Bogoliubov Type Recursions for Renormalisation in Regularity Structures
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Bruned, Yvain
Hairer's regularity structures transformed the solution theory of singular stochastic partial differential equations. The notions of positive and negative renormalisation are central and the intricate interplay between these two renormalisation procedures is captured through the combination of cointeracting bialgebras and an algebraic Birkhoff-type decomposition of bialgebra morphisms. We will revisit the latter by defining Bogoliubov-type recursions similar to Connes and Kreimer's formulation of BPHZ renormalisation.
 Thank you for the invitation. I'm very honored to be part of this event. And like Martin, I mean, I was mostly interested in SPDs. And it turns out that if you want to deal with the general KPSZ that Martin presented in the previous talks, you need to use algebra and you have to use this conchimer of algebra. And in this talk, I will try to go through these two renormations, the one which happened in SPDs, the one which is used for recentering monomials and the other one which is for renormalizing them, which is more equivalent to this VPHZ renormization. And I want to interpret these renormations in the framework of bogery, both type recursions, in the sense that I want to give a precise meaning that we obtain in the context of SPDs some algebraic Birko factorizations. So this is a joint work with Kourush, Ebrahim Iffard. So before going to the algebraic setup, I would like to present very briefly two applications or two fields where you can actually use this tool. So first, the one which is more singular SPDs. So you start with here and just write an equation. So you see an example with generalized equations. So I have a PTU minus the fashion of U equal a non-NIT depending on U and its derivatives, but also on the side, which is a space-time noise zone. And the whole idea of a very structure, trying to solve these equations, just give a lot of descriptions of your solutions. So your solution will be locally described by Y equal X plus a sum. Here you have some of the data expansions, where here you sum some combinatorics, as Martin presented in the previous lecture. And every tree here you interpret it as a monomial, so it will be small in Y minus X, where you use some map pi X. And you have some coefficient in front of it. So given by this epsilon tau X, so the coefficients come from when you do your Picar iterations in the equation above, you produce a perturbative expansion and you can grab some coefficient depending on the non-NITF. This is a Taylor expansion, so I truncate, and I have some reminder, which would be a small one, X is close to Y. So what is very important here is that I have a map here, pi X, with a base point X. And this map sent decorated trees without this combinatorial structure to actually recenter the region intervals. So it would be stochastic integral, would be a recenter around the point X. It's like a car on my decorated trees if you want your fame and rule. And this is the map that you will be interested to construct. So pi X is a recentering map, and if you want to have the so-called model that is being used in register structure, you need to add another map, which would be depending on two points, X and Y. And this map allows you to go from an expansion around the point X to an expansion around the point Y, doing with this transformation. And the copal, I mean, pi and gamma constitute what we call a model in this context of a single-paraspid. It happens that in the context of speedies, gamma is strongly determined by the pi X. So the recentering map actually determines your gamma. This was the first, I would say, renormidation or recentering operation. Then you have a second operation, which is more familiar to you. Actually, this pi X star, which are stochastic iterated integral, they could be ill-defined because you have some products which are ill-defined. Of course, these products are there. I mean, they are in this right-hand side of the equation because the noise is a distribution, so you have problems to define these products. And so also, this term will not be confirmed in that you need to construct it with a suitable renormidation. For that, you need to work with some renormidation maps. We'll build in our maps on your set of the corrected trees. And we do it in a way that we have a simple actions on this pi X, where we just apply first renormidation M, and then we apply this pi X, this recentering map. So obtaining these simple expressions really comes from the algebraic perspectives when you have the interactions between the off-algebra which gives you this recentering map and the one which we normalize. And these nice interactions allows you to write this simple formula. There are two renormidations at play in these PDEs, and I would like to give you all you can see then using this algebraic Birkoff factorization. Now I move to another application, which is more recent, and this was for numerical analysis. So imagine that you have some dispersive PDEs, which are all in the form. So here we have some differential operator, here we have P, which would be a polynomial in MIT. Now the issue doesn't come from the fact that you have a singular noise, but you can add rough initial data like this V. And the idea is that you want to derive an efficient scheme for this dispersive equations at lower regularity for your initial data. And it turns out that if you want to write a numerical scheme that I developed, I wrote with my co-author, Catherine H. Ross, what we wrote in actually you can pick up an approximation of the coefficient of your solutions, and it will be given by a sum of a trisor like in a like for a PDE zone. And you still have the same type of coefficients that will depend, this coefficient depends on the initial value. And you have some integrals, iterated integrals, but here the numerical, so it will be not the exact iterated interval that you will construct by doing the P-characterations, but you'll be an approximation of this iterated interval. And here we are taking a sum of our trees, the creative trees, but because we are doing this in Fourier, we have to incorporate Fourier coefficient inside the trees. So that's why I put this is a different set of decorated trees. And as before, we have a character PR, which moved from the decorated trees and the speed out and gives you an approximation of iterated integrals. So this has a main character. It turns out also that in that context, that from that character, you can build another character using some type of algebraic Bercow factorizations. And this allows you to perform the local error analysis in the sense that you want to know locally if your schema is good at approximating well your solutions. So these are the two applications. And now I would like to move into the algebraic framework. So now we switch a bit from the applications and we come to the algebraic setup. Okay, so I start with introducing what they mean by a decorated tree. So we have the first set of non-planar root trees. Then we have a set of finite sets, this geographic logic L. And then I have a set of decorations will be this all times and to the power of d plus one. So this n, okay, what are the interpretations of this decoration? So here, for instance, this finite set will parameterize the differential operator if you have a system of equations, so then they will be associated to some propagators if you want to encode the propagators. And the end d plus one here is the directive that you can put on your propagators. Okay, so you want to dissociate the propagator and the derivatives. And so now we're considering trees, decorative trees on this set of decorations. And there will be all that form. So I will have a node decorations, which will be this map. This map we take value into n to the power of d plus one, meaning that I allow to have some polynomials like classical monomials inside my intervals. So like x to the power. And I have an edge decorations that will encode actually all the propagators and their derivatives. So it would be a map from the edge of the trees into the set of decorations. Okay, and I take the spanner of this, the great trees and I call it H. So I need to have a provide here product. So the product that we use when I talking about having characters will be the tree product between the credit rates. So at the level of the non-planar root, it is just to join the root, the root of the two trees. And then you do the Djun sum between the decorations. So I want to just below you see an example of multiplications. So here I have one degree trees, so I represent here on the nodes here are the node decorations. Here you have the edge decorations is this D and with I do the joint with product, what happened is that, okay, you glue these two roots together. And on the way you add the decorations. So you have L, you add it to M. So this would be our product for for the three products that we use. So these are the set of decorative trees. Now there is a symbolic notation, which I was actually introduced by Martin Eau and this foundation paper on the structures. Is that when you have the decorative trees, we want to represent them with symbols. And then you have some graph-tingual operator, this I subscript A here. And you take a decorative tree and what you do and you just graph this decorative tree with an edge, the grid by A on this node. This is equivalent to is a similar thing to that's the B plus operator that we use for the conchimer of algebra. So you have the grafting of the trees and if I add this just a dot decorated by K, identify it as a monomial that will denote X to the power of K. And now my, sorry, and now my tree product, I mean, if I want to represent that tree using the set, this is two symbols, I actually give you these expressions so I can go it slowly. So here you have at the root, you have M. So this gives you this X to the power of M. Then if I go for the first branch, I have a calligraphic A, so