 So perfect. So can you upload? Are you going to use slides, sharing your, I guess you're sure. Yeah, slides. So they send it to you. And you actually be able to, if I have the rights to share your screen. Alex, I made you a co-host. You should be able to share your own screen. Okay. The bottom there should be like in the center green button. Let's see. It looks good. Yeah, it looks good. Okay. We still see the chat. Maybe you kind of close the chat. I can tell you when whenever there is a question if you want in the chat. Okay. Does this like go away? Can I hide this? Okay, well, I'll just put it on there. So. Yeah, it looks good. You want to have the questions right at the beginning during your talk or at the end? During the talk is fine. Okay. It looks good. Okay. So thank you, Alex. So the next talk is going to one in the session of lambda terms, and we're going to hear from Alex about the distribution of parameters in certain fragments of the linear and planar lambda couples. So thank you, Alex, and go ahead. Thank you. So yeah, my name is Alex Alex and I'm presenting you some joint work with Olivier Bodini and Norm Zenberg here. So let us start. So what is the lambda calculus? Well, I imagine like everybody already knows a bit about it and the previous talk also illuminated some of the basic notions but I'll just go over it again very fastly. So the lambda calculus is a universal formal system for expressing computation. It's universal in the sense that anything computable under some appropriate notion of computability is computable by lambda calculus. So this means that it's computable by, for example, a universal machine is also computable by a lambda term. And its terms are formed by the following simple grammar. We have that variables are valid terms. So we will have an infinite supply of variables which we will denote by lowercase Latin letters. And if x is a variable and t is a valid term, then so is the abstraction formed as lambda x dot t. So we say the abstraction has variable x and body t. And finally, if s and t are valid terms, then so is the application of s to t. And I guess it's t to s. So the lambda calculus also provides us with some tools to transform the terms and the most important one, the most famous one also is the notion of better reduction. And what better reduction does is it tell us how, how to reduce terms which look like the application of some argument of function. So if we have a lambda term lambda x t and we apply to it some term s, then what we do is we go into the body of the lambda term go into t and we replace everything with every occurrence of x with s. And so in this talk, we will not concern ourselves with the transformations of terms we will restrict our attention to the syntax of the lambda calculus. So we can see some example of such syntactic terms of the lambda lambda calculus. So first of all, we have the very well known, sorry, combinator where we have lambda x x applied to x and again lambda x is applied to x and we see that if we keep better reducing this one, it keeps reproducing itself. So the lambda calculus can have very complicated behavior, despite its apparent simplicity. And on the second term we see the notion of variables does not come with the restriction on the number of times they can appear so we can have lambda x lambda y x applied to x applied to y. And we see here that we can use x twice. And finally, we can even discard x as we in the last term so we are discarding x and instead we have another free variable now which is not bound by any lambda which is this z. So we can have free variables we can have duplications we can have discarding in the general lambda calculus. And so the combinatorics of the lambda calculus is what we are going to talk about. And in general, the terms of the general lambda calculus I presented before, we know that they are quite complicated in fact their growth is super exponential and the general functions involved are not analytic for some appropriate notion of a size which is given recursively by assigning size one to variables and assigning size one plus the size of the constituent terms in the applications and the lambda term case. And so the asymptotic number of this general terms is still unresolved the form of it. I put here a little question mark because I know, Professor Olivier Bodini and Cinco Evang they have some results towards this direction. And so we will focus on mainly linear terms, which are restricted the there are some sub system of the general under calculus and these linear terms. So that's that the bundle variables they must appear exactly once. So for example, the xx apply to x is not a linear because we have a duplication of x, while on the other hand, lambda x lambda y a a applied to y applied to x is linear, because every variable appears one once and of course even the free variables do so. And we will also consider planters which what which are the terms we also saw in the previous talk. And these are such that the bound variables must appear in the order they are introduced. So if we read the term from left to right we see that lambda x lambda y y applied to x is not planar because we have this y appearing before x, while on the other hand lambda x lambda y a applied to apply to y it is planar because they have to write the free variables come first and then x comes and then why exactly as they are introduced. And so the lambda calculus is very important in logic but of course it's very interesting from the point of view of combinatorics. And so there are various connections between lambda calculus and various combinatorial objects like we saw before the beta 01 trees. But what is important in our talk will be the connection between lambda calculus and maps. And so what are maps them, their graphs embedded in an oriented surface without boundary. So we care about not only the graph structure of the map but also the way that around its vertex, the health edges are oriented. And so we also consider these maps up to some deformation of the underlying surface some appropriate notion of homomorphism or ambient isotope. And the close linear terms, they are especially combinatorial intriguing because they correspond to a very specific class of maps which is widely studied and it's that of rooted connected trival and maps. And this connection