 OK, so welcome everybody to the first lecture of this course. My name is Camillo de Lellis. I'm not Emanuele Spadaro. So our course is organized in 12 hours, six lectures. I will actually give the lectures of the first week, and Emanuele Spadaro will give the lectures of the second week. OK? So the purpose of the course is to give you an introduction to a fairly well-established area of geometric measure theory which is called the theory of area minimizing currents. So let us start with the basic problem that we have. So it's a very famous problem which was formulated by the Belgian physicist Plateau. So Plateau's problem is the following. Say that you have some k-dimensional surface gamma in some euclidean space, but you can also imagine to have a Riemannian manifold over here. So this is a k-dimensional surface without boundary. So then we look for k plus one-dimensional surfaces, sigma such that the boundary of sigma is actually gamma, and sigma minimizes the k plus one-dimensional volume. So of course just imagine for instance you have a curve in R3 and you have a surface which is bounding this curve. You cannot achieve something which has as small area as you want. There's going to be something which has minimal area. So it's a classical problem in the calculus of variations. And actually for Plateau this problem was interesting when the dimension was three and k was equal to one. So if you're actually trying to solve the problem for surfaces in R3, having a certain given boundary which is one-dimensional, and in this case you actually get a very nice physical model for it which would be SOP films, which was his original motivation. Okay, so this is of course a very famous mathematical problem. And so what is going to be for us a solution or what actually we're going to be interested in? So an existence theory, also a regularity theory. And that's going to be what is most important for us. Well, so why do I stress actually on existence theory? Because one thing that you should first agree upon is what kind of area, I mean what kind of surface do we allow? So can they have corners? Can they be non-smooth? Well, so this actually opens a lot of questions, opens upon not a box of questions. So the way actually you should approach the existence theory is not uniquely defined. So what we are going to see is one possible answer for the existence theory, and that is called the theory of integral currents. But there are also other possibilities. Now we will see that actually the theory of integral currents has a lot of appeal because it has a lot of geometric content. But for instance, if you were interested in the original problem of plateau, the theory of integral currents have some disadvantages. And then for the specific physical problem you might actually want some other existence theories which maybe allow some type of singularities that the theory of integral currents do not allow. But we are not going to give you a panorama on how you actually define other existence theories. We are going to actually focus on this. And although I would be most interested in the regularity theory, at the beginning I will have to give you at least the basics of what is the theory of integral currents. So let me give you very quickly the main achievements of this theory and maybe a little bit of history also. So integral currents were defined first by Federer and Fleming, defined by Federer and Fleming in the 60s. And actually in general, but there was a pre... there were some early works by the Georgie in the co-dimension one case. So let me give you immediately the definition of what an integral current is. So first of all, what is a current? So a current is, I mean, a Federer and Fleming current because we would see a more general version of this. So a current is a linear functional. The space of smooth, compactly supported differential forms. Actually a current as a surface comes with a dimension. So let's say a k-dimensional current is a linear function on the space of smooth, compactly supported differential k-forms. So the basic idea is that if you give me a smooth surface, over this surface I can actually integrate k-dimensional forms. And I can look at the action of this integration on the space of k-dimensional forms. So if, say, sigma is a smooth surface, I can think of it as a map from the space of k-dimensional forms, which we are going to denote in this way. So these are the smooth, compactly supported k-forms. And here I just integrate over my surface, sigma, the form omega, which gives me a number. And of course you see by the definition of integrating over a surface that if I dry omega in a smooth way, this is actually varying continuously. If I take a nearby form and I integrate something, a form which is nearby, then I get a nearby number. So this map is some continuity properties, which is what actually is required on the definition, on the general definition of a current. Of course you see immediately that that allows you to do a lot of things. I mean if you are in such a general situation you can for instance take a notation for this which will be later on this one. So we use this funny parenthesis just to identify the current. So this funny parenthesis computed on the particular differential form is actually giving me this number. Of course you see immediately that if I have such a huge degree of freedom I can take for instance the following linear functional. I decide for a real number lambda in front of it. And this is also a linear functional on the space of k-dimensional forms. OK, with such general definition of course the first thing that you would like to understand is what is the concept of boundary. And for you the concept of boundary is implied by the Stokes theorem. So you know by the Stokes theorem that if sigma has a smooth boundary gamma, then the integral of an exact form, I mean you can integrate an exact form by parts. So since Stokes theorem tells you this, actually in general what you're going to take as definition of the boundary of a current is this