the correction on the H. And on the top, I have this N. So this is X to the power of M. And then I go on with that branch, it would be a calligraphic B, X to the power of P and so on. Of course, here the product is commutative, so I could have written in different ways. So we have like decorative trees, we have a nice B plus operator, we're using these symbolic notations. And now I can provide you the expression of a conchimer type of product, which should be appear in the next slide. So first, before that, I need to say that we are considering trees, but we have some truncations in the sense that we will not consider all the trees, we may sometimes just want to consider trees which actually will give you ill-defined iterative intervals. So you want to have a way to truncate, so you define what is the degree of a tree. So it's more related to the analytical aspects of the SPDs. So one, you have one assignment which is actually defined on the propagator, so basically give you what type of regularity you earn by converting, for instance, by the each kernel or other kernels. Then you have a second map which actually will be on the polynomials, so it depends which weight you put on the polynomial or which other space you want to use. So often in SPDs, we use a 211 scaling in the sense that times can double in comparison to the special components. And then the degree of the tree, it just that you take the sum about all the node decorations, those will be the polynomials one, so actually they increase it because they are out of regularity. And then you have to take the sum of all of the propagators, so here I have the things that I earn by completing with a kernel, minus I need to subtract the initial derivatives that could happen on this propagator. And then because we will be in the finite, I mean, this will also the space of positive decorative trees, which are that form, put in brackets because you will see in a minute, so that trees are that form. And in fact, I am asking that all of the branches which are exciting the root are to be of positive degree. So this will be in the sense that this I will have can associate to it a nice projections, which will be this pi plus. And this pi plus projection would be multiplicative for the tree product presented before. So this is the space that it is useful for doing, for instance, the recentering. And now I come to the main map, which is a complete reform picture of the primary product, which is given above. It looks very similar to the, to the point of the product in a way, in the sense that you can see that this calligraphic idea gives you the same expression that you will obtain with a deepless operator. But there are some subtleties actually, which makes it study a bit more steady, more complicated. In the sense that here I play with my polynomials, in the sense that these polynomials are primitive elements, okay, like even here, these monomials. And I play with decorations here, I have a sum over L in N to the power of D plus one. And this sum has to be understood as if you are doing as a level of the algebra, a Taylor expansion. So I'm taking derivatives on my branch or my calculator. And then I've x to the power of L divided by L factorial will be the classical monomials that you see in a Taylor expansion. And in fact, so this was not done in the original paper of re-destructures, but we did it later with Martin Ehrer and Lorenzo Zambotti. In fact, you can consider this product with an infinite sum, okay, you can keep the infinite sum. But if you want to give a meaning to this infinite sum, we use a big rating. So as a big rating, what you mean, what you will need to measure, of course, you will have the size of the tree. This will be one complement in this big rating. But you also have to take into account the data, for instance, the age decorations, because here when you do that sum, you increase the age decoration by putting these derivatives. So you can give actually a meaning to this co-product in this using the big rating, having this infinity. Okay, so now I'm saying that this is deformed. So this is a recent work with Dominique Manchon. Actually, you can really identify where the deformation happened. I mean, the deformation being the fact that you add all these extra terms, having this x to the power L, tensors, these derivatives on your on your plant tree here. And this is quite involved actually, it's not straightforward that we need to use another product, which is not exactly the grafting of a tree onto the other, but the plugging of a tree into another. And then you can apply, I mean, you can deform this product using these data expansions, apply some algebraic procedure, like the gangou don, which give you a necessity product. And the adjoint of this associative product actually will be this co-product. So it's like the same procedure when you start with a, that you apply it to a pretty product, construct an associative product, and then you can get your co-product. So this explain or you can see this co-product as a deformation of the original co-prime or co-product. Okay, so these are just for commenting on this map when you have made many this infinite sum. And what happened when you look at applications in SPDs or numerical analysis, okay, you will not use this infinity. I mean, you will truncate because they are constrained due to your applications. So the first truncation, which is these corrections, basically what you do, you truncate here. So I put a projection here, my pi plus. So in that way, I don't want this tree to be of negative degree. So by putting this direction, actually, my degree is going down. So at some point I have to stop. So the sum will be finite. And then the subtraction will be determined by the degree. Okay. And if I want to co-product, because this would be just corrections, because here I will be still on H. I need to project also on the trunk. So on the trunk also I have to put this projection pi plus. And this gives you a co-product. So we're not going to details, but if you think about the application I gave on numerical analysis with also these expansions in terms of trees, these projections in pi with pi plus will be different. Because in numerical analysis, you are interested in having an approximation up to order R. And so you will truncate, for instance, you will remove all the trees, which are too big, because there will be of order R. Okay. This is a different projections. Just to tell you that depending on the context, actually, you can play with this co-product and adapt the projections that fit more your word. I mean, the constraint that you have for NDPDs or singular NDPDs are not the same for as for a numerical analysis scheme. So that's why I mean this is very central. And you can really tailor it for your own applications. Okay. So now I will present the main results, which are in SPEs regarding this co-products. Actually, in the sense that you can able to construct an NDPD for this co-product. And the map that presented here, this one when you project on the right, I mean, on your make your expansion finite, it will be a right commodule for this H plus this off algebra of positive tree, of trees of positive degrees. And it turns out that what the map one of the most important map in in SPDs, the one which is used actually for recentering your object in a recursive way is a map called a twisted antipode. So you start to be in H plus, but then you exit and you go into H. And the definition of this twisted antipode also looks quite similar to what would be the definition of an antipode. But then you see that you have a sum, the same type of sum that show up as a level of the co-product. And you have some truncation according to the degree. And you apply it on what should be normally the formula of your of your antipode. And so the idea, and it's what we claim at that time with Martin and Lorenzo, is that actually this twisted antipode give you an equivalent of these algebraic Birko factorizations. And now in the next slides, I would like to make this statement more precise and really try to match up the two languages. I mean, the one you use with this twisted antipode and the one that you will use if you try to develop the classical algebraic Birko factorization or Bogolubov recursion. So here I'm going through what I will consider as my algebraic Birko factorizations and see how one can apply it to the context of this twisted antipode. So I just put here what is a framework. So I start with a connected graded of algebra with a co-product delta. I have the right commodule structure of H given by H-art and this dot. I will consider character over H and also over H-art and they will take value into some commutative algebra A. I will assume that I have some rotorbackstorm map Q from A into A. And this corotabackstorm map needs to satisfy this identity. So this is an identity for a rotorbackstorm map. And this rotorbackstorm map will induce splitting of A into two sub-algebra. So one would be A- when you apply Q and the other one would be A- when I apply the entity minus Q. So this is a equivalent for like if you have theories in Epsilon you just keep the remove the pole part. I mean this corresponds to this type of projections with your rotorbackstorm map. So this is basically the different objects at play and now I'm able to stay with what is my meaning of a bug-lubov type of retrosion. So here we pick up a character from H-art into A and then there will be unique azebramorphism one from H into A minus which would be basically what we call I think it's called in the literature the contraproms and the five plus would be the renormalized character from H-art into A such that for every element in H so here I think I forgot to say but this is an injection of I think it's here I have an injection from H into H-art. So if I pick up an tau in element of H and apply some preparation map called phi bar and apply make sure I got my minus and then my phi bar is constructing with some type of recursion. Recursion where I use three-dimensional notation for reduce