was first made by Bodini Garde and Jaco, and was elucidated also by a paper by Professor Noam Selvigir. And so we have that the subclass of linear maps which is a linear term which is the planner times they also correspond to rooted connected planner trival and maps. So we have this plan restriction of lambda games corresponding to the planner restriction of the maps. And if we also allow for open terms, then we allow not only for trival and vertices but also some unique balance. And we have to make a little comment here that I'm going to draw all these maps by having the route being actually of degree two, but there is a very simple bijection between trival and maps where the degree is the is an exception that it doesn't have a degree three but degree two. And general trival and maps rooted connected trival maps where everyone has a degree three. And this bijection is very simple to describe you just essentially dissolve the vertex so you just smooth out the vertex to just leave an edge connected to half edges. And so this connection between the lambda calculus and maps can be seen very easily in this pictorial representation I have here. So we have that a half edges correspond to variables free variables. For the case of applications we have that. If you have an application TS, then we draw them up flipped in some sense in that we put the map which corresponds to the subterm s on the left and the map was corresponds to the sub sub term T on the right. And we connect the routes with the new route, which is this little vertex I have here on top. And finally, for the case of lambda terms. We have lambda x with body T being represented by a map who's who's a great component here is the sub the map which corresponds to the subterm T. The variable which occurs in T, the free variable legs which occurs in T is connected back to a new route, whose other child is the old route of T. So this gives us a correspondence between lambda terms and maps. So we can have here an example of a term and corresponding maps we have lambda x lambda y y apply to lambda z dot z applied y to x sorry. We have put little annotations near the vertices I have not annotated the half edges corresponding to variables, but I have one of the vertices which correspond to lambda terms and abstractions. Sorry to abstractions and applications. And so we start here with lambda x and read reading the term from left to right we see first lambda x we see lambda y second. We see then an application whose one component is why so one subterm is why which is this half edge which goes off to the right so we see here we have why on the left, but the half edge goes off to the right. It's this flipping we talked about. So we have this why going off to connect back to the lambda y and the other subterm is lambda z z x. So again we have an application. On the right so we have a half edge corresponding to x going off to the left and connecting back to the initial lambda which correspond to lambda x. And finally we have lambda z, which is essentially this little loop here where the Z connects back to its own lambda. And so what is the purpose of this work is to understand how typical and by typical I mean random and of large size linear and plan terms we have. And that's our questions to help us elucidate this, this notion of how they behave is, for example, how many free variables do they have, and how often is a typical term abstraction or how often is it an application which is the complement of this question. And also finally it is to use and develop new tools from from analytics to obtain this to obtain some parameter distribution which will help us elucidate this question of how do these typical linear and plan terms we have. So in this talk will sketch the following results. I have divided them into general categories on the left hand we have the linear lambda terms, which are usually differential algebraic their equations. The equations which, which their generating function satisfy and the sequences themselves they are divergence in the sense that the power series does not the corresponding power series does not convert to the interior of any open disk in the complex plane. And so for this class we will see some, the limit distribution of free variables. So we'll see how the typical structure of free variables is for a big linear lambda terms we will see the limited distribution of identity subterms in closed terms. We will understand in typically closed linear lambda terms how many identities of terms there exist and will describe what an identity subterm is but in, in fact, they correspond to just terms which look like lambda x dot x. So functions which take an argument and just return it as it is, they do nothing so they are identities. We will also see the limit distribution of close up terms in close terms. And finally, we will see the probability that the term is an obstruction. On the other hand, we have planner lambda terms, whose general functions satisfy algebraic equations, and the corresponding power series they are analytic so these are much more classical in a sense. They are much more well behaved and standard tools of the of analytics can apply here to give us very nice results on the limit distributions. So here the case for a limit distributions of free variable for regular and bridges terms, and the probability that regular business open terms are an obstruction and of course this by compliment gives us the probability that they are an application. So let's start with the case of free variables in closed linear terms. And this case will be a bit unorthodox, we'll see in a little bit why. The variables as we mentioned before are those variables which appear in a term but they are not bound by an obstruction. For example, here we have lambda x ax here is not bound by any lambda structure so it's free. Now, that the limit distribution of free variable similar lambda terms of size n is Gaussian with mean and variance cubic root of n and so the cube root of n is a bit of a strange parameter. It is not often seen. And so the general idea that the proof sketch is to start with this functional equation we have here. This is a bit unorthodox because this is the only case where we will not have an algebraic differential algebraic equation but we'll have this here equation for the exponential function