a normal subgroup, namely the inner automorphisms, which are sort of the boring automorphisms of the group. And these automorphisms are just given by conjugation of elements in the original group. And the group of outer automorphism is now the quotient of the group of inner automorphisms. Or this is now the quotient group where these inner automorphisms are portioned out. And so I forgot to tell you that if this is too basic for you, please wait until the end of the talk. There's going to be some new results that I didn't present before. So also if you have heard this talk by me before, which could be at some occasion, then bear with me. In the end of the talk, I will give some new results that are very fresh. All right. So we have this outer automorphism group of a group, and we want to apply it to a specific group, namely the free group. And the free group is something you also very know. You also probably very well know. So it's just the group of n generators that can be thought of some lectures that you can just multiply by writing them in a string. And there's one identity that just says that the inverse of some letter times the letter itself is nothing. And therefore you can just form a group by multiplication of two elements. And you can also form inverses. So it's sort of the simplest groups, a group you can imagine in a sense. And this automorphism group of this free group is our main object of interest in this talk. This object is also something very concrete, so you can write down some generators. I think Karen even actually wrote them down on Wednesday in her talk. So for instance, you can take as generators the the isomorphism of the group where you just replace the first letter by a product of two letters, and another generator of the auto-automorphism group is replacing a letter by its inverse, and you may permute the letters non-trivial. So this is sort of a generating set for this outland group. Another example of an auto-automorphism group is the mapping class group, which is the group of homomorphisms of a close connected orientable surface of genus G. And you can think of this as the auto-automorphism group of the fundamental group of the respective surface. I mean, you also probably know this kind of these kind of objects where you think of the symmetries of some surface, but you can think of this also as automorphisms of the fundamental group of the surface. At least for surfaces, this works. An example is this group of homomorphisms of the toros, which is of course very important, the modular group essentially. And it has two generators. So you can cut the toros in two kinds of ways. They are indicated here in the as red and blue circuits that you can hopefully see. And you can cut the the toros along the slides and then turn it by 360 degrees. Then glue it back together and you get a generating set for your mapping class group of the toros. And yeah, you can identify this with as a 2 z. Right. So part two, so now I spoke on groups. So now why spaces? And the reason to talk about spaces is of course a very basic philosophy that you probably also all know, that the question, how do you study such objects, such groups as the mapping class groups or out of N can be answered. And for physicists, this is probably more intuitive even then for mathematicians that to study sort of a group, it's always best to realize the group as some set of symmetries of some concrete object. And this is the philosophy. So we want to study these groups by inventing some object, which is symmetric under an action of this group. So the idea is to realize the group as a symmetries of some geometric object. And this in this case, so the group would be either out of N or the mapping class group. And of course, this is an idea that has been developed by many people. For the mapping class group, this construction works by the Taichmuller space. So for a Taichmuller space, what you need for as a data is some close connected and orientable surface s. And you make a space from this data by taking the space of where each point is the Riemann surface x. And at each point, you also have a marking. This marking is just a homomorphism from the surface, which is given as data to the Riemann surface. So you can see in this picture that so you have to imagine like this big space, each point is a marking and a pair of Riemann surface and this marking, which identifies the surface with the Riemann surface. So an alternative way to think about this is just to think about each point to give some kind of metric on the surface. But you can also think about it as a marking, just a homomorphism. Okay, now the mapping class group is asymmetry of the space just by composing with this marking. So you can just modify this marking by a homomorphism of the initial surface, which it starts with. And so you permute the points in the space in some way. And this is this action of the mapping class group. And of course, this action would be interesting if it wouldn't have a bunch of very nice properties. But this is just the basic idea behind this construction. For out of M, this approach is sort of mimicked. And this was invented by Mark Keller and Karen in 1886. They sort of invented