 So the invitation is here in this presentation. And I'm talking about the work which I was doing the last year and it's not finished. So I hope that people will come out in a couple of weeks. But so far it's sort of not completed and there are some little tricky points and the important. So, the idea is to use algebraically. I think that's one of the things that we need to do. To apply to ADSE people and comments. I don't think that's everything. But there's a lot of obvious things I'm going to try and that is the motivation. Actually, the original motivation was wrong. So that was the motivation and the setting will be in the moment. I think that's why they are the same as me or them for the people. Which has been under intense communication for them yet. And it's 5 o'clock to be one. The way I'm looking at. And so I've watched the problem from the bottom you see it. And that involves some of them. And then there's a wonderful formula. Which is very useful for addressing this. And there will be some important. So this is set up. So there's this F1 and it's five system. And it's a new rise of the net is given by an ADS space as it's known. And it lacks number one. By spring number one. In a way, in particular. And otherwise present continuous spectrum of long strings is right on boundaries. And there was tremendous progress over the last 20 years by other people. And they were able to show that the battery at this point. It has a different version of the spring model. And most of the work was done before actually. And it has a different realization and changes all in the sense exactly. And the boundary period related to this. And it's generally known that this boundary period is a secret model. But the package space is the automatic product, the default automatic product. And it's a parameter which depends on the child is just five with one, and that is sort of essentially end of the run and then it's a different form of large and. That's what you're going to do. And then you have the function of your life here in the land and have a boundary and then their correlation function involved in search and call the operator that the boundary. And we will spend quite some time in discussing what the are now, but the idea is that you have endless three wraparound boundary. And then you have the combination points. And with higher genus in the general. This thing. And the complication of correlation function. And comparing correlation functions. And that has been, you know, an ongoing work for like over three years. Very good position and effort computation has been done, and in particular, I mean, fantastic agreement in the large ending. However, that's something Nancy will find the story. And the motivation to look into this. So we have to see a model of the boundary which has a huge target base. This is an engine with them and we have a trailer manifold. The point is that there's a more than life base. And that's a lower parameter, of course, all before a single idea of this. And such that it is finishing the time of the rule, the period is at it's local 10 minutes of green point. And the point is that there's a really powerful engine of things. So this is in a way that I'm brought in theory in the higher dimension ADS. The main theory. And then, of course, coming from the plant, the question would be what's happening. And generally believe that the theory should go into some kind of gravity machine, which from this part of the public. And that poses a little. The official engine to the one person, because somehow it's the object of this one. It is the large distance limit provides an income spectrum of higher things. And the last distance. Well, as it is one couple. The theory of the model is, he wouldn't have a higher impact. More like, maybe a month to take a spring. But he's the other way around actually week is how to handle this. I found it interesting to look into this in detail and at the point is, so far these complications have been done very close to this region of this model is safe because people use from form of innovation. We call it. The top color is the order by order that work of for the few others. But we keep the coming from the calabiao world. All order and form of preservation is no problem. It's called mirrors in this exact exact doing some up in the water for the foundation by somehow. And by the magic of how it works and then it will work the contact. The idea is to get a handle on this model that space from the foundation by committee, where to get an all order global activation extension here. However, really there's a topic that all the price to pay me to be topological. And there's always one thing that will be long for that. Yes, all before singularity. And then, then there's a lot more. There's a law that's a few one you can go up to it. You can add to the presentation. And that's that's the primary. Yeah, yeah, the order also all before it is the, the is essentially going to be unique. Normally in this in this business would say okay that we are doing. And all this question of the US mean that that's what it. But from this framework, we actually don't know really what happened. This frame of people to actually all the area is a lot of typing. Because they say, let's meet the cover the vision for the public. But I was the same stuff to all the. So my major vacation is what follows. This point is a descent of the moderate phase and it's all of us with other things. And their property is going to be patient. And