 First, many thanks to Karen and Eric for first, for organizing this event in three different versions. I think it's a good compromise. It really works well. Then thanks for the invitation to speak here. It's a great honor for me. And as many other speakers, I will go back to history to March 98. I just arrived at CPT Marseille at that time as a postdoc. And then a larger group of us went to a conference on the community geometry in Beatrice and Mara in South Italy. And of course, there was a contribution by Alucon. And Alucon reported on really fascinating groundbreaking work by a physicist, Joe Kreimer, who discovered the termization confit theory, which was a mystery for many mathematicians, is actually encoded in Hopf algebra. So it was absolutely remarkable and even more remarkably is that the same Hopf algebra in a slightly different flavor appears in another work. Alain was doing with Orymoskvici on the local index formula for with certain hyperlipidic operators on space affiliations. So of course we didn't understand immediately all the details, but it was clear to everyone that there's absolutely great development and we have to understand it. Well, so it had enormous impact to our group in Marseille. So we had seminars on it, we stopped all other work and wanted to understand it. So, okay, I work with Tuma, Tuma Kruski, our next speaker on this, and we soon understood the generic structure, but we had problems with overlapping divergences. And, okay, I think it was Bruno Yukon who invited Joe to Marseille for the end of May and as preparation for discussion. Tuma and I made our notes a little bit on archive. And, okay, here that's more or less the first part of our paper, and I want to highlight one sentence, which is to result already of discussion with Duke. So here we describe how Duke treats overlapping divergences. And there is a story behind this sentence. The story is more or less this. So I found on an old computer my email correspondence with Duke. So this Duke's message from May, and so, okay, he explains what to do and what I want to emphasize is here that so he points out that to understand overlapping divergences. It's assumed that the reader digested the use of stringer-dyson equation and refers then to the amputation thesis where okay, this can be found. All right, so I stop history here at this point because I discontinued working on the algebra. There are other speakers who can do this much better than I can. But for me, these divergences equations, they are of absolute central importance. Back in 98, it was new for me, but in the meantime, they really become, I would say, my most important tool. So and then I would like to show this impressive page. So this impressive number of 11 papers, authored by Duke, which have divergences equations in the title. So just in title. So I'm sure there are many, many more divergences equations are the main, one main subject, but already in title, there are 11 papers. So free with Karen, but many, many more. So, okay, therefore, I think contributing something on divergences equations is something well suited for this description. Okay, one can say really a lot about divergences equations. You can lecture series and so on. So I will be very short and give some general idea and then we go to to a specific instance. So desert equations are some equations of motions for green function from three three. One can understand graphically as the collection of five and graph series of same external structure. So they form blocks and the equations are of that type. So, for instance, one is interested in a connected two point function. And then one follows here the first line on the left. What can happen? Either a line goes through nothing happens. That's a free theory or it hits somewhere. So then you write the verdicts and then you ask what else can there be. So in this case, it will be a connected four point function, which then Well, What happened? So, Yeah, so, okay, there's a very, this picture is very easy to understand when collecting five and graphs, but the point is, somehow this assumes that the series of five and graphs converges, which is absolutely not clear. So, but there's another remark. So, such an equation can be completely rigorously derived without any reference to bed vision theory. So that's an exact equation. Which can be solved by pedperon theory, but it's not needed. So, therefore, in fact, these digital equations are known for different equations. The problem is, it's always sometimes often hard to solve it. And the problem you see already here in the structure. So you want to know something on the two point function. That's what you want to study. But this equation involves here the four point function. So that's the typical feature that somehow you need to information that you don't have it. So, either one has to solve all the situations simultaneously, which of course is difficult or one has to work. So, and, yeah, I'm working particular on concrete theories on finite dimension approximations. Another word is made model. So in these in this setting, it happens that these equations somehow disentangle. So we get equations where this relation sign is reversed. So the interesting endpoint function and then we only need functions where the number of legs is smaller, not larger than N. So the price to pay is that this question become nonlinear, and that's also difficult. So, okay, now I come to the subject from three theories of particular type on matrices. So, all right. So starting point is fundamental theorem in harmonic analysis to go now. Who proved that whenever we have an inner product in the vector