 Thank you very much for the introduction and I try not to write smaller than that. I hope you can read this. So my story begins like many other stories in this conference with Dirk in the 1990s. We met, I came across a paper by him and David and he had this fancy idea to associate knots to some interesting numbers. And for the first time for me it was the idea that numbers are more than just numbers, which is sort of a very modern view on numbers. So I was absolutely fascinated by these numbers and these ideas and I tried to calculate them right away. But to cut a long story short, I found myself some 10 years ago in Berlin sharing an office with Francis Brown and actually nobody except Dirk was really distinguished for a few years at that time. So we started working and the better part of the 10 of the last 10 years I spent setting up theories that allows me to actually calculate these numbers. And these theories, there are actually three theories, so we need theories. And the first theory is which I call graphical functions. And this is recently, this is with Michi Borinski and I should say that there is pretty independently these functions have been developed by many people in super young mills theory and they do really amazing things there. But I have a purely mathematical theory on these functions here. So this is theory one, you need a second theory to express these these are amplitudes. You want to express these amplitudes in terms of functions that you can handle. And these functions are generalized, single-valued hyperlocks and G, S, V, H. And the third theory is sort of an add-on, it's a dimwreck version of this dimwreck extension that you may want to have if you have sub divergent graphs to handle. And these are the three theories and they're now being published. And I should say, this is a sort of a special setup, so you can't do every quantum field theory calculation in this setup. But I think if it is possible to do a quantum field theory calculation within the setup then it's typically more efficient than other methods. So it's kind of a special tool. And what did we achieve the results? There was one most prominently is this Galois Coaction, the Coaction Conjectures or Principles or Cosmic Galois Group. This is what I got after you do these calculations and you immediately see that there is this structure going on. So everything what I did in this direction is just doing these stupid calculations and look at them and then you immediately see that there is this structure. And there was even a calculation at 7 nubes which was missing, which I couldn't do with my methods. I had calculations up to 11 nubes, but there was one particularly interesting 7 nubes of missing and that was done by Eric. And I should say that all the credit for trying to understand these things is due to Francis Brown and now many others. So I only stumbled over these things and understanding them is due to Francis Brown and many others. But at least I got him thinking about these things. And then you can prove something sometimes. There's the proof of the Sixa Conjecture. And this is a conjecture by David Broadhurst and Sir Greimer and it took 17 years to prove this so you can really do some mathematics. And then if you use this third part, this add-on, then you can do renumberization functions. And I started, before I actually did these calculations, I was telling it's possible to do this. It's just straightforward. And then I thought it should, as a proof of concept, I should at least do some six nubes, five and the four. But the tricky thing about these calculations is they're kind of weird sort of fun. So when you start doing these, you always try to do more and more and more. So I had these six nubes very quickly and I thought, okay, I do seven nubes. That was not that quickly anymore. So I had seven nubes, five and the four, which means I have the beta function, the gamma function, the mass gamma function, I have the self energy. I started recently doing some work on eight nubes, beta, eight nubes, gamma. I don't really know why I actually do this. I think I'm mostly doing this because I can do this. So it's just, you know, you know what to do. And so you just start doing it whatsoever. And there's more. Recently, with Mimiki Bovinsky, it was possible to extend these calculations to higher dimensions. So we looked at five to the three in six dimensions. And we got five nubes very quickly. And six nubes, yeah, maybe in the future. But I won't do this alone. I need Mimiki to do this with me. And then there was a proposal on QED, six nubes. That was only a proposal and it was rejected because it wasn't interesting enough. So there's a smallest cancel. But I could definitely try to do these things. You only know in the very end if you succeed, because if there's a single graph that you can't do, then you're stuck. Okay, so if you have interest in doing these things, I don't think that I do this. This alone in the near future. But maybe if somebody's interested, it would be good to help somebody doing this. And what else that we have for results? We had something else. So I won't write it down because there's no space on the blackboard anymore. We had the C2 environment, which has nothing to do with these things. So we proved that with Francis Brown, we proved that there is a K3 in 5 to the 4 and there are high dimensional geometries and there's modularity, such things. And there was other spin-offs, like the single bias multiple C2 values, which is also due to Francis Brown mostly. But I got him all the way to look at these things from my data. So I want to spend the time to explain one, two, and three. But I won't do it in a very mathematical way, because I don't have enough blackboard space to actually write down all the details. So that will be a birthday talk, a bit casual. And I'll try to explain the concepts and how they emerged and where they came from. And although I do these graphical functions now for some 10 years, I still remember the very second when