I would go here and then. This point here is a penniless point with hands and it sounds like. But the geometry of this is where the idea is one is the same as. It's a highly current function. And I think there is no notion of the top of God. It's highly current. It's a miracle is that you have a couple of different. It could be reconciled. Well, it's like that. And that's. But it's not. It's a very particular point. And I already anticipate that even if one can send these up into all others. Everything breaks down here. There's no obvious way to do something. So, this is now the idea. So far a few orders and petitions we have done and that that was all complicated coming up. But just from, you know, general gas building from. There should be a million methods, which focuses on a car or be a subsector. And then I can do genetically method. And the center piece is, it's. All of this in Hilbert cars. I always take out. And it's very present this picture in the paper. So, actually it's a multi-dimensional. There's more corners on this. And so here we have a signal model. And here we have a little bit. Talk about this here. And so if you see this, and now a DS, then. Well, we should apply. And the idea would be that this P1 here. This P1 will be the boundary of ADS. And they had invented a machinery because of the branch hovering over the design and the concept of covering that. So, this is actually the content. So the idea is to put it a bit. And identified the office for the law of law. And the counting of the covering maps, then we will. So let's say some few words on the table. That's funny. So we have correlation. The gamut of biology classes. There's a very simple construction for them. And the presentation that he's invented by that. Essentially, you apply with writing operators here on some corner. And add another comedy. And then there's a kind of twist. Which is another language, a winding number or spectrum flow. So in general, these products elements can be represented by starting from the empathy, then application of the operators where each one inserts an option element of the. And then you'll have sort of twist. And this database go in here. And it's combined to begin in your eye on what was called a partition of the. But because of the next to the skill tree. And they come together paired with some comedy. One has a list of its mind tradition and that's homology related. That's changing. And there's a total degree of charge. So this is so far this was homology classes, but they are all about the homology and that's very well known. It's actually dichotomy right in the 90s. The statement is that the age to it's just the same. Except one extra element. And this is the exceptional law. And in this language of this. This is just a point glasses. And then we have this order to know connection to keep itself. This appears of course only. Then we consider also a curve in the three. Such. So we have this entity because we want to count maps. So it's just one element. And we can package them together. So this is a class of page two, which is labeled by age. Okay, the case does. Well, k times the distance is essentially. But so now we want to make public. And once we count. All the maps into the target space. This extra insertion, you know, it is in particular the boundary of the ACI. Incidents relations and incident conditions. So in a word, you want to. You want to count the net into. This project. Subject that they intersect or touch. And there's of course, a charge rule, which says, okay, we must sacrifice another day. However, now, and he is a major obstacle. This is the biggest career. The point is that they are not classical. In physics, I would say this is just like in perspective. What's the impact. The correlation function. So what to do. So let's see what is the reason why, why, why, why the main issue. Is that the conditions for my data in this way that you can pull them out. Curves in one one. But the existing information. Move them to outside of this. This invariance must vanish before to come. On the other hand, we're mostly going for completely invariant otherwise. Otherwise, this is for the variant. And this means they must be able to identify. So the way out, the mathematician of the following is to modify the definition of it. And the mathematical logic is to change the structure. And in a more physics way, a way to view this is the computer because we as a fiber of a lot of people. And this means the definition that is picked up the automatically. That's one one. And therefore these are going to find me more. So one can see the case three as a fiber of a college a caliber of people. In the limit, where the days of this. But they can become very large. And one really is really committed only in 20 properties. That's not really the kind of a touch. When you work, you know, one can say blah blah blah, but in the end it is a restricted situation. And therefore I'm not sure in the end if I'm not. I'm not sure what it actually means. I think this is certainly called curves and count the states in some sense, but to make a map for correlation functions, maybe as I don't know. Then they think they're doing. But by say, looking at asymptotic situation where they keep the activity. It loses information that that is one one chart that don't perform with these. In this way, count can you make it actually all these curbs are doing. Yeah, yeah, this is one way of. But so far this is. My eyes are the ones I was, I was shown because