 So, yeah, so I'll talk about the accessibility for big panel labels, in a sense. So this is some drive work we've already done, which also, my study staff work in progress, so that we're still fitting the optimal way to really present results with the right spark for good, in a sense. I also mentioned that there is some quite related work by Dalash for example, that will be talked about in this conference. And also, I'll speak about this a bit later, but some similarities have also been brought up by Antonio Giuseppe, and he kindly, I'll be, I'll be back. So this is the first talk, I'll start with some history. So the starting point is that we have a compact flex, and we want to understand some conditions that ensure when our carotidometrics relates. So, a carotidometric is a carotidometric, which actually means that it's proportional to its curvature. So for a carotidometric to exist, we first need some of the logical conditions to be satisfied. So the expression plus of kx, so kx here is the mechanical bundle, which is minus the expression plus of the mechanical bundle. So this plus must be signed, because there cannot be, because the kx has to be equal to minus 1, 3, 1. And so, I mean solving some equations. And so, the information in this equation is that we want a carotidometric media, which is the curvature of the right side of the sun, which is the right side of the sun. And it's equal to some normalizing constant, which will exist. Next, this thing, which I call the exponential of the right side, which is really a volume form, so we're not going to need to really just take individual potentials of the matrix taking exponential of that, because this matches very nicely to a volume form. So, a carotid equation is solving this. So, strictly, the kx not like was 1, so we can paperize kx, which is that kx is not all. That was solved a long time ago, so that was solved by Yoban, so it's 6 and Yoban is 28. And so in this case, there are no obstructions. There always exist the carotidometrics. On x equal to 0, so the Calabria case, Calabria against those part, when he got his physics medal. And he also showed that there are no obstructions due to the existence of carotidometrics. So, Calabria case, such a matrix, it always exists as well. This question is much more related in the family case. So, on that, this makes one, which is that kx example. There are obstructions, they're not always carotidometrics. And furthermore, these obstructions are algebraic two-metrics in nature, which has been solved now, so the problem is that we understood in the case of xkx anyway. So, we have solved by many people, but the main contributors, as there are Tian, and then Chen Dawson soon, that's built the final structure. And so the final result is that so with my xkx example, then there is a carotidometric on x, if and only if x matches the x, is capable of stable. So, I'll actually never define this as a capability, but the thing that really is... The important part is that this capability will eventually be solved very quickly in nature, whereas the carotidometric the question is something that I like to use to complete the useful. So, the statement really means that we can check this quite difficult existence of PDE problem using some easier by nature algebraic condition. And so, with the method I talked about, I mentioned the disability. So, the disability is a thing that we find, right, when there is a resonance of yd in this case as well. So, I'll show you that. So, if there exists a current in x when x is things to be stable, and if there are more of these which are unique, then uniformly they're stable. So, I'll explain this if some people say it finishes. And, firstly, if x matches the x, if you currently think stable, then there exists a carotidometric on x. So, the problem, everything is unknown. So, there are multiple directions that we can look into next. So, one thing is to change the improvisation, so to not look at my SKX, but to look at any of them to see if there are cck-metric selects. And the right direction is to weaken the positivity condition of my SKX in a sense. So, that's the direction that my name is going to go. If L, so in this case, as my SKX, these are real facts. For example, then L is the eventually basement tree. So, I have powers of L, there are no points at which it points. That's the dimension of the space of the sections of the powers of L at maximum. So, this image, if you call the volume L, which is the dimension of the space of the sections of L, exists so it's in actual limits and it's also positive. Really means that's a function which is the sequence of dimensions of the space of the sections. It's a big goal of an enterprise and so it's a maximum of a single group. Maximum example is the train to space. It's identified by alpha-s, but there are less than a part of alpha-s. Okay, what we do is I define it as a function so we can say that L is the big if this volume exists and it's really big. Which means that the space of sections of L really have the maximum number. But that's all based on it. So, one of all the nice things we have actually is the space. It's important that we have base points. So, all the sections of the power of L are in Spanish and also we have the synthetic base of this which is really realized as part of L. So, we have this sort of scheme of the next image. All the sections of all the sections of all powers of L are present in this case. So, sorry, yeah, in the strictly big case. In the big one, that's something I hope we'll do. Yeah, yeah, so this can be an internal field. And we're going to want to look at their effects or positive effects in that sense. And we'll also see the other things about this synthetic base because it exists. So, we have to deal with taking as infinity some set of x to define the sort of weaker notion of the case. It actually gets in the way. But that's a strictly big case also, right? So, yeah, so I say it's strictly bigger than the big one. So, yeah, so, why is this definition in the correct way? Yeah, if... So, look at the current same problem in the case where plus minus k is big. So, we can make sense of the problems of this due to the heat papers