they were born. So it's the graphical functions here. The very second when they were born, I was walking with Francis to the S-Bahn in Berlin. And he was, I was thinking about these numbers. I wanted to have numbers. But he had a very parametric mind. We saw that he still has sort of a parametric mind. And he explained to me why this is good to have a parametric mind is because even if you look at pure numbers, it's good to have parameters. Because with parameters, you can do more things than with pure numbers. You have differential equations and stuff like that. And if you're interested in numbers, you can specialize values and you get the number of specs. So it's always good to have numbers. But I liked my position space approach that I had beforehand. And I wanted to stay as closely as possible to quantum field theory. And this parametric space always seemed a bit artificial to me. So I didn't want to use real tricks like this, this, this string of trick. I just wanted to stay as closely as possible to quantum field theory. And I personally liked position space, because the primary rules are the simplest in position space. So I said to myself, okay, Francis, you want a parameter, I get your parameter, but I stay in position space. So I started with zero scales. I say, okay, in one scale, but one scale doesn't do the job because one scale is trivial. It's just, it's just a scale dependence. It's just a power. So there's nothing new happening. So I need a second scale. I need a two scale processes. And that's it. I have just one, one parameter, not for every edge of parameter, just a single parameter. And that's the birth of the graphical functions. So set this as follows. We have position space, space. In position space, you rather want to have the mass equal to zero, in particular, if you want to do many new calculations, because it's just too complicated and the propagator is not algebraic, for mass not equal to zero. And you have a dimension, D, which originally it was four, but now it's four or larger than four, but it has to be even. Oliver, someone, one of the attendees has raised their hand and Federich also suggested you could erase your title if you want a little more space. But so, race my title. Yeah. And Jonathan, do you have a question? Well, maybe not. Okay. I don't know. The question will have to appear in text form later and I'll pass it on if it does. So I'm in position space. I have a two scale process and two scale is two real numbers. And the idea is that these two numbers should fit into the complex plane because the mathematics doing things in the complex plane is always an advantage. We can see this geometrically. There's two ways to do this. You can also have invariance to fit in the complex plane or you have it geometrically. Like in position space, we have vertices and we have three vertices to be fixed to have a two scale process. And the three vertices, which are fixed, they span a plane. Wherever you, however you arrange these three vertices, they always span a plane. And at this plane, I want to be the complex plane. And I move the complex plane and I choose the scale in such a way that one of these three dimensional vectors in these three vertices is the zero. The other one is the one in the complex plane. And the third one is my parameter, my only parameter, which is now a complex number. Yeah, so that's the idea. And I have this, to have this graphically, I have the three vertices get the labels zero, one and set. Although as an amplitude, they are d-dimensional vectors that could still be the origin in d-dimensional space. This is a unit vector and this is an arbitrary vector. And I give you examples right away to know. The first non-trivial example is this graph. So you integrate, you just calculate the amplitude of this final graph in position space. You integrate over this internal vertex. You don't integrate over zero, one and set. And then you interpret the result as a function on the complex plane. And the result is four times i times d of z over z minus z bar in four dimensions. Well, this is not this d. This is the Bloch-Mignen dialog. Bloch-Mignen dialog. This is in four dimensions. So that's a bit confusing now. This is d equals four. The dimension is four and this is the Bloch-Mignen dialog. Well, that looks like an interesting function, I thought. I should say that because John had it yesterday in his talk by uniqueness, that in six dimensions, you can calculate this in any dimension. In six dimensions, the result is even simpler. So in six dimensions, you get z, z bar, z minus one and z bar minus one. And this is uniqueness. And you can calculate in eight dimensions and 10 dimensions and 200 dimensions, whatever you want. So I have another example. What happens if I add an edge between zero and z in this graph? I get an extra position space propagator and in four dimensions, the position space propagator is the absolute value of difference squared. And if I translate this to complex numbers, what I get is z times z bar. And here you see one of the benefits of working in the complex plane. What in d dimension is a quadric, like the propagator becomes a pair of straight lines in two dimensions. So you want, generally, you want to get things as linear as possible. And this definitely helps. So yeah, let's say in six dimensions, you would get a square of these atoms that probably get this. And so you want to know what are the general properties. That's what's the first question that I have. Okay, are these functions? How will they look like in general? And just let me say it again, it's important, although we only work on the complex plane, the graphical function has the full information of the, of the amplitude, because they can't happen anything else. Everything happens on this plane, which is spanned by the three external vectors. So what are the general properties? So I erase the example. There are three general properties of different complexity to prove. So there's one. This is what you get right