 Alright, thanks Eric. And thanks to the organizers for putting together this conference. It's been very good. So I want to talk about some recent work with Mr. Burinski and my student Max Meinig, and I'll also be mentioning some work done recently with a video costume. There's a mathematician who's done a lot of work in the theory of resurgent trans-series. So the motivation. I want to put together, try to put together two things. One, we all know very well this underlying hop-helper algebra structure from perturbed of quantum field theory renormalization. And one of the advantages of this from a practical point of view in quantum field theory is that it enables doing perturbed of computations and understanding perturbed of structures at very high order. It's very hard to access in other ways. I wanted to think about combining this with a Kalos theory of resurgent asymptotics, which is a very general theory, but we can think of it here as a way of extracting non-perturbed of information from formal perturbed of series. And the advantage from the physics point of view is that there is physics beyond perturbed of expansions, which we call non-perturbed of physics. The question is whether that has any natural home in the world of hop-helper. So the idea is to try and use these resurgent trans-series structures to decode some non-perturbed of information about quantum field theories from their perturbed of hop-helper structure. So just a few words because people may not be very familiar with resurgent trans-series, although a lot of people here are. The function functions can be described in many ways, but one way to think of them is that they're closed under essentially all operations in analysis. And the operations will be interested in things like rural transforms, analytic continuations, inverse rural transforms, things like that, monodromes. In applications so far in quantum mechanics and quantum field theories and in matrix models, the general structure looks something like this, that you have some formal series in some small parameter X. But you include also these beyond all orders, exponentially small things, which we would call instantons or non-perturbed of terms, and also powers of logarithms. And the essential result is that iterations of these three basic structures of X and e to the minus some constant over X and log X, generate this huge class of so-called analyzable functions, which is much more general than standard analytic functions. And we also know from quantum field theory calculations and various asymptotic expansions in large masses or large momenta or strong coupling, weak coupling, that these are exactly the objects out of which these asymptotic expansions are built. One of the advantages of this resurgent approach is that this trans series, this generalized series, actually encodes analytic continuation information about the function you're trying to describe. Whereas if you just have some formal asymptotic series, it tells you something about the function, but it doesn't tell you everything about the function. And if this is achieved, if you can generate such a trend in order to reinstate this analytic continuation property. So this was developed by Kyle in the context of dynamical systems, and it's been explored in great detail in differential equation theory, difference equation theory. And it's been applied in quantum mechanics, matrix models, quantum field theory and string theory. The things that are best understood are in the world of differential equations and difference equations. The word resurgence, if you're not familiar with this, here's an image to think of that there are some critical points in your problem, think of them as singularities of a Burrell transform of a formal interpretive series that's asymptotic. And so the reflections around these singular points. And the idea is that this, the behavior near any given singular point encodes information about the behavior near other singular critical points. And so these functions in some sense resurrect or surge up again research near these other critical points. So that's the general perspective. And you can make it much more concrete once we're in a particular realm of talking about an integral equation or a difference equation or what. But the basic advantage from the appeal of this formalism from the physics point of view, is that it means that there should be some quantitative relationship between the fluctuations around different critical points, which if we think in terms of a functional integral would mean different saddle points of the function. So what about going towards quantum field theory so there are many examples at this point in metrics models and quantum mechanics, although having said that there's a lot of rigorous foundations to be solidified here. Once you go to quantum field theory the fact that we have to deal with renormalization makes this whole story much more interesting. Because the expansion parameter is not just some small expansion primary it's actually running according to some well defined renormalization group structure. There's been a lot of recent progress in the last five to 10 years for specifically regularize quantum field theories at some in some semi classical regime, and also in lattice quantum field theory. So what I want to talk about here is something different, and the idea is to try and see how this research instructor fits together with this, the Hopf-Algebra structure. And so the starting point is to try and find the simplest possible example where you can explore what this looks like. The question I have in red is that if we think of the Dyson-Schringer equations for the quantum field theory, which we usually formulate in a formal perturbative sense. An interesting question is do those equations contain all of the information about the quantum field theory. If it happens to be a quantum field theory that has non-trivial non-perturbative features. If the answer is yes, then you can ask further how can you actually extract or decode that non-perturbative information from the perturbative formal