 So it's a pleasure to be here. I was here in Park City before a long, long time ago, I don't remember. But anyway, let's try here. Thanks to the organizer for allowing me to do that. So I'll try to talk about minimal sets and cones. I decided to do things with some proofs, but this first lecture we're not going to see that. I will try to introduce the questions by talking a little bit about the plateau problem. And then by the end of the lecture, I should give you an idea of what I plan to do in the next ones, and I will not finish my transparencies anyway, as usual. Feel free to ask questions. I can see you, I think. Okay, right. So just to be on the good mood, here's a picture of Joseph Plateau. So this is the guy for a plateau problem. We'll say it's probably because the photography was not, yeah, it was great, but anyway, that's him. Okay, so let's see. Introduction to the plateau problem. And now you find out that I have to read it, and okay. So Plateau was interested in soap films. He was also interested in actually having two liquids close to each other, but you cannot mix with respect to each other and looking at the interface. And I saw descriptions of how you manage the experiments so that you can see something. I'll say something about the plateau problem many times. Let's see. In principle, soap films are obtained, they are described, and that's what I'm being told, so that's why I say apparently, but. So this is the soap film. Usually it looks better than that. You have little molecules, and I never remember whether I should say tail or head, but anyway. Those two guys like to sit close to water. Those two guys like to sit away from water, and they sort of align themselves like this, and it creates a film. And then people wonder what could be the energy of this, and the energy, they usually come up with the answer. It has to be proportional to the surface, okay? And so if you want to minimize the energy, you minimize the surface, and that's the way you start talking about minimal surfaces in that sense, okay? Right, so the plateau problem is the following. In general terms, you give yourself some boundary set. I will try to call it gamma all the time. You're looking for set E, which is panned by gamma, or it's attached to gamma, in a way that will have to make precise. And you try to make the surface measure of the set E as small as possible, given the fact that it's attached to gamma in the way that I didn't describe, right? That's the general plateau problem. I will say soap films for this problem, and from time to time, I will talk about soap bubbles or things like this. So what happens with a soap bubble is that there's air inside, so there is a pressure. So what you started with minimal surfaces when there is no pressure, and when there is a pressure, it's constant mean surfaces, and they look like spheres locally, okay? Right, so I should have learned my lecture better, because okay, so here is an example of a piece of soap film and some soap bubble inside. I claim this gives you already an idea of what the behavior of a soap film is. I want you to notice two things. The first one is that this set has singularities, okay? And second, so the, I don't know if you see it here. So here there is no pressure, so I think should be flat, those things here. Here there is a little bit of pressure inside, so I think is sort of round this way, which is not so shocking, so it's the mean curvature story. And then small bubbles, they have more pressure in them, so they have more mean curvature, and they are less regular in that sense. So that's okay, so essentially what I would like to do is do a whole theory of those things, arrive to the conclusion that the things in question look like what you see on the image, and comment about that, okay? Here's another example, so anyway, this is just the singularity that you'll see, the next, the simplest singularity, which is a Y-shaped singularity that you will talk about a lot of time. Okay, and then I decided instead of trying to talk about the plateau problem in general, to try to amuse you with one-dimensional minimal sets, or almost minimal sets. So in one dimension, the simplest version of a plateau problem would be the following, you give yourself a bunch of points. I thought they had names, but I think I called them AJs, but points like this, and you try to connect them in the shortest possible way. So in this case, connecting, it's a simple notion, right? You look at a set which is connected, contains those three points, and has the smallest possible length, so I guess here I would probably do something like this. Unless I get a better idea, which would be maybe, okay, the next time I find a color, maybe this is better, because if you look at the length, it might be smaller than what I wrote, and if this is the case, this is called a Steiner point, and this is the main point of a theory of one-dimensional minimal sets, but there exists Steiner points which solve the problem. So let's see, what did I say? So for this problem, it just turned out that in dimension one, connectedness is a strong enough regularity property so that host of measure restricted to connected set is lower semi-continuous, which means that it's not so