 Well, clearly this is quite a pleasure, quite an honor actually, to be able to give a talk, an opening talk at this place. To me it also feels like a throwback to my beginnings. The fact is that 40 years ago, almost to the day, René Tom, the late René Tom and Ivan Kupka, who is very much alive over close to 90, invited me to this place to hold forth about resurgence and related topics. The two gentlemen were kind enough to severe it in my lucubrations and even offered practical support for René Tom offered to get my things typed by the secretary here at the institute. There was even Denis Sullivan on that day in the room. He showed his appreciation in his own way by reclining on the floor full length, but with eyes wide open, mind you, so that he could follow my scribblings on the blackboard at an angle. So nice to be here and nice to reconnect with these memories of long ago. Before starting, for good, let us get a few reminders quickly out of the way. Research and functions, of course, live in three models, in the formal model as formal power series or more general things, trains series, a mixture of power series of powers of z and exponentials of towers of exponentials. And then with the convolution model, with the operation being path convolution. And then in the geometric model, as sectorial germs at the infinite. So we start from the formal model, we want to get to the geometric model, but all the work and all the difficulty manifest in the intermediate model in the convolutive model. So the singularities in the convolutive model are responsible for the divergence and more or less they carry important information because they carry the Stokes constants. So it is important to measure them precisely. And the tool for doing that is the so-called alien derivations, which are the hallmark of resurgent functions, actually resurgent functions or functions which can submit to alien differentiation. The, the alien, they have indexes, indices, omega, which can, which belong to C or actually the C, the ramified version of C. And the, by definition, the alien derivation of a delta sub omega applied to phi of zeta is a weighted superposition of various determinations over zeta plus omega. This is defined with our difficulty close to the origin and then we move, it is extended. Thank you. I'll use that. This is extended in the large, by analytic continuation. So they are defined like everything else in the convolutive model, but by pull back the, extend to the multiplicative models. And in those two models, we have the additional invariant alien derivations denoted by this, by the double-struck delta. They're defined as the usual alien derivations with in front of them, sorry, an exponential factor. Then those invariant alien derivations commute with ordinary differentiation relative to z, which is very useful. So and of course they have no equivalent in the, in the convolutive model. So the most two important notions, objects, that can be attached to a given resurgent function are the active alien algebra and the display. Take a resurgent function or an algebra of such functions. Then take the completion of that with respect to alien and ordinary differentiation. Then take the ideal of all alien derivations that kill that completion, that completed algebra. This is a bilateral ideal. And then take the coefficient of all alien derivations by this ideal. This by definition, this object might actually reflect the concrete mode of action of alien derivations on that algebra. So to that extent it is essential. And the thing is that although by themselves on their own, alien derivations are totally free, totally unconstrained in concrete situations, they are anything but free. They are, most of the time, they actually turn out to be equivalent to be isomorphic to algebra, algebras of ordinary differential operators. So we'll see many instances of that. And then there is a display. It is a sort, as you can see on the formula, it is a sort of alien tail formula. But the main property and the main use is actually that of extending any relation between any number of resurgent functions. Take any relation r involving the plus, minus, composition, ordinary derivation, et cetera. And it automatically carries over to the displays, which is extremely useful because this second identity involves a huge number of constraints. So it is extremely useful to prove independent theorems and that sort of things. So now let us start in earnest for orientation and for perspective. Let us comment on, let us examine on a double column the main similarities and differences between equational resurgence and co-equational resurgence. This is going to be our main topic today, especially the second column. Equational resurgence is relative to a critical variable, the variable of an equation. Co-equational resurgence is relative to a critical parameter. In both cases, the activity in algebras are isomorphic, as I said, to algebras of ordinary differential operators. But in the second case, for co-equational resurgence, this algebra actually splits into two parts. Correspondingly, for equational resurgence, there is one set, one system of allium different equations that describe the whole situation, one bridge equation. For co-equational resurgence, there are two of