 Oh, it's a pleasure to be with all of you, even if it would be nicer to meet in Bure and have some talk under the trees and around some real blackboard and not only virtual ones, but... Nevertheless, I want to share with you some impressions I have about our relationship with Duke. So, the one who knows me, I'm post-credit data extraordinaire, and I don't really give anything without a deadline to the image show. And in some sense, I hope that having two deadlines, the first one for the first date, and this one would allow me to have some polished and refined talk to present you. But in profession, I just finished the presentation you see a few minutes ago. I think that this comic page image has it all. I've quite worked out the problem just yet. I'm hoping to think around the problem and have epiphany eventually, and to show that it has worked for me. In some sense, to just wait a little more on things suddenly become clearer, and just find it a nice way to present the things I've thought about for a long time. For this talk, I could have spoken about our recent work with Enrico also. But at the time where the conference should have taken place, we were just not ready. I could have switched subjects to talk about this recent development, but somehow I found it nicer to stick to the initial project. Yes, one other thing I could have shown is speak about resurgence and all those things that I learned studying the work at Efectile. I was seeing that they can make tremendous opportunity in field theory to resolve some long-standing problems. But nevertheless, I will try to have some other things. In fact, what impressed me and what Derek has done is that he's been quite all over the world since his stay in Prismaya, which is maybe one of the farthest part of the world. I still can do physics, and myself I began physics in what could be and really thought of another world. When I was thinking about a series subject, it was 1983, but it was quite another world. In France, at this stage, lab, quite a big lab, would only take one student a year because they would like to be able to provide for a kind of opportunity for the guy. And so many people would start physics in a lab and then merge from the place for their entire life. And so, in this year, in 1983, I was just 22, 21, I went into the office of André de Veuil, which was some sort of informal advisor in Ecole Normale d'Iberia. And I said to him that I wanted to do something about non-perternative problems in quantum physics. And he had not quite big things to say for me. And he said, oh, it's a bit early. And then at the end of the year, it was too late, and I was just admitted as an additional student in Ecole Normale in the theoretical physics lab of Ecole Normale. And it just began some wondering on different thematics. I would not bother you with the lift. It's quite long on the way. I've done a number of things. I've been quite happy with it. In some sense, it was just when I found this article of bothersome karma, the possibility to go up to 30 loops in some simple quantum field theory. A problem that I went back to this old ambition I had. Now maybe there is some tool to go beyond the limited things which people have been able to do up to now in quantum physics. I must say that before I was quite intrigued by the relation with notes that is developed in this book. In fact, I must confess that I had the book on my desk in the lab for a few months. And it was just at the moment where the conference was announced to be delayed that I finally decided to open it. Now David will allow me not to speak about what I think about this relation with notes because he said it quite clearly in a way that I would not dare to use. But in some sense, it was the first time I heard about Dieg. In the next installment, I was quite disappointed. Why do we need this hope alzheimer structure? If in fact you can reach the exact same conclusion without using anything more than differential equations. In fact, it's something I would meet again later. In fact, the real meeting with Dieg was when Alan Korn exposed the common words in his course in Collège de France. I followed the course for some years then and I would be two participants up to the end. And it's sure that I missed this yearly encounter in the room 5 of Collège de France with a bunch of people that would become familiar along the years. And it was quite a revelation. I had been quite frustrated during my graduate studies in Paris that much would be said about renormalization but nothing very clear about practicalities. And in fact, nothing really beyond what is one local realization and all these quite important things that a good theory should be renormalizable. And it means renormalizable up to any other in perturbations theory. Nothing was said about how it could be done. And so, with Alan Korn, we learned about how to do begolubal recursions and use the hope alzheimer of craft to solve the difficult combinatorial problems which are there. And at one of his talks there was some person present which we talked with Alan and I learned that it was there. I would not speak with him immediately but at this point he stopped to be some man but a real person with some physical characteristics. And so at this point I was just decided to work on this stream but it wouldn't have to wait some more years. I proposed to a colleague in our lab to plan in the subject but he was not so hot and so we just went to do some numerical work on quantum integrable systems. And so the starting point would have to wait for two things. I don't really know which order. Maybe I said the personal one before. So a conversation with Fidel Scharbovnik encouraged my wife. I decided to pack and go to Argentina. And it had to be for one year and it was two years and it was a good thing because in the first year we could not achieve much. A little because Fidel had a serious health problem. What could have been the benign theoretical integration turned out to give him some nukosome injection which he had some struggle to clear. And so with my Argentinian collaborator we discussed and said what can be done with this hop-fragima. And the idea emerged why not try