action on forms. So if t is a current which means I know what actually is the action of this current on k-dimensional form, then the boundary of t is the current which is defined in the following way. So the boundary of t is the k-1 dimensional current given by the following formula, the action of the current on some form omega is defined to be as the action of t on the exterior differential of the form omega. OK, so this gives you already a very good starting point. So now if you have a general say, if you give me for instance a curve in R3, I can define the current which is associated to this curve, and then I can consider all the currents which have that current as a boundary. Now to formulate a plateau problem though, I have to tell you what is the generalization of the k-dimensional volume for this object. OK, so for this we need the notion of mass and commas. So if we consider a k-form, then the commas of omega at a point x is given by the following thing. So it's first of all a number, and this is simply the supremum over all e1, ek vectors such that the modulus of, OK, so define it in the following way, such that the determinant of ei dot ej is less or equal than 1 of omega x evaluated on e1, ek. OK, so remember the k-form is just an alternating, I mean for each point x, a k-form is an alternating linear form in each of the entries. OK, and I'm taking the supremum over all possible choice of k-table vectors such that the determinant of this matrix, the modulus of the determinant of this matrix is less or equal than 1. OK, so actually this you can check. So this commas is really a norm on the space alternating linear k-forms. Maybe I should put a c over here. Now check the following interesting property. So if you want, this is an exercise. If sigma is a smooth surface, the k-dimensional volume of sigma is actually equal to the supremum over all forms omega such that the commas norm is point-wise less or equal than 1, the integral over sigma of omega. OK? So that's for instance very easy in one dimension. Right? So if you actually take a curve in one dimension, just check the following computation when you're actually looking at the action of a curve gamma on a one-form omega. It's very easy to see. This is actually equal if the curve is embedded in the integral over gamma of omega computed on the tangent vector. OK? And now you see that the thing is defined in such a way that the commas of omega is always less or equal than 1 in modulus. So the thing that you're integrating over here, right, is always less or equal than 1. If you want to maximize this thing, actually what you want to do is to choose omega in such a way that this omega of tau is exactly equal to 1. OK? And that's where you actually achieve your maximum and the maximum is going to be the length of the curve. Of course, the thing is more complicated when you have more dimensions. And then you have to work with the klinearity and so on, determinants and aria formula and so on. OK? So this gives you the definition of the mass of a cavern. So now once again is the pattern that I had before. So I use this observation to just say, OK, I take this nice identity on smooth surfaces. In general, for caverns which are a sort of abstract object, I actually take this nice identity as a definition of mass. OK? So the mass of a cavern T is given by, so the mass of T is simply the supremum now over all forms which have commas less than 1 of the action of the cavern on this form omega. OK? So this is actually called the mass of a cavern. And that is a space of caverns which is called the space of caverns of finite mass. So in new definition, a cavern has finite mass if, well, the mass of T is less than infinity. OK? So now you notice actually something very interesting. So the mass is something, I mean, the space of caverns is a linear space, meaning that since you're talking of linear functionals, you can take not only multiples of the caverns with real numbers, but you can take really linear combinations. No, no, no, no, no. So this constraint is true for every point, right? This is true for every point. But this gives you a number, OK? So it's, I mean, in the case of, I mean, in this case it's like you're integrating over the whole surface, right? So the constraint is the constraint for every point, but you're integrating over the whole surface in here. I mean, that's the over, I mean, that's the underlying idea. Of course, this is not necessarily defined as an integration over something, OK? So you can then check somehow another thing, which is also very easy functional analysis. So the space of caverns with finite mass is a Banach space. This number, which is the mass of the cavern, gives you a norm. Well, OK, so it's a Banach space with this norm. OK, now let us go back to our original problem, right? So we have a general notion of boundary, which I have a general notion of mass. So now, given a certain boundary, you can ask what is the current which has least mass, which is spending this boundary. So given, say, a current s of dimension k in Rn, look for k plus one dimensional currents, t such that the boundary of t is equal to s, mass of t is the least possible. And it's not difficult to see that this is a problem which has existence out of simple functional analysis So somehow it's an exercise in functional analysis to actually see that there is always a solution. Well, OK, so if the mass of the current is equal, I mean, if there is no mass, I mean, no current with finite mass that has this property, then anything is a solution. The mass is always plus infinity. But if there is at least one current which has finite mass and which is spending your boundary, then this problem is always a solution. And is a simple exercise in functional analysis, meaning you see that your norm over here is defined by duality against something, OK? So your Banach space has some weak compactness properties, OK? So exercise, this mass minimizing current always exists. Now, we are not going into the details, but there is one drawback