 OK, I guess we can start. So first of all, let me add a couple of things to the lecture of yesterday. So one first thing that I forgot to introduce are zero dimensional currents. OK, so in the case of zero dimension actually, and this is going to be important although. So it's way much easier than k dimensional currents for k bigger or equal than one because essentially we, everyone saw them at least but on the other hand, you're going to play a very important role later on. OK, so of course I have to define you in some sense what are zero forms. So d zero of e is just going to be the space of f from your metric space into r which are lip sheets and bounded. OK, then a zero dimensional metric functional is a linear d into r. And of course it has finite mass if there exists a constant c such that anytime you evaluate t of f on a function, you get actually the inequality that this is less or equal than a constant times the supremum of modulus of f over e. And of course now from this definition you just recognize that you're looking at just the dual of continuous function so there's a well known theorem by Reitz that this is actually given by Radon measures. Under suitable assumptions on the metric space e, which of course we were making even yesterday somehow, suitable assumptions this just means is, I mean, has finite mass if and only if t is a Radon measure. And of course you will also recognize then our definition of the mass of t is then just the total variation measure. The action on a function f can actually be given by the integral. So here you will have the total variation measure. Then you have f. And then let us put here some, OK, maybe something like nu. Nu is a Borel measurable function which is taking, which takes values plus and minus one, OK? So and of course you can just think that this is all together just a sine measure which we might denote by t in this case, OK? So that's going to play actually a very important role later when we will study general currents by the procedure which is called slicing which is essentially trying to make induction arguments on the dimension of the currents when you're actually trying to prove something. OK, so that was one thing. And the other thing is that I actually remarked that I gave you at the very end about currents with finite mass and the equivalence with the Federer and Fleming theory. So let me actually just be more precise about that. So I made a mistake yesterday because I forgot an hypothesis. So the conjecture of Ambrosio and Kirchheim is that their currents of finite mass are equivalent to the currents of finite mass which are flat in the Federer-Fleming theory. So I forgot to say this word over here which maybe for some of you does not make that much sense, but at least I want to state it correctly, OK? And one direction is actually quite clear, and it's in fact this direction. So this one is simple. And that one was what we were talking about yesterday. So this is known in some dimensional case thanks to the work of Price, essentially, but more in general is still open, maybe solved by some reasoned results by Sjoernier and Jones. So maybe soon solved. OK, so very good. Now let me start the lecture today by giving you now some examples of currents, right? So let us go back to the zero-dimensional case. Sorry, to the Euclidean space. So let us now assume that E is equal to Rn, OK? So we just observed zero-dimensional currents of finite mass are equal to measures. So that's an easy remark. N plus one-dimensional currents are necessarily equal to zero. So I mean, if you want, they are the empty set, or the linear functional identically equal to zero actually does make, I think, for our definition an N plus one-dimensional current. And the reason for this is because you remember they are alternating. So it's just a simple exercise on the fact that there is no alternating N plus one linear form on Rn, OK? So the next thing which might be interesting to understand is what would you expect from N dimensional currents, OK? So you remember that we have this chain rule property that we didn't prove actually yesterday, but we stated. So if you remember the chain rule property, that kind of suggests you one way of constructing N-dimensional currents. So the chain rule told us, if I am trying to compute T of f d psi one of pi wedge d psi N of pi, I actually get T of f times the determinant of wedge psi composed pi d pi. And this is true for T, N-dimensional current, and psi, a map which goes from Rn into Rn, right? So now if you are in the Euclidean space, you actually notice therefore that you can reduce everything to compute something like T of f d psi one of x wedge d psi N of x, where now I just denote by x the identity matrix. This is actually going to be just T of f determinant of grad psi d x one wedge d x N, OK? On the other hand now if you actually fix, somehow if you fix this, right? And you look at the map which is giving from G, gives you T of G d x one wedge d x N. So if you sort of keep this part of the form constant and you actually only override this part of the form, then you realize that your definition of current of finite mass is giving you a linear functional on the space of functions. And you just realize that for the linear function on the space of