the passivity. I'm still not sure if there is a good notion of a graded version of the hop-fragima of graphs. I've learned that it could be useful if you want to work with bearer symmetries. In this case you would need some other graphs to describe the non-linear parts in the bearer transformation. To make them into a full-fledged set of data identities. But for me it's still a work to be done. Pierre Clavier has done something about my first student. I've done something about the battle that will go with formalism. And I hope that maybe something along this line could be made but it did not pan out. Finally we had just the idea to look at the simplest symmetrics. In field theory the western model in the fourth dimension. And first we just started the linear licensing equation in the vein of Hordes and Kymer. Which we were quite disappointed that the equation when obtained was simply the one apart from some rescaling that Hordes and Kymer had obtained in the Yukawa case. However when trying to apply these things to the non-linear licensing equation. Something quite interesting with Haven. First with respect to the things which were proposed but not really solved in Dirk and Karl and the article. We would have only one propagator correction to consider. Because Scholar, Fermion and even the auxiliary field would have the exact same correction to their renormalization. Furthermore the well-known renormalization property of this model means that it's not a further approximation to forget about vertex correction. They are just zero. And so we embarked and I did not present the paper which results in 2008. And we were quite happy to be able to present the result up to 200 loops and 200 at this stage. A few more loops were not so difficult to obtain. Then the question surely was, but what to do with all these numbers? It's one thing to have a long list of coefficient for a perturbative series. It's another one to be able to deduce from this truly non-perturbative information. For this I would have worked to wait for some additional information and I will not dwell in it now because it's not really the thing for which words of Dirk have a direct relevance. What has been truly relevant for our most recent works is this quite fine paper by Professor Barron and Dirk Reim. I would not say it's an easy read. I must confess that as a referee to the paper I tried to obtain some clarification. But nevertheless, sometimes later when I needed the ideas I found that there were some pretty nice ideas there. What is the point? Well, maybe I will try to save some more time. I will show you something on the blackboard. I will not do what I've done in the president conference. I let my chair block the view on the blackboard from my computer. I will stop the tension so that you can hopefully have a nice view of the blackboard. I'm just following you since I said that I will not speak about my recent work with Enrico, but this is all about this work with Enrico. What do we do with Enrico? I will try to go beyond this mathematical model in which I felt quite stuck for many years. Because generalizing equations for general theory models are not so nice. Because you may only remember that when you look to the correction to the propagator, you have this quite symmetric situation where you have one bare vertex and one full vertex. But you must subtract infinity for the parallel vertex, so you cannot use this to do a deficient solution which would require quite a lot of regularization. So we used an old idea of the world. From which I learned through theology that it's more than six years old. You just take a derivative and then everything is fine. But there is still a problem. You survive to a diagram of this kind. But how do you compute this thing? A general vertex has a complicated dependence on this and it. And you cannot just, it's difficult to evaluate such a diagram if you presume that the vertex depends on all the invariants. But it's there that the ideas on a timer would be there. It's maybe older than what they done. But through them that I learned about it and whether I find that there are shortcomings in the exposition. But that is quite clear. You just forget about the situation until you do the bold, bold first thing that you will completely avoid. You have three vertex, five, three point vertex. And so you make the vertex depend only on one scale. And then it's something quite similar to the. And the evaluation of such a diagram is just another form of one of the diagrams. And so you can go with your computation. It's sure that it's not the end of the world because you are doing some approximation by shifting this in some way. But the nice thing is that in fact, you can write the correction to this approximation as some. You just do the difference. And the nice thing is that this difference is more. And so we can go back. We can use all these things to. I will go back to. To share. And you should be. So. Maybe now I can. I will catch the point that whether. Why I go to the maybe the not so. Hidden motive of this. This talk. Why did we cannot and I not go to the next. Yes. This is a part of the work of Derek that I've not used up to now. And maybe I will. You will. Allow me to ramble a little. Why. Why do I think that it's important. It's that you will we. It's not going to. To study. Unitary. Unitary questions in. In contemplation. This is at the level of. Single daggers. And I would like to do this. At the level of. But in terms of. I think that it's not. You that. We. Go back to. To mean koski space. And the. Unitarity and. Analytical problems. Which have been. A little forgotten in past years. It's. Some old story. Maybe. Even old. When. I began. Knowing. A. A theoretical physics. But in. In the 60s, there was. It's big. Excitation. A. A bootstrap program. You. You know that. The. Contemporary. Some. Pretty. Strong. Concerned of. Unitarity. And. Analytical dependence. And a bunch of variables. And. Even if quite mysterious. And. Crossing symmetry. And. At the time. It must be. It could be sufficient to. To know everything about. Strong integration. In terms. It was. Not so. Not so far fetched. I did. Since. The. The bond. The