series in a practical possibly numerical way. And possibly the most interesting question is given that this Hopf-Algebra structure is so natural and so elegant and so efficient at the perturbative level. Is there some natural way that it fits into these so-called bridge equations of a car, which are the relations that relate all these different critical points. So these different critical points are associated with different non-trivial features. So three papers from Doug, that at least in my thinking have been very influential in how to understand Dyson-Schringer equations. So these have some very beautiful features in them that we can take advantage of here to start thinking along these lines. So one way to phrase one of the conclusions of these papers is that for certain quantum field theories, when you combine the RG equations with this Hopf-Algebra structure, you can reduce some part of the problem to a set of nonlinear ordinary differential equations for the enormous dimensions of the problem. And this is the realm in which resurgence is best understood. There are a lot of rigorous results and practical results for understanding resurgence once you're in the world of talking about nonlinear differential equations. So it's a completely natural place to start. I emphasise that it's just the beginning, it's not the general picture, but I think it's worth having some very concrete examples to understand before trying to formulate the more general case, which would be aiming for something in the world of cage theories. So there are some well-studied examples, and I think we'll hopefully hear from Mark tomorrow, I guess. Maybe he'll say something about this beautiful work on the Western Mino model. And recent paper with Michi, we studied the four-dimensional Yukawa model, and there'll be soon hopefully a paper about this six-dimensional Phi cube theory, which also Mark and Enrico have a recent paper about that. So just a summary of what's known for nonlinear ODEs, and using the notation from famous paper by Ovidio Kostin, you can write an n-thorder nonlinear ODE in first order form, so you make this an n-component vector. So y prime is just some function of the variable and y, and under certain conditions that are not too strenuous, you can form this into a normal form, where there's some formal Potobotus series, and then these are the eigenvalues of the first derivatives of f, and then one over z expansion, and the rest is suppressed at large z. Once you have it in this normal form, it's completely clear that the solution has the form of a formal series, an expansion here, here z is like one over h bar, so an expansion in inverse powers of z. But then because of these terms here, you get exponential contributions, and here you get inverse powers of z depending on the eigenvalues, so you can diagonalize this with eigenvalues beta one, beta two, up to beta n. And then this nonlinear stuff generates some formal series here. So this is an example of a trend series. Here we have just a formal series in inverse powers of z, but it's complemented with these exponentially small beyond all orders terms with some pre-factor fluctuation terms, and also multiplied by formal for the formal series. And there are very rich bridge equations which relate the structure of these various fluctuation series to one another, and to the original formal divergence series, and these are called the algebraic bridge equations. And if you form the singularity, the Braille transforms of these functions here, then the singularities of those Braille transforms label exactly the non-particular physics terms, which are these terms beyond all orders. The parameters here, sigma one up to sigma n, are the n independent parameters given in the boundary conditions of this nth order differential equation, and they're called trend series parameters. And here's a quote from this paper of Kostin, one of the interesting consequences of this structure is that given one formal solution. So you can actually generate from that a whole family of solutions beyond the formal asymptotes. So what I'm going to do is explore how this actually works in some particular examples of Dyson-Schringer equations. So the easiest one to start with is this example from this paper of David and Dirk of the four dimensional massless Yukawa theory. I don't think I need to spend terribly much time on this. There's a simple Dyson-Schringer equation iterating these self energy terms, and the anomalous dimension in a momentum scheme defined this way in terms of the renormalized coupling leads to a RG equation that's this nonlinear first order differential equation. So this is the result of this paper. And now given that we can study the formal expansion of this. This is done by David and Dirk and it has some very interesting combinatorial structure and asymptotic structure. So just rescaled a little bit just to clean things up. Instead of a function of alpha, divide alpha by four. And here's the nonlinear equation. Again, you see it's quadratic in this function C. C is essentially the rescaled in almost dimension, but it's first order in the derivative. So just rescaling has been done so that the expansion coefficients here are integers. And this is interesting combinatorially because these numbers correspond to the expansion of the generating function for connected chord diagrams. These numbers also have interesting large order asymptotics. So this is the factorial n plus a half and there are sub leading corrections going like gamma of n minus a half gamma of n minus three house. So this is the typical generic type of factorial divergence that is found very often in quantum mechanical and quantum field theory, asymptotic expansions. And this is not either full solution to this differential equation because there are no boundary condition parameters here and this is a first order equation so there must be something else beyond this formal series. And this is a simple example of this idea of a car that we generalize this formal series to a trans series with one trans series parameter it's a first order