another construction for out of M. And this construction replaces the surface, which is data or which is given as input data, with a very simple graph with the rows and with M panels. So it's just a self loop graph, which has N self loops or N tadpoles. It's the basic tadpoles graph. And so in this case, in this case, we have here R3, which is the rows with three patterns. And then you construct a space from this by taking each point in the space to be a graph with the length assigned to each edge, just like in the talk by Francis on Wednesday. And at each point, you also have an additional data, which is a marking, which is in this case a homotopy from the rows to the respective graph. So the marking now gives a sort of a deformation, or how you have to give sort of the information, how you have to deform the rows to get the get the graph. And just as in the mapping test, we picture out of N now acts on the space, which is therefore called outer space, by composition with a marking. So there's a tricky point here that I'm sort of skimming a bit over that you have to identify your element of out of N with an element of the auto automorphism group of the fundamental group of the rows. But the fundamental group of the rows is just the free group. It's quite easy to see. So this works this way. All right. Okay, so this is a picture out of space, just as an illustration by Karen to fund it article by her. And you can see that it's made up of these graphs. And so they are different cells. And in different cells, they can be the same graph. And in the different sets, these graphs, they only differ by the marking. So you can imagine now out of N, permuting these cells, but always mapping like the graph on the same graph, but changes this marking. And in the boundaries, so in the smaller dimensional phases of this thing, there are then the graphs of one of the edges is contracted. Right. So you already heard about a couple of applications of outer space. So of course, you can apply this to study the group out of N. And there's also applications to the modular space of punctured curves or tropical curves. You can use it to study some invariance of some practical manifolds. And in Marco's talk, you heard about applications in mathematical physics. And Francis talked about applications to graph complexes. All right. So now we want to apply all this stuff that I have been discussing now. And the application is, of course, we want to study some invariance of those groups. And what we want to do, we want to study the cohomology, of course, of out of N, for instance. So that would be one invariant that we want to study a topological invariant. And the trick is there that we can identify the cohomology of this group with the respective object of the quotient of outer space with the group. And this quotient is this modular space of graphs. And this works because outer space is contractible, as Karen and Marcela showed. So one of the simplest invariants that one can imagine that one would like to study this, of course, the Euler characteristic. And yeah, so we can study this Euler characteristic using this quotient. All right.  I mean, so why is this Euler characteristic especially interesting for the case of out of N? The reason for this is that there's a very natural map from the free group to the free a billion group. So FN is the free non-a billion group, of course. And ZN is the free a billion group. And this map, of course, just works by forgetting that the generators of the free group, they are not commutative. So it just slashes them all together. And what you get is effectively an element of the a billion group. And you can derive from this homomorphism, you can just get a homomorphism of groups, also from out of N to the outer automorphism group of ZN. And the outer automorphism group of ZN is a very well known group. It's just GLNZ. And now one is interested in the kernel of this map. So the outer automorphisms of the free group, which go into GLNZ are sort of the non-interesting or the commutative automorphisms of the free group. And one wants to study these inherently non-commutative automorphisms. And these are in this mysterious Torelli kernel here, which I called calligraphical TN on this left hand side. And one can just make this into a little short exact sequence. And yeah, so like I said, this non-a billion part of out of N is now interesting. This is this Torelli kernel. And by this short exact sequence, one can relate the Euler characteristic of out of N with the product of GLNZ and this Torelli kernel just by the vibration property of the Euler characteristic. And so this Euler characteristic of out of N here on the left hand side of this equation down there, it gives sort of a leverage to look into or to peek into this interesting kernel. But there's something really interesting happening there, namely that the Euler characteristic of GLNZ is zero for all N larger than three. So with some very basic arithmetic, you can see that there's something funny going on, except for the case where the Euler characteristic of out of N vanishes. So if the left hand side also vanishes, this is all fine. And then it just means that the Torelli kernel Euler characteristic