the community got delivered. These are the things with city and the boundary. Actually, this is now a different. I mean, the other thing that comes to this later but normally speaking, these are the operators. These are the, and which are branch points, which I do with the bias sheet together. Because, you know, the great branch points. And it's higher to me to me to me to me to me to me to me to me to me to me to me to me to me to me to me to me. So, but even can count something and somehow it would be the states but I cannot really say okay magic. And now this strategy has been really busy. Before being in the world as a family. Okay, now, the centerpiece is this equation by overbeat. And that's a very important interest. We have here. Okay, now the centerpiece is this equation by overbeat. And that that's a very important interest. We have here the other side, you have the conversation comes to be this discussion. We are later like this for me to get the questions. And I count maps from zero individuals and K, I remember, I recall with this, it was the law class, they are taken away and the summit of a right to display that it is a government. And on the other hand, on the side, these are from different brands, or one of the one is some fixed point, they are relatively. So, if you want this other location of this. So, and they are a genius G. And it's a branch covering. So one counts. Maps. And this, this is exactly the story of what was created. And the interesting thing is that there's a map between the fly and the new, an exponential here. It's shifted by region. So, it's, it's, it's all the point which is quite. So what we do here is an exponential expansion. And here we have a power like the most expensive and counting on the most expensive. So this, this is, I can actually put the formula and also, of course, because it's not about the entity does for any deal that he doesn't need to be large. So on the left hand side, it's things where they're going to add the petitions but what, what is it on the right hand side. So, you have to go back to the definition of what, what are the quantities, looking at the gammas, and they were labeled by pairs or components of conditions together. And the translation game this language can be one piece that they are becoming essentially. This is. And now this is sort of an example of picture. We have a few one which is really a version. That's the bottom line here and then, and then, well, we have four sheets so I can't be noisy. I identified one year, and then we have this insertions with the full dots. And then we have on the form of the glasses age one of the three points. And then they get here. And then just tell how do we go to the link. There are the solid lines. We have a lot of ramifications. We've got a lot of ramifications. For that, for example, that he seems identified. What she can do together here on this one. And there is the next dot. I together this. So there's one cycle here then because I could hear and what I could hear this one is just a representation of the floor. This is a six point function. One, two, three, massive states and then massless killer model I would say plus one muscle. Default. So this is a mean zero. Now, these go a bit and variance. It counts maps. Point. And I can go on from this because now they have the very famous. We developed by a conquer. Who all this form of it. And what they say is that essentially, there's a duality or correspondence. And it's a discovery. In terms of the logic gravity, essentially by identifying. So that's a cancellation of gammas into a linear combination. And it's a native specific fire. Essentially, it translate this into product. So you match the whole thing into property gravity. It's probably an advantage. I mean, because in principle, you have to focus in the section behind and because in relation to the mirror. So at the end, it's much simpler than the other computation. So from this correspondence, at the end of the basic of it, they will send them. And it enters as a perturbation. Okay, so one can assemble this into a generating function. So one has to come with invariance and be also now takes a kid another with power and this powerful enough to be working with the dependent. Just a fiber class. And then the sum of your view and then the family. I mean, I actually I thought that this form but it's kind of instant. Tell me if you look at the different but more or less speaking, there's something like that. And then you have these versions, then you have some parts that should be included before. What's behind this. formulas. Rather to move them to make this one vanishing entirely in a general. The important thing is that in one way or the other, this is maybe not written very correctly, but in one way or another, you need a number of and how he won insertions in order to make it really right. This means we have this situation here. We have to evaluate this. And this rule must. Well, it must be produced, I mean the number of versions of twists and not work correctly. In general, there's a deficit. This is described as Venus. They have to be 100,000, then you need 100,000 version of the one to make it one way machine. And one can view this as a generating function with boundary conditions described, you know, by this gamma. And the boundary. Actually, in this ADS literature become evident. This is an opportunity to independent with the situation, which is very high twist means, then these things really extend more to