of the open linear lambda terms with you tidying free variables. And this comes from the definition of combinatorial maps, computer maps are a way to to represent maps as systems of permutations where we have three cycles corresponding to the half ages around a vertex fixed points here which correspond to univalent vertices. And finally we have involutions which tell us how to glue half ages together. So if one sees the definition of another maps essentially it says that the computer map may be represented by a permutation and an evolution effects when free evolution on the same set of half ages. So this is exactly the symbolic way to define this. So we have this which corresponds to not necessarily connected maps, so we take the log item here to get connected maps because we're interested in connecting maps and finally we root them to get rooted connected trivalent maps, trivalent or one valent map, maps, and we have of course some initial conditions here. And so the proof sketch is that starting from this equation we perform a certain point analysis on the Hadamard the exponential product which appears here, which leads us to these two. So the results for the equations of the corresponding exponential functions for the coefficients of the corresponding exponential functions. And finally we are we are interested in translating the information about these disconnected maps into information about the connected maps and we do this by taking the logarithm in the symbolic part. And we have a problem here because this function which appears in the interior of the composition it is divergent. So we have to use some, we have to use Bender's theorem in fact, which tell us how to compose divergent power series with some convergent ones. And so by an application of Bender's theorem for R equals to. We have these following results with tell us how to connect asymptotics of the connected, sorry the disconnected maps here with the asymptotic of the general connected ones. We have a little change of variables to fix some periodicity issues in our functions. And this finally yields the desired result. Next we have the distribution of identity sub terms in closely in our terms. So identity terms, we discussed them a bit before but they are essentially terms which are alpha equivalent to lambda x dot x and alpha equivalence is essentially a way of describing the naming variables in such a way that it does not create any new bindings. So we say that essentially we are looking at terms which look like lambda x dot x up to appropriately naming. And so for example in lambda x x lambda y y, you have that this little sub term here in red lambda y y it's an identity sub term. And so they appear as loops in the corresponding map we saw one example before, but here we have highlighted these two. I did it sub terms and we can see in the corresponding map that we have these two little loops corresponding this one corresponds to lambda x x and this corresponds to lambda w dot w. And so, for this case we have that the limit distribution of identity sub terms in close linear lambda terms is personal of a lambda equals one. It's a bit perhaps surprising because it means that if you take a huge map or a huge lambda term uniformly at random, you will not see many loops, you will just see maybe one two or three. And of course it's a constant number which is also surprising does not scale with the size of maps or the size of terms. The proof sketch here is to use moment pumping on this functional equation we have here, where G counts closed linear terms with you tagging identity sub terms. In this case we have an ordinary energy function, not as before where we had an exponential one. And so let us start by giving a bit of justification for this different functional equation. We have that dysfunctional question essentially tells us that tells us that terms are either identity terms, which is this case here we have a jet squared, we have a little correction, we will not cross ourselves with but it's a net case. So it tells us that a term is either an identity term. So it is either of the form lambda x dot x. Or it is an application. Or finally, it is a lambda term which is formed. Equivalently by their considering as a lambda term as we discussed before or by considering as a term which is this body term here T, where we have a little loop, which we have pointed out. The projection between these two versions that lambdas correspond to pointed the loops is that this point we can essentially we can cut it open and glue it up into a new route. So this correspondence is invertible. And it tells us essentially that lambda terms are the same thing as taking a body term and finding some loop in it and tagging it. So from this differential equation, we may start pumping moments to find the moments of the corresponding limb distribution. And in general, we can see that the case derivative of this equation may be written in this form, where we have the case derivative of G minus some term which here is of this form and depends on the parity of K minus two times this product between Z G sorry and the case derivative of G equals plus one derivative of G and essentially the idea is that here this S and this to Z G derivative of G do not contribute asymptotically. And this is because of the factorial growth of these coefficients. And so essentially because here we have always a multiplication by Z we have a shift in the coefficients. And because they grow factorally, if essentially we have that the coefficient GN is sorry the coefficient GN minus one is dominated by the coefficient of GN and so maybe neglected asymptotically. And so in the end we have that the case the coefficient of the case derivative equals the coefficient of the case plus one derivative of G and so we get a portion distribution. And the next one, the next distribution distribution we'll talk about is the distribution of close up terms in close linear terms. So close up terms are a bit. They include the previous case but they are a bit more complicated and this is a work in progress so I will not talk about it a lot. And I will not give much of a justification for it but I have it here for the sake of letting people know that there is all the. A bit so close up terms they are sometimes which have no free variables. And in the map or they