a co-options. So this reduced correction which is defined here and I just remove the primitive part of my correction and so I do the recursion with phi and with phi and I define this with phi minus and if I want to get the renormalized character I just take the convolution here between phi and phi minus where for the convolution I'm using the co-action and basically what why is just called a bookly-worth type recursion because here I'm using a co-action if I replace my co-action with a co-product I get an algebraic okay then depends what is your definition of algebraic the alcohol factorization. So here's a recursion is built on these reduced co-actions and what is so important is to have this rotaback format this Q map which actually do the projections. Okay so this is what I'm calling this bookly-worth type recursions and it turns out that in the cases or the application that presented it fits well to at least two applications. So for instance when I was talking about the renormization of the model meaning that you want to recenter your specific processes so this would be done okay you need to extend a bit your expectations but you can think of Q as the expectation and in numerical analysis Q would project according to the frequencies so for instance if you have zero frequencies then you want to remove this this chance so this Q will project according to that. So in that context it works well because for renormization of the models it's an extraction contraction of algebra which is connected and also in the recal analysis the of algebra will be connected so you can basically apply not okay almost directly the bookly-worth type recursion that I've presented before. Okay now it turned out to the of algebra that I presented the one which is h plus this one of trees with positive degree unfortunately it's not connected because I have this x to the power k I mean all these monomials and one needs I mean one needs to find what should be the reduced question in that case because it's not only subtracting the the the permitting part but what you need to do what you have to do to do in that case. Also remark that I will not expand further is that while you have these co-interactions between two renormulations this one recentering and this one for renormulation and we see that they correspond to actually you have two twisted antipodes you have two bogliuble recursions and in sense is that you have some co-interactions between the two bogliuble recursions in the sense that you can start by applying the one bogliuble recursion produce a character and then from these characters you can actually apply the second bogliuble recursions as the co-interactions tells you that actually you can switch between the two bogliuble recursions so this is also something we can be a nice interpretation of these co-interaction properties. Now for the rest of the talk I will go to the complicated example the one when you are not connected and all you need to deal with this problem in this in this application. We need to find a suitable create for a reduced question and then we will be able to perform a so the reduced co-product should be the primitive one in our case because we have these polynomials we need to subtract a bit more so in the sense that this reduced correction will be given as such so tensor 1 and then I need to subtract this big part basically what this big part says that in fact I put all these branches all the part of my branches on my trees are on the right inside of the co-product and then I have all the play with the decorations in the sense that I can extract some x you can put xk which come from the data and I put the sum over li which are the potential derivatives that I can put on the different branches and that's the combinatorial factors here so what I want I want to subtract all this part I want to remove all this combinatorial to all this part which makes sense because these terms are produced by these deformations and they are made the term you want to remove when you consider a reduced map. Okay just one one justification of this choice is that at least on these terms which were at the beginning this is why your offer is not connected at least the reduced map is zero on that terms so the reduced map has been designed in such a way that it is zero on x to the power k to the dot when you have these k decorations. So now we are equipped with this reduced correction and now okay what I want to do I want to introduce this splitting and also introduce this reticulant. So for the splitting we look okay we work as with the spruce models we mentioned that you replace your singular noise by some approximations and reflections so here we are looking at infinity because one into r and what we do is that we fix the base point the x in our d plus one and you get the following splittings where f plus x contains a function that vanishes x and x minus one would be the polynomials which whose coefficient depends on x which are a function of x. Okay and then you have the natural rotor backstorm map even below where I pick up the infinity functions and what I do here I have some telejet between y minus x so I just take the telejet of f and I have some truncation here I mean I just truncated because there is an reticulant object I truncate and this truncation actually reflect in the algebra when we truncate the tele expansions according to the degree. So this is this telejet operator and actually it's well known that I don't know actually with fancy in the literature but you have this rotor backstair identity playing with these parameters when I have alpha and beta and so I have this rotor backstair identity and I think it's been understood in the literature that you can actually extend this result of the Bougou-Roubault recursion with a rotor backstair identity to a family of rotor backstair maps. So this is actually gives you a family of rotor backstair maps and that satisfies these rotor backstair identities. So we already put the framework so we have this splitting space splitting we have the rotor backstair identities we have our reduced fractions all the tools are in place for formulating the Bougou-Roubault recursion. Okay so here we will instead of we consider a family of of characters from H into into A these cn3t functions and here these characters will be indexed by a base point here this x bar so we will see it in the next slides but you have to understand this x bar as some a priori recentering. You know that inside these trees I have some polynomials these monomials x to the power k and analytically I mean I can just interpret this as a polynomial functions or I can say that maybe I want to have a priori recentering of these polynomials so I can try to recenter I'll say that they're recenter random point x bar and see if you cannot get some invariance for that point x bar. So here I have a family of characters depending on this point x bar and then I can set up the Bougou-Roubault recursion exactly the same way as in my main definition where now I'm using my Taylor jet operator so I have always my preparation smart but now it depends on more points so here I have the x bars reset the a priori recentering I have x and y and so this gives me this recursion in terms of phi and phi minus and here this sum this is all three of the annotations for my reduced corrections and actually if I want to compute my phi minus I need to subtract here I'll apply my family of Rota backstorm app and this is where here you have you have the degree I mean the parameters of my Taylor jet is determined by the degree of the tree by the degree of of town so this is where this is this match the the properties that you will see in your co-product when you take you truncate according to the degree and then I take the convolution product between phi and phi minus to get my phi plus delta plus so it looks familiar I mean close to the to the Bougou-Roubault recursion one need to be cautious because here I've introduced several base points one base point was it is x bar that I hope to get rid of at some point because I want maybe to have some imbalance property and there is another base point which is this x which would be the the point x in which I want my my current I mean my renormidation to be recenter around the around that point x so this gives you this nice Bougou-Roubault recursion so actually what you can do with that so we construct these subjects and then okay you have some assumptions okay so you cannot take any any characters depending on x bar which this is quite reassuring because if you look at like this the assumption that you have when you try to design a model in registrations one also needs assumptions on the characters so one assumption which is very natural is the one that okay we want to say that this x bar actually is recentering on your monomials so I have x i would be yi minus x bar i okay so this is what I tell then you have another requirement you want that when you tweak the decoration on the edge when you tell or expand it in the algebra this actually corresponds to derivatives in the analytical part okay in the analytical counterpart okay so you want to match this when you add the decorations this amounts to take a derivative on your new character so these are very natural assumptions but actually they give you an interpretation of these decorations and if you have these two assumptions and using the rotabax stir identity that you've seen before actually you can prove what you obtain in the classical Bougou-Roubault recursion is that this phi minus is a character from h plus into a minus x and the phi plus would be a character from h into a and very nice result we also show that this phi minus it's actually obtained by phi by applying the twisted antibody so this is actually what you see in a register structure what you this what we claim to be the the the algebraic bit of factorizations so these formulations by recursion actually make appear actually match the recursion that you will do with this algebraic object which is the twisted antibody and you can go a