them, two systems and very different ones. Then for co-equational resurgence, we have a real Stokes constant that can assume any complex values and they are usually transcendental. With co-equational resurgence, this is perhaps the main surprise, the main difference. Instead of that, we get discrete Stokes constants, which we call as a very specific object. They assume only after renormalization, there will be integers, we call them the tessellation coefficients. Then equation resurgence, as I said, all the work, all the proving has to be done in the convolution model. But with equation resurgence, convolution is ordinary convolution. For co-equational resurgence, it is not a binary operation, but a multiple convolution and a weighted one. There are weights attached to it and it is a much more, quite different operation. And then, as I said, with equation resurgence, the differential operators, which are equivalent to which the active value in algebra is isomorphic, they can be anything, they are not constrained. With co-equational resurgence, they are heavily constrained. They verify, they kill a certain two-form, which we call the isographic form. And then, with equation resurgence, the sums in the geometric models are ramified at infinity. In the, with co-equational resurgence, they are either not ramified at all or finitely ramified. And so the, the, the exhibit, this very unusual combination of divergence, of being resurgence in various sectors, and being an entire function. We call them, and there's more to that, we call them attack function. And then the last event, of course, with co-equational resurgence, everything is ten times simpler and more straightforward than with co-equational resurgence. Okay, so let's go. So very briefly, and let us review, equation resurgence, not for its own sake, but as a counterpoint to what is, to co-equational resurgence. Make any equation differential, non-linear, differential difference, composition equation, or general functional equation, and then form its full, its full solution with a maximum number of parameters in it. Then the rule of thumb, you can say that the existence side by side in this equation of powers, inverse powers, z, and exponentials, either of z of powers, z points to the existence of resurgence. Actually, there will be as many critical times as there exist blocks, and let us say, powers inside the exponentials. Mercifully, in most situations, there is only one critical time, but if there are more than one, then we can live with that. There is an apparatus to deal with that, simply there are many, we have to consider to go through many boreal planes, and we first to classify the critical times from slower to faster, and then we go through the boreal planes by means of operators, which resemble Laplace operators but with a different and more sophisticated kernel. So, and then there is a few, once we've got hold of the formal solution, there is a purely formal, there are purely formal manipulations, which tell you which alien derivations are going to act effectively on that, and for forming the homogeneous, differential linear, homogeneous differential equations, which the alien derivatives are going to verify. And following this line of reasoning, we arrived very quickly at the bridge equation, which consists of three things, you see on both sides, the full solution with the maximum number of parameters, on the left-hand side, the invariant alien derivations, and on the right-hand side, the ordinary differential operators in the parameters precisely, in the parameters of the equation, and sometimes also in the variable itself. So, these operators carry the source constants that they ask coefficients, otherwise they are subject to no other constraints than making sense, that is to say they must bear off a similar exponential on both sides. So, you see this reduces the part of analysis to the minimum, it makes everything more or less formal, and we arrived at, so in this case, the nature of the active alien algebra is obvious, it is simply the native alien algebra generated by these sub-omegas, these ordinary differential operators. So, in this case, just two remarks, in this case the display, it looks like a magnified, actually, full solution, because the power series in it are going to be more or less the same, but they are two different. First, they carry in front of them Stokes constants, and then that indexation, see I recall the definition, instead of being by simple frequencies, by simple omegas, the indexation is by strings of omegas, so it is a much richer object, and then it has this property, which I mentioned a minute ago, of extending to all relations. So, this is, it is an object which has no classical equivalent, and which is extremely, I cannot overemphasize this enough, and this is extremely useful. And then, just a remark, a funny remark, I mean, resurgent functions, meaning in this case, in the case of, in all cases, in fact, they have a cohesion, which ordinary solutions of differential equations do not have, and cannot have, in the sense that knowing one part of the full solution can constructively lead to the reconstruction of the whole solution, even of the original equation itself. You can think as an analogon of an irreducible polynomial of