of this general definition. And the drawback of this general definition is what we saw at the beginning. I give you a current, and you can do this nasty thing. I give you a nice surface. But you can multiply this nice surface by a real number. And that's going to be your general surface. So I give you a curve, and you might decide to take e times this curve, or square root of 2 times this curve, which is a bit disturbing. So you might get actually in this way solutions which are not so nice. I'm not going to give you any detail about this, but it's a non-phenomenon in higher co-dimension that, for instance, if I give you a curve in R4 and you apply this algorithm, you might get a funny solution as your area minimizing current. You can get, for instance, half of a nice surface solution. This might happen. So the first examples, I think, go back to Fleming. So this might give you bad solutions. So this simple approach might give you bad solutions, for instance, half of a surface. There is also somehow, of course, getting something which is singular is maybe sometimes something that you would regard as bad. But this getting half of a surface is really very bad because it will give you some kind of gap phenomenon, meaning the following. So if you sort of look at your classical plateau problem, meaning I give you a smooth boundary, and you look at the infimum of the area among all smooth surfaces that have this boundary, it might actually happen that this infimum is a certain number, say 10, and it might happen that the minimum of this generalized plateau problem is 5, so that there's really like a gap between the minimizer over here and what an infimizing sequence in a classical sense might achieve. So this is called a Lavrentia Phenomenon. Let me give you a silly example of Lavrentia Phenomenon. So I decide one surface, which I like a lot, which is my surface, Camillo's surface, and I define the area of the surface to be 0. I just define it to be 0. It's silly, but I define it. Then there's always a minimizer, and it's my surface, which is not what you would like to pick up. So of course, and there is a gap, say, the minimizer you get by looking at in sort of normal definition is 1, and what I get is actually 0, and I get it in a silly way. Of course, this is much more interesting because you don't get it in a silly way. You get it in a way which looks very natural. Unless this is what happens. Okay, so the reason, I mean, this kind of phenomenon is the reason why you actually introduce something more complicated which are called integral currents or integral rectifiable currents. So therefore, we want actually to restrict our class of admissible objects a little bit more. I will give you later on maybe a more formal precise definition. So let me give you a first definition. So an integer rectifiable current is a current T of finite mass for which there exists three sequence of objects. So these objects are going to be a sequence of C1 surfaces, embedded surfaces in Rn, and the most subsets inside, well, they have to be oriented actually, integers with the following two properties. Well, first of all, this series is finite, and second, the action of the current on any form omega you recover from this converging series. So the upshot is I allow you to take any C1 surface. I allow you actually to take any closed subset of C1 surface, and I'm integrating, of course, over this in the sense of Lebesgue, for instance. And then I allow you to take linear combinations of these objects, but these linear combinations have to be integer combinations, not real combinations. And then I'll allow you also to take an infinite series, as long as I have this requirement over here. You see that this requirement over here will guarantee the convergence of this series. Oh, EI, sorry, EI, yes, EI, sorry, EI. Otherwise the whole purpose of introducing the EI is empty. Thanks a lot. So the current, for instance, might be, I mean, it might be that I give you a very nice surface, but allow you also to take scattered pieces over their surface. So let's look at a practical example. So let's give a practical example over here, in one dimension. What I allow you to do, which you will like, obviously, because you will recognize it's a very natural thing to do. So, for instance, I allow you to take two curves, gamma 1 and gamma 2, and then I allow you to take two pieces, so that's going to be one piece, and this piece, right? And now your current is just the union of these two pieces. OK, this is a current for you. And the action of the current on a form is I integrate over this line, I integrate over this line, and I take the sum of what happens. Two copies of this and maybe three copies of that. Why not? But I don't allow you to take a square root of two or half a copy of this and one-third of a copy of that. OK? So your coefficients have to be integers. OK? So now, of course, you recognize that a classical smooth surface which bounds something, whether it exists, is an integrative current in this sense. And now I would like to actually minimize the mass in this class. But now it's not any more an exercise in functional analysis. You see, before I had the currents of finite mass, which was a Banach space, because I was allowed to take real combinations, I mean linear combinations with real coefficients, but now I'm taking a nasty object because I'm telling you that you're allowed only to take integer coefficients. Right? So now you're not any more a Banach space, and the question whether the corresponding formulation of the plateau problem is well-defined or not is a very subtle question, and this was the achievement of Federer and Fleming. So this thing over here. So formulation of the plateau problem number two, which is the formulation we're going to take, well, it's the generalized plateau's problem. OK? So consider now an integral rectifiable current, gamma dimension k, with the boundary of gamma equal to zero, and look for k plus one dimensional integral rectifiable currents, such that their mass is