equation. The first power of this parameter Sigma would just be this formal series, but the higher powers would correspond to these exponentially suppressed. So in practice you simply it's extremely simple to implement you simply insert this type of unsat into the differential equation, connect power collect powers of Sigma. And you see that you have a non non linear equation for C zero the perturbative series, but all of these other ones satisfy linear but inhomogeneous equations. And you can solve them it turns out with a little bit of work and here's the first term here it's actually expressed as an exponentially small contribution involving C zero. So this is an example of this feature of resurgence that the, this exponentially small contribution not a bit of a contribution corresponding to say the one instant on correction to this formal series is explicitly expressed in terms of the formal series itself. Moreover, you can make an asymptotic expansion of this quantity here at small x, and there's this exponentially small instant on factor, and then it's multiplied by another formal series with certain coefficients that you can simply generate very easily, since you know this. And now you look at these coefficients and go back and look at this large order behavior of the formal series coefficients and you notice something very interesting that these numbers here are just identically equal to the coefficients of the sub leading corrections to the large order behavior of the perturbative series coefficients. So this is a generic feature of resurgence that the large order behavior of the expansion on a given instant on sector here the zero instant on sector show up again in the expansion around another instant on sector. And these coefficients here in C one this one five halves this year have also some interesting common ethics associated with them recent work of Carolyn Ali. So we can continue this we can actually look at these coefficients of the C one and look at its sorry their large order growth, and we generate another set of numbers here. And then we can go to the differential equation and look at the next exponentially suppressed term. And again we see the same feature, these coefficients here show up in the low order expansion of the fluctuations around a higher exponential In fact, you can continue this to all orders in the expansion and generate and completely all orders summation for the full trans series. So just to remind you what we've done. We've generalized this formal interpretive series solution to this differential equation this distance ringer equation to a sum of what I'd be calling non perturbed of instant on terms, and each of these it turns out can be expressed in terms of C zero, the lowest one. And not only that not only can each of them be expressed in terms of C zero, you can completely re sum to all orders with this generating function here little f is some function of two variables and you notice that this is exactly the structure of this C one solution here. So this itself is just exponentiated again, and that gives you the all orders complete non perturbed solution of this station stringer equation. So this is an illustration that we can indeed implement this resurgent asymptotic structure of a car this general structure. And since this is a first order differential equation in this case even though it's not linear, we can carry out all of these steps and complete generality. I would also like to mention that another way to do this calculation is using this nice alien derivative structure that Mishi developed, where you use the fact that there's a functional relation satisfied by this function C of x and applying this alien derivative which is the thing that gives you this structure of bridge equations on all the singularities of the broad transport, you can generate exactly the same all orders resummation. So just to summarize here's the picture again. Imagine we started here with the formal divergent asymptotic expansion coming directly from the interpretive hot found structure. Using that non linear equation derived by David and Duke, we can actually generate all of the other non perturbed terms directly and show that they are. They correspond exactly to the singularities of the Braille transform of the perturbative series and that they're all expressible in terms of the original formal series. So this is reassuring that one can actually do this in complete detail. So this hence the conclusion would be that indeed the station stringer equations have all of this non perturbed information, you just have to do a little bit more work to extract it. So having done that can turn now to a slightly more complicated and more interesting quantum filter. I've also studied by David and Dirk in their papers, which is five cube theory in six dimensions. I'll use this expansion parameter alpha the natural expansion parameter alpha. So this theory is interesting because it's asymptotically free six dimensions is the critical dimension of this theory. It's a non Lepatov style instant time. So that's a non perturbed feature of the theory coming from a subtle point of the functional integral, not directly from some diagrammatic perturbed expansion. It's also interesting because it has a renormal chain structure from chains of these bubble diagrams. So these features here that have listed hint at the possibility of interesting non perturbed physics in this system. So in their work now 10 years ago, David and Dirk uncovered a third order non linear differential equation for the anomalous dimension in this theory. So I've done a little bit of rescaling again to just get things in sort of natural units. So here's the third order equations you see there are three derivatives. It's fourth order in non linearity in C is very interesting factorized type structure. And David and Dirk studied the perturbed solution to this equation you just expand in small powers of X and with some appropriate rescaling here, you have integer expansion coefficients and I've listed the first few of them here. And as far as is known at this point, these