functions that you're looking at, right? You actually have exactly the assumption of the Radon-Nikodym, I mean of the Ritz representation theorem. So you remember then you would have for the definition of current of finite mass of G d x one wedge d x N. This thing over here is less or equal than a constant. Then you would have the Lipschitz constants of x one times x N. And these are all equal to one. And then you would have the supremum of modulus of G over E, right? So if you just forget about the dependence on these guys, you just see that you have the usual thing which allow you to apply the Ritz representation theorem. So you immediately actually conclude that T of G d x one wedge d x N must be something like the integral of G d mu for some measure mu. And then you might try to apply the chain rule that you have over here, right? And therefore this simply suggests that when you evaluated T of G d psi one wedge d psi N, actually you would like to represent this guy as the integral of G determinant of grad psi d mu. OK, so this is what you would like to write, but now you run somewhat into troubles because the determinant of the gradient of psi is not everywhere defined. So this wouldn't make actually up to a proof that your current has to be represented by this thing, because you know that psi is just a Lipschitz map, so it's differentiable almost everywhere. So if this measure were anyway a Lebesgue, I mean something times the Lebesgue measure, so if this measure were absolutely continuous, then this would make sense and would be defined almost everywhere, OK? So now you also see what is the connection maybe for the few people among you that know this work of Price and Sonier and Jones. So one of the questions which are around are the questions for, I mean it's the following question, for which measure mu can I define sort of this object in such a way that it has nice properties, and the conjecture is that the only measure for which it makes sense to define this integral is the Lebesgue measure, OK? And this is what it has been proved by Price, say in some situations in the two-dimensional situation, in one dimension it's easy, in two dimensions it's fairly difficult and it has been proved by Price, and that is what needs to be generalized to go back to this conjecture of Ambrosian key chain. OK, so we are not going to bother about whether this fact is true or not. We just take it somehow as an inspiration to define now an n-dimensional current in the following way. We just say if you give me a function g, which is in L1 over n, I define the current in the following way, so I have to define the action of g on something like f d pi 1 wedge d pi n, when all of these are Lipschitz functions, and this is by definition going to be the integral of g times the determinant of d pi times f in the Lebesgue measure. That's one definition, and now you might maybe wonder what happens if I actually ask that the current is normal, so when is g a normal current? Sorry? You don't know how to define this, right? I mean, this is defined point-wise almost everywhere. Of course, let us make for instance the following thing. You could say the following, so one way around this may be you regularize your function f, you get its c1, and then you define this linear functional for an arbitrary Borel measure. Then, of course, you would like to pass into the limit in the approximation. Is this thing well-defined, for instance? If this thing is converging to something, it gives you a well-defined linear functional. So the kind of question that I would like to answer is when I define this kind of functional with an arbitrary Borel measure on the space of c1 functions, when can I extend it to Lipschitz functions in such a way that I get something continuous? And this actually is what price proved in two-dimension. This has a continuous extension, if you want, to Lipschitz maps if and only if the measure is absolutely continuous. Remember, out of our definitions, we had to satisfy some continuity hypothesis. Of course, you might wonder why, for instance, if I define my current in this way, why when I have a sequence of Lipschitz maps which is converging to pi, I can actually prove that these things behave in a continuous way. OK, so that is actually a nice problem. It has to do with what is called weak continuity of determinants. So the weak continuity of determinants in the space of Lipschitz maps, which, by the way, is w1 infinity, actually implies that this g over here satisfies the continuity property of the theorem yesterday. OK, so now you can actually ask, when is g a normal current? And, well, let us just imagine how you want to compute a normal current. So remember what is the boundary of g? So the boundary of g, for us, was defined as, I mean, acting on some form, say, for instance, f d pi 2 wedge d pi n. So the current g acting on f d pi 2 wedge d pi n is nothing but the current g acting on the form for the f wedge d pi 2 wedge d pi n. And now when you go back to definition, this is just going to give you the integral of g times the determinant of the map by pi 2 n. And now you see what happens. So if, for instance, I use pi 2, pi n equal to simply x, x2, xn, then this determinant is very easy to compute. This determinant is just df dx1. And