hard to find minimizers because of that. We'll discuss this problem again here. I don't want to spend too much time on collapse theorem, which is the thing that says that host of measure is lower semi-continuous when you restrict to connected set. So essentially I'm saying if you have a minimizing sequence of connected set that goes to the infimum, then you can take a subsequence that converges, say, in the host of distance, and the limit is going to be a minimizer, okay? Right, and there's not so much more to say about the regularity of the minimizers, so minimizers are just composed of line segments that can touch each other and connect only at points which are called standard points with three equal angles. No problem, okay. So this is the exercise. So connect the points with the smallest possible length, and now I essentially gave you an idea of what the answer is. You put two standard points and you connect it like this and I'm claiming you cannot do better than that, okay? Right, okay. So this next slide is just, I feel a little bit bad, a little, sorry, I wouldn't do it. I don't feel so well because I will talk about set all the time and most of the time when people talk about minimal surfaces, they talk about currents. And I imagine many of you did not know about currents before, so this was my attempt to do currents without doing current. And maybe I'm wrong because last week you saw lots of currents, but okay. So instead of doing exactly what I've been doing, minimizing length, I can try to do networks. So again, I will put a certain number of points, AJ. I will see them as sources of electricity, say. So let's say here I put plus two, which means that electricity goes out, minus one, minus one. I put, the sum has to be zero because of, you know, if you sum everything at the end and you're looking for a net, so I'll not try to do something optimal yet. I will put intensities, let's say from here to here, so I have to have intensity two going out of here, and I will tell you already what's the best choice probably. So I have, this is a source. It sends intensity one and one. And then my picture is too simple, okay? Here, okay, here I get, so here I get a source, I get minus one, and here I get a source, I get minus one. So in fact, I mean, this would not be useful, okay? And then maybe there is another part of a, so let's say I should have prepared my examples before. Right? If I have a node here, which is not, and here I would have to connect it somewhere here. And here somewhere here. These are supposed to be line segments, but this is the way you do this when you didn't prepare. At a point like this, you should have that the sum of the intensities coming in is zero. Okay? So it was probably better up there. So that's what I call a net. So a net is just a connected set, probably connected. If there is a non-connected part, you forget about it. It's composed of line segments with intensities. They are oriented, the intensities are oriented, and there is two rules that I have to give you. The one which is this rule here, which is the Kirchhoff rule, which is at a point, which is a regular point. All the intensities coming in add up to the same thing as the densities coming out, okay? And at the sources, it's not exactly the same, you add plus two or minus. And you can ask again, what is a minimal connected graph given the initial points A, J? So in fact, in this case, usually I advise not to read the transparency and look at the picture. Here maybe you should do the other way, okay? And you try to minimize something. And in this setting, the something is not maybe as obvious as you would have thought because in the previous slides, what we wanted to minimize is just the length of a segment. Here, it makes sense also to minimize the length of a segment multiplied by the intensity because maybe that's an interesting quantity, okay? So this is my description of our current. Okay, so now you sort of guess that I don't like currents so much. And again, summing the length of a segment amounts to what I call computing the size of a current and summing, multiplying by multiplicities. So by the intensity first and summing afterwards is coming as the mass of a current, okay? When you multiply by multiplicity, okay? And this picture that I did badly, the sources are just, you want the boundary of a current to be equal to a sum of Dirac masses with multiplicities. And so my net is a current whose boundary is the given story about sources. And again, you could try to minimize. Okay, correct. Right. Okay, so that was the bad first introduction to currents. I'll give you a definition of currents afterwards, but, okay. I decided to say a few words about the initial ways to solve a plateau problem. Maybe one thing that I should say right away, I've been talking two or three times about the plateau problem, and that's an enormous mistake. There are essentially as many plateau problems as you want. Okay, so you pick your own, you prove a theorem, and then you decide that this was the plateau problem that was interesting. Okay, so anyway, the first guys here, in particular Jesse Douglas, were thinking about a plateau problem with parametrizations by a disc. Okay, that's what happened. So