interesting integers do not have any direct combinatorial interpretation. So that's an interesting open problem. Whereas in the Yukawa system, the corresponding sequence of integers had an interpretation in terms of these connected chord diagrams. So David and Dirk generated, I think 30 or so of these coefficients and found indeed factorial divergence establishing that this was an asymptotic expansion. And they also studied the Boral summation of this. So with some more data and we publicly thank David here for sending us a huge file of coefficients for us to play with. You can generate more refined asymptotics. So first of all, this is the end just from the natural rescaling. But the factorial divergence is in fact not gamma of n plus two it's gamma of n plus 23 over 12. And you can actually generate the sub leading corrections and there are these particular coefficients corresponding to gamma of n plus 1112 and gamma of n plus and minus 112. So this has very similar form from compared to the asymptotics in the previous Yukawa case. However, this was a third order nonlinear differential equation. So from the theorem of a video constant, this means that there are actually three missing boundary condition parameters, not just one of these sigma trend series expansion parameters, there must be three of them working somewhere. And to find them is actually quite simple. Just insert an ansatz for the form of the expansion with this exponentially small term beyond all perturbative orders, with some as yet undetermined pre factor power, multiplied by a formal series in powers of X. And just inserting that into the differential equation is very easy to see that there are only three possible solutions and their combinations lambda and beta one lambda equals one lambda equals two lambda equals three with corresponding rational values of data. And you should recognize this 23 over 12 here as this 23 over 12 here and there's a very natural reason for that which others playing in a minute. Another interesting feature of this is that these lambda parameters which are remember these are identified with the locations of the singularities of the barrel transform of this formal solution of the formal solution to this problem. They're actually resonant, they're multiples of one another whereas for a generic third order equation. These three eigenvalues lambda would be at any points in the complex plane. And this is related to the fact that this equation factorizes into this factorized form with one two and three here responsible for this one two and three. So what I'd like to talk about now is to explore this non perturbative structure with these three different types of non perturbative terms. So what it means is that the full trend series structure has the formal perturbative series remember this is the thing coming directly from a formal expansion of this alphabet structure. But the full solution to the differential equation can be expanded to the form of a trend series with three trend series parameters sigma one, sigma two, sigma three these correspond to these three basic solutions from this trend series and that's and powers of the corresponding instant on factors here with these strange rational factors. And each of these is multiplied by some formal series. And then there are plus dot dot dot where you mix these various times. But these are the basic series and seed non perturbative terms. So given that you have an explicit differential equation, and you have the form of what the trend series should look like. And if you have access to something like maple or Mathematica, you can simply insert this into the differential equation and start generating many terms in this series so it's not something you want to do by hand. But it's fairly straightforward to do this using some mathematical manipulation program. So for example, you can look at the first few terms of this series here at the K equals one level so the first instant on term with the lowest. So the dominant exponentially suppressed terms and the first few coefficients are some rational numbers here, which you can get from the differential equation. But those numbers here should look familiar. They're actually the terms appearing and the sub leading corrections to the large order behavior of the expansion of these of this formal series. So this is again an example of this generic resurgence structure that the larger the behavior of the coefficients of this formal series research in a natural form in the low orders of the expansions here around these non perturbative terms. So this is just one manifestation one example of this infinite set of resurgence relations, implied by a callous bridge equations. So there are all these relations between the different exponential contributions which remember correspond to the different barrel singularities. They're all related by some intricate algebraic set of relations. And this is one of the examples where we can actually see it in in full detail to go beyond this. So this you can do just by things like studying the large order growth of the coefficient so you can do ratio tests and then you can accelerate this with with things like Richardson acceleration. But to go beyond this since there are these three different barrel singularities to go beyond this it's really necessary to invoke not just this sort of combinatorial treatment but it seems to be much more powerful to go to the barrel plane and start using more analytic properties. So remember what I'm calling non perturbative information. These are directly related to these barrel singularities so remember what a barrel transform does it is it takes a formal asymptotic series such as would be produced by this half algebra. Schwenger equations and converts it by factoring out the overall factorial growth converts it into a convergent barrel transform, but the barrel transform function is convergent but it has singularities in the barrel plane. And those singularities correspond to these exponentially small corrections to the formal series when you do the inverse barrel transform which is