then what you have just discovered is that the boundary of g acting on that particular form is just the integral of g df dx1. OK, on the other hand, this was an arbitrary choice. I mean, it's just that pi 2 is equal to x2. So say, for instance, that you put pi 2 equal to x1 and then pi 3 equal to x3. Then here you would have another partial derivative. OK, so say. Fine, for instance, do this. So now the current is normal if and only if you have a bound of this in terms of the leap sheets constants of these things. So dg of finite mass, this is true if and only if you actually have the dg of f d pi 2 wedge d pi n is less or equal than a constant times the product of the leap sheets constant of the pi i's. And this is going from 2 to n times the supremum of modulus of f. And now you apply it to this particular situation. Then the x2, xn, the other leap sheet functions with leap sheets constant equal to 1. So what you actually discover is that your current T or your current g is a normal current if and only if the integral of g df dx i is less or equal than a constant times the supremum of modulus of f. OK, so but now here you recognize something that you should be familiar with. So what is this? Right, so g is actually just an L1 function. So this is just the action of the distributional derivative of g on the function f. And of course this inequality over here, again by Ritz representation theorem, is just telling you the distributional derivative of g should actually be a radical measure. So this means just that g must be a bv function. OK, so this characterizes you normal currents and actually if you look at the feather and framing theory is exactly the same thing, it's actually easy to see that you can go through the same proof. So this is for instance what is, I mean, this is the main idea which is telling you it's not that difficult to prove that in the Ambrosio-Kirchheim theory the bv functions are actually, I mean, the normal currents are equal to the bv functions and therefore they are equal to the feather and flaming normal currents. OK, so this is the idea behind the proof of Ambrosio and Kirchheim of the equivalence between the two formulations when you actually have normal currents. Of course this is just a particular case. You would then have to use push forwards and things like that to sort of use this kind of idea which proves you the identity for n-dimensional currents in Rn to prove the same identity for n-dimensional currents in a Euclidean space with a larger dimension. OK, so but this is the idea behind it. Of course then somehow the idea is that in some sense when you are normal you have some kind of a priori estimate on the derivative of this g, right? So it's like for instance if it were represented by a measure mu you would ask what are actually the measures whose distribution of derivatives are also measures and it's very easy by an approximation procedure to see that this is equivalent to be a function which is bv, right? So that shows you why this situation is easier than the general one, OK? So now let go also one step further. So usually, right, you would like, I mean, OK, so usually you would like to say to establish some kind of identity between certain type of currents and usual submanifolds, OK? So what are submanifolds? So what are n-dimensional submanifolds over n? So n-dimensional submanifolds over n are just open sets over n, right? So if I take an open set omega and I take as g the function, the characteristic function of omega which is equal to 1 if x is in omega and 0 otherwise, right? Then this guy over here gives me, sometimes we will denote it in this way just to be faster, right? Gives me the current which is naturally associated to your submanifold, OK? So now, integer, so what we said is somehow yesterday is that we like to take integral combinations of this but we don't like to take real combinations of this, right? So it sounds that for natural just to say that a current is integrable rectifiable of dimension n on our n if this current t is representable it gives some integration with respect to some function g where g is not only l1 but it also takes only integer values. Now what is nice about currents is that I can push it forward, right? So I can take an n-dimensional current and I can take as a map into some larger dimensional space and I can push it forward, OK? Now if you remember the classical if you remember the classical Stokes theorem actually this gives you locally any type of integration on a submanifold, right? So how did you define the integration of a submanifold of our n? You just take a local chart, right? And then you just pull back the forms from the manifold into the local chart, OK? So that is exactly equivalent in our language of taking some patch of our n which is going to be the integration over some omega and then look at the map which is putting this Rn into your Euclidean space in a nice way and look at the push forward of the corresponding current, OK? So it doesn't surprise you, therefore, that we actually give a definition of, so, a general n-dimensional integral rectifiable current in some metric space e is a current of finite mass for which, and now you're going to see the magic three countably many objects which we introduced yesterday. So