away. I call these graphical functions f indexed by the graph. And what you get is a residual symmetry. You get a reflection symmetry. That bars the complex kind of get upset. You should have said this before. And it's also a real function, of course, because everything is real, but that's not so important. That's probably one, you get this for three. Property two is already a little bit harder. FG for any graph, for every graph, FG is a real analytic. So it will never be an analytic function. It's a real analytic function. And it's a single function on C with two singularities, zero and one. And when you think of numbers and you see all these fancy numbers with all the roots of unity and stuff like that or elliptic or K three services coming up, it wasn't surprised to me that you for every graph G you only have singularities is zero one and you have one at infinity. So this is due to master gods Eric and myself in 2012, I guess that's a mathematical proof. And then something else, which is very important, although it looks a little bit technical, you get something, you get, you look at the singularities, zero, one and infinity, and you get something that I call log log on expansions, single by log log on expansions at zero one. And you also get one at infinity and having an expansion at infinity means that you actually not don't leave live on the complex plane, but on the Riemann sphere in some sense, it's one of a meromorphic property on the Riemann sphere. And how do they look like I only write down for set equals zero. So you get a sum, a finite sum and it goes through the number for the internal vertices, but that doesn't matter. It's a finite sum. And you get infinite sums and m and m bar are just integers and they are bounded from below by some m zero to get some coefficients c naught, l, m, m bar and these are real. You get because it's single value to get a log z, z bar. You can't have a log z alone that would never be single value. And you take this to the power l. And then you have just a typical Lorentz expansion. You get z to the power m and z bar to the power m bar. So it's a little bit of a bad notation. z bar is the complex corner gate of z, but m bar is not the complex area of m. It's just two independent integers. And it's very important to have this lower bound on m and m bar. So if I do some examples, give you some examples to see what this property does. It is, for example, what does not have the property is something like 1 over z minus z bar because if you expand and set, you get more and more negative powers in z bar. So this will never be, never have this property three. And what is okay actually, but you have to think about it, is the Bloch-Wiener dialogue over z minus z bar. This has the property three. And this is actually in a recent paper, it will be in a recent paper with Michi Boinski and myself. And the proof is more or less finished except for interchanging some with an integral, which is a tricky thing. So it's still a little bit in progress, but it's clear that we have this. So that's a very important property. So not everything that you can write down with say z minus z bar denominator has this property. You need to have a numerator that cancels the singularity on the real axis in the denominator and the Bloch-Wiener dialogue is such a numerator. The next thing that you want to know is how do I actually calculate this with these numbers? How do I calculate that? But before I do this, I want to show that you get these numbers that I was interested in from the very beginning from these functions. And how do I get periods? There's the trivial way you can set z equals zero, one, or infinity. And that's what we did for the zigzag conjecture, which actually used a very recent theorem by Don Zagny. But there's a better way to do this. This is actually a complicated way to do this because you have to calculate the function which is sort of too complicated. You don't have to have because if you substitute, you did too much and then you restrict to a parent and then you lose something that you have calculated. So the better way to do this is to integrate over z. So you have a two-dimensional integrate over z and this is an effective integral. It is over the complex plane, but it's not the normal integral. You get a special measure from projecting everything down to the complex plane. And there's this example that we have, we calculate this graph in the function. And the period, if you integrate over z, you get the period. And this is the integral with this effective, this effective integral of the log-needle dialogue, which we had in the previous example. And now we have the z minus z bar. And then we have the z bar from this edge here. And we have z minus one and z bar minus one from this edge. And we have an effect of one from this edge. And you see everything is linear. So you may guess that you can do this integral and this is six zeta of three. That's the real three spokes. Another very nice example, which is very easy with this method is the K34. So you have four vertices on one side and three on the other side. And you join them in such a way that everything is connected from left to the right. So these are the seven vertices here. This is the K34. And now you have several choices to choose where you want to have your one, zero, one and z. But here it's very convenient to have this here. And you immediately see that the period gives d2z, this effective integral over c. And if you look at this amplitude, like you integrate over this vertex, but it does not connect to these vertices here. It only connects to zero, one and z. So this is actually the fourth power of what I did beforehand. It's a four i d of z over z minus z bar to the power four. Maybe you have to think about it for a while, but you see that this vertex is not directly connected to this so that the function factorizes. It's a fourth power of the example that I had before. And the result is something which has a ceta 35. Okay, so I was glad that I could do