can be anything. But computations, they actually show that the Euler characteristic is not zero for most cases. And one can actually prove or I mean, we actually prove eventually that it's not zero for any N larger than two. But from this one can actually deduce information still on this kernel, namely that it has nonfinally generated homology for N larger than three if this Euler characteristic of out of N is not zero. So this is interesting that one can leverage information on this Euler characteristic into this kernel and deduce that it has nonfinally generated homology. So the conjecture that John Smiley and Karen had in actually seven is that the Euler characteristic does not vanish. And they also conjectured that the Euler characteristic grows exponentially or the magnitude grows exponentially for N to infinity. And they performed some initial computations of this Euler characteristic up to N to 11. And these computations, they were later strengthened by Saugier up to 100. But yeah, so the question was of course if one can prove this in general. There's a related conjecture, a stronger conjecture due to Magnus, which states that this Torelli kernel is not finitely presentable. And this says in topological terms that the second homology group of the Torelli kernel is infinitely generated, which of course also implies that the Torelli kernel is not finitely generated. It does not have finitely generated homology. And also Bess Wiener-Bachs and Magalit in 2007 proved that this Torelli kernel indeed does have a non-finitely generated homology. But they didn't touch any of the other two conjectures up here by doing this. They used a different argument to actually show that it has non-finitely generated homology. So the upper two conjectures remained open. And yeah, so the result of Karen and my work was now to make the upper conjecture, the initial conjecture on the Euler characteristic into a theorem. So the result is that the Euler characteristic of Audifin is not zero. And yeah, that could be the end of the talk, but I will explain some details in the rest of the talk on this result and give some further aspect and newer results too. So the theorem explicitly is that the Euler characteristic is always smaller than zero for n larger than two. And that it has a specific growth rate, which is given by the gamma function for n to infinity. And there's this weird log squared term here in the denominator, which I don't really know how to interpret. But this is how the awesome topics actually look like. So this settles the initial conjecture, but there are a lot of immediate questions. For instance, the large growth rate of the Euler characteristic indicates that there's huge amounts of homology and odd dimensions of these groups. And nobody knows where this homology is coming from, because there's only one odd dimensional class known in rank seven. And this is due to a large computer calculation by Bertoldi in 2016. And the question is, of course, what generates all this homology? So I will talk a bit more about this later. But first of all, I will talk about how we obtained the theorem. So we proved this based on an implicit expression for these numbers, chi out of n. And this expression looks as follows. It's a bit complicated, but bear with me. So it's an asymptotic expansion of a function that you see there on the left hand side, this two pi i square root times e to the power of minus n times n to the power of n can be expanded asymptotically for very large n in terms of the gamma function. And if you do this, you find that the coefficients, they encode these numbers, this Euler characteristic of odd n. And this is what we proved. And one can deduce from this that these one can deduce properties of these numbers from this theorem. So there's an analytic argument, which is quite complicated, that goes from this asymptotic expansion expression or this implicit expression of these numbers to the actual result that the Euler characteristic is always negative and the growth rate. But in the rest of the talk, I will mainly talk about how to prove this implicit equation here, because it's connected to quantum field theory. And one can also use this Hopf-Valduber of Dirk to actually prove this. So just to put this a bit into perspective, an analogy to the mapping class group or the analogous calculation of the mapping for the mapping class group was performed by Hara and Zergier. And in this case, things are much more beautiful, in a sense, because you get an explicit result for the Euler characteristic. And they just found that the Euler characteristic of the modular space of curves of GNSG or mapping class group of a surface of GNSG is given by the Bernoulli numbers, divided by some fractions, divided by some stuff. And yeah, the original proof of this fact was due to Hara and Zergier, also in 86. But there's an, there quickly came an alternative proof based on quantum field theory or topological field theory methods later by Penner. And this proof was then eventually not still simplified by Konsewicz in 1992, who also used topological field theories. And this simplified proof was sort of the blueprint for our proof of of the Euler characteristic for