the last transform. So, what from the quantum field three point of view the locations of those barrel singularities correspond to what we would call instantons or renormal ones. And the nature of those singularities whether they're branch cuts or if they're square branch cuts or logarithmic branch cuts or whatever type of branch cuts. So those facts tell you something about the gas inflectuations around those non perturbative objects and the Stokes constants which are the constants appearing in front of this overall factorial growth. These Stokes constants tell you something about the strength of these instanton terms and how they relate to one another through these bridge equations. So this is interesting and important information that you would like to be able to extract from the differential equation and in fact extracted from just this formal series solution to the differential equation. So to do this, you need to use first of all work in the barrel plane where you have some convergent series to deal with. And it turns out that to do this to the required and desired precision requires some new methods of barrel analysis. And this is the work I've been doing recently with the video custom. A very common method used by both mathematicians and physicists is so called Burrell per day, which is once you go to the Burrell plane you have a convergent series and you would like to understand the singularities. But of course in practice you don't have necessarily all the terms of that convergent series you might only have 10 terms of 50 terms or 100 terms. So in order to study the singularities of that function which is in the end just a polynomial. One possible approach which is very powerful is to make a party approximate to that truncated expansion and the party approximate represents that truncated series as a racial polynomials. And so it's singularities are necessarily only poles. And this is both good and bad. It's very easy to implement and it gives you some hint of where the singularities are and what they might be like, but it doesn't give you much more detailed beyond that. So it gives you a good starting point approximate information. So for example, how they struggles to represent the branch cut because it's inherently a representation in terms of rational functions. What it does do is if it identifies the leading singularity in the Burrell plane, the T plane will be the Burrell plane. It would represent a branch cut by putting a whole sequence of poles along this axis accumulating to this branch point. And if you had detailed information about the distribution of these poles and also zeros, you would be able to tell what type of branch cut it is. The problem with this is that in a nonlinear problem such as we're dealing with here, if there's a Burrell singularity here, there's necessarily a Burrell singularity repeated at all integer multiples of that. This is this generic truncated structure. So it's hard to see these because these are further singularities, but they might just show up as one of another pole hidden under these line of poles that are trying to be a branch cut. So there's a way around this which is to make a conformal map from this cut plane here, bring it within a unit disk, and a feature of this conformal map is that it resolves and separates these singularities. So this singularity is placed here and it has its per day approximation will have a series of poles accumulating to that point, indicating the nature of the branch cut. And the second singularity which is hidden is actually completely separated up here also with the line of singularity of poles accumulating to it. And so, so you can resolve this higher order research and structure. And in fact, a theorem in fact in this recent paper with a constant is that if you use instead of a conformal map a uniformizing map. This is actually the optimal procedure to resolve and probe these structures. Because after all what we want to know is not just where these things are but what type of singularity they have and what is the corresponding Stokes constant. So when I say optimal in practice what that can mean is something like here. So this is an example of the Burrell transform with a singularity t equals minus one. The behavior is known to be one plus t to the minus 35 over 12. And so if you're interested in the coefficient of the that singularity simply multiply that by this and it should go to a constant. And so here we're approaching minus one and notice the scale here. If you use a conformal map in blue, you can get to several digits, the Stokes constant multiplying this behavior, but if you use a uniform as a map, the precision is much, much greater you get many, many more digits of precision. So that's useful in getting high precision in the Stokes constants, but it's actually more important in other examples or applications. So for example, if you look at the, not just the leading coefficient, leading singularity here, if you try to probe this sub leading coefficient, so you're trying to probe this point here. We find if you look at the asymptotics of the coefficients, it has a dominant contribution here, going like gamma event plus 35 over 12. So the sub leading corrections to that indicating that there's a branch cut at that point, but it turns out there's another singularity on the positive real braille axis, with exactly the same location so the same distance from the origin. So the radius of convergence here of the braille transformer say one normalize it to be one, but both of these singularities are the same distance from the origin. Which means that if you just look at the asymptotics of these coefficients, you have a competition between these two terms, one of which is alternating in sign and the other which is not. And simple things like ratio tests are not good enough to disentangle these two competing terms, let alone this sub leading corrections. However, if you use one of these uniformizing maps. So you map from this doubly cut plane using some particular elliptic function. Remember that you can use elliptic functions to map the three