these things very easily. But in general you have to be able to calculate the graphical functions and more complicated graphical functions and what I had beforehand. So you want to have a strategy how to calculate these graphical functions. If I couldn't calculate these functions, then the theory would be pretty useless. So lots of effort went into the finding ideas how to calculate these graphical functions. But there's a very straightforward way to do it if you get started of graphical functions. So there's some trigger steps that you can do like adding an edge between zero and z or between one and z or zero and one. You can even commute zero, one and z and you know what's happening on the function side and there's completion which I skip here. So you can use conformal symmetry to add another external vertex in infinity. But to get something substantial you need to do some integration and in position space and that's also a reason I like position space you can use the kind Gordon equation. Because we have zero mass it's only a Laplace equation. So you want to solve an equation like Laplace, a d-dimensional Laplace of some function is some other function. And in practice this would be a more complicated graphical function and this is a simpler problem. This is the way you if you think of an amplitude the plus d-dimensional Laplace makes a propagator collapse to a delta function and the edge shrinks. So this is the way to contract edges or if you solve this equation then you can grasp with the extra edges from the smaller one. Because we work in two dimensions this gets down to an effective equation. So we get an effective Laplacian and there are some trivial rescaling that you want to do. So I skip this but essentially what you need to do in the complex plane is to solve this differential equation. This is the Laplacian and now you have to solve a similar equation where these are slightly reskate functions. That's what you want to solve. And because there might be some other medications to listen to this talk I although I don't really need it I want to say something about this Laplacian because there's some nice mathematical structure going on. This is interesting enough. This is the Laplacian on the hyperbolic space the upper half plane model of the of the hyperbolic space. So there is a factor minus 16 times this Laplacian on the upper half space of hyperbolic space. I don't know what graphical functions have to do with hyperbolic space. This is the only connection that I have. But more interestingly the whole effective Laplacian is actually a casimir minus d minus 3 the dimension minus 3 squared of a si2 c c le algebra which you get if you consider these differential operators z to the n dz plus z bar to the n dz bar for n equals 0 1 2. So these three differential operators they generate an si2 c le algebra and if you calculate the the casimir this is the casimir of this le algebra then you see that you can express this effective Laplacian in this form. So you expect at least the homogeneous solutions of this differential equation having something to do with the representation. But we only knew this after we have already solved this differential equation. So the trick is you want to find a general solution of this differential equation delta effective of f equals g for any dimension by only taking primitives and normally I thought this is hopeless so I for a long time I stuck to d equals 4 then this is gone and this is trivial you can just integrate with respect to z bar and then you integrate with respect to z and you're finished. So d equals 4 is trivial and I thought it would be impossible to go beyond d equal 3 but then Michi Burinski entered the game and he was ingenious and stubborn enough to try this I didn't even try to look for a solution because I thought it was hopeless and what we achieved is a closed form for the solution the general solution closed form for the general solution in the space of single valued functions with a single average it's zero one and three solution of this delta f f equals g well that was a surprise to me and this is I have to say it again this Michi Burinski played a very important role in finding the solution so this but this is only the first step if you want to use this the triangle order equation for calculating grabbing function this is the first step you want to have a general solution but that's not enough because you want to have a special solution so you have to control the kernel there's a huge kernel here in this differential equation if d equals 4 then the kernel is just the sum of a homomorphic function plus an angiomorphic function in general it's more complicated we know the kernel and you have to control the kernel so the next thing we did is looking which function of in this kernel has the property the general property is g1 g general problem one two three of of graphical functions and the big surprise is that there's only a unique graphical function in this uh pre-match of this differential operator so the kernel is trivial trivial in the space of functions with this property three it's very important technical property that we have for graphical functions so because I know that my my function my my graphical function has this property three I have no problem with the kernel and I get a unique answer solving this this differential equation and the third step is the third step is find a function space where you can do this calculation you need a function space and this function space is the generalized single variant hyperbox to do actual calculations and this I want to do the last 15 minutes or so um tell you something about the general single variant hyperbox there's just two more things I want to say about graphical functionary just say verbally um there are more identities that you have there are many more identities and there are still identities to be found there's this amazing fishnet conjecture by Benjamin Basso and Lance Lixon which I can't understand so far mathematically and the other