Kai Autophen. And the only thing that we sort of added was this inverse philosophy that comes from the Hopf-Valduber approach of Dirk to quantum field theory. So how does this proof by Konsewicz of this Euler characteristic for the mapping class group work? So the first thing that Konsewicz did was prove this identity, which relates on the left hand side, the the mapping class group or the Euler characteristic of the mapping class group with with a sum of graphs. So this is MGN. So you also seen this in this talk in this conference already. So it's not the mapping, the modular space of surfaces of genus G, but there's a generalization with n punctures. And you probably all know that. But you can order the ingenious thing to do is to form sort of this weird linear combination of different numbers and package them together in this generating function with the strange 2 minus 2g minus n factor popping up there. And if you package it up this way, you can calculate the this linear combination of numbers in terms of a of a generating function that are in terms of an expression, which every physicist immediately knows how to evaluate. So it's just a sum of a connected graphs. And I have to add that. So, so there's the so you have to take all graphs which are connected and which have neither 0, 1 and 2 valent vertices, but three or higher valent vertices are allowed. And for each graph, you take the parity of the number of vertices and you divide by the automorphism group of the graph and you take into account so that you mark the Euler characteristic of the graph, which is just the number of vertices minus the number of edges of the graph. And so the difficult part, of course, is to prove this identity and considered proof that's using a combinatorial model of mg n based on this model by Penner, which is a model from ribbon graphs. And you probably also know the story and how this works. So here's a little picture of this. So I unfortunately don't have a pointer, but let me see if we can do this. So on the left hand side, you see this ribbon graph gamma. And you can convince yourself that it has one boundary component and it has Euler characteristic minus one. So from this data alone and follow set, you can embed this ribbon graph on the torus and stretch out the ribbon graph such that there's only one point left on the torus. And so therefore, you get sort of a model of a surface of genus one with one mark point. And this gives you sort of the model for m11 for the epineclase group, genus one with one mark point of the modular space. All right. So consider to use this model to prove this identity. And then the easy part for physicists, maybe for mathematicians, it's a bit more complicated is to actually evaluate them with some on the right hand side, which you do by a topological theory argument. And if you do this, you see immediately that you get these negative zeta values there as a result in each order and that. And then sort of to recover the Harazaki formula for the epineclase group, all the characteristics, you just use a short xx sequence argument, which relates modular spaces with different numbers of punctures. So you can just forget one puncture on your surface. And this is a nice map. So you can use it to relate the other characteristics of these spaces. So this is the proof of the Harazaki formula in sort of one slide. Due to conceivage, it's pretty nice that it is so short, in a way. And the proof of of our result sort of works analogously. The only thing that we added is this sort of this this renunciation argument. So to actually make this argument work, it's useful to take sort of an algebraic perspective, just in the philosophy of Dirk. So we take H to be a vector space, which is spiked by all kinds of graphs. And so all the graphs, they are connected and three valent or higher. And you can just now formally write these kind of objects as this linear combination here of the order characteristic of the modular space as sort of an evaluation on an infinite sum of graphs, which is sort of an element in the formal ring of power series on this vector space. And this map phi, it maps from this vector space to Q. So it associates to every graph just this alternating sign on the number of vertices, which you can see below. So this phi is very easy to handle via topology-coffee theory. But if you look into the analogous picture for Artifan, you have sort of a similar, you can write it down in a similar way. But this map that you need to evaluate on all graphs is much more complicated. It's not just the sign on the number of vertices of the graph, but a non-trivial sum. So here on the top, in the top, I wrote down this generating function of the order characteristic. And you can also express this as sort of an evaluation of such a character on this formal sign. I call this character tau.  