bit further you can also have a nice expression for phi plus with the preparation maps will be careful that this map will be multiplicative will be evaluated on plenty trees so a bit will be multiplicative here on on meter so actually you can from this recursion starting from this family of characters you can show the character property that you expect on phi minus on phi plus and then you recover the expression for the twisted antibody okay so now maybe we want to get rid of the x bar so getting rid of the x bar so this you ask some invariance property so here you need a bit more assumptions so basically you need to not only interpret the decoration on the edge of derivatives but you need to make a pair what should be your your calculator or your kernels for instance here I make a pair the two index of the decorations for these two because one give me a kernel okay and the second one will give me the the derivatives on my kernel and here I will put the convolution with the character apply so you need to have another prescription or what should be your Feynman rule you need to have the convolution structure when you are on the on the plenty tree and if you add this assumption on the top of the two previous ones actually you can show some x bar invariance in the sense that for phi plus you can actually remove x bar doesn't depend on x bar then you have some invariance for the preparation the book rule of preparation math phi bar and phi minus but they are weaker that's for phi bar this would be x bar invariant on plenty trees and for phi minus would be x bar and on trees with zero node decorations as a result of some restrictions I mean you get fully the dependence on x bar but for five five plus you get it so this is something you can you can show and now I would like to finish my talk with a final remark is that if you use mostly the language of relative structures what we were considering it's like a character which I denote by this called face pi which for this character indexed by this point x bar and normally if you were in the context of the of the work we did with Lorenzo and marching what you do is that you start with this character you apply a tree student code and then you express it with a convolution with your delta r plus and you construct that map by x to one of the beginning to construct result interval and we show that in the process that actually the map is independent from x bar and this will be the model I mean the the py axis would determine this map and finally what you can say that these maps here which are used for constructing the model they can completely have one-to-one correspondence between the map you can see in the in global recursion so these five plus we give you the pi x and this will be your counter terms even by this five minus and you get the x bar invariance for the model so this really matches the two languages so this the conclusion of this of this of this work that actually is this algebraic bicofactorization or bugly above recursions show up I mean is really in the framework of spds and one too can use this language to reinterpret the fundamental objects at play in in relative structures and all the the renormulations but it's not only limited to stochastic partial differentiation even can be seen the level of pds when one has to deal with numerical schemes so this is a very nice applications and started research I mean I was interesting mostly in PDE's or spds and it was really unexpected but very nice for me to be in touch with these tools this con primer for product and little by little I realized that it was actually a central central object in both spds and maybe next in in this numerical scheme for PDE's so thank you for your attention thank you Ival for your nice talk other questions for Ival from from the audience yes I would have a question about these two bugly above recursions in co-interaction is do you have a general framework for this do you need only I mean a commodule bi-algebra and a character with value in rotabaxter algebra for example I like what you need is like of co-interacting bi-algebra which give you two character group of characters which are can take the semi-direct product of them okay and then within that you can actually write a bugly of a bugly of recursion on this on semi-direct on this semi-direct product of these groups and you can try first to do a as you start by the the recent ring and then you do renormization or you can switch this has not been really formalized it was just a more general idea or an observation that we we got in the paper that PDE maybe can try to push this formalism or this framework actually on the co-interactions yeah looks very nice yes thank you I have a question even though I'm surprised that I'm asking this kind of question so going back from combinatorics to analytics you know in physics one of the consequences of the the coprera and these these these recursions is that it has some influence on on the analytic behavior of the function right so there's something like a leading log expansion or a normalist dimension so by studying these combinatorics to learn something about some features of the actual solutions that are underlying this procedure that it is describing so so I was just wondering maybe also to Martin in these two talks we on one hand this is analytics side on the other hand there's a combinatorial side but is there any feedback in the sense that so far we have seen that you use this to to define what the well-defined to normalize solution is to these kind of equations but is there anything beyond that can you can you learn something about behavior when things become singular when when arguments of these things come close together or anything like that I'm just curious if that makes sense so you what if you can infer from the algebraic part some analytical properties on your solutions or I mean the simplest example in field theory is this thing called the leading log expansion yes so if you have a if you have a greens function suppose it just depends on the momentum and then you want to know how it does it behave as a function of momentum and then if you order it by powers in the momentum then you get corrections by by the log of the momentum and you can for example tell that the highest power of the log like the biggest correction at each order is related to some simple co-product of some simple graphs that enter your combinatorial structure and there is like a there's a way to order everything in terms of these kinds of corrections and logs for example so I'm just curious if anything anything like that is exists also in this spd applications but I have no idea what the analytic question to be asked would be I mean I'm just curious if there is anything I mean as a practitioner we're all like you have some problems you have these analytical features and you want to encourage them at the level of the algebra because you want to organize your computations nicely and to be able to derive general results but I think I've seen recently maybe in the work of Martin was for the support theorem was using some equations on the characters to infer some analytical properties maybe Mark could comment but this this I think was nice perspectives from the from the algebraic part or even he was able to show that the as if all given by some if you have the constant this is that corresponds to some of ideal your for algebra so but it was it seems to be a huge effort trying to connect this what should be normally analytical properties and to see them at the level of the of the algebra it's really a truplici for at least for us I think for everyone but but the kind of result is that for example you can relate the the coefficient of a log I mean it doesn't tell you what the coefficient is so it doesn't really does the analysis for you but it does tell you that a certain coefficient of log is the same as another coefficient of like somewhere else so it tells you some information of tells about how some information gets redistributed due to the nature of subtractions that are underlying the definitions of things but I'm sorry for the super vague around the question unless yeah I actually might have a question do sub hope for a give us play a role for you for example root trees which have at most k outgoing branches and every vertex for if k is one these are just linear root trees which never split so to speak does such sub hope for a give us play a role for you how are you just have linear treat like with no branching at all that would be the simplest one or if you say I just take root trees which have always two branches at most outgoing ah two branches you want to talk about by any way would put such sub hope for you plus play a particular role for you because under the co-products this is closed under the co-products such trees goes into such trees tends to such trees and maybe maybe there is something to be learned from that no for some reason I don't see what I mean I mean when we can't construct this regularity I mean we have this notion of rule right that sort of very yeah that's essentially gives you right because so so in these regularity structures you naturally don't have all possible trees showing up but you know you only have somehow because they're the trees are it's really more like the Feynman diagrams right so in the Feynman diagram you sort of fix yourself the interaction so you give yourself a number of types of notes somehow right and so here here the analog of the Feynman diagrams would themselves be trees already and then the analog of that would also be so if you give yourself rules about how things can kind of come together and then these sort of Taylor expansions they are the ones that look kind of like the conchrima hopf algebra also with these sort of cuts and then in this case they would in our case also consist only of trees that have certain structure right so there wouldn't be all possible trees but they would only be the ones of you know depending on the degree of your non-linearity and these kind of things okay thanks thank you if there are any further questions now then let's thank Ivan again and all speakers of today