the interior. And the only thing that's really at the boundary of the drama. This is a partition function, where you can do this partitions here. This is a thing called insertions here, you can do them as a background machine. And then detectivations in pure and new and are they then remain interested. So, how does it connect to physics. Somehow you would like to think that this is like a spring company. You cannot do this. Because even if it's always for zero, that's already an alternative expansion. So how does it tie together. So, my part of you would be that this geometry here, the decontamination defines a bear curve. You know, it can be negative. Related to this journey to exist at zero, you might apply a way this by by this problem. In this way. You write it in this way, and then you make the usual map, you know, a large map which sort of it limits the large and the full large and limit. And this string company is a prefix, because it's not just a composite equation. I mean, this is a new horizon. You know, rising to the general thing is think of the flow, you know, fix for him and he is, he is a fix number. But it will end. You make this transformation and you end up with this formula. And this term is exactly what, what the idea, if you could say it's a think of it. So it's a lot. The new ingredient here is that you have the extra parameter. And then that has into the taking the whole time. So you start from the Belgium, we are going to leave one round early. And then you can switch on longer and make it in order. So an example. This was most trivial, but also a more significant example is a trivial. This is this guy here. Just equally. Another time and then just one more and that essentially only need to make a picture of the automatic. That's really important. So don't have to be far too long point correlation function here, but it's completely degenerate because all these sheets are separate. They're both one potential of strings. It's done couple to each other. If the compilages into many that completely different. And now the question is what happened. Well, this is a different function. And now the extremely important point. The reason is that this came to talk to all orders, which involves a couple of them, which are in the next slide. The first is in particular cases, namely one of the drops one of our data. That's usually the outside formula and the material. This is a kick me formula. And this formula appears in many publications. And I'm not going to generalize on it. There's a lot of research about the polls and whatever it means. Yeah, that's actually this positive power. So they are not holds here. But also we had to deal with you doesn't really make sense of this company. So in this context, actually, this factor has nothing to do with the extra gp. So this is the physics point of this. But that it is this function appears. So this is a total form can be written as a function. And it's a standard standard to be generator. Model of a minus two and then do the same. And that's all the double theory. And at infinite case three fiber volume to zero. And then this function trans just in the square of this. It is signed function that appears in all these animal problems. You can do this as a building block. And you can divide this in this way and then this way everybody knows that this is then this is the perfect. And then in the church here. Okay, good point. Can you access again again. Okay, so the question is what happens now if we expand on the other. And the idea is of course that in the absence will link these is connected together which can be used as a finding for that. And it means, and if just petrified high enough. And then these sheets are all put together this is not this today in the morning, get connected to the surface, which corresponds then to a bound state of originally the new web screens, the one we kind of know. So naively run it would expect that this expansion. And then I figured out what was going to be the picture I just want to say, this would be the picture, the picture will be correct. This is the first one. But the geometry is more degenerate because the collection actually all look like. That's the reason why this expanding. It's not given by by who it's numbers, but just by a part of the time. You need at least these, it was, that's what it is we have four sheets. Yeah, and if you add one, one test fields here. But if you add to the other one here, then this link these two sheets together. Well, these original two sheets are driven here because we're not going to have a patient, the ad s people people we started something we already and and then we're going to complete with some of all of this. Then we start with a big region one of this, but they also do that then you have only a finite number of. So, whenever you start with such a situation, you don't determine your genus six. So you need to switch on on the view in order really to get them. So there's a difference between this one's coming out here. Yeah, it can be written in terms of who it's not in this way, and there's a lot of corrections. And so I think we're doing. But the more interesting potentially more interesting situation is this one way to go from phoenix. Because that that is somehow one, one, one, one such as for them. And because we have this jacobi form, a priori it looks like a priori, a priori it looks like a priori, and every every song so many shifts, you get to a new wiki couple points and you can make new