 So let's say this is, whoa, okay, let's see, where am I? Yeah, this is one with a disk here, it's empty space and again, the two things can connect. You have two disks here and there would be other ones that I forget and this one is just the dual way of trying to do things with nothing inside, sorry. Okay, and again, I mean the slide up there says that unfortunately, if you try to minimize size among currents, there is no general theorem that says that there is a minimizer, okay? In fact, even in dimension, so for two dimensional sets bounded by curve, the existence of size minimizers is not known at this time, okay? Right, and then I have a list of partial results or things like this, but I think I will skip some of it because otherwise I will be even more late. There is one little problem with this description still is that it likes orientation because currents like orientation, so mobius strips are harder to get this way. They could be, I mean you can suddenly organize a soap film which it looks like a mobius strip but it's hard to model it the way that I exactly said but you can use other things like currents, we value wherever coefficients are in a group or you can think about currents modulo two which in this case would be the true and it would work but there might be more complicated pictures where you would have to think sometime before you know exactly what's the right model with currents, okay? So that's the second complication that I have here and okay, I think I'll, yeah. And again, the typical game for trying to prove exist of size minimizers is to use size plus epsilon times mass because this way you can apply the compactness, I mean the reason why the usual proof for mass minimizing current does not work for size is that you could have a minimizing sequence whose size tends to a limit but with mass goes to infinity and then the usual compactness results about currents don't apply because the mass goes to infinity, okay? So one way to force it not to go to infinity immediately is to add epsilon times mass and make sure that it stays. Then you can minimize this problem, okay? And you can try to prove some regularity for the minimizer of this guy. Try to prove that it does not depend on epsilon and try to go to the limit, okay? But you know, so far only partial results anyway. Okay. Last one, so since today is the day of Reifenberg, so Reifenberg homology minimizer. So this is a very nice way of setting the boundaries story. So it's not differential geometry anymore. We're talking about sets but then we're going to do a little bit of topological algebraic topology, sorry. So this is the definition of a set being bounded by a curve by Reifenberg. Unfortunately, you have to work with check homology groups and not singular homology or simpler things but that's because it goes in this homology goes to the limit better and you're going to say that a set is bounded, let's say, let's talk about the curve only, all right? So here's my curve. The set would be bounded by the curve. If the following thing happens, you look at the elements in the homology of a curve, okay? So in other words, just think about the curve itself. Run at speed one and once, okay? You turn around. But now you try to say, okay, good but in my new set E that contains the set gamma, this curve becomes zero in the correct homology class, okay? And in this case, in the picture I've said here, yes, it becomes zero because you can even contract it to a point, okay? So the best way, you know, I should say the set bounds if I can contract the curve to a point but I don't say that. Instead, I play it more clever and I say if the curve disappears in the homology of a set. This is Reifenberg's definition, okay? And you try to minimize the host of measure of dimension two of a set that bounds the curve in that way. And what's, so let's say, long story here. Reifenberg proved it, so, sorry, homology goes with a group of coefficients. Reifenberg proved it with one of the group of coefficients is a compact group like Z two. I think also another good thing to try is R over Z, that's ours, okay? Then in some cases, I think there was a result by Dupeau that says that you could work with the homology with coefficients in Z. I mean, people like to take coefficients in Z, so that's okay. And finally, I think there is a result of Fung somewhere saying that any group works and also probably more, less smooth boundaries than the usual thing, okay? So this is good because there is a good, I mean, at least there is an existence result. It really represents lots of beautiful soap films, so it's very nice, okay? Probably by the end of the lecture, I will give a hint of a proof of that, but it's only going to be a hint. It's still a hard theorem to prove, and I was told that the original Reifenberg proof was even more horrible, but yeah, okay. Right, and I have a last one, which is the one I like, so you're not supposed to complain yet. So I'll talk about the sliding plateau problem now. And I'll try to give you the definitions because I will use it all the time, okay? Here we go. So far, I mean, it's a little late to ask questions about the past, but again, don't hesitate because, yeah. Okay, so the thing looks a little bit horrible from here, but it should not be so bad. So we are, again, we're giving a set which would be, yeah, which is E0, which is here, okay? And we will try to minimize host of measure among deformations of E0. And I will have to explain what I mean by a deformation of a set E0, okay? And also there is this boundary curve gamma, okay? So again, you start from a candidate, and you try to look at all possible deformations of a candidate, and the deformations will be obtained as, you know, the following way. So there will be images at the end of a one-parameter family of deformations, if you want, of a set E0. So what's with this thing? So again, you have this one family of mappings phi t, they are defined on the initial set and they are mapped in R3, okay? Everyone is continuous, right? X to t, mapping is continuous both in X, which is normal and a one-parameter family is always continuous, so that's the second thing. So here you're not shocked, okay? You start with the identity, you're not shocked also. You