minimal, they bound gamma. And now it's not any more an exercise. Well, if you're able to solve this exercise, you are Federer and Fleming, so kudos to you. It's a pretty difficult exercise, especially because the proof at the time was very complicated. The proof now is still complicated, but kind of much less complicated, but also because there's a lot of technology which has been introduced meanwhile. So the proof is less complicated, but you have to know much more stuff. So it's still a pretty complicated thing. But let's say so the Federer and Fleming theory has in some sense three cornerstones. So the first one is what is called usually compactness of integral currents. I will state this compactness actually later on in the course in a more precise way. This compactness of integral currents gives you actually the existence of a solution for the plateau problem in this person. Then there is a second cornerstone which is called deformation lemma. I'm not going to tell you exactly, actually, even in the course what the deformation lemma says, but one of the effects of the deformation lemma is that there is no Lavrentiev gap phenomenon. So one effect of the deformation lemma is if there is a smooth surface which in Rn is the minimum of your plateau problem in the classical sense, then that minimum is also in the solution of the plateau problem in this sense. Whereas with the previous first formulation you might have had a nice solution of the classical problem, but it was not a solution of your generalized problem. So the deformation lemma gives you any time you have for some reasons a smooth or reasonable solution in the classical sense of your plateau problem. This is also a solution in the generalized sense. OK, so actually this is a consequence of the deformation lemma. A true statement of the deformation lemma would be a density statement. So the smooth surfaces is dense in an appropriate sense in the category of integral currents, although this is not, literally speaking, really true. I mean, you have to take a class of currents which is a little bit more complicated, which are called polyadrol chains. But anyway, the deformation lemma, if you want, is something like the density of smooth functions in a subolive space. So one good way of reading this theory is to draw the following analogy. The integral currents, or, I mean, the theory of currents stays to function, I mean, the theory of currents stays to classical surfaces as general distributions stay to currents. The theory of integral currents stays to classical surfaces as subolive spaces stays to classical functions. And this deformation lemma states the density of classical functions or of classical surfaces as you have density of classical functions in subolive spaces. So that's more or less the idea. OK, so that is a third statement which I'm going to tell you in the future. It's also the other cornerstone. It's called boundary rectifiability theorem. I don't know if I'm actually going to state it, really, but, well, let me tell you just that it exists. So maybe later in the lecture I will give you a precise formulation of this. OK, so far so good. Now you have an existence theory which gives you something reasonable when the reasonable solution exists, but which might in principle give you a very, still a very rough solution of your plateau problem. I mean, you might have you have smooth surfaces, that's fine. You have integral coefficients, that's fine. But you have, first of all, countably many of these surfaces which might intersect in a strange way. And you have this nasty thing that you have closed subsets. OK, so closed subsets of a surface might be extremely irregular. So you might have in principle an integratifiable current which consists of pieces of C1 surfaces which are nice, so you might have a structural irregular closed subsets of your surface. OK, so, therefore, there is a very natural question you ask how regular can actually this solution be? Right, a similar thing, for instance, you encounter when you define harmonic functions as a minimizer in W12 of the Dirichlet energy. A priori, a Sobolev function is very bad. But the minimizer is going to be an harmonic function which actually is analytic, real analytic inside. OK, so this is the question that we want to answer. What is the regularity of a solution in general? And in principle, there is a fantastic complete answer. Well, this is not even in principle. There is a fantastic complete answer which gives you a very nice regularity theory which is also optimal. So answers, co-dimension one. So, for instance, the problem real analytic surfaces except for a singular set which is closed and has half-storey dimension at most and minus eight. So, for instance, up to seven dimensions you take a six-dimensional surface and the plateau problem gives you a real analytic surface. In eight dimensions it might happen that the solution has a point singularity. OK? And the statement is optimal, meaning in eight dimensions the following surface which has a singularity at the origin is area minimizing anytime you intersect it with a sphere, for instance. This surface is locally always area minimizing actually the unique area minimizing current. Of course, this surface is infinite so it doesn't have a boundary. What you do for instance is you take a sphere of radius one, you cut this surface with the sphere of radius one and you get a product of two S3. OK? And then you look for the solution of the plateau problem rather than flaming is this cone and this cone has a singularity at the origin. So, this is a very this is a very nice answer in co-dimension one. In co-dimension two that is an equally nice answer. So now let K be less than n less or equal to n minus two then any dimensional solution of the plateau problem is p o analytic surface except closed set of dimension at most K minus two. So this time you might encounter in singularity in a much I mean of a much larger dimension than before. So remember here you were considering hyper surfaces so the dimension