experiences around this in the sense of the time with the jacobi, which is anything would be interesting. However, because the index of the jacobi form is one zero, you gain extra anomalous time issue, and they become diverging. So this means that this function which you want to move we go to infinity, even if you go to infinity which is y equals zero, you can write it in a different way. So that's a limit, because infinity which is y equals zero, y equals zero, you can write it in a different way, so that's a limit, because infinity which is y equals zero, if this condition satisfies and then it's kind of smaller than one, so then you can come up with the option. So in other words, it's a very bad singularity, there's some kind of obstruction, there's something keeping pattern, but nevertheless you end up with a whole other singularity, and there is no way there would be a simple duality concentration. And I think this is exactly fitting to this philosophy about resurgence in the next model, because, and this feature appeared also all the time with a lot of things. Typically, this thing coming down to S being an elliptic parameter, which I can say is the first entry of the problem, and it fights nature, and this question, what happens to infinity, it's the same and then there's a universal action in the field, and it needs to be cared for by completion. And I think this question, I mean, the whole basic question, what happens that large would be handed to you, what happens is a large number as a new business, the same question. And the question is, that's the non-partuitary completion, the real resurgence, what does it mean? No, no, even here they're not called, because they're probably from a different part of the business, but the problem is the same. I mean, it's not the problem, it's really actually as I learned from Thomas, this is just a sign function. Sign of you in the limit of large, if you go to infinity, it's a monotone function. Yeah, so it's very simple limit, yeah, sign of you, you go to infinity, it's a monotone function. So somehow it must be verified. Also from the perspective of people, people are so strange in COVID theory, and then there's lots of work on on artists, sector curves, whatever they are, which sort of was really large, well, probably larger. Usually models are non-compact because one looks for only full kind of compact, but at least, I mean, this is clear that these questions are out there. Sorry, no, no, there's a conclusion. So the aim was to understand the global structure of this modern state, which is a common concept, a total parameter. Then it's a blockchain toy model for something which is anywhere already simplified here in the U.S. And you use this from a little bit more common. So we will match this in the U.S. expansion of the parameter and the basic example is exactly tolerable, except it can be and the caveat is, I'm not sure what you could really use in the U.S. And then there are other questions. And then of course, then, okay, thank you. Yeah, the problem is that this a priori, if it's a physics, it's immuno-like. It's not like an ordinary. It's the sort of writing them all, except after you're looking at the interesting. But however, it enters here in the formula. So as long as it's not zero, but even provides an upper bound for the coupling. And then this is a non-linear phenomenon. But the question is, yeah, it's immuno-zero, and that has to do with the U.S. Of course, there's ADS. And then nobody did all the things about this, and there's a very funny story. I mean, there's a partition function, you find it again, one of a type 10, which is really tough, usually not. And so what is tough, again, much of the point is that even when people run, when you have really one thing you can do, there's not going to be a U or Y line. The funny thing is, they are able to do the same thing with this formula. Yeah, so if you get a Y line, it's an extra term. If it's an extra object, then you can provide some good context. And then there are also with at least this reconciled. And then they have been losing work. Partition functions. Partition functions, you know, some of the work I've done, and there's another, because the boundary itself is it has an interior. I mean, it's a boundary of ADS from the inside. So, it's not really that you can't curve next. The what sense you have I want you to curve. And thereby arguments that we pleased in the limit is really a stick to the boundary. But then they are always the boundary of the two different versions. So even if you go to the high level, you have this limit level, you can imagine that the boundary is very large. Now, the verge is still looked like a provider which the page is in boundary and on the inside. And probably only the probability of the branch from the mechanism is there. But that could also be a different story. But if I see how they come across and how competing them comes, it's the same thing. Yeah, yeah, there's these operators which actually, well, this is a set of data. We need the data on which you write the partition, which is that the partition and the key elements. And they are given in all of our language. This is actually here, so we have, say, sigma two, sigma two, and sigma four here, and one here, and one here. But in this whole context, these things translate into the same thing. And there's a complicated map. I mean, this is a complicated method. And in this way, you can translate the termification from this into section two.