And this tau maps again from the vector space of graphs to Q. And to each graph, we associate a sum over all forests of the graph. So a forest is just a subgraph with no cycles. And for each forest, we take now the alternating sum over, so we take this alternating sum of the forest where we take into account the number of edges in this forest. So this character, or this function on this vector space is harder to evaluate. But the Hopf-Valdubar comes to the rescue. So this is not directly approachable via t with t. But yeah, so and also, so I forget to mention how to actually prove this. So this was already proven by Smiley and Karen, but on Smiley and Karen. And you need this forest collapse construction, which is due to Mark Haller and Fogman to prove this kind of identity there. So to make progress there, one can employ this Hopf-Valdubar of graphs. So we can we can take, we can promote this vector space to an algebra by just allowing for multiplication and just freely multiplying graphs. So we also have to take disconnected graphs into our vector space now. And we can define a co-product. And probably you all know this co-product that takes a graph and maps it onto the formal sum over tensor products of the graph times the respective tensor products of subgraphs.  And in this case, it's the set of our bridge of the subgraph that you want to sum over. And if you follow this construction through, you obtain the core of the algebra of graphs. And Dirk invented this or generalized his renormalization of algebra in the sense in 2009. And of course, this core of algebra is very related to the to the normal renormalization of algebra. And it's even the renormalization of algebra of quantum gravity. So this is an example of a co-product calculation of this algebra. So you can see that if you hit the wheel with free spokes with this data, then you get a formal sum of the graphs. And in each tensor products sum, you sort of end up with with respective pre-factors giving you the number of these kind of graphs that appear there. All right. So, so the important thing now with the software device that using this construction, you can now, you now have a group structure on these kind of linear maps from the graph to something else. So linear maps as this very simple phi map for the modular space, of course, are this more complicated tau map that I had before for out of N. And so, so under this map or under this, this group of multiply, so, so you can multiply them, but they actually form a group under this multiplication. And the interesting thing now is that this map phi, which is associated to the modular space of curves, and the map tau associated to out of N, are mutually inverse elements under this group. So this core Hopf-Fargeber or this Hopf-Fargeber of bridges graphs, it sort of duly relates these two characters. And this is sort of key to this implicit formula for our Euler characteristic. So in a way, this means that tau is the renormalized version of phi or as the respective counterterm to phi. So we can write this down in a more physical way. So remember that we can evaluate this, the strange linear combination of Euler characteristics for the modular space case via this topological field theory here in the top on the right hand side. So you can see that, so you integrate over an action or a functional sort of thing, it's just a one dimensional integral in essence, but you can interpret it as a quantum field theory. And the action is this one plus x minus e to the power of x up here. And you take the log of this whole thing to get the connected graphs. This gives you an explicit formula for the Euler characteristic of the modular space but this duality between this phi and tau map implies that these chi out of N terms, they are encoded by the renormalization of the same topological field theory. So by renormalization, I mean that you can set up an implicit renormalization condition, so to say, where you say that this topological field theory is supposed to vanish if we add certain counter terms and these counter terms, they are just encoded by our numbers chi out of N. And this is the trick. And this way you get this implicit asymptotic expansion for our numbers chi out of N. All right, so this is it. So if there are any questions to this initial calculation, please ask them. I will still briefly talk about some extensions of this work. So as an outlook, so what I talked about was just the rational Euler characteristic of these groups, but you can of course also just go into the naive Euler characteristic, which is just the normal alternating sum of the better numbers of the group. But this thing is much harder to analyze than the rational Euler characteristic because it does not behave nicely under isomorphisms or homomorphisms. But there's an explicit formula for this naive Euler characteristic, which is due to Kenneth Brown. You can express it as a sum of finite order elements of the initial group out of N in this case. In the normal, the rational Euler characteristics of the centralizers that correspond to these finite order elements. And so our investigations so far, they indicate very strongly that the quotient of this naive Euler characteristic with the rational Euler characteristic, it approaches a finite constant for N to infinity. So this would then automatically also prove that we have that this indication of all this existence of homology in odd dimensions is actually there. So that would prove this existence of this homology. And yeah, we didn't quite prove this yet, but we are almost there. So it's