there exists countably many. Well, actually there are going to be only two objects. So gi in L1 over n, psi i, the Lipschitz maps from Rn into E, such that you can formally write, OK, so gi with gi taking values in Z, such that t acting on any form omega is given by this formal series acting on omega. Now, of course, you might wonder whether this thing over here converges at all. Well, it turns out that it's no, so it's not, I mean, it's a requirement which makes the series converging, but it's a requirement which essentially is harmless. So maybe this requirement should go before the identity in such a way that you're sure that things converge. So you just require that the sum of the Lipschitz constant of Ci to the power n times the L1 norms of the functions gi, they're less than plus infinity. So here, of course, you remember yesterday in the Federal Enflaming Theory, we had three objects, right? So we had the integration over submanifolds and then we had integer coefficients. You see the integer coefficients are just hidden over here, right? So, of course, you just could say that if we integrate each gi as a series of integer coefficients functions times the identity, times the characteristic functions of some sets omega ij, right? Where omega ij, now, they're going to be just general Borel sets. So yesterday we actually introduced, just say, omega ij have to be closed instead of being just Borel, but it is a simple exercise in measure theory. Of course, just to redefining the omega ij correctly, you actually can actually get omega closed. The only other difference with respect to yesterday is that we insisted that these maps, psi i are c1 and not Lipschitz. Actually, it turns out that it's an effect of a well-known theorem in real analysis, which is called Whitney's Extension Theorem, that if e is the Euclidean space itself, then you can approximate Lipschitz functions efficiently with c1 functions, and therefore the definition is really equivalent to what I gave you yesterday. So yesterday we had the main difference being that the psi i were required to be c1. But, of course, then if psi i is required to be c1, then the metric space has to be Rn, and e equal Rn, the two things, are equivalent thanks to Whitney's Extension Theorem. Very good. So now let us come to the important points. So, of course, now you can introduce a notion of convergence for currents, right? And it's a weak notion of convergence. Well, it's what you do usually, right? So you have a certain space of test functions, and your objects are defined by duality. The weak convergence is essentially equal to the point-wise convergence when you fix the text function, okay? So the sequence Tk of currents is converging weakly to T, if on each test we have Tk of omega converges to T of omega, okay? Now you will notice one thing which is also elementary in usual functional analysis, since the mass of T is defined by duality supramising over a certain set of functions, it's not surprising that the mass is actually a lower semi-continuous quantity along weak convergence, okay? So this is also a very simple exercise. So if Tk converges to T weakly, then the mass of Tk, the limit of the mass of Tk is bigger or equal than the mass of T. And the proof is extremely simple. I'll just catch it. So remember the mass of T is the supremum over Lipschitz constant of pi i's less or equal than one and modulus of f less or equal than one of T evaluated on f d pi 1 wedge d pi n, right? Now of course, if instead of being a supremum it would be a maximum, right? If this would be attained, right? If mT is equal to T of some such f d pi 1 wedge d pi n, then of course you just would test this T on the sequence. This would be the limit as k goes to infinity of Tk tested on the same thing, but then by definition this would be less or equal than the mass of Tk. Now of course this is a supremum, so it's not achieved, but for every epsilon you fight something such that this one is m of T minus epsilon. And by this of course you have done that the limit is bigger or equal than m of T minus epsilon and then you let epsilon go to zero. Very easy. Of course now you are in a very good shape. You just need some compactness to solve your plateau problem, right? So say that you consider a certain S, a certain current S without boundary, assume that exists T with finite mass such that the boundary of T is equal to S. Then take a sequence Tk such that the mass of Tk is converging to the infimum overall possible of R with boundary equal to S of the mass of R. Now what you would like to know is can I extract a sequence or a subsequence which I still label as Tk which is converging to some T bar well being a boundary is easily seen to be preserved in the limit just because being a boundary means I have to test on the correct forms so in the form you're also a boundary if I can get the weak limit then I get the solution of the plateau problem. I just get the minimum by lower semi-continuity. So the real question is this one. Can I extract a subsequence? So now let us go back to what I told you yesterday if you're looking for normal currents so if you allow actually real coefficients instead of integral coefficients it's a simple functional analysis exercise. Well maybe it's not that simple but somehow