of the surface is n minus one. The singular set has dimension at most n minus eight. So in the surface the co-dimension of the set is at most seven. In here in the surface the co-dimension of the set might be two. That's also an example which shows optimality and the example is way much easier than before. You just take any holomorphic curve in C2 and you get it. OK, so let's I mean I will give you more general examples anyway later. So let's take say ZW in C2 which you identify with R4 such that Z squared is equal to W cubed. This gives you a solution of the plateau problem, the unique solution actually once you intersect it with a compact subset. So is R4 always the unique minimizing integrated rectifiable current and obviously has a point singularity at zero. OK, so as you see there is a dichotomy between co-dimension one and co-dimension two. I will be able to explain why or what is the reason that there is behind this fact. I mean why this is actually area minimizing and has a singularity that's easy. Whereas it's way much more difficult to actually prove that this is area minimizing. Well thanks to Widow actually now there is a very nice proof but I mean it was originally way much more difficult. This is a simple observation. So it's much easier to give you counter examples to regularity but of course as you might expect it's much more difficult when you have easy counter examples is then much more difficult to prove a regularity theorem than it is when you have less counter examples. OK, so this theorem in co-dimension two is way much harder than the theorem in co-dimension one. OK, so the theorem in co-dimension two so the result in co-dimension two is due to Angren proof is a book of one thousand pages. Actually his original manuscript which was type written so he did it with the typewriter because there was no tech was a thousand seven hundred pages. But then tech allows you to put much more stuff in one single page, right? Then a typewriter does. And I've actually been involved together with Emanuele who's giving the second part of the of this lecture. We've actually tried for quite a long time now to give a simpler proof of this although we are still using basically the ideas of Angren so we have just put everything on the web now and we have essentially five papers which all together give you a simpler proof of this so five these and papers give you a simpler proof and this is joint work with Emanuele and somehow the idea of this course is to give you some bits on how this can actually be proved. Now one has to say also following I'm not going to talk too much about the co-dimension one case in a sense it's also because the co-dimension two case contains as subsets some of the main ingredients to have the regularity in co-dimension one. And also because it would be too long of a story so the part of co-dimension one which is specific to co-dimension one would be a story for itself. There are actually very nice references so for co-dimension one you can look at the book of Giusti but there's also a more recent book by Maggi. So the classical references on the theory of Carrens, I mean the most classical reference is Federer which is a thick book which not everybody considers as a nice bedtime reading so there is Federer and this is somehow in some sense a traditional book so the title is Geometric major theory much better and is what I actually learned when I was a student he's a book by Leon Simon and this is called Lectures on GMT it actually contains more than the theory of Carrens but of course it contains actually less than Federer when you go to the details but recently well there's also part of the book by Jacinta Modic and Sushek but recently on this theory and you would see something of it in the other courses as well I don't remember where the sign on the C goes, I think it goes like here but due to recent developments which we'll see also in these lectures there's actually quite a nice reference by Carrens and Parks okay on the regularity theory of course as I told you there is in co-dimension two I mean all these books they contain something I mean most of the things about the co-dimension one regularity theory I think Leon's book actually and Federer's book they are surely complete, I'm not sure about this too, Giusti and Manji they are also complete of course about the regularity theory in higher co-dimension as you might imagine since these five papers they are very very recent there is no book and no nice reference okay so the attempt of this course would be to give you some feeling for this regularity theory and of course you can take the papers of Emanuele and myself as further reading if you want to go deeper in the theory so essentially in this course now in the first in the next five lectures I will give you a little overview of a more general approach to the theory of Carrens which is given by Ambrosio and Kirchheim this approach is not only more general but it's also slightly simpler I mean much simpler actually than the original approach on Federer and Fleming is so the goals for the next five lectures or hours is so go through I will give you the basics the approach of Ambrosio and Kirchheim to Carrens which will be useful also for the lecture of Stefan Wenger next week actually I will not be able to give you two detailed proofs of anything but at least I want to give you an idea on how nowadays the compactness theorem is proved this compactness theorem will also give you a nice motivation for some tools which is used later in the regularity theory so some ideas on what is called slicing theory for Carrens and compactness then I will give you an idea on how the regularity theory might be approached and what is giving you a first step a first introduction to what is called nowadays a large regularity theorem which actually for area minimizing current is essentially an idea which goes back to the Georgie yes in fact that's why we will I mean I will give you an idea and the idea is essentially already going back to the Georgie if you want to look