thing is I do have a thing in Euclidean space but you also get the same result from Minkowski space because in the Euclidean space you have z and z bar is a complex conjugate and in Minkowski space they would z and z bar they would be real and independent so with the result you have both worlds the Euclidean and then Minkowski world and for some consideration it's even good to have z and z bars independent complex variables and it starts also with the story and most prominently the story was Francis Brown started it was a kind of very lucky coincidence to analyze functions which are single-variate and are multi-poilers so he had a theory of single-variate multi-poilers was in a paper which was never even been published with a beautiful paper and he used generating functions so it's extremely beautiful you have their horror action in there but as beautiful as it is it's not very practical because in generating functions you have the whole lot the whole universe of these objects to do a single calculation so I thought if I have a single-variate multi-poilers which I used first to do these calculations I need a procedure to get a primitive without having the full generating function so the idea was to corner the single-variate multi-poilers by two errors and one is the interval the single-variate interval itself and the other one is the complex conjugate I want to write it as a single-variate integral and a single-variate equal to victory z bar so here's something like dz single-variate multi-poilers and here is dz bar of single-variate multi-poilers and I need some space here of this again and the idea the obvious idea is to have a commutative to have a commutative square at that this is z bar because you want to kind of undo this integral your vz bar and then you get a weight drop and then you move up doing the single-variate integral vz so this should cancel this and this should give this that's the idea that should commute more or less but it does not really commute on the nose or what you need you need two more errors that's a hexagon to make this commute and you need projections on the residue free part and this is the anti-residue so this kills residue phi pi bar they kill anti for the bar residues and with this commutative hexagon it's very easy to construct these single-variate multi-poilers explicitly so start with the function here subtracting the residues no problem and here the whole mystery of single-variate integration is the anti-holimorphic part and the whole mystery of single integration with vector z bar is the homomorphic part so if you intersect the mysteries you're left with a constant and that solves the problem but the single-variate multi-poilers is not what I wanted to have because it's not it's not enough you know you get more functions you want to have more functions you want to have differential forms like d z something like a z z bar you want to have pi linears b z plus z z bar plus t and so one example would be this d z over z minus z bar and this d d of z z so we get this quite frequently and you you also may have may use the fact that you have this log Lorentz so you get log Lorentz expansions with a three this property three you may impose this for this function that you want here this differential forms you have a log log one expansions at the singularities and then you want to find primitives and it took me a while to figure it out but the solution of this problem is to ignore it so you just use the same the same thing here and you just replace single-variate products by general single-variate hyperboles and the thing works but this is one of the lucky circumstances where you can solve a problem by ignoring it so I feel obliged in the current situation to say that that is not always you could approach but here it worked and then you get a very efficient way to calculate these general single-variate hyperlocks so one example would be this if you if you integrate this function we saw that this is in three this gives if you integrate this with respect to d z take the primitive then you get a general single-variate hyperlock whereas this if you integrate this it's not good because there's no three and this is not a general single-variate hyperlock you can extend these objects to these things but they are sort of single-variant sort of meaningless because they kind of is attached at the real axis where there's a situation so that's what I wanted to say about about generalized single-variate hyperlocks and if it's possible I would use another five minutes to say a little bit about dim break to sort of close to the side and coming back to jerk at the end of the talk go ahead yeah it's sort of straightforward so I can't really tell the story about these things because we just do it the way you naturally would do this and the first thing that you observe that this delta effective in z you get an extra term plus two epsilon z minus z bar d z minus d z bar and to actually prove this this is a naive answer that's what you get naively just by substituting the dimension in the form that I've worked at before to d minus epsilon but you have to prove that this works and master goes proofed in this master's thesis in parametric space where everything is well defined and then you just expand in epsilon and everything goes through so what you do is you get a theory you get Laurent series in epsilon every order in epsilon is looks like a graphical function an ordinary graphical function so you use the standard use the standard properties of graphical functions of graphical functions and the next thing you use is some extra stuff which works only for some orders in epsilon that's actually quite powerful these standard properties of w in this work all order like this solving the client order equation but there are some you can do is sort of approximation you take a sub graph replace it by a simpler graph which happens to have the same expansion epsilon up to a certain order and if you're only interested in some orders in epsilon and if