it's really functional analysis and soft arguments essentially. So the space of normal currents you could say it's a Banach space and this weak convergence which is in duality with some other Banach space and this weak convergence is really just the weak topology that you put when you're looking at duo. The weak stuff topology maybe. I always get confused with this. And then there is a general theorem which tells you that the bounded sets I mean bounded closed sets are compact. So for normal currents you have the following theorem. So if Tk is a sequence of currents such that the mass of Tk plus the mass of the boundary of Tk is uniformly bounded then there exists a subsequence which is converging weakly. So now how would you apply that to that situation? Well this situation is extremely lucky because the mass of Tk is converging to the infimum so the mass of Tk is not exploding. The mass of the boundary is actually just the mass of S which is fixed. So it's a constant. Then you apply the theorem you get your minimizing sequence which is converging and you apply the direct methods of the calculus of variation. So now what disturbs you of this theorem is that as I was telling you before for instance an integractifiable current as I've defined but I take half of it that half of it is admissible it's in this class. So you actually don't like this. So what you would like to do now is to actually take a sequence which is converging to the infimum but the sequence is required always to be a sequence of integractifiable currents. So if I add here integractifiable of course if I have integractifiable dungeon then I just have a subclass. So I have a subclass to which I can apply this theorem. So the Tk is converging somewhere but unfortunately the somewhere where you are converging might have real coefficients in principle. So now the question that you would like to answer is assume I have a sequence of integractifiable currents and I actually know this thing over here. Is the limit going to be integractifiable or not? So question. If the sequence in the theorem is integractifiable is T also? So this is actually what funnily enough in the literature is called Federer and Fleming compactness theorem so the answer is yes the limit is also integractifiable under this assumption but more than a compactness theorem the closure theorem because the compactness comes out of the functional analysis. So this is the question that we want to ask that we want to answer in the next hour so let us make a consistency check first. So let us apply it to the easiest situation we know so let us apply it to n-dimensional currents in Rn. So a sequence of n-dimensional currents in Rn is a sequence of BV functions for which I have a uniform bound for the BV norm and which take integer values. So is the limit also going to take integer values almost everywhere? Yes? Because actually by Rellich BV embeds compactly locally in L1. So if I look at the functions the fact that I have a uniform BV bound it tells me actually the functions are converging L1 to a function in the limit to take integer values as well. So this gives you another simple exercise for n-dimensional currents in Rn the theorem is pretty easy. So for n-dimensional currents in Rn the closure theorem is identically equal to the BV compacted bending of BV of functions of bounded variations in L1. It looks very optimistic apart from the fact that if you look at the theorem of the compacted bending of BV in L1 it's way, way, way, way much easier than the closure theorem for integral currents. So as I told you this is one of the corner stones of the Federer and Fleming theory. But then I told you there are two other very important theorems inside. So they are answering two very interesting questions. So I told you about the boundary integrality theorem or rectifiability theorem. So the boundary rectifiability theorem tells you the following fact. If you have an integral, I mean if you have an integractifiable current which is normal its boundary is also integractifiable. And actually such nice objects which have boundary, I mean which has rectifiability of both, I mean the boundary and the current itself are called integral currents which currents are called integral. Then as I told you the third cornerstone of the theory of Federer and Fleming is the so-called deformation lemma that is the possibility of approximating currents with I mean under suitable assumptions with nicer objects. Now that of course I cannot tell you that it's going to hold metric space. Because if the metric space is very nasty actually the chance that you can approximate any I mean for instance say if your metric space allows only essentially Lipschitz maps without any kind of notion of what is C1 then you just see from the start that you don't have a nice way of approximating Lipschitz functions. So you cannot expect to have a nice way of approximating currents in general. Although of course what you can imagine a lack here when you have a metric space which does not allow too much nice things maybe the the rectifiable currents are simply not there maybe it's just void. I mean maybe they are just trivial. Anyway the deformation lemma gives you the possibility of approximating currents in the euclidean space so approximating