 Thank you for the invitation. I'm very honored to be part of this event. And like Martin,  I mean, I was mostly interested in SPDs. And it turns out that if you want to deal with  the general KPSZ that Martin presented in the previous talks, you need to use algebra  and you have to use this conchimer of algebra. And in this talk, I will try to go through  these two renormations, the one which happened in SPDs, the one which is used for recentering  monomials and the other one which is for renormalizing them, which is more equivalent to this VPHZ  renormization. And I want to interpret these renormations in the framework of bogery, both  type recursions, in the sense that I want to give a precise meaning that we obtain in the  context of SPDs some algebraic Birko factorizations. So this is a joint work with Kourush, Ebrahim  Iffard. So before going to the algebraic setup, I would like to present very briefly  two applications or two fields where you can actually use this tool. So first, the one  which is more singular SPDs. So you start with here and just write an equation. So you  see an example with generalized equations. So I have a PTU minus the fashion of U equal  a non-NIT depending on U and its derivatives, but also on the side, which is a space-time  noise zone. And the whole idea of a very structure, trying to solve these equations, just give  a lot of descriptions of your solutions. So your solution will be locally described by  Y equal X plus a sum. Here you have some of the data expansions, where here you sum some  combinatorics, as Martin presented in the previous lecture. And every tree here you  interpret it as a monomial, so it will be small in Y minus X, where you use some map  pi X. And you have some coefficient in front of it. So given by this epsilon tau X, so  the coefficients come from when you do your Picar iterations in the equation above, you  produce a perturbative expansion and you can grab some coefficient depending on the non-NITF.  This is a Taylor expansion, so I truncate, and I have some reminder, which would be a  small one, X is close to Y. So what is very important here is that I have a map here,  pi X, with a base point X. And this map sent decorated trees without this combinatorial  structure to actually recenter the region intervals. So it would be stochastic integral,  would be a recenter around the point X. It's like a car on my decorated trees if you want  your fame and rule. And this is the map that you will be interested to construct. So pi  X is a recentering map, and if you want to have the so-called model that is being used  in register structure, you need to add another map, which would be depending on two points,  X and Y. And this map allows you to go from an expansion around the point X to an expansion  around the point Y, doing with this transformation. And the copal, I mean, pi and gamma constitute  what we call a model in this context of a single-paraspid. It happens that in the context  of speedies, gamma is strongly determined by the pi X. So the recentering map actually  determines your gamma. This was the first, I would say, renormidation or recentering  operation. Then you have a second operation, which is more familiar to you. Actually, this  pi X star, which are stochastic iterated integral, they could be ill-defined because you have  some products which are ill-defined. Of course, these products are there. I mean, they are  in this right-hand side of the equation because the noise is a distribution, so you have problems  to define these products. And so also, this term will not be confirmed in that you need  to construct it with a suitable renormidation. For that, you need to work with some renormidation  maps. We'll build in our maps on your set of the corrected trees. And we do it in a  way that we have a simple actions on this pi X, where we just apply first renormidation  M, and then we apply this pi X, this recentering map. So obtaining these simple expressions  really comes from the algebraic perspectives when you have the interactions between the  off-algebra which gives you this recentering map and the one which we normalize. And these  nice interactions allows you to write this simple formula. There are two renormidations  at play in these PDEs, and I would like to give you all you can see then using this algebraic  Birkoff factorization. Now I move to another application, which is more recent, and this  was for numerical analysis. So imagine that you have some dispersive PDEs, which are all  in the form. So here we have some differential operator, here we have P, which would be a  polynomial in MIT. Now the issue doesn't come from the fact that you have a singular  noise, but you can add rough initial data like this V. And the idea is that you want  to derive an efficient scheme for this dispersive equations at lower regularity for your initial  data. And it turns out that if you want to write a numerical scheme that I developed,  I wrote with my co-author, Catherine H. Ross, what we wrote in actually you can pick up  an approximation of the coefficient of your solutions, and it will be given by a sum of  a trisor like in a like for a PDE zone. And you still have the same type of coefficients  that will depend, this coefficient depends on the initial value. And you have some integrals,  iterated integrals, but here the numerical, so it will be not the exact iterated interval  that you will construct by doing the P-characterations, but you'll be an approximation of this iterated  interval. And here we are taking a sum of our trees, the creative trees, but because  we are doing this in Fourier, we have to incorporate Fourier coefficient inside the  trees. So that's why I put this is a different set of decorated trees. And as before, we  have a character PR, which moved from the decorated trees and the speed out and gives  you an approximation of iterated integrals. So this has a main character. It turns out  also that in that context, that from that character, you can build another character  using some type of algebraic Bercow factorizations. And this allows you to perform the local error  analysis in the sense that you want to know locally if your schema is good at approximating  well your solutions. So these are the two applications. And now I would like to move  into the algebraic framework. So now we switch a bit from the applications and we come to  the algebraic setup. Okay, so I start with introducing what they mean by a decorated  tree. So we have the first set of non-planar root trees. Then we have a set of finite sets,  this geographic logic L. And then I have a set of decorations will be this all times  and to the power of d plus one. So this n, okay, what are the interpretations of this  decoration? So here, for instance, this finite set will parameterize the differential operator  if you have a system of equations, so then they will be associated to some propagators  if you want to encode the propagators. And the end d plus one here is the directive that  you can put on your propagators. Okay, so you want to dissociate the propagator and  the derivatives. And so now we're considering trees, decorative trees on this set of decorations.  And there will be all that form. So I will have a node decorations, which will be this  map. This map we take value into n to the power of d plus one, meaning that I allow  to have some polynomials like classical monomials inside my intervals. So like x to the power.  And I have an edge decorations that will encode actually all the propagators and their derivatives.  So it would be a map from the edge of the trees into the set of decorations. Okay, and  I take the spanner of this, the great trees and I call it H. So I need to have a provide  here product. So the product that we use when I talking about having characters will be  the tree product between the credit rates. So at the level of the non-planar root, it  is just to join the root, the root of the two trees. And then you do the Djun sum between  the decorations. So I want to just below you see an example of multiplications. So here  I have one degree trees, so I represent here on the nodes here are the node decorations.  Here you have the edge decorations is this D and with I do the joint with product, what  happened is that, okay, you glue these two roots together. And on the way you add the  decorations. So you have L, you add it to M. So this would be our product for for the  three products that we use. So these are the set of decorative trees. Now there is a symbolic  notation, which I was actually introduced by Martin Eau and this foundation paper on  the structures. Is that when you have the decorative trees, we want to represent them  with symbols. And then you have some graph-tingual operator, this I subscript A here. And you  take a decorative tree and what you do and you just graph this decorative tree with an  edge, the grid by A on this node. This is equivalent to is a similar thing to that's  the B plus operator that we use for the conchimer of algebra. So you have the grafting of the  trees and if I add this just a dot decorated by K, identify it as a monomial that will  denote X to the power of K. And now my, sorry, and now my tree product, I mean, if I want  to represent that tree using the set, this is two symbols, I actually give you these  expressions so I can go it slowly. So here you have at the root, you have M. So this  gives you this X to the power of M. Then if I go for the first branch, I have a calligraphic  A, so the correction on the H. And on the top, I have this N. So this is X to the power  of M. And then I go on with that branch, it would be a calligraphic B, X to the power  of P and so on. Of course, here the product is commutative, so I could have written in  different ways. So we have like decorative trees, we have a nice B plus operator, we're  using these symbolic notations. And now I can provide you the expression of a conchimer  type of product, which should