a Q, if it is reducible, I mean, knowing one root will not tell you anything about the whole, the other roots, but if it is fully reducible, of course, you can, you know, the full picture, you can reconstitute the full polynomial from one root. So, they, they go here, in the, they have this property. Now, let us, to better understand the interplay between equational and co-equation resurgence, we are going to focus on a problem, a model problem, which manifests both types of resurgence. See, we start from the equation, the blue equation with a small t, time variable, system of new equations, non-linear, and with, in front of the system, and possibly in the other coefficients, perturbation parameter epsilon. Now, it is convenient to switch to the critical variable and to the critical parameter and to work at infinity instead of the origin. So, we move from t and epsilon to z and x, and the system becomes the red system down here. So, let us take along, we are going to work on that for the rest of the talk. So, let us take a long look at basically a system, it's, I reproduced it here in any case, and just one remark actually, we can assume without actually losing much that the perturbation parameter is absent from the coefficients b, it's present there would, wouldn't add anything of substance to the analysis, it would simply complicate notations. And then a second remark, there is a special case in which a very special case, a special, in the coefficients b that reduced to negative power, meaning to 1 over z, to a constant of a z. In that special case, the two, the variable and the parameter coalesce into a product z times x. So, in that case, obviously the two resurgences coincide. So, something of this sameness is going to survive in the general situations, but on the whole it is fair to say that the differences are going to dominate. So, instead of instead of working with a full solution, actually, we are going to work with an object which is information equivalent, but more flexible. It is a formal automorphism which the normalizer, which takes the normal form of the equation and it's a trivial normal solution to the equation, to the system itself and its non-trivial solution. So, this normalizes, direct and reverse, consists of two ingredients, ordinary differential operatives, d, that encode the Taylor coefficients of our system. And then the main object from the point of view of resurgent of analysis, they are bi-resurgent monomials. So, you see down here in blue they have a double indexation, a two-tire indexation, the things in upstairs mean their weights, complex numbers, and downstairs the v's are simply analytic terms at infinity in the, on the Riemann sphere, with analytic continuation to the whole Riemann sphere and possibly any pattern of singularities there. So, again, I reproduce, they are going to be our main object. And the, so just a shorter side, I mean, this is Yusuf's terminology, these normalizers, they are just said that they are automorphisms. And this is, this reflect there being the contraction of ordinary differential, ordinary derivations, the d's, and a mold, I mean, these bi-resurgent monomials which are symmetrical with respect to the indices. Symmetriums simply mean that they multiply according to the shuffled product. And the algebra, I mean, equivalent to that, I mean, is alter-node. Instead of multiplication, you get zero here. And this is contracting symmetrical with derivations, you get an automorphism, and contracting alter-node with ordinary derivations, you get, again, a formal derivation. So, this is all our indexed object are going to be, to fall into one of these two categories, and we are going, I mean, every time to contract them, so as to get either automorphisms or derivations. So, very quickly, these, as you can see, above the bi-resurgent monomials, which I should say, I mean, this has to be justified, but actually the fact is that they carry, they are simple objects, but they carry the whole diversions once, and the resurgents, once they have been understood, I mean, in a sense, everything has been understood. So, we are going to focus on them. They are defined by this equation two, and in this case, I mean, coefficient, in their case, in this instance, equation resurgent is simplicity itself. Borel takes you from z to zeta, the conjugated variable, and the recursion equation simply actually simplifies. You can see equation two. So, this equation three, I mean, immediately give you all the information about the the equation resurgents in this case. It tells you exactly which alien derivations are going to act on that, and how they are going to act. They are going to produce elementary, sorry. So, equation resurgents is a trifle in this case. Now, co-equation resurgents is quite, you see, in this case, the recursion equation instead of simplifying gets messier. We get equation eight and so on. I mean, when we perform Borel from the x to xi, the conjugated variable, it becomes a partial differential equation, which is suitable limit conditions. And it is in the case of a depth one, I mean, the solution is obvious, but for larger depth, there is no simple formula for solving this recursion solution. So, we need a new operation, which is precisely the weighted convolution. So, actually, again, for