do that often for one-parameter family is here. And the main condition is this one here. Not only you deform continuously, but when some point of the set is on the boundary, it stays on the boundary, okay? It can move along the boundary, but you're not allowed to pull it out from the boundary. Usually I like to say, you know, think about the shower curtain, right? You can move the curtain as you like, okay? But I mean, you have to keep the upper part on the rod, maybe if you move it sideways, but you don't pull it off the boundary. So that's the main condition here, right? This one, okay? The last one was introduced by Armgren when he was given this definition away from the boundary. And it can be convenient, it does not disturb, so if you want, you can either drop it or keep it. So the last mapping, you assume it's leapshits. But you never assume bounds on the leapshitness, it's just leapshits, okay? So that's a deformation of a set, and what I have in mind is exactly that. You have this set, you move the points of a set in a continuous way, and you just make sure when a point is on the boundary, that it stays on the boundary. Otherwise you do what you like, okay? No question about that? We'll use that a lot, so it's better if you go slowly. Well, okay. And I have a certain number of comments and I will give them to you in disorder, but anyway. So again, there was this last condition while Armgren don't pay too much attention to it. If you were playing with currents, it would be good because it's a way to say that you can take a current and push it by the last mapping, phi one, to get another current. Whereas if the form of mapping is not leapshits, it's harder to define. But let's forget about currents for some time, okay? The second remark that I could do is that here, I've been insisting on the fact that it's really a continuous deformation. Most of the time you don't need to say that because you could always, given the initial map and the final map, extend the one parameter family by just convexity. So in some cases, you can do that. So that's the reason why you don't see the one parameter families in some definitions. Here I think it's better to have this one parameter family. And it's important here because there is a sliding boundary here. So you want to make sure that even if the final guy is on the boundary and the initial guy is on boundary, you want to make sure that the point stays on the boundary along the deformation. And that's maybe not going to be satisfied if you just extend linearly between the zero time and the one time one, okay? There is, so the plateau problem that corresponds to this is just you take this initial set E zero as you like and you look for the deformation of a set which has smallest host of measure. That's a simple question. So in other words, you define the class of your initial set E zero and you try to minimize in the class of deformations of this set, host of measure, and this is the plateau problem, okay? And it's a plateau problem that depends on the initial condition, but that's okay because you could even say that soap does something like this, right? You have some initial configuration. So try to retract on something with small boundary so that I'm happy with. It seems realistic to me, but I don't know if it's really that realistic, but anyway, it sounds fine. It looks simple in the sense that I don't have to know about algebraic topology or current to set the question. The question has a bad problem with it is that for each two there is no general existence theorem. So we would also have to prove existence theorem, okay? And the last comment that I'm supposed to make, I think, is you might be disturbed by, you know, I have to give you an initial set E zero. I claim this is nice, I mean, it means that you might have, I mean, there were examples that you've seen before where given a boundary, there are a few different solutions to Plato's problem, and here, the few different solutions would be obtained by different initial values, and that makes sense, okay? Of course, the best way to define an initial value problem is to look at the soap film, take this soap film as an initial value, and then you win, right? But, okay, and this is essentially the end of my overview on Plato's problem, so there are many of them, and I claim the more related to soap films are the ones that are not solved, okay? Except in dimension one, right? Okay, plan for the future. So far, and then I should look probably so that I, okay. So there are two things that I try to do in these lectures. A certain number of them is going to be giving you proofs. This is because I'm sort of tired of giving lectures where I show you these bubbles and I say there is a theorem behind the bubbles, okay? But I will cheat because it's not always so easy to find proofs. So in fact, what will happen is that I will be able to give you lots of proofs, but they will always be the same. So there will be a hero, the hero is the Federer-Fleming projection theorem. So I will be spending a lot of time on the Federer-Fleming projections, okay? Right, so that's the general goal, and since I don't want you to stop at the situation of maybe 10 or 15 years ago, I will try to describe a little bit what happens at the boundary. So the goal of the lectures should in principle be try to describe the way a soap film is attached to its boundary when the boundary is a curve, okay? And this story is fairly new in the sense that there was maybe some years ago a first attempt of descriptions of all possible singularities, but I don't think it was so serious an attempt, okay? But I don't have a complete solution either, so, okay. All right, so that would be the goal. So anyway, what I intend to do is give you more definitions, I'm sorry. Then try to talk about the first regularity properties of soap films, I will call them minimal sets or almost minimal sets, and then talk about the, so, of course, here I will give you statements, then I will introduce Federer-Fleming projections, I will use them to essentially prove those things here, to prove that cost of measure along, let's say, for instance, minimizing sequences would be, correct minimizing sequences would be lower semi-continuous. Then there is