correspond to bridges. So we have here the term lambda x x lambda y y lambdas easy. And we see here that this little subterm. It contains no free variables, if we consider this subterm of the biggest of the bigger term and of course it corresponds to this little grayed out component here, which if we cut this bridge here it becomes a true connected which doesn't contain the root. So the larger context here lambda x x and the application become the outer connect component and then the one is corresponds to this subterm. And so we have this correspondence between bridges and close up terms in these terms. And again we have the result which shows that the limit distribution of these close subterms in close linear lambda terms is possible parameter lambdas course one. And so the idea is to use the moment pumping agent this time on a much more complicated differential equation we have here where w counts closely in terms with v tagging identity subterms. And this is a bit of a work in progress or you cannot go much into it but the idea is that here this combinatorial it corresponds to the composition of terms by considering again some cases of the form identity or application or lambda. And in this case the lambda is not a real nice bijection like the previous case but it's a bit more complicated. And on the other hand, for the moment pumping process, we have again the same idea that we find a general form for the case derivative but it's a bit more tricky to localize the main contributions. Now, finally we have for the case of close linear lambda terms the probability that asymptotically such a term is this is an obstruction. And so we have asymptotically almost surely that such a random close linear lambda term is an obstruction. And the ideas here is that is very simply follows directly from the factorial growth of the linear the close linear lambda terms. So the idea is that we compare the coefficient of L and to set the fall derivative with respect to Z of L which corresponds to the case of lambda terms. So this corresponds to the case again of identities this corresponds to the case of abstraction and this corresponds to the case of application, sorry, this is the case of absurd applications and this is the case of applications and so we can find in this equation the case of applications with a general case, and by taking the appropriate portion we find that asymptotically these terms are always an obstruction. Finally, we turn to the realm of the planner terms. We can give a bit of a justification for this equation I pretty much I already did. But it is again that we have that a lambda term is either a identity term, or this is an application or financial lambda and how do you form a lambda, you take a map, you pick some vertex so we have this pointing here so you pick some vertex, and you choose the edge above it to split it by introducing a new vertex in an application and one of the half edges which goes off the application can either go off to the left or the right so we have this to here. And this goes back up and connects to a new vertex which becomes the root of the whole map and this gives us exactly this term and exactly a way to form lambda terms for closely near terms. And so finally we turn to the realm of planner terms and here we have a distribution of free variables in plan and business plan terms. So we hear we see here a very different behavior for the case of planner and business planner terms. So the limit distribution of free variables in planner terms is Gaussian with mean n over eight and variance nine and over 32 so they are linear in the size n. And the same is true for the business planner terms will have a mean of n over five and variance of nine and over 25. So perhaps I didn't mention what the business planner term is but it's the name is very descriptive it's essentially planner terms have no closed subterm so no bridges. And so this both the proofs gets here is that both results follow similar steps. So we start with two equations because of time limits I cannot go much into it. But the idea is again that a term a planner business planner term is either a free variable or an application or their lambdas. And here we have to take some care to have for example here in the planner case we have to choose a term which has at least one free variable which we're going to abstract over. And here we cannot that we have no derivation or no derivatives. So if you have a planner term you have no choice as to where to create your next lambda, you have to pick the appropriate free variable which doesn't create any crossings. Right and so this leads us from the remove different equations to the remove plain algebraic equations. So again, you tied in free variables. And the idea is that we can use elimination and quadratic method here on these two equations to obtain a close form solution for p and q. And finally we can proceed by applying some standard theorems from safe laser lens edge week on the perturbation of algebraic singularities can mass etc. And see here that the close form solutions are not very pretty so this is for the bridge less planar terms. And here's the one for the general terms. So they exist they are here but they're not very pretty, and they're not very illuminating. And in fact here it would be interesting if so we have by elimination or by the quadratic methods we can, in fact find a close form for this p z zero, and this you want coefficient of q z you, but it would be very interesting if we also find some functional equations because this will allow us to also, for example here talk about close the planar terms which we do not talk about in this talk. So, for we also have the probability that an open planar or bridge less planar term is an obstruction. And here again this follows, plainly from the analyticity of these generating functions. So the idea is again to estimate the case of the abstraction and compare its coefficients with the case of the general terms. This corresponds to the number of it's the general function for the lambda abstractions for the planner in the business planer case and here in the denominator we have the general case of all plan terms and all business plan terms. And so, since these functions, they are very nicely behaved, we have that the closed planar terms and the you want coefficient. We