bond. Unitary. Confections. Due. To. Unitary. Concerned. Is. Pretty much saturated in. Hadron interactions. You. And when you. Look at. Amplitude. Process. With the production of. Many parents. Finally. The. The. The phase space is. Quite. The. The. The most important cycle. In fact. Nevertheless. It. Was too short. In the information. And the. The idea. Would. Would. Be appeared. All the more. So then. The first steps. Into the. Standard model. Say that. Contemporary theory. Had not said is. That's what. Nevertheless. Remarked. One side. Effect of this project. With that. Proposed. An amplitude. With. And. Sometime later. People recognize that it was. Some. Amplitude. For. And the rest is history. Like. When we said. And so. I. And so in some sense. The. In. The usual swing like. Station. The. The. People had burned what they. Worshipped. On. All the more than. And. With an approach to the normalization. Group. And. Good. And. Would give a prominent place to the. To the. To the. And. Contemporary theories. With. The. With the ability to be. Subjected. To. Some. Regular. And. Approaches. Of. And. What I would say. Of. Constructive. And. And. Never the less now. I think that it's time to. Go back to. To. Considering. This. Question. And I would. It's my. The. I would like to. To say. Every time I can. That. You. You had. You have to add. Another dimension to the. It's the. In the. With which. You. Can have. The. Door. To. To. Is maybe. Subject for. For the talks. I will. I think that. Will be quite time for me to. To stop. I may. Just show you. A number of. Things. I've done. Recently. Is it. wanted to show you this. So what you see there is the quite huge number which was in the appendix of of the paper by Brogles and Kramer. So it's what you get at order 500 in the when solving the linear synchronization equation for the scalar field in six dimension problem. And so I just tried to, you can see the progress in computers. It took only a few seconds to obtain this number. With my MacBook, which is far from being a top of the event machine. And this is just a first step to a full uh, uh, resolution analysis. But uh, there are some quite difficult points to to have. The other thing I would like is it just what can be done with, with, um, no, in, in, um, um, in the simpler, uh, case, uh, Geralt explained you that in this case you, you can have a quite complete, um, uh, resolution analysis. But this used quite special properties. So I wanted to see what could be done, uh, using, uh, more generally available tools. And so, uh, the idea is that you, you can compute, uh, uh, terms in the, in the, um, the transition equation. But if you convert the exponential terms in, uh, in, um, in what Euclid calls, uh, uh, uh, transponomial or reversant polynomial, you, you, you can see the nth order, um, so, uh, on the, on the upper scale, it's, uh, uh, perturbative order. And so I'm fitting the perturbative order. Uh, the, um, uh, the perturbative series up to some perturbative order with the help of one terms, two terms, or three terms, as, uh, uh, the interesting point are these ones. The, the yellow one correspond to, uh, not too well optimized, uh, computations. I'm able to compute, uh, uh, the coefficient which appear in the, in the terms, in the, uh, in, um, in the bridge equation. So the coefficient which will give you the exact island derivatives with better and better precision. Uh, in the vertical, uh, uh, scale, we have the log of the difference between the, uh, the number of, I can compute using, uh, a number of, uh, series coefficients and the limit. So you, you see that using more and more terms, you can have faster and faster convergence. And this may be, uh, a good thing for quantum fields, general quantum field theory where even with the tremendous work of research as we were exposed by Oliver Schmetz, it will be difficult to, to have full computation between, beyond the five, six, seven loops or eight. So thank you for your attention. I just might say that it was, uh, uh, a wonderful thing to be able to, uh, build upon the, the, the nice work of their, his collaborators. And I, and I really hope that it will be possible to, to, to, uh, to have, uh, all this, uh, uh, ideas that I advocated in the last part of my work be fulfilled with cooperation. Thank you very much. So, and, uh, it's really, uh, uh, a good Thursday. All right. Thank you, Mark. Thank you. Are there questions for Mark? I have one, if I may. Um, you mentioned this, um, in the West Sumino model where you replace the vertex correction by just a kind of one scale correction by rerouting the external momentum, right? And, uh, uh, yeah, what did I, what I, what I heard on the, on the, on the, on the word, yes. Yes, exactly. So, so I was just wondering, um, if I remember the, the paper by Francis and, and Doug correctly, the angles and scales. Um, I mean, there's quite some procedure, right? Where you, if you have sub divergences, you have to reroute the, the momentous and the sub divergences and make everything one scale in a recursive way. So I was just wondering if you do this on the generating function level, like you illustrated, um, is, is there a nice way to, to implement this without having to look in, into the individual graphs and so on? Uh, uh, in some sense, the, the point is that, uh, whenever you, uh, when you, you, you look at the, at this kind of corrections, since it's, um, uh, it's behave at, uh, uh, order to primitive graph, it can only contribute to next to leading log, uh, uh, correction to, to the scale behavior. And so you have some, uh, uh, systematic way to, to push, to, uh, uh, higher on, uh, uh, uh, uh, to next to next and next to next to next, uh, uh, exact to have to end to the K log order. And, and it's quite, uh, for me it's quite important because I, I have some, I have not, it's not formally, uh, written, but the, the, in some sense, in, uh, Schwenger-Eisen equation, uh, leading log behaviors of the, of the propagators and, uh, are nearly equivalent to, uh, leading behavior of the, uh, perturbative series. So, thank you. Thank you, Mark. Are there other questions for Mark? All right, let's thank Mark again. Thank you. And I think these