 So I'll be a little bit general this. So I guess that some theme of the conference is they have a convex analysis of complex analysis coming together, and that fits pretty well with some of the work we're talking about. So let me start by one situation, very classical, where this is well known. So this is about going to have a complex upgrade for us. But this doesn't mean like C star to the D in this case. So let me see how well is this big enough to? So this does a copy. And then you have a map down to rd. So if I have coordinates here, i, just take individually, say, the law. I'm just going to solve this by ti. So this is a, let's call this map pi. This is a proper map of how this dimension one, the three images here are just circles, or more generally, they're real tori. And again, I have an action, let's call this s to the power d. So for I mean, my s is the unit circle. So that's acting on these fibers. So this kind of situation. And this allows me to, I mean, in this situation, you have quite a nice point of view in terms of a complex analysis to illustrate this connection. See if I take, say, u inside rd on x. Right, so on the one hand, if I look at, say, then it looks. There's something like that, but that's not what I'm interested in. So again, able to see with the, so at least the last two or none of these. In this case, you can define essentially the same picture. So what do I need to change? I guess this is, well, I'm just going to define k to the d. Okay. I'm actually going to have the work image in terms of here. It's a multi-plicated norms. Semi-validation. These are the ordinate series, but just go wrong. Natural psychology. This is just a point wise convergence. So I demand that, if I could call the d, it's a couple of space. Now I can just, maybe I should say one thing here. So it's not equal to k. So if I always have an embedding of this space here, k start at the rd, it's like that. So I just make elements of k the same. I just put the norm here. And I still have an action where s to the d. Now this is not some work of it. You can do that too. We'll be sort of circles, but it's all right. I'm going to look exactly like that. So we'll just change. So we're going to look at complex stats and something else. Something I just said. I want to write two things. I said this is a couple logical space. You can make a program with a spreadsheet to define what are an open subsets. And if I look at an open subset of this, I'm going to look at a series of the pre-image of you, a complex that is in fact the same thing. So these are going to be convergent or raw series, where you have to suggest that we're already speaking to this in the same way. And this also is my, that you can be continuous, blur-cell, my functions here. So here, this is good. And this is also going to be continuous. These functions is going to satisfy this. And both of these, this was various iterations of this that have been done, but I think it's in the setting. I guess the general theory is that I plug these functions and then you want this to be closed under, comes together via this, this here. Okay, say a smooth, right, this study of, you talk about, for cleric potential here, maybe I'll say global, or geometric, or symmetric, some more, but I have spent a lot of time on this. We're in long series of other pieces that find a way to solve this equation using the variational method that we should try minimizing or maximizing some kind of equation. The question again, what's that mean? The picture here was the iterations come together and also formulate a theory which just doesn't really make it complex. And I don't have to be in this situation. Complex are here, right? So what does it mean here in the complex? It means that we have a global, almost a zero form and a wedge, it means conjugate, that's what your structure is. It's a kind of simplicity. So you're saying about the solution, I mean, we still have to play. Yeah, yeah, yeah. People are still at it. We should, in this case, right? So we've been one, right? So if I want to solve, now as opposed to solve, I want to compare, I guess, to solve this with a continuous solution. So there's some sense on this state that if R in this state goes through the solution, it doesn't have to do it. It's not in my state. It's to say that, say that it's R. So T is this parameter, and F is T. This is just kind of a generic and homogeneous polynomial. In that situation, say what exactly it means to solve the problem. It's got a global version of it.