a bit technical to do this kind of stuff. What this also leads to and how it connects also to Francis talk is that this is sort of all part of a bigger picture of these graph complexes of these considered graph complexes. So there's a trinity of graph complexes. There's associative graph complex, the commutative graph complex and the lee graph complex. And I already indicated here in this table that the associative graph complexes of a lot of belongs to the world of modular spaces of curves. And it's so it's these very course invariance as the Euler characteristics, they have been studied by Har as a gear already, or one can already associate them to this. There's this commutative graph complex, which is quite, which is already a bit more mysterious. So the rational Euler characteristic of this is really simple. It was written down by Konsevich in 93. And Wilwacher and Zivkovic, they computed the integral Euler characteristic of some up to some order. And this commutative graph complex is also the one that Francis talked about on Wednesday. And yeah, as you learn from there, so it has also very complicated homologic structures, and it's kind of mysterious where the cycles come from in this object. But there's another graph complex, the lee graph complex, which is sort of related to autofen in a sense. And you can calculate the rational Euler characteristic of this graph complex, also with Konsevich methods, there's a bit of a different expression for it. But the integral Euler characteristic for this is unknown, or there's no formula for it. There's only some calculations by Morita, and we used a supercomputer to calculate the integer Euler characteristic up to order 11. But recently, I figured out a way with Karen, we figured out a way how to actually also write down a closed formula for this Euler characteristic, for this naive Euler integer Euler characteristic. So this is the formula, and so it's sort of explicit. It's an infinite dimensional integral over an infinite dimensional space. And so you can expand this and actually calculate these numbers. And when I did this, when I implemented this formula, I was able to get like 14 coefficients out. And I was really happy because I got more coefficients out than the initial Morita calculation up to order 11. But then I showed this program to Jos van Maseren. And he sent it back to me and it was much faster. And then it was 40 coefficients. And yeah, just before the talk, Jos emailed me and told me that he found another trick to make the calculation faster. And now there are 70 coefficients of this integral Euler characteristic norm. So this is quite some progress there as well. And this is in preparation, so to say. Yeah, so this brings me to my summary. So the short summary is that the Euler characteristic or the rational Euler characteristic of our defense nonzero. And there are lots of open question exactly on all this homology and where it comes from in this group. So this rational Euler characteristic itself, it indicates that there's much homology in odd dimensions, but it's not known where this comes from. So and our investigations into the naive Euler characteristic, they support this totally. And but the question, of course, remains what generates this? Yeah. And so there's also other questions. So can one explain this kind of weird duality between these two things? So there's already in the original paper there by Konsevich, he indicated that this is like a casual duality. But it would be interesting, I think, to try to understand this like on this very graphical level on this on this level of the software. And of course, yeah, we're not can also think about generalizing this to to things like right-angled a team groups that that Karen was talking about in her talk. All right. So thank you. I also would like to take the opportunity to also still thank those two people here in this picture for organizing this great conference. So this is a rare sight of them from the front. So usually if you go hiking with them, you only see them from the back because they are so fast usually. So yeah, thanks, you two. You are great for organizing this great. All right, let's thank Mishi for his great talk. Are there questions? Yes, I have a question if I'm allowed to ask. Absolutely. You have explained that Konsevich formulated his theory by this formula where he some over takes the sum over graphs and these graphs are ribbon graphs, right? Yeah. And later you formulate the renormalization, the hop-free renormalization also with ribbon graphs or is it with other graphs? And all the trick there is that you can forget about the ribbon graphs. So this is sort of this trick how this this identity works there. So you take this non-trivial linear combination to be able to forget about the ribbon graph structure. But you're right. So you go from ribbon graphs to connected graphs. Okay, because ribbon graphs and the other graphs have different automorphisms and the groups are different, right? Or is it? Yeah, but you can project down to graphs if you forget about. So there's an easy map from ribbon graphs to normal graphs by just forgetting about the orientation of the vertices. Okay. So if you just so this is a nice function.  