integral rectifiable and more general integral currents nicely with smoother objects let me just tell one effect which will be very important later on next week so a corollary of the deformation lemma is the following iso-parimetric inequality so there is a constant C which depends only on M and N such that for any closed so for any current T with the boundary of T equal to 0 that exists a current S of course this is a current of dimension M with the boundary equal to 0 that exists a current S of dimension M plus 1 such that the boundary of S is equal to T sorry the boundary of T is equal to S and the mass of T is less or equal than a constant times the mass of S to the power M plus 1 divided by M and in addition if the current S is integractifiable actually you can choose the current T to be integractifiable as well right yes sorry I am getting confused on the notation yes okay so let's draw a picture and then say T here I think now it's correct okay so if T is integractifiable then there exists S as above which is also integractifiable right and this kind of closed the discussion on the plateau problem now say you give me a smooth manifold which is closed in Rn I apply the deformation theorem and I find a first integractifiable current which bounds it and which has a bound on the mass so I know that the set on which I want to actually apply the direct methods of the calculus variation is non-empty and then I apply all this machinery and I get the minimizer which is an area minimizing current with that given boundary okay so now let's make a ten minutes break and then of course I'm not going to give you a proof of the closure theorem but I'm going to give you some ideas on how it can be proved in an efficient way using modern techniques which have been introduced in the last ten years so let me just say the I mean the first proof by Federer and Fleming of this theorem actually relied on a very hard study of rectifiable sets in particular it relied on what was called on what is called the Bezikovich Federer the ability criterion which is a pretty hard theorem in functional analysis in real analysis okay so there's one thing which I am forgetting here maybe in the iso-perimetric inequality I didn't tell you what is this n of dimension less than n in Rn okay so we were talking about Euclidean currents and actually if you want I mean the best proof up to now of this iso-perimetric inequality gives you that the constant does not depend on n in fact it depends only on m and the worst possible thing for this constant is the boundary of an m plus 1 dimensional disc so it's also a computable constant and it's a word by Andren okay but the original proof of Federer and Fleming just keeps your constant also on the dimension of the ambient space sorry the current T has be an integral current no no no if it is integractifiable you get an integractifiable current if you multiply by lambda right no okay sure you're right you have to get rid of the you have to get rid of the homogeneity yes okay I think you need to be bound from below or you put the sides yeah if you have real coefficients the minimizer is exactly lambda times yes you are correct no you have to put integractifiable I mean even on bv is false in this way you have to put integractifiable you are correct yes okay there's something that you can do for normal currents as well but now I don't remember exactly how you formulated but he is correct that you need integractifiable so in the top dimensional case when you have a bv function you can of course write down what is the Poincaré inequality but the Poincaré inequality wouldn't have the gain in the exponent okay yeah thanks good so let us now go to the compactness well to the compactness theorem okay so our first case the extremely easy is prove it for zero dimensional currents okay and it's of course very easy because you can just well of course for zero dimensional currents in principle I have not defined to you what is an integractifiable current well the definition of zero dimensional integractifiable currents these are simply a combination of Dirac deltas with the integral coefficients the boundary of a zero dimensional current is zero by definition so in this case you just have the following property the metric space E is locally compact then what you reduce to prove is simply that you have a sequence tk of some Dirac mass you don't know how many and of course if you have integral coefficients you can just decide you can just you can just decide that you can just decide that the sign over here is either plus or minus and then of course without loss of generality you can assume extracted sub sequences that the qk is equal to a certain constant q the xik are converging for k to infinity to some xi and that the yik are converging to some yi okay and this would actually give you that the tk is converging weekly to the sum of the dx i minus the sum of the dy i of course in general it's locally compact or if you're a general metric space if you have a sequence tk with the mass of tk which is uniformly bounded which consists of zero-dimensional currents and moreover the support of tk is contained in a compact subset of E then there exists a sequence such that the tk is converging weekly to t and t is as well integractifiable