everything is not singular around this sub graph then you can replace this and get a simpler graph and then you can hope to cover the simpler graph or graphs with the standard methods and what you get is a reduction in the number so many of the graphs you just do right away without thinking and the complexity of the of the of the of the graphs that you need to think of seven lives by the before you have some whatever 10 000 graphs you want to do for the beta function you calculate 95% of it right away and then you have five percent problems with these methods you can't get anywhere with graphical functions and with the standard properties and with these substitutions but the problem sort of still reduces the graphs to some sort of kernel and well this is one of the major benefits of using graphical functions but these calculations they just come collapse to something much smaller and this much smaller just they are much more appropriate to use some external techniques if you want and there's high print by every which is sort of hard to use because it's normally time and memory consuming but it's still very very helpful if nothing else works and then what you should do what I haven't really done yet to a full extent is you should you should renormalize the graphical function and what I mean with renormalize you can calculate them as power system extra but it's more efficient to sort of subtract sub diverges imagine you have such a sub graph in the graphical function above all this diversion four dimensions so you want to add and subtract a graph where you reroute an edge you take this and you add this again and so you haven't changed anything but now you have reduced the problem because this is simpler here because you can integrate out this this vertex this is simpler and this is less divergent and less diverges means that you have less orders and epsilon that you want to calculate and the number two becomes much more powerful on on these less divergent graphs so you really gain a lot to doing this and then you're back at con trimer uh i think from our renormalization which was very prominent in this workshop and you also have this paper by Dirk and by Francis how to reroute these edges to get rid of the of the singularities and it's it's important to know that even if you don't fully solve the problem even a partial solution is very helpful you don't have to solve the problem altogether every step that you do in the right direction is good you also have other techniques and uh some very important property of this clubbing function is that they are very complementary to other techniques so what is hard to renormalize is easy to calculate graphable functions right away with techniques one or two or what is hard for hybrid is easy to do uh for graphing a function and the other way around that's that's sort of the fun of it so i come to my conclusions and the conclusions us thank you and thank you to everybody else in the community all right well thank you for your talk thanks all of them so uh there are a number of questions in the q and a over the course of the talk some of which are still sort of live and so i would start with tibaud amour asked if you have results in odd dimensions notably d equals yeah that's that's a big problem we don't have anything in odd dimensions so far but if you look at this uh as i to say li algebra so there is maybe some hope if there's a way that the the fact that you can solve this differential equation in even dimension has to do with the with the li algebra structure then you have a hope to extend it to odd dimensions but we weren't able we haven't really tried very hard uh because we don't need it in quantum theory but for classical problems it would be very hard to have something in three dimensions but we weren't able to it's not easy it's not a trigger to solve you make your even solutions all right uh slava richkov had a few different questions um and i've allowed them to speak in case they would like to just ask them directly yeah go ahead yeah we hear you yeah uh hi thanks for a nice talk i had uh so what are the questions so i got some answers on on q and a so i'll only ask those which i didn't grant answer that so when you deal with the high order graph there may be a few choices for what for who to call z yes that's what's power that's the power of the technique that you can use set everywhere you want like you can see your one and set so you have many options there's not a unique way you have to calculate the graph so yeah choose the best way this is very helpful okay sounds good and okay i finally i have a general question since you mentioned the cosmic gala group i don't really like have a very good idea what it is but i heard some people say that one should think of this cosmic gala group as some sort of new symmetry over quantum field theory um so i was wondering if well if if this were the case then it better have some meaning beyond perturbation theory yes i was wondering if you have any comments about that i think how should i think of this cosmic gala group is it just some purely perturbative thing or does it have some non-perturbative meaning i don't think it has a i don't think it really has a physical interpretation but it is a very practical thing to have if a gala co-action and you have some amplitude of a period the co-action principle says whatever you calculate in quantum field theory you want to know the co-action and you get something which is something again well these are not spaces or functions or numbers and you get something wrong and here is these are the galovar conjugates and the the co-action conjecture on the principles of the cosmic galovar group says that this is very very restrictive what you get here you get only very few galovar conjugates so it may be a symmetry but because it's the dual of a symmetry it in practice it's more like a restriction you're very limited what a quantum field theory result could look like because you know if you calculate the galovar co-action you have