space, so find dimension for simplicity, for instance, the space of separate joint and by matrices. Okay, so it's spring. So, okay, the theorem is whenever you have an inner product on such space, then there is a unique probability measure on the dual space was fully transform is x of minus one half inner product. It's completely general. So, we take a very specific inner product on space of matrices. If we give ourselves n positive numbers you want to be in. So this summer stand for the energies or the spectrum of the plus operator truncated to n levels. So when we build a scalar product, well this one. KL and prime k divided by EK plus EL. So that's Gibson with a product and okay, we take this one. So, as I said, it has something to do with Laplace and other geometries. And, okay, here that's sort of Gaussian measure and everything is now, of course. Okay, we will not stop here for the next step and also next step will explain why we are interested in this particular inner product. Okay, the reason for choosing this inner product is the consequence model. So the consequence model is a deformation of the measure I just introduced for free theory by a cubic potential. So e to the minus lambda and discovered that. Okay, if you do where computations with this particular measure so vacuum graphs and to understand them as function here of these vector values you want to get. Then this gives rise to what the numbers which you get rational numbers. There are in fact intersection numbers of total logical classes on the movie space of complex curves. So it's very deep result. And so there are many, many developments connected to this result. So it's an incubate model related to the KTV hierarchy. So in fact, all these relations were already suggested by written short before and conservative proof by inventing this particular measure. But it's of course not a real measure. As you know, if I cube is unbounded, you know, so you better have lambda purely imaginary and then it's not a measure, but formally this. So, okay, and it's connected to something which is very professional. Now to put a question also talk by tomorrow. We'll deal with it. And so. Okay, so, and, okay, this everything is known. So we are interested in the deformation by aquatic potential so far to the fall instead of five cube. This is a very small step from point of view of concrete and very natural step lambda five to the fall is much more nature than lambda five cube. But the mathematics is completely different. So there is clear logic by conservators fight to buy cube. And you may find cube has something to do with simple zeroes of stable differential on human surface. And you cannot place it by fault, which would be quadratic zeros and but these are exceptional so the generic case is really fight the cube and there's absolutely no relation mathematically between five cube and five to the fall. Also pet Bayesian theory is completely different. It turns out, however, that that's the recent result of previous year that nevertheless we find structures which are completely similar. Mathematical structures has something to do with topology recursion. And well, this is the subject of the talk. But one remark. You get something from three. So it's a toy model. But it's something where we have really exact solution. Exact and company constant. So, okay, that's the content. Okay, so let's start. A starting point. Well, is, we will derive the equations. So we consider now the full transform of the measure. And that's the full measure. So depending on the spectrum values, you want to be up again. And the company constant. So that's the 40 deformation. So then it's a completely elementary exercise to derive these equations of motions. For the full transform. So, and these are important. Because if you remember, I try to go back to the picture. So, okay, these external legs in this formulation result by differentiating the full transform with respect to the source matrix. So if I have any derivatives, then I get an external X. So derivative with respect to M brings the five downstairs and then graphically, this is the end point function. All right. So, and the problem is then in this equation, we have too many extra legs and these equation help us because they allow to express here a second derivative. So it gives two additional legs. Second derivative of the full transform by something which has only one leg or in this case no leg at all. So, okay, we can express these derivatives by something which has less derivatives. Or maybe the same same number, but okay, that's the key point, which is not available into the concrete theory that's special for matrix models. So that's the key, but it can go much further than for normal. Right. So I will show some of these equations then which results. So the idea is you differentiate some more times and then you get identities and these identities are the dysentery nitrations. Okay, so these different questions, they, they have them. They inherit these matrix indices. Of course, and then we have a finite number. Distinguished by finite number of matrix indices of these equations. So there's a key step. Namely, it turns out that one can complexify the resulting lines to equations. So that they become equations between mirror morphic functions in many complex variables. Of course, they are given by a finite number of points and complexifying them. Well, it's not unique, but there's a natural way to do it. So stress that there's a very natural way to obtain the equations for mirror morphic functions. And then we can use the whole power of complex analysis to tackle and to solve them. So in doing this, so