punctured sphere into one of these so called geodesic triangles here, reflected to form a geodesic quadrilateral. So this point here is mapped to here, this point is mapped to here, and these higher singularities are mapped to the boundary of this geodesic quadrilateral region. And once you're in this region here that's the region in which you can optimally analyze the structure of the braille transformer. With that information, you can identify all of these coefficients here of these leading and sub leading corrections. And just to illustrate what this means in the combinatorial language. If you just took the dominant singularity behavior so this is the fastest factorial growth, which corresponds to this braille singularity here. And you took the actual coefficients relative to that leading factorial growth. This is the type of behavior you would see this a solitary behavior in this wide band. However, if you include also this competing non alternating growth from here and the sub leading corrections. You see that you get this ratio tending to one much more rapidly and to much higher precision. And the reason that there are still these sub leading oscillations is that the sub leading corrections of this alternating behavior have a factorial growth which is extremely close to the strength of the factorial growth of the leading non alternating term. There's only a difference of 23 over 12 versus 25 over 12. And working in the regime of just using the expansion coefficients. It's extremely difficult to disentangle these. So with a braille plane, you have it your, you have access to more analytic tools for extracting these independent terms and their coefficients using these analytic continuation techniques based on conformal maps and uniformizing maps. So to summarize what we learn about this scalar phi cube theory, it has a much, much richer non perturbative structure than the carl model which you can trace back to the fact that the corresponding Dyson Schwinger equation as a third order structure and a fourth order non linearity. Independent non perturbative structures which nevertheless interact with one another because of this resonant structure of the parallel singularities line on top of one another and integer multiples. Nevertheless, we still see evidence of these large order loader resurgence relations that deflectuations of the formal perturbative solution that was generated by David and Dirk. If you look at the sub leading corrections to the larger growth, those same coefficients show up as the lower the coefficients of the first non perturbative correction to the formal series. And we also see just from the differential equation. This is just a concrete example of the result of Costin's theorem that all of the non perturbative terms are generated from linear differential equations which have an inhomogeneous part which are all encoded in terms of the original formal perturbative series. But all of these non perturbative terms can ultimately be generated from the formal derivative series. All right, so that brings me to my conclusions. So there are more questions in the conclusions than outcomes. But let me just remind you what the goal of this. The idea is to ask whether it is indeed possible to learn something about the clone field theory, starting from the perturbative Hock-Valgebra structure, which is formulated in the language of formal and in practice asymptotic expansions. And to use the techniques of recertan analysis, recertan asymptotic analysis, to extract from those formal expansions some non perturbative information, which when combined with the original formal expansions, give you the complete non perturbative solution of the Dyson-Schringer equations. So given you two examples, they're probably the simplest examples you can possibly think of, but the phi cubed theory is richer than the Yukar example, but we see that it's possible and it's actually fairly systematic and quite easy to implement. So that's sort of the good news. Now, the more interesting news is the more interesting question is, is there a more efficient way to do this? So, you know, this is a bit of a roundabout way to do things. So we've taken the natural Hock-Valgebra structure of the formal series, used the fact that there is this Hock-Valgebra structure, which interacts nicely with the renormalization group equations as a means of generating this perturbative information. And then we apply, say, Borel analysis to this to extract the non perturbative trend series structure. However, it would be much nicer if there are a way to directly understand this non perturbative trend series structure in the same Hock-Valgebra context. And there are strong hints that this should be possible and they must be possible, just because in the end, a Karl's bridge equations, which relate which are the resurgence relations, which tell you how to extract the non perturbative information from the original perturbative series, those are inherently algebraic. And so I strongly believe that there should be some nice fit between that structure and the Hock-Valgebra structure. However, the bridge equations are extremely general and extremely formal. And it's hard to be very concrete about them, unless you're in the formalism of dealing with couple non-linear differential equations or difference equations. And so that's why I would motivate this type of questions here as the right place to start, because it's probably the simplest venue in which you'll be able to recognize how these two algebraic formalisms could fit together. Well, another possible direction to try and make this work is to use this very nice result of Michi from his thesis, where by studying the ring of formal divergent series, he was able to find a very natural and extremely explicit implementation of these bridge equations and alien derivatives of a Karl that's much, much more concrete than in the completely general resurgent case. And I think that is very likely to be a useful approach to this first question. So in the Yukawa case, we're able to do this in complete detail, but even in this Phi Cube