be appear in the next slide. So first, before that, I need  to say that we are considering trees, but we have some truncations in the sense that  we will not consider all the trees, we may sometimes just want to consider trees which  actually will give you ill-defined iterative intervals. So you want to have a way to truncate,  so you define what is the degree of a tree. So it's more related to the analytical aspects  of the SPDs. So one, you have one assignment which is actually defined on the propagator,  so basically give you what type of regularity you earn by converting, for instance, by the  each kernel or other kernels. Then you have a second map which actually will be on the  polynomials, so it depends which weight you put on the polynomial or which other space  you want to use. So often in SPDs, we use a 211 scaling in the sense that times can  double in comparison to the special components. And then the degree of the tree, it just that  you take the sum about all the node decorations, those will be the polynomials one, so actually  they increase it because they are out of regularity. And then you have to take the sum of all  of the propagators, so here I have the things that I earn by completing with a kernel, minus  I need to subtract the initial derivatives that could happen on this propagator. And  then because we will be in the finite, I mean, this will also the space of positive decorative  trees, which are that form, put in brackets because you will see in a minute, so that  trees are that form. And in fact, I am asking that all of the branches which are exciting  the root are to be of positive degree. So this will be in the sense that this I will  have can associate to it a nice projections, which will be this pi plus. And this pi plus  projection would be multiplicative for the tree product presented before. So this is  the space that it is useful for doing, for instance, the recentering. And now I come  to the main map, which is a complete reform picture of the primary product, which is given  above. It looks very similar to the, to the point of the product in a way, in the sense  that you can see that this calligraphic idea gives you the same expression that you will  obtain with a deepless operator. But there are some subtleties actually, which makes it  study a bit more steady, more complicated. In the sense that here I play with my polynomials,  in the sense that these polynomials are primitive elements, okay, like even here, these monomials.  And I play with decorations here, I have a sum over L in N to the power of D plus one.  And this sum has to be understood as if you are doing as a level of the algebra, a Taylor  expansion. So I'm taking derivatives on my branch or my calculator. And then I've x to  the power of L divided by L factorial will be the classical monomials that you see in  a Taylor expansion. And in fact, so this was not done in the original paper of re-destructures,  but we did it later with Martin Ehrer and Lorenzo Zambotti. In fact, you can consider  this product with an infinite sum, okay, you can keep the infinite sum. But if you want  to give a meaning to this infinite sum, we use a big rating. So as a big rating, what  you mean, what you will need to measure, of course, you will have the size of the tree.  This will be one complement in this big rating. But you also have to take into account the  data, for instance, the age decorations, because here when you do that sum, you increase the  age decoration by putting these derivatives. So you can give actually a meaning to this  co-product in this using the big rating, having this infinity. Okay, so now I'm saying that  this is deformed. So this is a recent work with Dominique Manchon. Actually, you can  really identify where the deformation happened. I mean, the deformation being the fact that  you add all these extra terms, having this x to the power L, tensors, these derivatives  on your on your plant tree here. And this is quite involved actually, it's not straightforward  that we need to use another product, which is not exactly the grafting of a tree onto  the other, but the plugging of a tree into another. And then you can apply, I mean, you  can deform this product using these data expansions, apply some algebraic procedure,  like the gangou don, which give you a necessity product. And the adjoint of this associative  product actually will be this co-product. So it's like the same procedure when you start  with a, that you apply it to a pretty product, construct an associative product, and then  you can get your co-product. So this explain or you can see this co-product as a deformation  of the original co-prime or co-product. Okay, so these are just for commenting on this map  when you have made many this infinite sum. And what happened when you look at applications  in SPDs or numerical analysis, okay, you will not use this infinity. I mean, you will truncate  because they are constrained due to your applications. So the first truncation, which is these corrections,  basically what you do, you truncate here. So I put a projection here, my pi plus. So in that way,  I don't want this tree to be of negative degree. So by putting this direction, actually, my degree  is going down. So at some point I have to stop. So the sum will be finite. And then the subtraction  will be determined by the degree. Okay. And if I want to co-product, because this would be just  corrections, because here I will be still on H. I need to project also on the trunk. So on the trunk  also I have to put this projection pi plus. And this gives you a co-product. So we're not going to  details, but if you think about the application I gave on numerical analysis with also these  expansions in terms of trees, these projections in pi with pi plus will be different. Because in  numerical analysis, you are interested in having an approximation up to order R. And so you will  truncate, for instance, you will remove all the trees, which are too big, because there will be  of order R. Okay. This is a different projections. Just to tell you that depending on the context,  actually, you can play with this co-product and adapt the projections that fit more your word.  I mean, the constraint that you have for NDPDs or singular NDPDs are not the same for  as for a numerical analysis scheme. So that's why I mean this is very central. And you can  really tailor it for your own applications. Okay. So now I will present the main results,  which are in SPEs regarding this co-products. Actually, in the sense that you can able to  construct an NDPD for this co-product. And the map that presented here, this one when you project  on the right, I mean, on your make your expansion finite, it will be a right commodule for this  H plus this off algebra of positive tree, of trees of positive degrees.  And it turns out that what the map one of the most important map in in SPDs, the one which is used  actually for recentering your object in a recursive way is a map called a twisted antipode. So you  start to be in H plus, but then you exit and you go into H. And the definition of this twisted  antipode also looks quite similar to what would be the definition of an antipode. But then you  see that you have a sum, the same type of sum that show up as a level of the co-product. And you  have some truncation according to the degree. And you apply it on what should be normally the formula  of your of your antipode. And so the idea, and it's what we claim at that time with Martin and  Lorenzo, is that actually this twisted antipode give you an equivalent of these algebraic  Birko factorizations. And now in the next slides, I would like to make this statement more precise  and really try to match up the two languages. I mean, the one you use with this twisted antipode  and the one that you will use if you try to develop the classical algebraic Birko factorization or  Bogolubov recursion. So here I'm going through what I will consider as my algebraic Birko factorizations  and see how one can apply it to the context of this twisted antipode. So I just put here what  is a framework. So I start with a connected graded of algebra with a co-product delta.  I have the right commodule structure of H given by H-art and this dot. I will consider  character over H and also over H-art and they will take value into some commutative algebra A.  I will assume that I have some rotorbackstorm map Q from A into A. And this corotabackstorm  map needs to satisfy this identity. So this is an identity for a rotorbackstorm map.  And this rotorbackstorm map will induce splitting of A into two sub-algebra. So one would be A- when  you apply Q and the other one would be A- when I apply the entity minus Q. So this is a equivalent  for like if you have theories in Epsilon you just keep the remove the pole part. I mean this  corresponds to this type of projections with your rotorbackstorm map. So this is basically the  different objects at play and now I'm able to stay with what is my meaning of a bug-lubov  type of retrosion. So here we pick up a character from H-art into A and then there will be unique  azebramorphism one from H into A minus which would be basically what we call I think it's  called in the literature the contraproms and the five plus would be the renormalized character from  H-art into A such that for every element in H so here I think I forgot to say but this is an  injection of I think it's here I have an injection from H into H-art. So if I pick up an tau in  element of H and apply some preparation map called phi bar and apply make sure I got my  minus and then my phi bar is constructing with some type of recursion. Recursion where I use  three-dimensional notation for reduce a co-options. So this reduced correction which is defined here  and I just remove the primitive part of my correction and so I do the recursion with phi and  with phi and I define this with phi minus and if I want to get the renormalized character I just  take the convolution here between phi and phi minus where for the convolution I'm using the  co-action and basically what why is just called a bookly-worth type recursion because here I'm  using a co-action if I replace my co-action with a co-product I get an algebraic  okay then depends what is your definition of algebraic  the alcohol factorization. So here's a recursion is built on these reduced co-actions and what is  so important is to have this rotaback format this Q map which actually do the projections.  