to have galovar conjugates which are in a very restricted space and typically you know the space because it's from lower orders you have a feeling what it should be for example if you fight over four periods you just know that you get again the fight over four periods and there's not so many around so this is a huge restriction almost everything is ruled out by this co-action conjecture so the way if you want to look at it practically then it's a huge restriction of what type of results you could get if I can add just one thought I think from my experience all we have done so far and what we have seen was always on a perturbative level so you compute some time and graphs you compute some amplitudes and n equals four things like that it's a perturbative origin because that's where you have the motivic periods so you have this motivic galovar theory and and I think the transition to the non-perturbative setting is very interesting but at the present the kind of question where we don't really have much information because the mathematical structure changes right you I don't know what the nature of these numbers is you get from resumming the entire series I mean as we've seen in talks yesterday right this is a very different world of of integrals and special functions and so on but at least I mean maybe one glimmer of hope to get some understanding in that direction might be recent work that has been started also by physicists and also mathematicians are working I mean Francis Brown is in the audience maybe he can comment on that that there are ways like a perturbative series right as a formal series that incorporates many many periods and all of their coefficients and then you can look what does the Galois correction do on coefficients but there isn't there's a program going on right now to define galovar theory for these kind of full perturbation series without having to do the expandant I mean this is dreaming very wildly and I don't really know if that will give you a physical application for the resummed physics problem but I was saying that there is work going on in mathematics that might give a meaning to a Galois correction of a perturbation series could I could I chip in with a very quick comment sorry to to draw out but it's a comment that was made in Alain Kwan's talk and and Galois called his theory the theory of ambiguity and the idea is that if you say if you write down square root of two that doesn't make sense because there are two square roots of two there's plus square root of two and there's minus square root of two and that ambiguity naturally gives rise to a group action and that group is the set of all all ways of permuting the answer or all possible ambiguities and and sort of thought logically anything in quantum field theory that is not where there has been some choice made where there's some intrinsically some ambiguity will necessarily give rise to a group of symmetries which is the group that permutes all the possible answers and so I have no idea of course how to make the the Cosmic God of Group as you understand it act non-partuitively but I think the spirit of it the idea of it is very encouraging that perhaps something like that does exist non-partuitively are there other questions for Oliver? I also want to say that your seven loop results on on the fight to the fourth I really have a lot of impact in the in the community which cares about the critical exponents so if you can do the eight loops then you said you don't really know why you do it but I think I think you're really showing the way that this epsilon expansion calculations which have been stagnating for many many years decades that you're showing the way that they can actually be pushed and even though there are other techniques nowadays to compute the critical exponents it's great that the realization group approach and still is not running out of steam so I think this is one of the motivations that maybe will allow you to complete this group work. As far as I understood they were even the second loop result was not so easy to get. Yeah still some hope I think the idea was the best so the gamma function maybe has a certain value to calculate. David has a question? Yes well really directly following on from that my understanding is that you don't need the hot val algebra of the iterated subtraction of sub divergences for the beta function if you can calculate the bare diagrams you just set them all up. The great thing about the BPAC is it gives you in the old way that we did it gives you one loop for free I mean you know what I mean by that so might you find yourself using our star at eight loops? There is an idea by I think Mikhail Kompaniac wanted to kind of implement our star to graphical functions calculations but this is a very crude thing that I did and I have a very crude maple implementation I do these C7 loop calculations. I never did more than I actually had to do to get the results but it's not done in any nice or complete way so whatever you use to do these much more complicated momentum space calculations you could also use for graphical functions and that makes it even more powerful and the way I use this conkran hop algebra it's not for remuneration because I just calculate the actual periods to the order and epsilon that I need so I took a very very naive approach to remuneration but it's on the other hand a very practical tool for calculating these graphical functions so I just use it not for remuneration but just as a practical tool to get more results to get rid of problems and that's the good thing about this is that you don't have to have a full understanding of it because you know you have other tools and you only have to lift the problem above the barrier where you can do it with other methods so even an incomplete even like isolated divergences if you handle isolated divergences the simplest thing that you can do is very very helpful for calculations and I've done very little in this direction thank you all right let's thank oliver again