case, it's not completely clear yet how this works. Further interesting questions more from the quantum field theory side relates to some of the very nice work that John Gracie, that these Phi Cube theories become a lot more interesting, and especially in six dimensions, when you're just more structured than just having one five feet of them. And these, these types of generalizations would be extremely interesting from the quantum field theory point of view. Another question that's still open is how does this connect to these results of, say, Lepatov and of Brennan, so these are other sources of non-perturbative physics, and it's not obvious yet how these other sources of non-perturbative physics fit with this half algebraic picture. Instant times are usually understood as arising from the generic factorial growth of the number of diagrams, and renormal arms are usually understood in terms of the behavior of iterated structures of diagrams. So it's it feels like they should have some natural place in this discussion, but it's not fully understood yet. Other slightly more technical questions maybe other renormalization schemes beyond the momentum scheme might be more efficient. And then ultimately to look at some more complicated quantum field theories involving real gauge structure. And so that brings me to my real conclusions, which are today, happy birthday, looking forward to many more years of exciting physics and mathematics. And also thank you for your wonderful ideas that have been extremely rich and influence many people in many fields. So thank you very much and happy birthday. Thank you so much. Thanks, Daryl. In the chat, there was a program that you want to ask the question. Yeah, okay. Yes. Yeah, hi, hi, Daryl. Hi all. And unfortunately, I had to come late to your talk, but I have a question to the very beginning of the talk, as I'm sure it was there. So, could you go to the slide where we are starting ODE is for gamma. In which case in the UK case. Yes, in the cover case. So here, here it is in terms of alpha, but let me let me show you the next slide. Wait, I don't think we can see a slide. Stop sharing. Okay, hold on a second. Okay, do you see it now? Yes. Yes. Okay. Let me go to the next slide. Yes. Yeah. I see. All right. Yeah, but can you go to the, to the previous slide? Yes. Yeah, because the different I have my question is regarding the differential equation for gamma for the anonymous dimension. And I'm asking what, I mean, this differential equation is obtained by by David and Dirk and the paper called exact solutions and my question is whether this you have to put in more than just the Dyson Schwinger equation to obtain this, right? So you need that it's scalar. So that's in six dimensions and that is scalar. This is the Eukala. This is the Eukala. Yeah, the Eukala. I can go to the scalar one if you like. No, no, I just wanted to know whether there's more than just the Dyson Schwinger equation to obtain this nonlinear ODE. It's a conversation group equation. Yeah, but the, you know, no, no, the renonization group equation you get that the higher gammas, the gamma two, gamma three are recursively given by gamma one by this. There's a recursive formula, but this is from the renonization group equation. But I can't see this. So, you know, this was a short talk. I didn't go into the fact that you can generate the Green's functions and so on. But yes, I keep it nice and explicit here. Yeah, I was talking to Michi yesterday during the evening and this morning I tried to obtain this equation using the renonization group equation and the Dyson Schwinger equation. And I've got the feeling that I'm missing something. And I think that something is about how the explicit integral is built. So there's this remarkable duality between these two. And I think that's the key. Yeah, so there's something missing in a, yeah. Yeah, just to, if I remember correctly, it is indeed true what you just mentioned is that this is particular for this insertion into this. So you write down this integral, and then you compute the balance transform and that goes into this equation. And in the other case you have a different balance transform in the six dimensional scalar case and you get this third order thing. So it is true. This, this particular equation really depends on the particular integration kernel. Yes, exactly. Okay. Yeah, thank you. Thank you. We have also a question from David. Yes, thank you again general for this wonderful exposition from my work in the 1990s and engage theories I know that, at least at large and I had a much worse situation with with no half plane analyticity. And how to undo the Burrell transforms as Laplace transforms because there's intrinsic into ambiguity. Now in the Phi cubed example, you've pointed out in these higher instant on corrections that you don't have half plane on the city. So is your result intrinsically ambiguous if I give you the three boundary constants is there still a unique answer if you haven't got half plane analyticity. Yes, but that would take a long time to explain that there is the answer is yes. Yes, there is a, so this is in this 1998 paper of costing. I can, I can send you some more details about that, but there is a way to define a unique parallel a call solution to the differential equation. Thank you. Mark belong. Mark, did you want to ask a question. I would, I would just point to, to uncover that using the formalism of mailing transform is quite easy to to obtain the, the equation that for the linear equation for for data. Yeah, that's a very natural way to. One more question from Enrico also Enrico. Can you hear me. Yes. No, sorry, it was just the same thing that Mark wanted to say I wanted to, to answer to all of saying exactly that you have a natural way to right finger dice an equation you use mailing transform and then you take the first derivative with respect to L the log and you have an equation for the normal dimension in a general way. Well, okay, that's it. Mark said, yeah, thank you all. Thank you very much. Let's thank each other again. Thank you. Thank you.