So this is part of the trick. And yeah, you're right. I skipped over this because yeah, because of time. Okay. So there's some subtly there. That's true. All right. Well, other people are thinking of a question. I just can't resist to ask what does the beginning of the sequence for the integer Euler characteristic look like? Okay. Yeah, it looks, I mean, I can, I don't know if I it's a couple of numbers. I mean, it's not really there's positive and negative one. So it's not exclusively negative or positive. And it also vanishes, I think for a couple of initial numbers. I don't know. I think I have some coefficients here. Yeah, I can send you the paper about the with the with the initial 11 numbers by Mopita. And I can also send you a list, but yeah, sorry, I didn't include them in the presentation. And like I said, just recently calculated 70 of them. I think yesterday. Francis has a question. All right. Thank you. Forgive me if I've asked you this question before, but you had tall and five that were inverses under the convolution, convolution product. So I wonder, and so you've got, say, say tall to the power of one and tall to the power of minus one. But you can also look at tall to the power n under the convolution. Have you, do you know, have an idea of what that might mean? No. Sorry. No idea. Yeah, I think we, yeah. Yeah, I have to think about this. No, I don't, from the back of my head, I don't know anything about this thing. Yeah. So maybe there's something that I don't have any idea. Sorry. I was wondering, since you mentioned that you have a, you have this picture with the pie and tall for the out of n and the, and the MDNs, the modular spaces. So do you also have a way to, to the commutative graph complex and it's order characteristic? I mean, you, you mentioned the papers where people have computed this with other methods, but is there a way to formulate this in a similar way in your approach? Yes. Yes. Character in that case. The character would be, well, yeah, so, yeah, actually, I mean, it will be the same one for the rational order characteristic. It's the same one because it's surprisingly the associative and the rational and associative and the commutative graph complex, they have the same order character, the same rational order characteristic, but this is kind of an accident. So for the integral order characteristic, there you have to work more. And yeah, I, I also, because Francis was talking about this, I already started to look into this and try to actually extend these methods, these new methods for the, for the integral order characteristic, for the league graph complex, to the commutative graph complex to sort of get a similar expression that I showed in the back in the, in the end of the talk for, for the commutative graph complex. But yes, I, I wasn't able to do this now on the side of the conference. But yeah, I think it's totally possible. It should be actually not so hard. And it's just a question to get the science right eventually. So it's very just. Yeah, so it's, so this is sort of the challenge with this things to get all the factors of two and science properly. So this is the, the challenge. And yeah, and also, I mean, this works by, you can interpret this as, as some, as doing some representation theory on graph. And yeah, so one can totally also do this for the commutative graph complex. But they have already been different methods by, by conceivage and Zivkovich, who, who did this calculation, but they got sort of a different formula. And they don't have any asymptotic results as far as I know. Thank you. Are there any final questions? Can I ask a question, Karen? Yes. So do you have any sort of easy way to explain the appearance of this strange asymptotics inverse square of log n? Yeah, I mean, I, I know where it comes from. Because this is how I calculated it. It comes from the, so the Lambert W function. It has a singularity on the minus one branch. So the Lambert W function, it has a branch cut.  And if you take them the first derivative of this singularity, you get a one over log singularity. So it's sort of a log to the power of minus one. And this is the type of singularity that pops up there. It's a pretty funny one. And it has this kind of kind of, okay. It's always that Lambert W sneaking up, isn't it? Yeah. Okay, thanks. It's guilt by association, because I remind people that Lambert W occurred in a paper by Dakin and Karen when they made a toy model of beta functions. And it's the absolute of the essence of the, of the all of the summation that Reimard talked about. And now you see it here in this context. It's also in the stuff Gerald and Michi did together. That I mean it, I'd love to understand better. It's related to an idea of a video constant called trans asymptotics, where you re-sum these trans series in the other order. So you re-sum all orders of the instanton expansion for each order of the derivative expansion. And that naturally leads to this Lambert W because you're changing from exponentials to powers of the small parameter. So it's deeply buried in this story, as we all know. All right. Well, let's thank Michi again.


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