which have also come up in my thesis talk this morning. We can make sense of the motion by equation. We see phi super of n equals e to the power of 5 as before. So, in this case, phi has similarities, but we can really just define this as a measure in the same way, away from the normal case to measure e to the power of 5. And by instituting it here, I can solve the current same problem in the journal 5 case. So, on the x to the power of 6 is the senior or weak current same problem. So, in this case, there are some things that make the problem a lot simple to examine. So, the kind of covering is quite iterated, and there is a kind of equal apple model which has some similarities with the apple model. For x, in that case, we can solve it here. We see it here. And this is a fact that we use when I'm asking you to prove the existence of the current in that case. But if we look at the big fellow case with minus x, it's just big. We don't have a connectable model in this way of iteration. So, we have to solve it here. And in that case, it's even false in general, right? So, the other example. Yeah. Yeah. So, first of all, for things that make sense, we have to assume a singularity condition in a sense. As I said, there is work with singularity tricks. There is a subset of nicerly tricks which have minimal similarities. And we have to assume that for any such metric, the measure e-minus 5 is associated as 5 volume, which is not guaranteed. So, tie for it to this actually guilty conditions. It really just means that nicer, positive metrics have minor similarities. This is more or less a condition that you can make all too brick, but they're going to initially belong to the asymptotic base locus, or different base locus, which is ultimately advised. This is an alternate condition. So, the result of that measure is one direction of the utenal subjector in the big case. So, the work of the Tanashian QA dimension knows the other direction of the user from the national disability. Let's go into the if x is big and if x minus px is kLt in that sense, then A inter-exists a weaker geometry on x than x is least stable and inter-exists a unique worker geometry on x than x minus px is uniformly stable, which I will try to explain in the spectrum of actually defining kLt. So, one thing that we've not come out from this data, but that we don't actually have would be to understand if uniqueness of asymptotic bricks is related to the automorphism group. So, we really want to look at automorphism that prescribes, that preserves the singularities of three abled inseminative matrix. But, because this is alternate, this means preserving the main focus of xkx asymptotically, this should be just the automorphisms of x. We have a version of both here on strict positivity as the thing functional in that case, so we don't actually know if uniqueness of weaker geometry comes from the fact that the automorphism group is just real. So, we can still formulate the state of the state of the state. Yeah, we haven't really tried to work that out. Really, is it important to us to know if there is something that's sort of wrong? If there is something in this question as well, because in this case, why is there a case in here? And so, these indices are in the sense of currents. The singularities of the singularities of the currents, or is the equation just a singular set? And then some kind of a singular set, it's a literal standard equation. They also mean something about the singularities. Yeah, it really means that the... So, ultimately when you solve the equation, you're going to have something with the most similarities, and so this means that you control the similarities of the metric, and the singularities of the metric are also determined by some algebraic things in the TQX. It really means that the asymptotic base of the case is not to... So, to redefine the usability, I need to redefine the notion of test operations. So, the idea is that caseability is kind of the three-dimensional version of JAT stability. And JAT stability to test the stability of a point. So, to sign up some invariance at infinity, touch to it. And so we need to define a sort of notion of one parameter separate, but more... in a sense, because you can see just the point, in the general space. And the correct notion for that is the notion of a test integration. So, in the classical scenario, it's just a scheme of our family which is the scheme of our Q1. We have a system action which is the usual system action of the line bundle, which is unirrised. And we want that over the fibrat one, where the every fiber starts the restriction of your disproportion is identified with the... But so, when we work with big things, it's usually more trouble to work rationally, in a sense. For example, so it's usually good at each and every work also with irrational models of X. And the classical way to really encode all of these data compactly is to use the Zaris scheme in those spaces. So our project is going to make sort of all the irrational models of X. So, these things come up already by the work of the remuneration of the system, which is terrible terminology. Sorry. So we're going to have a sort of path of version of this structure. So we find part of the remuneration of space, and the rest of the space which writes X more which is also representation. So it's the predeterminates of the system these sort of objects are contractions sort of from some Y to X. So, just the quantitative morphism with native fibers. You want the exceptional quite responsive and also make us that's the predeterminates of the time model which pushes forward to write X. And we also want the volume of the process. We want the volume of this. We want the volume of X to move up here very soon. Let's actually find the objects into the system. So we want some sort of order which we will take to the next minute. And it's really a natural order that I say that this one dominates this one if there is a piece like this which is greater or not. So that's the angle here which is more into the angle here. So this domination relation in the system that we take to the next part of this. Yes. Excuse me. In the category of so the things that we come