at the real allard regularity theorem then you have to deal with very false that's absolutely correct although you don't have to do actually that much of a theory so if you want a complete proof of allard's regularity theorem look up on my web page I have lecture notes from a course which is essentially 20 pages and gives you starting more or less from scratch a complete proof of allard so further reading you can either take the book of Leon Simon but I have also some lecture notes then I will tell you hopefully what happens in higher co-dimensions if I or what is the main obstacle so what makes things difficult to answer to this problem I will give you some basic intuition of why you need so-called multiple-valued functions hundreds of multiple-valued functions and then from here on next week Emanuele should take over and give you some idea about how the proof goes on I mean in principle I have still another hour of lecture but maybe it's a good idea to make a five-minute break because it's like one hour and five minutes that I'm already talking at least I am tired so I guess you must be also so let's say we make a five-minute break and then I start over again so in this lecture we are now introducing the Ambrosio-Kirchheim theory of metric which goes back to 2000 so this is the paper which is entitled Carnance in Metric Spaces so first of all you will see that instead of defining things in duality with smooth k-forms which don't exist in general on a metric space we are actually going to build our theory in duality with Lipschitz functions so the first definition is in some sense a substitute for differential forms so then in our case so from now on maybe I should say over here so in this hour E is a metric space with a distance D so when k is bigger or equal than one the set of which we will denote in this way set of differential forms so this denotes the k plus one tuples omega pi one pi k real-valued Lipschitz functions where f is also assumed to be bounded we don't assume anything on the sub-port though so in the future maybe we will use the following notation omega so what we are thinking on the back of our mind is that this replaces the usual differential form in Rn which would be simply f d pi one wedge wedge d pi k so although in the Euclidean space we are actually used to consider these pi one pi k functions in this very general situation we don't know what a smooth function on the metric space E might be and even if we know it in general this gives you a much more rigid situation than in Rn so it's of course even maybe possible to introduce a notion of more regular functions than Lipschitz functions for general metric spaces but then this gives you very few test forms so to say the second definition gives you metric functionals so a k-dimensional metric functional is any linear functional or is any map t it cannot be linear because this is not a linear space sorry but it has actually the property that t is sub-editive meaning that t of for instance f plus g pi one pi k is less or equal than t of f so that's sub-editivity with respect to the first entry and actually it has to be sub-editive one homogeneous now one homogeneity is what you would imagine when you're multiplying by a lambda so with respect to the first entry so here I've written actually the sub-editivity and the one homogeneity with respect to the first entry but in fact for the metric functionals we actually ask that the same property is true for every of the entries over here so if I have t of f and then here I have pi plus something then I will have it less or equal than the modulus of t of f then the pi and here is the t of f dk of a I can define the operation of pushing forward so if I have a function phi which is going from symmetric space e into symmetric space f and it's Lipschitz and then I have omega which is f pi one pi k and belongs to dk of e the push forward of omega is actually defined as composing all the functions in the pitopole with phi and this of course is going to be an element of dk of f so that's the push forward and then I can define an exterior differentiation the exterior differentiation for any element omega in dk of e, yes? e yes, yes, yes, yes, sorry I'm getting ahead of myself so of course it is yes, so this is not the pushing this is not the push forward sorry this is the pullback of the form actually that gives you a push forward on the metric functional sorry, perfectly correct and the pullback is going of course in the other direction so I start from something which is defined on f and I get something which is defined on e not, okay then when we define the same object on the caverns we will have actually a push forward instead of a pullback we will see it in a moment so maybe let us say it immediately so this defines push forward on metric functionals, right so and actually right so so the pullback actually has the ds is the sharp above in fact standard notation, yes so the push forward metric functionals then defined in the correct in the other direction so if t is metric functional dk of e then the push forward of t is defined as a metric functional on dk of f and of course the push forward of t is acting on omega by simply taking t and act it on the pullback of omega so that is the first natural operation which of course corresponds exactly to the operation of pullback and push forward for usual forms the second operation is given by exterior differentiation and that gives you a boundary operator for metric functionals so the exterior differentiation goes from omega in dk of e a k plus 1 by 1 by k and d omega is going to be equal to 1 f by 1 by k and that's actually an element in dk plus 1 of e so of course what we have in mind as usual is that in Rn is this object well then d omega is nothing but df wedge d pi 1 wedge d pi k okay so this would be for standard smooth forms in Rn and then you read off that our definition is consistent because here you can just think that you have k plus 2 2 tuples with 1 in front where there would be the coefficient 1 okay and that's