perspective, let us say what we are saying in advance, what we are going to do. We need four things. We need a weighted convolution to express the bioresurgent monomials with respect to x. Then, to express their own alien derivations, we are going to need a second type of weighted convolution. This time, it will be a symmetrical, an alternate with respect to the index. And then, we will need exact formulas for finding the alien derivations of these two types of weighted convolutions. And then, to express, actually, the alien derivations, the alien derivatives, we need a new object, which is one of the styles of this theory, the translation coefficients. So, first, the symmetrical convolution is given by a very complicated multiple integral with a complicated multipath. And it is very nasty, in a sense, very contorted. But the fact is that all the calculations for quick resurgence are going to be based on that. So, we have to live with it. But we should actually mark a pause and describe this integral. You see, upstairs, there are weights, and downstairs, there are germs in the Borrel plane at the origin, which have infinite continuation, but usually highly ramified. These are the inputs. The output, this is a statement, is exactly of the same type. In germ, the analytic germ, they're the origin, with and as continuation and unimidication. So, we cannot describe this integral in more detail, but we can give more telling, actually, characterization. Just like ordinary convolution is the Borrel image of ordinary multiplication. In the same way, weighted convolution is the Borrel image of, let us say, a weighted multiplication, which is a simple interval kernel, this one here. And on this formula, although it is of no practical use, actually, all the work has to be done in the convolutive plane, the Borrel plane. But formally, it's a nicer picture, and it has the advantage, the merit, at least, of making a symmetry obvious, because the kernel itself is quite obviously symmetrical. So, now, as I said, we need a second to get a closed system that expresses everything. We need a second type of convolution, this time, alternate with respect to the indices. It can be defined in any of three ways, either as a superposition, as a finite superposition of symmetrical convolutions, or by a direct integral formula, which is even more complex than the last one we've seen, or, again, in the multiplicative plane, by a simple kernel. And this kernel, as you can see, I mean, the first convolution is called, we call a second wheel. I'm sorry, but we have to introduce these objects. We cannot do without them. And going from one to the second, from the first to the second, actually doesn't involve significant complications. So, again, I have to be content with their somewhat schematic survey. But the aim is to actually get to show that there exists a machinery for dealing with all situations, not so that we're not condemned to deal with special case after special case after special case, which is not a very inspiring way of doing mathematics, but I think something that deals with co-equation resurgence once and for all. So, what is the relevance of these two convolutions? Well, the first one, the symmetrical one, it's quite simply that the body's urgent monobules with respect to x can be expressed in the Borel transform in the xi plane can be expressed as weighted convolutions of what? Well, of these germs. Now, this is rather strange. You see, these germs, Ti, xi is in the Borel plane, but it is built from data, which originate from the multiplicative z plane. So, we have this very unusual and rather unpleasant jumble of two structures, multiplicative and convolutive. This is of the essence, actually, of co-resurgence, co-equation resurgence. This is how it is, actually. You cannot help it. And the relevance of the second weighted convolution is simply that we'll need that to express the alien derivative of the first type of weighted convolution and also the second type. This time we don't need a third one. The buck stops there. We get a closed system. Now, all this is fine as far as it goes, but it leaves the main difficulty unresolved. The thing is, how do we calculate the alien derivatives of these weighted convolutions? Now, the first idea would be actually, well, we might look at the integral formula and then take the endpoint close to a singularity and then look, see what happens. But, actually, it would be the height of madness, actually, to proceed in this way. Because just look, even in the case of ordinary convolution, as soon as we move away from the origin, the integration path has a way of getting impossibly contorted and well, convoluted. And this is because we must not only dodge the singularity themselves, but also the mirror image, images of the singularity with respect to the midpoint. To the point, mean halfway between the origin and the endpoint. And this is for ordinary convolution. Just imagine what it can be for a weighted convolution. Because it is, again, it is a binary, the r-ity, r, the number of the depth, is irreducible. Meaning that a weighted convolution cannot be broken down into a chain of binary operations. It has to be taken directly in its full horror. So, we must come up with something else. And