after complexification, we can relax a little bit. The conditions that these you want up to end were previously degenerate. So we can admit multiplicities. So our different eigenvalues are different values are little e1 up to ed and they come with a certain multiplicity. Not really important, but it gives us a lot of flexibility. So here is the simplest or the first equation which one gets. So we consider here a second moment of the measure with a two point function. All right, there are certain factors of N. We give it a name. G is two indices. Okay, if B is different from Q. No, that's okay. But here's something else, then it's zero. So that's the interesting object to study. So we're interested in large N expansion. So this is a formal power series and inverse powers of N. Turns out it goes in even steps. And a little number G, the parameter is or will be related to the genus of Riemann surface. So that's it. But okay, here it's just a name. But there's this form expansion. So as I said, we complexify it. So here everything is discreet. We complexify it. So we arrange that our G, PQ is actually the evaluation of a mere morphic function in two complex variables, sine eta, no, theta, theta and eta. The theta is EP. Sorry. And eta is EQ. So these initial spectrum values, if we evaluate our mere morphic function on them, we get back this, okay, the exponent coefficients. So, and the first equation is this one. So it has a long history playing with that because many, many years ago. So important thing is that's an equation which only involves the, this two-point function complex by two-point function. So that's nice to have. But as I said, the price is it's a nonlinear equation. So here's the linearity, which was really hard to understand and to solve. So it's a very general setting. If we go to the large end limit, then we could have divergence. So better have Bayer-Mars and Bayer functionalization there to have a lot of freedom. Well, so, okay. Well, for me, the breakthrough was in joint work with Eric, which we achieved during a school in Lisouche organized by two years ago, where we were able to solve this equation in a special case, namely that the multiplicity is one. And all these spectrum values are equally spaced. So this is a particular choice which corresponds exactly to the lambda for the form model or two-dimensional spatial space. There's a particular interest in studying this in large end. So it turned out that doing this, so all these sums become integrals and so on. And then, okay, you do perturbation theory, not for this equation, but okay, for some of you, this is an excellent equation. And remarkably, using Eric's hyper-end program, pushing far enough, we understood the first coefficients. So data model pattern, and we understood then that they are the first terms of a series, which is the series of the lambda function, lambda w function. Okay, so, well, that's something. Solving such a non-equation to lambda w was, okay, unexpected, and this opened the way. So it opened the door to take the general case. So the lambda function has special properties. And after understanding them, which took some time, we understood the general case. So take general e-case, general r-case, finite or infinite, whatever, we can solve in any case this equation. So a particular interest is the forward margin real space where we get, instead of lambda function, we get the inverse function of the hypergeometric function. So this will be covered in the over next talk by Alex Hawk, 1550 in Paris time. So he will present more details on this approach and very nice consequences of this example. Good, so I continue with the discrete case because here for finite d or finite n, there are extremely nice structures which appear, which has something to do with complex analysis and algebraic geometry. All right, so that's also a long history, but I will just give the final theory. So I will describe the solution of this equation in complete generality in case of finite, and finite d. So you have to do something. Remember, we are given these vectorist ek, multiplicities, and coupling constant. So define new parameters, epsilon k, rho k, which somehow deform from these data in a following way. So build such a function of z, z minus lambda divided by n. Okay, that's a fine sum. So here you should see in a certain regime, this will produce a logarithm. And that's z plus lambda log z is then, okay, the inverse function is a Lambert function. So there's a logic behind this, which can be guessed from the Lambert function. All right, so take this problem, function R, and okay, here you have epsilon k, rho k, and they are implicitly defined to satisfy of epsilon k is ek, and our prime epsilon k, rho k is rk. So that's implicitly defined problem. So then, so remember this was the function we want to have. So of this equation, hopefully. So you go to the pre-image, so zeta, eta are given, and then you go to the pre-image via r, so z and w. And you transform variables, and after transformation, the result is script G. So for this function script G, we have a formula. So here it is. So it's, well, it's a regular function that's important. And it involves the definition of r, and so r of z can be replaced by zeta. That's clear, but the point is we also have the negative r of minus w of minus z, where we can do it directly. So that's direct relation, so we have to go via r. So r's gone. Then notation, there's a little head. So the head means, okay, first go back here. It's, well, you see, this is a function, which is the ratio of polynomial of degree D plus one by polynomial degree D. So the D plus one pre-images