have a closed planar term with one free variable. They are both analytic at the respective singularities of the general functions p and q. And so, since we have the close forms for both p q and we know that the other two are analytic, we can replace everything with either a singular expansion or Taylor expansion at the corresponding singularities and obtain the desired results so we got essentially that this guy do not contribute and we have the z p z one over p z one and z q z one over q z one. And so this leads to the fact that the probability that the random open planner business plan term is abstraction equals its radius of convergence of the corresponding generated function. So to conclude, we have clear distinctions between the divergent and the differential algebraic case of linear and terms and on the other hand we have the well behaved algebraic one of planar terms. So it's very interesting to note that there is this essentially these are two realms in the in our study, which require very different tools to handle. And so we have a need for on the one hand more tools to handle divergent computer classes. So theorems like the theorem by Bender I showed you earlier or some theorems to help us for example linear linear I sorry, some of these differential equations I presented. Some of them are of course the cutting. And on the other hand, it would be very interesting, as I mentioned to have some other results for close planner terms which will allow us to find some equivalent parameters, as we did for the close linear terms. So for example, distributions of bridges and loops. So of course for future directions, it would be very nice to study also the dynamics of the lambda calculus or the notion of better reduction and finally of course a very important part of the lambda calculus which is the typing of it. So the linearly type lambda calculus connection logic, etc. And that's where it's made. Thank you. Thank you very much. So we have time for some questions are there any questions. So far they were in the chat. Can you hear me. Yeah, you will. Yeah, I want to know what the obstacles to extending the result to say, I want to say genius, genius one. With only one handle you could perhaps get a nice embedding that allows you to do surgery and reduce it to the planner test. I wonder if you explore that and how difficult it might be. So I didn't hear the class. Pardon. The class of the terms you mentioned. What is the difficulty of extending our results to what class. The class of those linear, I say linear terms embeddable in the tourist genius one. So, in our case, we do not have any restriction on the genius. In the case of the general linear lambda them so when we consider linear lambda terms, they are regardless of genius so we have crossings. On the other hand, we have the case of planning terms which are embedded in the sphere. And so you're asking in between if you can have some information about the classes which are specifically embedded like of genius one or something. Yes. I have not tried this. I don't know if it's what it corresponds to in the in the realm of London, I guess you're allowed you're allowed for a limited number of crossings. That's a very good suggestion. I'm suggesting the father. If you can, if you can do surgery on a single handle and reduce it from the Torah, the genius one between the zero, you may be able to do an inductive argument and do all geniuses. Yeah, that's a very interesting question. I have not studied it. Perhaps the professor knows more about it. Just to say, I mean, I think it's interesting that question that this, these kinds of questions have been studied in the realm of map commentaries and this is something that Alain, who has looked at from the map combinatoric side and so it could be interesting to try to to do some of these analysis for lambda calculus parameters for for restricted genius terms. I think it's an interesting question. It also would be interesting to see where the difference pops out first time because we see that we have this essentially like a phase transition between the, the general case of the linear terms and the planner case. So it would be interesting if it already breaks up for the torus or happens more gradually as you increase your genius. You go from the gosh and the linear gosh and distribution for for example to the one for the cubic root on the general linear terms etc. I confirm it's a good question. I know a lot about lambda terms but if everything on planar maps applies then probably if we are on limited so bounded genius then the results as I'm taught it will be roughly the same except perhaps some polynomial factor something like that, probably not not even that. But then we will probably see a phase transition when we go from zero so it's zero and fix genius to the arbitrary general case there would be a phase transition somewhere but it's still very, very unknown to even the in even in the map community. So it's something very interesting to explore I think. I have a small question. On your slide for you show the bijection between maps and you have an application as T and then to swap the T on the S for the map is there a deep reason for this or just. Yeah, so I think it's just a convention of how to draw it because you could as well decide that applications, they go the other way. And the another like ad hoc conversion was that I always was putting the half edge which corresponds to the variable of a lambda term going off to the left, I could might as well have put it on the right. And this would essentially mirror all the images you saw. So I don't know if it's for me and perhaps I'm wrong. But I think it's just a convention right of how you decide to interpret the lambda term as a map. But I would like a confirmation. Yeah. Okay, thank you. So I think even in some papers the conversions are a bit different. So I think between the paper of Professor Noam and the paper of Professor Olivier, maybe there are some differences in how the, the half edges go on how the sub the sub terms of the applications go. I'm not sure I have not looked at it in a long time but yeah. Okay. Any other questions doesn't seem the case. So thank you Alex again for your very nice and interesting talks. The other questions we spoon or if there are any to the discord channel. So I encourage you to.