the only salvation, actually, is to find a set of test functions which meet three conditions. There must be numerous enough, actually, to model the general ramified functions. There must be stable enough to self-reproduce under all the operations, convolution and and weighted convolution. And then, they must be, again, simple enough to lend themselves to alien differentiation in terms of themselves, again, to get a closed system. Fortunately, there is such a system is at hand, they are the hyper-logarism. But, again, I mentioned this feature, this interference of the multiplicative and the convolutive structure in the equations. And in actual fact, the hyper-logarism was specially suited for that purpose. Because they are stable with respect to convolution and with respect also to ordinary point-wise multiplication. But there is a, there's a hitch here. We need to differentiate, we will have to juggle two types of indexation because convolutions mean adds singularities in projection. Whereas point-wise multiplication obviously keeps singularities in place. So we'll need to, I mean, to use both the so-called positional indexation, which reflects the position of the singularities themselves, and incremental indexation, which measures, indicates the gap between one singularity and the next when they are on the, okay. Now, unfortunately, when everyone is very familiar with hyper-logarism, but here it's a question of, I mean, they are capable of many indexations and many forms. I mean, here we need a very special two indexations, and we cannot spend too much time on that, but the formulas are there. And there's a whole set of formula which we'll, we'll be of constant use in the stake well. The formula is for finding the alien derivations when they are elementary. You see, when under alien differentiation, when the hyper-logarism when taken in this basis, behaving exactly the way they produce elementary stocks constant, the red objects with, and they are the logarithms themselves, the hyper-logarism or symmetrical with respect to their indexes, and the associated monics, the stokes constants, are all general with respect. So that we'll need all the time, and then we'll also need to know exactly how they behave under ordinary partial differentiation with respect to the indices, there are formulas for doing that. And then the jump rules, the way they have been defined, these monics or piecewise analytic functions defined on the finite number of C to the R. And we know the hyper-surfaces which limit, which limit these various domains of analyticity, and we need jump rules to describe the passage from one domain to the next. So these are, these are the tools. Now, the first main set of formula, when we, when we consider a weighted convolution with the simplest possible inputs, ordinary simple poles, what we get is far from simple. You see, depending how I wrote here the first for the depths one, two and three, we get a number of terms, we can express everything in terms of a superposition of hyper-logarisms, I mean in total there are quite a lot of them. The so-called odd factorial one times three times five times, et cetera, two R minus one, and you see that the, the U's and the V's, the U's in blue, the V's in red, behave in a very specific way. There is of course an induction behind all this, there are actually two inductions, one which adds indices and one at the end and another which are forward going, which I think this is at the beginning, et cetera, what will be so, but this is not enough, I mean our inputs are not simple poles, I mean of course this would be enough to deal with the case of metamorphic inputs, but we want to deal with all cases, so we want to have a general hyper-logarithmic input. Now, there is again a formula for that, I didn't write it here, but it's in the notes which I posted on my homepage and in various papers, but there is an important remark here, the inputs themselves mean they are hyper-logarism and the output is also a superposition of hyper-logarisms, but the inputs have to be inserted in a positional notation, in positional indexation, whereas the outputs have to be noted in incremental indexation, whether there are tables in my paper, I mean the number of terms which you get is even larger, the formula in blue up there tells you how many, if each of the inputs has depth di, this tells you the number of terms, you see, just to understand how complex the whole thing is, how devilishly complex it is, just take a convolution, a weighted convolution of length four with inputs which are themselves hyper-logarism, each of them again of length four, what you get is close to 10 billion terms, of course there is a combinatorics behind it and the symmetry, the symmetry of all makes everything rather manageable, but this is just to say that I mean following the integration path, the multi-path wouldn't have got us anywhere, we have to develop singularity combinatorics, so there is a formula for calculating the weighted convolution of hyper-logarism and since they can approximate anything, we can take the weighted convolution of anything, so now here comes the main surprise, I want at least to be able to describe these very strange objects, I mean the source coefficient, the water places, the source coefficient, the