of every value. And the other pre-images except of the principle one they carry head. So this absolute head are then the pre-images of these deformed energies. Okay, so that's the solution. And so proving it when one knows what to do is very, very easy. So that's, you can give it as an exercise in complex analysis course. But it really took us 10 years to come from this equation to the solution. So, okay, remark, symmetry, although it is not manifest, it is automatic, can be checked. And yeah, okay, that's remark. And that's the point. Good. More remarks. So again, stress, that's the solution of nonlinear Dyson Scholar equation. We found it first with Eric by, well, brute force, I would say, preparation theory and a lot of luck that understood what we were computing there. But then we proved it. So what we got an idea and then the proof is beautiful. It's beautiful complex analysis. So then, of course, you ask, how is this possible that you solve an only a problem. And it's more or less clear, there must be something behind some deep structure, which we don't have it. But we are confident to find it. And this involves the fine equation, which I will then talk next. Yeah. Okay, one remark. It's very important to deform the data. So originally we had a case with my capacity lambda and in these variables, you don't see anything. It's absolutely important to transform by the inverse of such a relatively simple function. Then we see a structure. And okay, it's an empty statement, but nevertheless, maybe one should think about. Maybe something similar could be true also in familiar conflictings that we have to transform in a way which we do not know to other variables. And in these other variables, we see more structure. Okay, but I have no idea how to do it in nice examples. Okay, so now I come to the final questions. So there, well, by general theory, analysis, it's clear they are solvable, but there doesn't seem to exist, seem to exist any theory which could give us this equation explicitly. So the key idea was due to Alex. So he suggested to look at auxiliary functions. So namely, so here these are our two functions. And so, well, they somehow depend implicitly on the initial data, the energies and the coupling constant. And so you differentiate these two function with respect to the energies. Of course, you can do this in practice because we don't control the measure. But okay, let's introduce them and call them omega with indices a one up to the end which okay one a one special and the others are derivatives. And then again apply these genus expansions. Okay, so, and the key step is, we don't need to know. We can derive doesn't make question directly for these omega n's why not directly they are coupled to other functions which I will introduce. And then we can try to solve it. So what we need are the initial date, initial data, our function are and the function, but nothing else. Everything else is contained in directions. So the for me at least complete unexpected result was that these oxygre functions. Unexpectedly translate to something which has a clear mathematical meaning, namely they are object of blog topology proposal proposed by bro and charting five years ago. So I will say a little bit on this. Okay, here that's more or less the overview what one has to do so it is not we are interested in these omegas. So from them we later get then the moments here these which point to effort. Okay, they were oxygre function but we have to introduce more auxiliary functions, namely these two sorts of them. And then one has to go through such a triangular pattern. So here this first box is what we know the two point function function and then. Okay, for genius zero, it's enough to compute this chain in the upline. But if you want to increase the genius, then we also need here this strange other function. And then we won't go here then. Okay, this is the way to go. Okay. Okay, these for all these functions we have dashed equations. And so evaluating the tease is relatively easy, but the result is horribly complicated and you insert this horrible result into the equation for the omegas. And then many, many cancellations arise and finally the outcome for the omegas is extremely easy. So I will only show the results, not the equation and not way to compute it. So first result is quite mark points. So interest is this expression here in parenthesis, especially the right one. So this is. So the differences is the background kernel of topology in charge. So, having found this, we knew that we are on the right way. So, and they're always these prefectors. So the idea is we multiply by the differential of our and any of these variables, also lambda goes away and then we come to differential forms. So one form in each variable. And so, okay. That's, so we know it's true in the planar case, otherwise it's contract jet. We, we are sure that these omegas ends have a very simple post structure so they are more mere morphic. And we think we know where the ports are so we are sure in planar case but otherwise it's contract. So they have inputs at zero on the opposite diagonal and on special points, the two of them were the derivative of our benches. So these are the so called ramifications points of the ramified cover encoded in this rational function. Good. Yeah, okay. So mark these keys to function they are much more complicated. They have many, many more points. Okay, so here's the few solutions. Okay, looks also complicated by in fact, there is a key logic and in fact, these are relatively simple functions. I wanted to show maybe show first this. Okay. And so these are non perturbed results. So that's