Okay so this is what I'm calling this bookly-worth type recursions and it turns out that in the  cases or the application that presented it fits well to at least two applications. So for instance  when I was talking about the renormization of the model meaning that you want to recenter your  specific processes so this would be done okay you need to extend a bit your expectations but  you can think of Q as the expectation and in numerical analysis Q would project according  to the frequencies so for instance if you have zero frequencies then you want to remove this  this chance so this Q will project according to that. So in that context it works well because  for renormization of the models it's an extraction contraction of algebra which is connected and  also in the recal analysis the of algebra will be connected so you can basically apply not okay  almost directly the bookly-worth type recursion that I've presented before.  Okay now it turned out to the of algebra that I presented the one which is h plus this one of  trees with positive degree unfortunately it's not connected because I have this x to the power k  I mean all these monomials and one needs I mean one needs to find what should be the  reduced question in that case because it's not only subtracting the the the permitting part  but what you need to do what you have to do to do in that case. Also remark that I will not expand  further is that while you have these co-interactions between two renormulations this one recentering  and this one for renormulation and we see that they correspond to actually you have two twisted  antipodes you have two bogliuble recursions and in sense is that you have some co-interactions  between the two bogliuble recursions in the sense that you can start by applying the one bogliuble  recursion produce a character and then from these characters you can actually apply the second  bogliuble recursions as the co-interactions tells you that actually you can switch between the two  bogliuble recursions so this is also something we can be a nice interpretation of these co-interaction  properties. Now for the rest of the talk I will go to the complicated example the one when you are  not connected and all you need to deal with this problem in this in this application.  We need to find a suitable create for a reduced question and then we will be able to perform  a so the reduced co-product should be the primitive one in our case because we have these  polynomials we need to subtract a bit more so in the sense that this reduced correction will be given  as such so tensor 1 and then I need to subtract this big part basically what this big part says  that in fact I put all these branches all the part of my branches on my trees are on the right  inside of the co-product and then I have all the play with the decorations in the sense that I can  extract some x you can put xk which come from the data and I put the sum over li which are the  potential derivatives that I can put on the different branches and that's the combinatorial  factors here so what I want I want to subtract all this part I want to remove all this combinatorial  to all this part which makes sense because these terms are produced by these deformations  and they are made the term you want to remove when you consider a reduced map.  Okay just one one justification of this choice is that at least on these terms which were at the  beginning this is why your offer is not connected at least the reduced map is zero on that terms  so the reduced map has been designed in such a way that it is zero on x to the power k to the dot when you have these k decorations.  So now we are equipped with this reduced correction  and now okay what I want to do I want to introduce this splitting and also introduce this reticulant.  So for the splitting we look okay we work as with the spruce models we mentioned that you  replace your singular noise by some approximations and reflections so here we are looking at  infinity because one into r and what we do is that we fix the base point the x in our d plus one  and you get the following splittings where f plus x contains a function that vanishes x  and x minus one would be the polynomials which whose coefficient depends on x which are a function of x.  Okay and then you have the natural rotor backstorm map even below where I pick up the infinity functions  and what I do here I have some telejet between y minus x so I just take the telejet of f  and I have some truncation here I mean I just truncated because there is an reticulant object  I truncate and this truncation actually reflect in the algebra when we truncate the tele expansions  according to the degree. So this is this telejet operator and actually it's well known  that I don't know actually with fancy in the literature but you have this rotor backstair  identity playing with these parameters when I have alpha and beta and so I have this rotor  backstair identity and I think it's been understood in the literature that you can actually extend  this result of the Bougou-Roubault recursion with a rotor backstair identity to a family of rotor  backstair maps. So this is actually gives you a family of rotor backstair maps and that satisfies  these rotor backstair identities. So we already put the framework so we have this splitting space  splitting we have the rotor backstair identities we have our reduced fractions all the tools are  in place for formulating the Bougou-Roubault recursion. Okay so here we will instead of we  consider a family of of characters from H into into A these cn3t functions and here these characters  will be indexed by a base point here this x bar so we will see it in the next slides but you have  to understand this x bar as some a priori recentering. You know that inside these trees I have some  polynomials these monomials x to the power k and analytically I mean I can just interpret this as  a polynomial functions or I can say that maybe I want to have a priori recentering of these  polynomials so I can try to recenter I'll say that they're recenter random point x bar and see  if you cannot get some invariance for that point x bar. So here I have a family of characters  depending on this point x bar and then I can set up the Bougou-Roubault recursion exactly the same  way as in my main definition where now I'm using my Taylor jet operator so I have always my  preparation smart but now it depends on more points so here I have the x bars  reset the a priori recentering I have x and y and so this gives me this recursion in terms of  phi and phi minus and here this sum this is all three of the annotations for my reduced  corrections and actually if I want to compute my phi minus I need to subtract here I'll apply my  family of Rota backstorm app and this is where here you have you have the degree I mean the  parameters of my Taylor jet is determined by the degree of the tree by the degree of of town  so this is where this is this match the the properties that you will see in your co-product  when you take you truncate according to the degree and then I take the convolution product  between phi and phi minus to get my phi plus delta plus so it looks familiar I mean close to the  to the Bougou-Roubault recursion one need to be cautious because here I've introduced several  base points one base point was it is x bar that I hope to get rid of at some point because I want  maybe to have some imbalance property and there is another base point which is this x which would be  the the point x in which I want my my current I mean my renormidation to be recenter around the  around that point x so this gives you this nice Bougou-Roubault recursion so actually what you  can do with that so we construct these subjects and then okay you have some assumptions okay so  you cannot take any any characters depending on x bar which this is quite reassuring because if you  look at like this the assumption that you have when you try to design a model in registrations  one also needs assumptions on the characters so one assumption which is very natural is the one  that okay we want to say that this x bar actually is recentering on your monomials so I have x i  would be yi minus x bar i okay so this is what I tell then you have another requirement you want that  when you tweak the decoration on the edge when you tell or expand it in the algebra this actually  corresponds to derivatives in the analytical part okay in the analytical counterpart okay so you want  to match this when you add the decorations this amounts to take a derivative on your  new character so these are very natural assumptions but actually they give you an interpretation  of these decorations and if you have these two assumptions and using the rotabax stir  identity that you've seen before actually you can prove what you obtain in the classical  Bougou-Roubault recursion is that this phi minus is a character from h plus into a minus x and the  phi plus would be a character from h into a and very nice result we also show that this phi minus  it's actually obtained by phi by applying the twisted antibody so this is actually what you see  in a register structure what you this what we claim to be the the the algebraic bit of factorizations  so these formulations by recursion actually make appear actually match the recursion that you will  do with this algebraic object which is the twisted antibody and you can go a bit further you can also  have a nice expression for phi plus with the preparation maps will be careful that this map  will be multiplicative will be evaluated on plenty trees so a bit will be multiplicative here on  on meter so actually you can from this