before by duality once we have a notion of exterior differentiation on our dk e then we have a notion of boundary on the metric functionals for us the boundary of t acting on omega is just simply equal to t acting on d omega right because the second entry is constant of course as an object somehow it doesn't vanish but now you will see that we require certain continuity okay so as you see I'm not talking about alternating yet okay so the whole point actually that you will see somehow in a second is that there is no need of introducing alternating at the level of forms once you actually give the definition of alternation at the level of currents you will just have actually that 0 when you are computing on dt so by the definition of giving you dd omega actually is not equal to 0 because you've not introduced alternation in there but you will actually discover that when I introduce currents I will have that 2 times the boundary of t is equal to 0 okay the fact is that I've not yet told you what the current is I've just told you what the metric function is right so there are some axioms which are coming in a second okay so let me give you then another definition so a third operation that you can actually do is taking essentially wedge products if you want which then gives you a restriction operator on currents well let me just give you the restriction operator on currents so if t well on metric functionals first okay so if t is a metric functional dk of e and omega is in d j of e then t restricted to omega is a metric functional dk minus j of e defined by so t restricted on omega computed on u is actually equal to t computed on f g pi 1 pi j and then let's say rho 1 rho k minus j and that's if omega is f d pi and nu is g d rho and we can introduce the mass of a metric functional so a metric functional t has finite mass if there exists a measure mu on your metric space e such that anytime that you compute t on f pi 1 pi k you actually get that this is less or equal than the product of the Lipschitz constant of the respective pi i's times the integral over e of the modulus of f so mu actually has to be a finite measure so this defines what is somehow a finite mass of course you can imagine that then you can define the mass so the mass is just going to be the following so the mass of t at this time the mass is not going to be a number as we said before so maybe let us put just quotation marks over here so because the notation is a little the denomination is a little inconsistent with what we said before so this time actually the mass is not going to be a number but it's going to be a measure so the mass of t is the minimal measure this inequality here holds actually it's not so clear that such an object exists in fact this is slightly more subtle I mean it's an interesting if you want is a fairly interesting exercise to show that actually such a measure exists so from now on let's say this measure is denoted by this symbol so this is characterized by the following two properties well first of all the inequality E holds if you substitute this measure to mu and then second if E holds for some mu then in the sense of measure this mu is larger than the mass of t so that's the sense in which somehow this measure mu is the minimal mass which is satisfying that you observe somehow that the operation of push forward and restriction behave in a natural way with respect to the mass so first of all if I take the mass of the push forward of some metric functional t then I have the following inequality and if I take the restriction then I have this other inequality so these are both very simple exercises and another important point is that so when a current has finite mass so you can define the action of your metric functional even when the first coefficient f is simply a Borel bounded function and the reason is the following so of course what you can do in that case just take a sequence sorry a metric function so then you can define actually the action even when f is a Borel bounded function in this case actually I mean it's a straightforward exercise I mean there exists a sequence fI of Lipschitz functions which is converging to f say mu almost everywhere and also with soup of modules of fe which is less or equal than soup of f ok then you can show easily that exists the limit of the action on this t over fi pi 1 pi k I mean it's important that actually the fI I mean the f is Borel but the pi 1 pi k remain Lipschitz ok so there exists this limit as I goes to infinity and it is independent of the approximation check it as a simple exercise in measure theory very good so maybe which is worth remarking and this is also an interesting exercise an interesting exercise so when you actually want to compute the mass of an open set so the mass of E is actually the least constant such that the following inequality holds ok and consistent with actually the notation that we had in the previous lecture for us maybe before called the mass of t which was actually a number is just the total value that this measure has ok so far so good I mean we have not yet actually introduced what a current is maybe one last thing that I have to tell you which might be useful in the future is that another interesting definition is for a metric functional of finite mass is the support so the support of t is nothing but what is usually the support of the measure mu right the support of the measure mu is just the sets of all I mean the closed set of all points x in E such that of this measure such that the measure completed on the bowl is positive for every radius you choose x ok so as it was actually already asked before so where is the multilinearity structure that you usually see in forms so this of course has to hold for every f and every pi oh that has to be positive sorry yes so here the condition of course is I mean the points in the support are the points such that anytime you center a ball on the point you actually have that the measure as you would define it classically so we are now ready finally to give you a