a main ingredient for minimal surfaces and the sets, which is monotonicity of density, so I'll talk a little bit about that. We'll talk a little bit about limiting theorems, so if you take a limit of minimal sets, easy to minimal set, it had better, but you have to prove it. And again, the ultimate result in that respect at the interior is genitalis theorem that says that the soap bubble that you've seen before cannot be worse in terms of regularity, okay? And again, then I'll talk about the boundary regularity in your curve and I wanted to give an idea of how you could prove existence following a construction of Fevrier, but I don't know if it's going to be possible, but I'll try to put it in the notes. There is a last thing that I maybe should say, which I said before in the slides, but I forgot. It's important for me to have not only the minimal, good descriptions of the minimal sets, but to also have a description of the almost minimal set. So for me, what is an almost minimal set? So we'll see a definition. The definition will not be very interesting. You take the definition of a minimal set, you add a little mistake somewhere, and you say, okay, fine, that's an almost minimal set. Okay? But I think it's interesting in the sense that, you know, for instance, soap bubbles, I would like to be able to describe correctly. And soap bubbles are not minimal, they are almost minimal, okay? Or, you know, imagine that some extra force is applying to my soap film. I would like to still have a description of a soap film instead of saying everything destroyed as soon as I left the minimal set setting. So for me, it's important to be able to talk about almost minimal sets. There is another way, which is from time to time, it will allow me to cheat. Because minimal sets, they probably have some nice rigidity properties, and it might be very hard to describe exactly what minimal sets are. Almost minimal sets don't have this rigidity, so I can come up with a theorem saying, you know, the best theorem for almost minimal sets is that, for instance, they are C1 plus epsilon curves. And it's true for almost minimal sets, because I, you know, C1 plus epsilon curve, it is almost minimal locally. It's not true for minimal sets, and this way, okay, this way I don't have to solve the hard problems about minimal, and I'll just content myself of saying it applies to a more general setting, and it works, right? But this joke being put aside, for me, it makes sense to allow the most minimal set. And I'm saying this because there are ways to deal. So for instance, one of the nice ways to deal with minimal sets is to talk about stationery variables and things like this, okay? And then there's the next stage, which is very false for which the first variation is a measure or something like this. And this looks like a general class more than just the first variation is zero or something like this, but when you think about it, it's usually sort of strong as an assumption. It's really being close to minimal, and I think this sort of thing is a little bit too close to minimal. Here, we'll be playing with our hands. Is it a hint? I will be playing with our hands to some extent. Okay, since I don't know, it's safer here because otherwise I'll have to put my password again. Okay, so again, I mean, almost minimal sets make sense in this business. Okay, I still, so let's see. I know I will be a little late, so I think although I expected this to be the last minutes of the lecture, I will still inflict you the definition so that it's done with for next day, right? Okay, right, and anyway, you're not going to be too shocked. So, okay, so here I will do it without the boundary first. I will do all the proofs without the boundary. And from time to time, we'll say, oh, by the way, this proof works also when there's a boundary. When there's a boundary, you just have to do more complicated proofs but the idea is usually the same. So I decided to do it this way, okay? So no boundary. Competitor for a set in a ball is just a set which is the image. It's here by mapping phi. And again, here, the mapping phi is essentially the endpoint of my deformation before. Okay, and usually I ask it to be leaf sheets. And again, I ask this thing here which says that outside of a given ball, I don't change anything. And whatever I change inside the ball, I stay inside the ball. This is the definition of a competitor for the set E inside the ball B. Okay, all right. Now, you don't recognize the previous definition because the previous definition was more complicated because I wanted to include the boundary set. And but here it's the same. Again, because given this phi here, if you just interpolate linearly between the identity of this set phi, you get a deformation. Since the ball is convex, the quantities, all the relations are going to be satisfied in the meantime. And since I don't have a boundary to take care of, I don't have to be careful to check the point stays on the boundary. Okay, so that's the definition of a competitor. And so I have this comment about sliding competitors and the almost minimal set, again, plain, which means no boundary around, is exactly given by this definition. Okay, so I just give the definition here and stop. So the set is better than any of its competitor in the sense that it has less cost of measure, except maybe a small error, which is here. In this year, I have the scaling, which is R to the D, and some function of H, H of R that tends to zero, which tells you the size of errors. So it's just a function that tends to zero that's given in advance. I call this the gauge function and it's satisfied. I decided, I think, to take something non-degreasing from time to time, even require continuous, but okay, tending to zero and so on and so forth. Okay, so again, an almost minimal set is a set which is essentially better than all competitors in any both, and that's it, okay? And thinking about it as being some sort of a derichlet condition given by the set itself. So the set is standing, maybe it goes to infinity, but whenever I pick a ball, I cannot improve inside, given the boundary values there. Okay, so I tried to play the watch as long as I could, but yeah, it's maybe a good time to stop. No question?