up with building the big writer in our definition as well is kept in the category of schemes with and on all these ones. Is it the same as the model of the pullback or is it the same? Is it equivalent to the model of the pullback of minus k in the end? That's not the same. So before I say that in general, I know that we ask the egos that I will know a bit soon. Yeah. So if we push forward as a cycle we can think of it as a cycle. I see, so yeah. Are you not pushing forward the shift of sections? Yeah. You can formalize it in different ways. Yeah, so if you push forward the shift of sections but then you also ask that 280 is associated by model, so you have days invertible and this makes actually the weak minus kx. Days actually? Ah, sorry. If you want to see how wide as the shift and then you push it forward you ask that the forward is invertible and it actually makes the shift of sections to make the x sign. Okay. The function is subjective fractal dwarfism which has connected fibers but really you want to make a base of it. So if we have something like this then there is an observation which we did as a verification of positive matrix on the right angle of stairs and on the minus kx so we can work in the same way under a line under the minus kx and devise the matrix for mean multiplicities but that's kind of why we have to add this condition. So that's the variance of space and then we can define a test comparison for the variance of space. Okay, so if for each such Y we ask that there is a test comparison for Y which is super effective which really means limits of big things which is a minimal visibility condition such that there exists positive matrix there. It's compatible with the order and that's the line model currently LY pushes down to the line model currently LZ and the last thing that needs to be approval for this integration for X times kx which is just nice to work with because then we can pull back this integration and work with the variance of Y and there is an actual condition so we work really over some days and then we have asymptotic base so we want to say asymptotic base so we for some technical reasons we need to compare between these weights of size so we just assume that they are the same which is a function that's chili issues for example like yours would be these configurations as well. So here I mentioned the work with Christiane he was in the front of the case but with first comparison parity this is a senior optimistic metric on YSKX and he asks for characteristics in the weak sense that have the same similarities and in doing that he also basically comes up with the same efficient as this progression I just want to say that this is something that he also proves in the pantheon this paper is also as well as I said but it's basically so there is one part which is really the volume part for the asymptotic voting parts which is defined using the theory of intersection of mid divisors which is some of our Janssen and then it's been treated by also Amalfa so it's a sum of that and some local critical term with respect to the base of this and so we state that X is the probability stable if this mean of variance is not written as a uniform as you've done but X is greater than that so normal 5.0 should be followed as a normal project so that this is your oldest appearance this is what we call uniform possibility and we can actually define it yes it is algebraic and it is a positive quantity so it's actually it can be like a norm I guess same is the ability to be easily defined it's only just 0 yeah yeah that's the last interesting part yeah it's easier to understand but I think there was a complexity but also as a conjecture part about the torrid case so classically if X is a torrid file so there is the torrid section of X Y is uncle and linearized so then the Y is the conjecture was solved way earlier usually I've done some general case and so the issue that's a case to deal with if you want to really explicit non-extraetric condition so X Y is a torrid it defines a pointer and then it is case table if you don't have the associated pointer that's very center of the origin that's the complete conjecture entry description so we can ask the question the big case so in the big case there is a disruption of the pointer based on its release of this factory we can't, for example, reconstruct X from the pointer so X and the pointer so what I think might happen is that we go to the base loci of the of the MSKX and its powers and because the base loci has this subactivity property in a sense we have a direct system of polytips which should really be an inclusion of polytips non-indistability that's and so some work on that material the focus is sort of systems of polytips in a sense and so I think that the limit of the system of polytips to obtain it without should be convex poly so it should be limit of polytips just convex so is this part of the conjecture it's a fact is this part of the conjecture everything in green is conjectural everything in green is conjectural except so the other part is not it's a larger part it's a smaller part everything from here on sorry I didn't see it it's even written conjecture should have taken from here should have written so we have a limit of polytips so you just need convex poly to compact the excess which should be a underbody so underbody's appear when you try to do the other reconstruction you can't do it in action so you just evaluate sections against the motivation and then rescale but it should have the same volume as now because that's really pretty type of estimation and I was also thinking about this post-correctional urethase case and in this case this goes to the work of Wing Chun who defines the good bodies that work with post-correctional urethase this limit of system of polytips in that case you can recover some of this work or say that's in this story case we have the input visibility you can only if the body center of this limit of convex poly is actually the origin and if it's if this is not cured from the work of the terra corpus there is still a notion that there is center for limits of polytips and that's that we should still be able to relate some projections