definition of the current now a metric functional t is a current if so t has finite mass t is multilinear t is continuous with respect to the pi entries in the following sense so when I compute well it's continuous on the first entry that we sort of saw from the fact that there is this finite mass condition but we didn't have we didn't give any continuity on the other entries and the continuities are given through this axiom so if I have a sequence I mean if I have sequences pi i1 pi ik converging to pi 1 pi k then the action of the current passes to the limit and for this I have to tell you that in the following sense when a pi ij converge to pi j point wise there is a uniform bound on the liquid constants so then the fourth axiom is actually that t of f pi 1 pi k is equal to 0 i 1k the function pi i is constant on a neighborhood of the points where f are different than 0 so now we actually come to so who asked the question I think over there somebody yes you asked the question so you see in here somehow we're thinking about I mean if we go back to the definition of feather clamming this would be somehow acting on a differential form so if this f pi 1 pi k would correspond to a differential form on our end so if omega equal f d pi 1 wedge which d pi k is a differential form is a true differential form in the classical sense then this assumption over here just tells you the d pi i vanishes where f does not vanish which actually means that that would be equal to 0 and so this condition here is consistent with what we know usually for currents of course what we also know for currents is that they would be alternating on the various other entries but that's one of the interesting things of the theory actually to assume this alternating condition it actually is given to you by the other four axioms let me give you yet another definition although maybe first I would have to do a remark so t is a normal current if both t, the boundary of t are currents okay so this is going to be one of the quarter stones of the Ambrosio-Kirchheim theory so the extension of a current t to Borel entries or to Borel functions in the first entry satisfies the following properties so first of all you have the product and chain rules this tells you first of all t is multilinear in all entries and as you would expect first of all if you compute t of f dpi1 wedge wedge dpi k plus t pi1 df wedge hdpi k this is equal to t of 1d of fpi1 wedge dpi k so this is the product rule you can actually have the chain rule so the chain rule is actually telling you if you compute t of f and then you compose classical functions with some pi so here pi is a function from e into rk which is Lipschitz and then psi is a function from rk to rk which is Lipschitz actually this over here is equal to t times f here you would have the determinant of dpsi and then dpi1 wedge wedge dpi k so it's important actually to know that since you are dealing with Lipschitz functions psi is differentiable almost everywhere actually it's differential is also a Borel map so here you have to compose with pi which I forgot so pi is a Lipschitz map dpsi is actually a Borel map so this determinant of dpsi is also Borel map and so this object is well defined okay and then finally we have also the locality property no first we have the continuity sorry so if fi minus f is converging to zero strongly in the L1 space given by the mass of the current and ij is converging to pi i to pi j point wise with the uniform bound on the Lipschitz constants then you can pass into the limit on the actions of currents and then finally you have the locality property so the locality property is actually telling you that t of f pi 1 pi k okay so t of f pi 1 pi k is actually equal to zero if different from zero so the set where f does not vanish is contained in the union of some sets bi vanishes on bi okay so this actually is a consequence of the axioms we are not going to prove this theorem although the proof that you find in the paper is not that complicated you will maybe notice one interesting fact this chain rule over here this chain rule over here gives you the multilinearity so as far as I told you a consequence of the axioms of continuity locality and so on is actually that you have the multilinearity as a bonus so the reason why that actually implies the multilinearity you see it immediately so if you apply this identity to a map psi which goes from rk to rk and just permutes the components of the vector you will then discover that you just got alternating property so apply the chain rule maps psi which just rearrange the components of vectors in rk then you get that t is alternating so you may wonder why actually this happens so the reason why this happens is essentially because for the continuity I mean for the continuity axioms that we have given and for the locality axioms essentially the determinant is the only thing which makes this happen anyway another question which is maybe interesting is so how does this theory compare with the feather and flaming theory it was actually shown by Ambrosio and Kirchha in that a current in their sense corresponds to normal currents in the sense of feather and flaming if you are on the Euclidean space and it's also easy to see that a current of finite mass in the sense of feather and flaming is a current in the sense of Ambrosio and Kirchha but it was open whether a current in the sense of Ambrosio Kirchha is always a current of finite mass rather than feather and flaming so possibly the currents of finite mass in the sense of Ambrosio and Kirchha are larger but actually I think in the third week Marianna Axiornier will talk about this so it was known in some situations because it's linked to some deep result of price in real analysis and I think she has been able to extend this in general just in the last years so this should actually be now so the equivalence follows and some special cases of price and I think in general my recent work of Jones and Sornier OK so that's all for today, I'm sorry for going quite a bit extra time