in this case, you start from a curve, a smooth curve gamma. You try to find a surface, so the surface is going to be the image of a disc. Okay, so gamma is our source group. E is going to be the image of a disc by mapping F, which has a main property, which is that when you restrict to the circle, you go along the curve. Okay, and the second one should be that the area is minimal. And the way you compute area, since you talk about parametrization, you don't think about it too much, you just compute the area in the usual way, which is you compute a Jacobian of a derivative and integrate that and you get, by the area formula, you get some idea of the area, and you minimize this. Okay. So this is one way to set a plateau problem. What can I say about this? So the reason why I decided to do a slide on this is that there is a very beautiful solution to that from Jesse Douglas. Let me say one or two things first. Using parametrizations to solve a plateau problem has two drawbacks. And I'll just talk about the first one now, and which is the obvious ones, is that when you try to take a minimizing sequence and try to make it converge, if you are not clever, you will not have enough compactness to find a limit. Okay, because for instance, a given set might have lots of parametrizations. Some of those parametrizations will be really bad, and if you just take a sequence of parametrizations that are worse and worse, it's essentially not going to converge to anywhere. So getting limits, I mean getting compactness results for parametrizations, usually it's hard and that makes the problem difficult, okay? But if you're in dimension two, you can say, okay, so there were two ways. One, which was to try to use conformal mappings, because in dimension two, conformal mappings exist and they have some sort of, and then you get additional compactness because you chose nice parametrizations and you can try to prove something. Here what Jesse Ducas does is even more cute to some extent, so you're is going to try since we're looking for a parametrization F by the disk, it makes sense because such parametrizations exist to take a harmonic parametrization. You have a lot of choice and you can try a harmonic one. And if you try your harmonic one, then you have formulas relating the values on the circle to the values inside and you can try to compute area, you compute area and what you find out is this very strange formula here, which computes the area in terms of a boundary values of a function F. You still can't choose the boundary values and when you have a boundary values, you take the harmonic extension and you get something, you compute, you get this functional here. The FJs are the coordinates of the parametrization and then you find out that this formula is really, really nice. It has some convexity to it and essentially as soon as you decide that you want to minimize this formula, you can. You get a minimizer and the minimizer is the solution of a plateau problem by the glass. And it's really, usually when I go try to read papers in the other good old times, I expect that they'll have a very hard time, not understand anything at all and this one is really well-written and beautiful and this is what happens. Okay, so these were the good news, okay? The bad news is what I wrote down here, which is that unfortunately this way of describing so films, if you wanted to describe so films, does not work so well when the surface crosses itself. So with this setting, when the surface, when the piece of surface crosses another piece of surface, it doesn't care because you just compute the Jacobian and you add, okay? When you try to do this with soap, soap will find out that you're crossing and it will do something about it. Okay, so that's a little defect of the results that we hear. So this is just an image of a plateau problem which Douglas does not solve. I mean, the way Douglas would solve it would be, you would have a flat thing or almost flat, oops. Here, right? Something that looks reasonably flat here and then they would just cross almost perpendicular and they wouldn't see each other. And soap notices that they cross and again you have something that looks like a disk of soap here. It's not so easy to see but you have singularities of type Y and things like this. This is the real film and you can also do another one which is this one where in this area there is nothing and there is again a singularity of type Y here which is, I admit it's hard to see but it's not smooth, okay? So that's one of the problems of the Jesse Douglas solution. Also it's very hard because you could say yes but it works well with the mapping is injective but it's very hard to know when the mapping is going to be injective in advance, okay? And finally there is a second drawback about parameterization which I was talking about. I don't think I have a slide, no, okay. So there is a last drawback of parameterization which is we decided to parameterize by a disk but they are nice, beautiful soap films that are better parameterized by something else than a disk, like some sort of a torus or a disk with a handle or whatever. And of course then you would have to think in advance what you want to parameterize your minimal set by and