recursion starting from this family of characters you  can show the character property that you expect on phi minus on phi plus and then you recover  the expression for the twisted antibody okay so now maybe we want to get rid of the x bar  so getting rid of the x bar so this you ask some invariance property so here you need a  bit more assumptions so basically you need to not only interpret the decoration on the  edge of derivatives but you need to make a pair what should be your your calculator or your  kernels for instance here I make a pair the two index of the decorations for these two  because one give me a kernel okay and the second one will give me the the derivatives on my kernel  and here I will put the convolution with the character apply so you need to have another  prescription or what should be your Feynman rule you need to have the convolution structure when  you are on the on the plenty tree and if you add this assumption on the top of the two previous  ones actually you can show some x bar invariance in the sense that for phi plus you can actually  remove x bar doesn't depend on x bar then you have some invariance for the preparation the  book rule of preparation math phi bar and phi minus but they are weaker that's for phi bar this  would be x bar invariant on plenty trees and for phi minus would be x bar and on trees with zero  node decorations as a result of some restrictions I mean you get fully the dependence on x bar  but for five five plus you get it so this is something you can you can show and now I would  like to finish my talk with a final remark is that if you use mostly the language of  relative structures what we were considering it's like a character which I denote by this called  face pi which for this character indexed by this point x bar and normally if you were in the context  of the of the work we did with Lorenzo and marching what you do is that you start with this  character you apply a tree student code and then you express it with a convolution with your delta  r plus and you construct that map by x to one of the beginning to construct  result interval and we show that in the process that actually the map is independent from x bar  and this will be the model I mean the the py axis would determine this map and finally what you can  say that these maps here which are used for constructing the model they can completely  have one-to-one correspondence between the map you can see in the in global recursion  so these five plus we give you the pi x and this will be your counter terms even by this five minus  and you get the x bar invariance for the model so this really matches the two languages  so this the conclusion of this of this of this work that actually is this algebraic  bicofactorization or bugly above recursions show up I mean is really in the framework of  spds and one too can use this language to reinterpret the fundamental objects at play  in in relative structures and all the the renormulations but it's not only limited to  stochastic partial differentiation even can be seen the level of pds when one has to deal  with numerical schemes so this is a very nice applications and started research I mean I was  interesting mostly in PDE's or spds and it was really unexpected but very nice for me to be in  touch with these tools this con primer for product and little by little I realized that  it was actually a central central object in both spds and maybe next in in this numerical scheme for  PDE's so thank you for your attention thank you Ival for your nice talk other questions for Ival from  from the audience yes I would have a question about these two bugly above recursions in  co-interaction is do you have a general framework for this do you need only I mean  a commodule bi-algebra and a character with value in rotabaxter algebra for example  I like what you need is like of co-interacting bi-algebra which give you two character group  of characters which are can take the semi-direct product of them okay and then within that you  can actually write a bugly of a bugly of recursion on this on semi-direct on this  semi-direct product of these groups and you can try first to do a as you start by the the recent  ring and then you do renormization or you can switch this has not been really formalized it  was just a more general idea or an observation that we we got in the paper that PDE maybe can try  to push this formalism or this framework actually on the co-interactions yeah looks very nice yes  thank you I have a question even though I'm surprised that I'm asking this kind of question  so going back from combinatorics to analytics you know in physics one of the consequences of the  the coprera and these these these recursions is that it has some influence on on the analytic  behavior of the function right so there's something like a leading log expansion or a normalist dimension  so by studying these combinatorics to learn something about some features of the actual  solutions that are underlying this procedure that it is describing so so I was just wondering  maybe also to Martin in these two talks we on one hand this is analytics side on the other  hand there's a combinatorial side but is there any feedback in the sense that so far we have seen  that you use this to to define what the well-defined to normalize solution is to these kind of equations  but is there anything beyond that can you can you learn something about behavior when things  become singular when when arguments of these things come close together or anything like that  I'm just curious if that makes sense  so you what if you can infer from the algebraic part some analytical properties on your solutions  or I mean the simplest example in field theory is this thing called the leading log expansion  yes so if you have a if you have a greens function suppose it just depends on the momentum and then  you want to know how it does it behave as a function of momentum and then if you order it by  powers in the momentum then you get corrections by by the log of the momentum and you can for example  tell that the highest power of the log like the biggest correction at each order is related to  some simple co-product of some simple graphs that enter your combinatorial structure and there is  like a there's a way to order everything in terms of these kinds of corrections and logs for example  so I'm just curious if anything anything like that is exists also in this spd applications  but I have no idea what the analytic question to be asked would be I mean I'm just curious if there  is anything I mean as a practitioner we're all like you have some problems you have these analytical  features and you want to encourage them at the level of the algebra because you want to organize  your computations nicely and to be able to derive general results but  I think I've seen recently maybe in the work of Martin was for the support theorem was using  some equations on the characters to infer some analytical properties maybe Mark could comment  but this this I think was nice perspectives from the from the algebraic part or even he was able  to show that the as if all given by some if you have the constant this is that corresponds to some  of ideal your for algebra so but it was it seems to be a huge effort trying to connect this what  should be normally analytical properties and to see them at the level of the of the algebra  it's really a truplici for at least for us I think for everyone  but but the kind of result is that for example you can relate the the coefficient of a log I mean  it doesn't tell you what the coefficient is so it doesn't really does the analysis for you but it  does tell you that a certain coefficient of log is the same as another coefficient of  like somewhere else so it tells you some information of tells about how some information  gets redistributed due to the nature of subtractions that are underlying the definitions of things  but I'm sorry for the super vague around the question unless yeah I actually might have a question  do sub hope for a give us play a role for you for example root trees which have at most  k outgoing branches and every vertex for if k is one these are just linear root trees  which never split so to speak does such sub hope for a give us play a role for you  how are you just have linear treat like with no branching at all that would be the simplest one  or if you say I just take root trees which have always two branches at most outgoing  ah two branches you want to talk about by any way would put such sub hope for you  plus play a particular role for you because under the co-products this is closed under the co-products  such trees goes into such trees tends to such trees and maybe maybe there is something to be  learned from that no for some reason I don't see what I mean I mean when we can't construct this  regularity I mean we have this notion of rule right that sort of very yeah that's essentially gives you  right because so so in these regularity structures you naturally don't have all possible trees showing  up but you know you only have somehow because they're the trees are it's really more like the  Feynman diagrams right so in the Feynman diagram you sort of fix yourself the interaction so you  give yourself a number of types of notes somehow right and so here here the analog of the Feynman  diagrams would themselves be trees already and then the analog of that would also be so if you  give yourself rules about how things can kind of come together and then these sort of Taylor  expansions they are the ones that look kind of like the conchrima hopf algebra also with these  sort of cuts and then in this case they would in our case also consist only of trees that have  certain structure right so there wouldn't be all possible trees but they would only be the ones  of you know depending on the degree of your non-linearity and these kind of things okay thanks  thank you if there are any further questions now then let's thank Ivan again  and all speakers of today
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