tessellation coefficient and their essential discreteness, so you see in a sense we have everything, we have expressed the convolutional products in terms of hyper-logarism and we know how to alien-differentiate them, but here comes the surprise when we take the sum, the sum, huge sums and there's a gap in complexity and usually such gaps are attended by emergent properties and here the emergent properties is the emergence of discreteness, so let's say we have no time, let's take an example, perhaps at length 3, we get exactly 15 terms and they are all, they are hyper-logaring and all the indices on top of them have the same length, so if we apply an alien derivation of that, I mean u1 times v1 plus etc, u3 times v3, each of the terms is going to contribute something and we have to get the result, we have to replace each of these 15 hyper-logarism by the corresponding monic, so the x disappears and we have this sum, now and the surprise is we know how to partial differentiate these monics with respect to the indices and now take any u or any v and apply the rules for partial differentiation and you find zero, but this is not obvious on this formula, but this is what you get and all the way to infinity, but they are not constant, they are not zero, if we apply the jump rules we find this recursion equation in red, which I have to explain the notation, but I can take my word for it, they are a fine number of domains on which the function is constant and they are jump rules, which amount to explicit recursion, which can describe the whole thing and this leads to, again I cannot explain in due detail the formula, but these recursion rules leads to an explicit and simpler expression for the desolation coefficients, but they have one effect actually, they are highly polarized, they are simple, they are elementary in the sense that they involve only sine functions of elementary functions which are homographic in each of the u's and each of the v's, but you see this is, we have the choice here between their locally constant and discrete functions, these desolation coefficients, but we have the choice between expressing them as sums of hyper logarithm, which is slightly incongruous obviously, but in principle and an expression which is much simpler and more appealing, and also from the practical point of view, infinitely preferable, but highly polarized, this is a situation which is not unusual in mathematics, but it is a rather extreme instance of such a situation, and in fact the simplicity of these desolation coefficients is deceptive, actually they are rather mysterious and highly interesting objects, which are lots and lots of properties, just mentioned three, they are invariant on their their symmetry with respect to their indices, they are double indices, they are excuse me, they are alternal, and then they are invariant under a very important involution, this is called swap, which exchanges upper and lower indices, and that makes them be alternal, mean twice alternal, this is a very important property in arithmetic, and the study of multisitters and all that, and then again they are discrete, and for a depth one they are trivial, they are always equal to one, but at a higher depth, surprisingly they are not constant, but they are almost always zero, if you mean by any measure, and what else, well many other properties, so they are quite highly interesting objects, now we are in a position with the help of the these two convoluted convolutions, and with the help of these desolation coefficients, to express, to do what we set out to do at the at the outset, mean to express the alien derivations of any convolution product, there are formulas for this, they involve three things, the express, I cannot describe them in detail, but just I'll say it in words, we can express the any alien derivation, alien derivative of a witted product in terms of three things, a new convoluted, a new witted product with new inputs, which are either the translates of the old inputs, or alien derivative of the old inputs, and new weights, so this formula exists, I'll have to skip them whenever we should, you know anyway if you care for that, we deal in the in the slides, which I posted on my homepage, and now at last we can describe the, we can give a complete description of the two types of resurgence, the first, in the first, I mean in terms of not of the complete solution, but as I said of the normalizers, which are more flexible, the first bridge equation is quite simple, what is one of the standard forms upstairs, I mean up there, and it involves as usual ordinary differential approaches, which in this case depend on all parameters, including this parameter x, and it is a constant in z, but it is an entire function of x, so it describes everything because it can be iterated in terms of itself, so it gives all the information, not so with quick equation resurgence, because the first bridge equation, which you find the second one, looks formally the same, but actually the q's, which on the right hand side do depend on x, but not as an entire function, because here x is the resurgence variable, the resurgence variable, so to describe the resurgence in x of this new object, it takes the third bridge equation, which thankfully is closed, it expresses