then you try to parameterize by this set but the disk was, it was the obvious choice but it's not the only nice set which is bounded by a circle. Okay. Last attempt to try to do currents, okay? So I'll give you the definitions of currents that I know and if you really hate currents, well, one you shouldn't and second, okay, bear with me for two seconds, okay? So I'm only going to be, so a d-dimensional current, it's not so bad, it's just a distribution acting on forms, okay? And the typical examples that we'll get is not horrible distribution, it's going to be measures acting on forms. So in terms of regularity, it's going to be measure and it's just when instead of being just a measure with positive values, it has values which are individual of forms like the d-dimensional forms, okay? So the d-dimensional current that's here is a continuously in your form when it says the smooth d-forms, okay? dx1, dx2, dxd and then a sum of things like this, okay? And but I have two main examples. The first one is you have a smooth oriented surface of dimension d, nice object. With the orientation comes the fact that you can integrate d-forms on that surface. I don't tell you how, but it's differential geometry, okay? And this is exactly the functional that corresponds to the current I'm talking about. The current of integration of forms is just this thing. You have a form, you have this orientable manifold, you integrate the form on the orientable manifold and you get this thing here, okay? So that's, so I write this as the set would be s and the current would be s prime and it's essentially this thing here. I don't tell you how to integrate the form on a manifold of the same dimension, but that's differential geometry, okay? That's the simplest example. And the second example that we'll use is a rectifiable current on a rectifiable set, okay? And you can think that the rectifiable set, for instance, could also be a smooth manifold of dimension d. And I just give you the formula here. So the current is acting on a form, you integrate on the set, the set is rectifiable. You have an integer multiplicity or it doesn't have to be integer, but in our cases it's also always going to be an integer. So it depends on x, it's measurable, but it's always an integer. Then you have the form that you write down, but you should write down, but I don't write down. Then you have this product with the orientation here. So this comes from the orientation. And this thing here is my short way of saying that when you have an orientation, you can compute the integral locally of a form by taking the scalar product with whatever this thing here. In, okay, but I don't do the algebra, and then you integrate, sorry, and then you integrate everything against host of measure, okay? So the previous example was just m is equal to one and everyone was smooth, okay? And after all, it's not so bad, right? You don't even have to have a beautiful orientation on the surface, the surface is a rectifiable set. I'm saying it has an approximate tangent plane almost everywhere. And this plane, you orient it in a measurable way at every point, okay? Now, if you do it badly, you will get a horrible current at the end, but that's okay in particular in terms of, okay. Very fast. I'll be using host of measure a lot. So here is the definition of host of measure. So you take this set, you try to cover it by set dj. You sum the diameters to the power, you take the best covering that you can think of. You multiply the right constant so that at the end you get, for instance, exactly the Lebesgue measure on sets of dimension d. So anyway, you get this number here. This number is when you try to cover with small sets and this number is an increasing function, sorry, the decreasing function of delta. So it has a limit and the limit is the host of measure, okay? And the main property of the host of measure is just that it is a measure. It's defined on any borel set and when the borel set is a smooth surface of dimension d, it coincides with surface measure. That's what I need about that, okay? I'm sorry for, you know, because you probably knew that already. Okay, let me continue the story about currents. The way, so there's something beautiful about the current is that there is a way to talk about the boundary of a set or now it's going to be the boundary of a current which is just differential geometry and which, you know, makes sense and makes some definitions very easily. So that's my way to define a set E is bounded by a curve. For instance, it's going to be like this, okay? So there is a notion of boundary for currents which is this one. It's by duality with the exterior derivative on forms. Okay, so the effect of a boundary of a current on a form is just the current applied to the exterior derivative of a form. And again, if you're like me, you don't understand differential geometry and then this does not make so much sense. The main point is that when you apply the green theorem in this setting and you have a manifold with boundary, then the boundary of the current of integration on the manifold turns out to be exactly the current of integration on the boundary. Okay, so in other words, for smooth guys, the boundary operator that I'm writing