important. In principle, you can use yourselves and finding which five man graphs use a five man graphs contribute to such omega GM. Of course, there are infinitely many of them. And so they, the sum is convergent. And so there are certain graphs of a certain external structure, certain topology and, well, all they sum to such an expression. And of course the dependence on the coupling constant is a little bit hidden. Well, remember, we had the system of equations, the epsilon case, in the whole case, they are implicitly defined. And then we have the betas solution of a prime beta zero. And when this system has to be solved somehow about implicit function theory. So abstractly, it's completely clear we have a whole morphic solution. And also, as you know, we have good conversions results if one wants to find a solution. But one can also approach them via a Taylor expansion to the implicit function theory. And then this Taylor approach will coincide with the five man graph expansion of a certain topology and external structure. So our idea is that this is something which we would like to contribute to this sigma volume for Dirk's birthday. It's really nice to compare the five man graph series with the exact result going in particular through the stratification points. So few remarks on color code. On the next two pages, I will show that the terms in blue have a very clear mathematical meaning. So there's something very well known. So in particular here you see these rational numbers appearing, at least some of them have a meaning as intersection numbers of characters classes on the multi space of curves. Not all of them, but many of them. And then here this term in black with posts on the opposite diagonal. We understand them at least in for all planar functions so genus zero. Projected by this key. It's true. And so we are confident that very soon we will understand them completely. The idea is that they are just there for confusory but maybe they will not have a meaning in mathematics. But these terms here in in red puts at zero because zero is special. So you remember we had always this reflection that goes to minus that was here, either minus beta. It's very special for this particular model and use the fixed point of this reflection and these numbers 1816 and so in particular these combinations of ratios of derivatives are zero. So my feeling is they have a certain meaning. So this is something which will extend the result of conceived to something but I have no idea what that's for the future. So, okay, now what are the blue jumps. So what so here there's a general picture of the ramification points. Of course I can only draw a real picture. So, if they are simple zeros then the real paper is just that square function. And then you have said, and the reflection is called sigma offset. So, means offset is off sigma offset and beta it to come together. So this reflection is the local gallery. So, okay, now if you do the following omega zero one is special. So you define as minus of minus that a prime of safety set. So in all other energy and as before, and then well surprisingly but there will be some origin. We have so called loop equations. Okay, these omega gm they have poles very high order at beta, but if we sum the conclusion at set, and at, sorry, and at sigma sets the reflect value, then the ports cancel. And you get something also constant term advantages. So it's something which starts linearly about the ramification point. So the quadratic loop equation that. Okay, it's a very strange combination. So we take omega. Okay, one genius lower insert that and sigma for said, and then you share it in product quadratic and you share that and sigma set in both variables. Otherwise you distribute the other variables and the genius. And then, so again, these are ports of high orders, but in this particular combination, all the polls cancel, also linear term tensets and you get some things a quadratic form in Z, which starts quadratically about the point. So, okay, this is true in Plano case up to five and also for one one. And of course we can check child is can be an accident that all these points can so this should be true in general. So then there's a general theorem of topology recursion that whenever you have a family of differential forms were more of it, which satisfy these absolute look equations. Then you know what is the polar part the polar part part of these omni GM with poles at the ramification points. And this part is given by such a formula. Okay, it looks complicated, but anyway, but this familiar. So that's the famous formula of topology recursion. And so in the original version of topology recursion was just this. So that's the formula how you build from so here on the right. Everything is known. So you want to compute on the GM and all the others are already known you have them. And all you have to do is to evaluate just this residue and he has a certain recursion kernel. The Birkenkernel integrated. So this something easy. And the market thing is that many many structures and mathematics follow exactly this formula. In, in large class of examples, PZ omega is equal to omega so you can remove the speed. There are no other poles, but it's both by boring charting that's not the general case. So when you only require these absolute look equations, then you can afford more. And then these formulae describe the polar part. And then there's something else, age the whole movie part will move at ramification points and they, okay, they are formed by these blocks. So additional input. So here's a picture. So that's the picture of topology recursion. So, okay, we live