the alien derivation of q in terms of q itself, now this is in the case of holomorphic, for the first equation it is absolutely general, no assumption at all, for the second equation, for the equation two and three, we must assume that the data, the p's, are holomorphic, if they are not, then there is an added complication, there is a new level of complexity, instead of only q's, you have two series of new objects, mean new disurgent objects, q's and p's, and they are related by the tessellation coefficients, so again we would have to mean describe, to look at this calmly, and we cannot do this here, but the object again, it was to give a survey, to show that there exists a machinery for dealing with the general situation, so you see I'll have to, to skip the the end, we'll just say it very quickly in words, but let us suppose a little and take stock of the, what we've seen or achieved so far, you see, contrary to equation resurgence, which is simple enough, although it covers a huge, huge ground, I mean, but it is simple, I mean conceptually in terms of calculations, a coefficient resurgence is, there is a stratification of four level stratification, you see, the inputs themselves, they are, let us say they are hyper logarithms, then we have the weighted convolutions of these inputs, they are huge combinations, I mean huge clusters of hyper logarithm, again with emerging properties, quite unexpected, and then we have the Qs, which are again, I mean, large sums of the such clusters, and then in the general case, in the, with ramified inputs, Bs, we have a fourth layer, which again, I mean, the P's and the Q's and the P's up top, in the last layer, and the passage from the Q's to the P's is again mediated by the tessellation coefficients. So, then again some differences, I mean, what equation resurgence, we'd only to assume that the inputs B had to be jerks at infinity, I mean, no, in the case of coefficient resurgence, we had to assume that they were in the multiplicative plane, they admitted the endless continuation with, again, uniformity conditions. And then in the, for the first bridge equation, the index reservoir is simple enough, it's just, it is generated by the multipliers by the lambdas here in this equation. In, not so with coefficient resurgence, it is much more complex, it is the linear expressions of the weights and the singularities of the inputs. So, you see, but to make up for that, the source constants simplify, they become, I mean, the tessellation coefficients, which are, so there is a trade-off between the complexity of the set of omegas of active alien derivation and the complexity of these source constants. And then, well, many other properties, but I want just to say in a few words, there are only five minutes left, the two, what is the matter, actually, with autarchy and with isography and autarchy. You see, in the case of coefficient resurgence, we have, the q's are, the q's and the p's are resurgent functions. But you can imagine that they are, they stand in close relations, with the a's, the a's of omegas of the first bridge equation, because remember, there is a case when both coincide. And so, in the first case, they are entire functions of x. In the second case, they are resurgent functions of x. And they are not exactly the same, but they, there is a close relation. We go from one to the other, one system to the other. So, they have this double property of being entire and resurgent in sectors. So, obviously, this imposes on the active alien algebra certain constraints, because otherwise, there would be an effective ramification at infinity. If there is to be no ramification, or only a finite ramification, there has to be some constraint. But the, the interesting thing is that these constraints assume the form of this form in the each operator in the active alien algebra, each ordinary differential operator kills a certain differential two form, which I called the isographic form. Why? Because it had to be given a name. And so, I gave example, I gave three examples and I'll produce a paper, hopefully, with many more, but of a so-called, such isographic forms. And then autarchy. Well, autarchy is precisely this, this property of, well, I'll finish on that. Autarchy. Autarchy is the opposite of anarchy. Autarchy is usually spelled with a key, but I wanted it to rhyme with anarchy. The, you see, it is, you see, roughly the, the autarch functions or entire functions, who the syntotic behavior in the various sectors is fully described by resurgent syntotic expansions, which in turn generate on the alien differentiation a closed finite system, their autarchy relations. And so you see, despite being transcendental, the autarch function has a strong algebraic, I mean finite and algebraic flavor about them. And moreover, they are not freaks of nature. They are quite common, actually. If you take something at random, of course, it will be anarchy. But in practice, it will be, I mean, if it is something of any use, it will turn out to be autarch. And the showcase example of autarchy is, of course, the inverse comma function, you see, it's on its own, it's an entire function. But there is, on the right side, there's a syntotic expansion, which is well known, and which, I mean, combines with the first