down here is exactly the operator which sends the manifold to its boundary. And we want to say this is the same definition for more complicated sets. And that's the game we're playing. And I should have said before, the reason why we want to play these games is exactly the same reason as for solving some differential equations, PDEs. From time to time, you want to look at weak solutions and then after you prove the weak solutions exist, you try to see whether they were actually strong solutions and so on and so forth. And that's the game we're playing. We're playing the game of weak solutions because it's gonna be easier to find weak solutions. And our weak solutions will be currents. They're not going to be smooth sets because we don't expect the solution to be smooth immediately. Okay, so the classical, when you talk about the plateau problem for currents, you do the following thing, this thing you always do. You start with a nice current gamma. So for instance, think about the current of integration on the curve, okay, of dimension one. And then you try to solve the equation dt is equal to gamma. So in our case, gamma was a curve. So this was a current of integration of dimension one. You're looking for a current of dimension two and its boundary should be the curve. And you hope that it sort of corresponds to a nice surface, okay? And this is the equation. And then you try to minimize something on all the currents that satisfy this equation, okay? And I will give you just two and a half quantities that you like to minimize. Most of the time people like to minimize the mass. So the mass is just the norm of a current as acting on the forms with the L infinity norm. Since I'm only interested in the rectifiable currents before, in the case I had a rectifiable current like this, it was just the integral of the multiplicity on the set, okay? That's the mass, okay? It's the norm of the current. And the second thing that you can try to minimize is the size and then I will comment. The size is this. The size is just the host of measure of the support of the current. So you don't count multiplicity, you just count zero if you're not on the support and one otherwise, okay? Two different quantities. Traditionally people like size better because for size there are beautiful theorems due to feather, flaming, the Georgie, maybe in a slightly different context. Many other ones. And let me put the story short. Given a problem like this, where this guy is a rectifiable current with, sorry, with integer coefficients and whose boundary is zero, then there is a solution that minimizes size. Let's say assume that everyone is compactly supported and then it works very well, okay? And I added this that I had forgotten but that was on the slide. Since the boundary, so since the exterior derivative of the exterior derivative is always zero by algebra, it means that d composed with d is always zero. So if you want this to happen and if you want to take a second d here, it should force this d to be zero and that's the reason why you always ask this. So for instance, you would not solve a story for open curve. If a curve is closed, its boundary is zero and you can try to solve, okay? So again, you never, so this is necessary if you want ever to solve this, okay? So that's one thing, beautiful existence results and also nice regularity theorems. So it's not, things are not always regular but for instance in co-dimension one and if I don't compute, I mean if I'm not mixed up with seven and eight, all the minimal surfaces that you get so all the support of the minimizing currents that you get are smooth up to dimension seven, okay? Beautiful theorems but I'm not interested, okay? But okay, right, because I want things that are, okay. So I could tell you why I'm not interested. The first one is that I don't understand current. The second one which is more of an excuse is that of course since we've seen so bubbles that have singularities and we're in very low dimensions at least some sub-bubbles or some fumes will not be well described by currents, right? The currents are a very nice way to describe minimal surfaces but so fumes, there is a doubt, I mean some of them only, okay? Right, it makes more sense to minimize size. So size is this thing that corresponded in dimension one to the length of segments, okay? So this looks better. Let me draw pictures, okay. And again, you could want to minimize size in this problem and I announced you already some bad news because it's there. The theorems are not as nice, sorry, I'm being French again, as in the other story above, okay? Maybe by tomorrow I will be in better shape. Okay, so this corresponds to a boundary which is due to circles. The soap films will often be like this. It could also be two parallel discs but this is a nice soap disc. This one is the support of a minimizing current. You would have, oh, you would have multiplicity one here, one here, multiplicity two in the disc in the middle, 120 degree angles here that you don't see so well between the disc and the two faces and this is a minimizing current, so far so good. Mass minimizing, so this is just to tell you that the same curve can give you lots of solutions. All those guys can describe soap films.