here on space of lines, compactified complex lines to mark points on the remand service. And so you want to build such a particular form and that space and the idea is you distribute lines and general in all possible ways. And then you glue everything with recursion. So that's done on top of recursion. And in the block case, there is an additional input at every request step. So important point is that's an infinite space, but our omega g n select one particular part in this space. And question is what's the significance. Good. It's a sort of outlook, whatever. So it is known from the paper that whenever we have these absolute look equations, then these omega g m and code intersection numbers on the model space of stable complex graphs. That's a general fact. So, and they give formula. More or less the ideas we get in general several copies of the same intersections of the cyan copper classes so cyan churn classes copper among four classes as in conservative model. And, but they are not independent they are coupled somehow or describe what they give a precise description by the blocks. So, any case, I get something, but it's not clear that this of interest. So here, but in our case we have this global inclusion that to minus it, which places special order so that let's hope it's not proved. I hope that at least these polls about zero could have some meaning in mathematics. We do not know what, but it could be something. So this, so this global inclusion as an addition structure on the movie space of cars. So another thing is so you have endpoint functions forms. We don't have partition function itself. At some point, we would like to identify it. And then the interesting question is whether this partition function is a top function for a he wrote equation. So if so, this would be a precise formulation of instability. So in general top logic version, there's always this relation. There's always a top function. For block top logic version does not know. So it could be could also be not the case. So there's something which one should decide. Okay, I think I am, I'm time. All right, so someone Yeah, it was on first talk on the string equations. And so it's just from the fiber fiber grass where it's the tool to disentangle the problem of interverges is to things but I presented. So the string equations are central to dark ones to me. And yes, so at least I tried to convince you that the string equations are really, I would say the best approach to study from field theory and paternity free. So in this particular case of matrix models from fields on the spaces. And produce a complete understanding. And I'm sure, but I can prove it that also other concrete series should be treated in this way. So brings me to an end to do. So I wish you a lot of pleasure and success with your work. Thank you very much. Thank you very much. I have a question for him. So what, what would be your advice, how do you use topological or plot topological the curve. And I want to go from software to quantum fields. Well, this I would say this relation is very special. But So, so I'm sure there's some instability and you can expect this for normal concrete series. But I'm sure that the idea to to look for to disentangle the problem. When you have a concrete theory that you split, you understand or understand it as the convolution of function with the inverse of another function. Part of the difficulty you throw in this problem of determining the inverse. As you have seen for the Lambert function. That's, it gives all the logarithms and equals then the knees and polygorithms and so on. And it's the inverse of a very, very simple function. So I'm sure something in this direction should be there also in other concrete series. I'm not sure. God, I'm going now sitting next to me so I can actually try to get something from him. That's that you will have to. Thanks. There are further questions, be free to raise your hand if you're a viewer or just speak up as a speaker. I have a question when you when you showed your formula for the, I mean these omega g and they're all everything's rational right there's no transcendental numbers. This one. Yeah. Yeah so so I was just confused because if I remember at some point, as you just mentioned there are transcendental things right there's logarithms and maybe some data values or poly logarithms. I said you can somehow match some classes of Feynman diagrams to these individual omegas. And I just wonder where does the transcendental score. Well, so it's to do this. So, so. Well, that's a total model of conflict. There's no organization. Everything is, it's a regularization. So everything is fine dimensional for conflict theory you have to go to the large end limit into infinity to infinity in a certain way. And this produces here integrates. And in the case which we had this was the logarithm. You almost see it here that this converges to logarithm. In other cases so that will be the presentation by Alex in in one hour or so. This will be hypergeometric function. And then you have to produce the inverse. You see it somewhere here after this transformation. And even if you have a very simple integral here. For instance, the inverse of the hypergeometric function is not clear to me that this is, well, of course it exists. It's a monotonous function. So it must have an inverse. But what it is, it's not clear. Expand it and you get all the hyper logarithms. So there's a step which is hidden here. So I described the fundamental case for conflict. We want to have the large end limit. Yeah. Thank you very much. That's very clear. All right. I suggest we we thank Reimer again for his nice talk.