ramified factor, actually, to destroy the ramification and to produce something which, despite resurgence, turns out to be entire. And then the showcase, the instance of anarchy is, of course, the xi function, which is associated with the zeta function, the Riemann zeta function, because, alas, there is this strip, which defies accurate description. So, and the two things mean isography, isography and autarchy, they are closely connected. And as you mean, I gave three examples. But the first one, in keeping with our philosophy, we deal with the bioresurgent monomials. We want, I mean, systematically, as a matter of principle, we want to study the end to solve the difficulties at the most basic level. And so the first example is devoted precisely to the bioresurgent monomials and to this double genus-faced nature, being entire and being resurgent. So, one more minute, if I may. So, I wanted to say something about the other types of resurgence, but we have no time for that. I just want to mention, in 30 seconds, a spin-off, I mean, a fallout of this theory. When we describe the weighted convolutions of hyperlogarism or the tessellation coefficients, you observed, certainly, that the U's, the weights, and the V's, the singularities in positional notation, interacted in a very specific way. I mean, the U's got added cluster-wise, the V's got subtracted pair-wise, with additional constraints. And this gives rise, actually, to an interesting theory. This gives rise to the so-called flexion structure, which is actually made very informal. This is the sum total for all interesting operations, which you can obtain by using four basic flexions operations, which are four flexions, which combine the U's and the V's. The W's are two-tire indices, meaning V's and U's, and the flexions can combine them, transform them in precisely this way, adding the U's. And this flexion structure contains, I mean, the joule inside the flexion structure, is a set of a certain algebra and a certain group, which are Adi and Gadi, which have the properties of preserving double symmetries, which is extremely useful for in arithmetic for studying what I call dimorphi. Dimorphi, you can think of it as the property for a set of transcendental numbers of a Q to be stable under two different multiplication tables, which are difficult to combine, actually. So the multisiters and the hyperlogarithms fall precisely. They are highly interesting and highly useful, fundamental transcendental numbers, I mean, and they are dimorphic. And this structure, the flexion structure, is extremely useful, actually, to study that. And they mostly over the last 15 years or so have been basically into this dimorphic business. And so I wanted to mention this spin-off. But actually, these transcendental numbers are precisely the typical ingredients, the building bricks of the Stokes constant for equational resurgence. So in a sense, we have come full circle. We have here a nexus of objects, which is very closely knit, and in my view, rather harmonious. So time to stop. Thank you. Do you have any questions? I have a couple of questions here. This is a photographic form, which you mentioned. Is it a non-degenerate balliniform? Yeah, it is actually, even in the case when the number of variables is odd, even in that case it turns out to be a symplectic form. But it is not given naturally in that way. It is not. This is interesting because there are quite different. I mean, there are also resurgence when you consider symplectic systems. In certain cases, we're on top of the resonances or the standard resonances implied by symplecticity. You have other resonances. Then resurgence gets grafted into this. But the underlying symplectic structures get grafted onto that. And to that, the ease, you see, of the bridge equation, they derive from a potential. But they are real, they are continuous. So in that case also, we have the similarity superficial because in that case also you have resurgence constants which derive from a potential. But they are continuous. They are indexed. There is no discreteness about them. But there is an interplay. But yes, there is a symplectic structure behind it. But it presents itself naturally in an unusual form. I gave three examples in the notes. There is also a short question. There is also desolation coefficients which are zero plus minus one plus minus two, yeah? Yeah, after a lot. Yeah. After a realization, you see that very often zero can be back measured as a function of some complex parameters. Let us say, well, in every sense, you can think of, actually, if you take the coefficients at random, it would be zero most of the time. Or if you consider them to be a function on the Riemann sphere to time in, and you take the Borel measure, it would be, although I did not actually compute the exact Borel, but this is at least spectacular actually. I mentioned the values for the first depth four, actually. They are almost always zero. Again, the simplicity of these creatures is highly deceptive. I have to say, well, at least I do not understand them fully, although they are there. I mean, they are described. But we would like actually to get other formulas for them and to know more about them. So there is a lot of work to do in this line. One more question. Thank you.