so zero-dimensional integractifiable currents is integractifiable as well of course without this assumption you see immediately that there's a problem the tk I mean some delta for instance in Rn might escape to plus infinity and since you're not okay you're not testing with compactly supported functions if you want and then you can still talk about some weak notion of convergence but in general in a metric space E in which we are not making assumptions like being compactly supported if something is escaping at plus infinity then you don't have a chance to capture it so in fact in the theorem about the compactness of normal currents which I told you there is one assumption missing that the supports are not escaping to plus infinity so maybe I should better say something like that so let me actually get the correct assumption which I forgot so in addition to the hypothesis that you have a uniform control on the mass of the current and of the boundary you actually have to put also the following hypothesis so for every say in Euclidean zero there exists a compact set such that mass of what is outside is fairly small so for instance on the Euclidean space this would tell you the amount of mass which is escaping to infinity is essentially zero anyway for the problem that we are looking at for instance for the plateau problem in Rn one first thing that you could remark is that if I give you you can actually take the convex hull of this boundary and without loss of generality you can assume that your minimizing sequence belongs to this convex hull so if it's not in the convex hull of course you can just re-project the entire Rn with the Lipschitz method on the convex hull and this projection is actually decreasing the mass so in all the cases of interest you just get this extra hypothesis by some additional remark very good so let us therefore simply from now on assume that this is not an issue so let us assume that E is compact and see one thing that we would like to understand is how to use this simple observation that zero dimensional currents are compact to conclude the same thing in higher dimension so for instance how am I going to conclude this with two curves which would be one dimensional objects so from now on say E is compact and one thing that I would like to understand is how to use zero dimensional the zero dimensional statement to infer the same compactness argument in higher dimension so this is answered by the so-called slicing technique which was introduced already by Federer and Fleming which has been recast in the metric space by Ambrosio and Kirchheim so of course the idea is kind of simple, the idea would be that if you have a curve for instance and if I take a curve and I intersect it with a number of parallel lines the intersection between these lines and the curve is typically a zero dimensional set and if the curve is actually smooth the Sartz theorem is telling you the points where you don't intersect the points down in which the intersection is actually not a finite number of points so something where you would have for instance a touching over here or where you would have infinitely many crossings this is actually a zero dimensional set down there so the basic idea of the slicing theory is how to embed so how to make rigorous this idea of chopping a current into smaller dimensional section so one first observation that I want to give you is the following so if I have a sub-manifold without boundary right? and I chop it say with hyperplane the section of this is nothing but the boundary the restriction of your sub-manifold to the half space observe if sigma is a sub-manifold without boundary and more in general you don't need actually to take a hyperplane pi from rn to r c1 function, smooth function t, a regular value for pi if you have this identity the intersection of pi to the minus one of t with your surface sigma is nothing but the boundary sigma intersected to the open set pi less than t so let me draw a picture over here so say this is the on this side you have pi less than t and then here for instance you have a curve right? and the intersection of this boundary with the curve which is going to be this point and you recognize that this point is the boundary of this manifold over here of course this is not true for every t this is true for almost every t right? and that is actually so it looks very tempting to just say that the slicing of your current by for instance parallel planes simply is equivalent to take the boundary of the current restricted to the upper half, I mean to the lower half space for instance of course this has to be corrected so assume that your current t assumed that your current t instead had a boundary so assume that your current t had also a boundary over here some boundary points of course then if I take the boundary of the intersection of your sub manifold with the lower I mean with this portion of the blackboard I actually get two points in the boundary one point is the boundary I like and the other point is the boundary I don't like but what is this? this is just the boundary of the manifold restricted to this set right? of course if there is a boundary over here then I'm not counting it by that operation okay? so this suggests the following slicing procedure this is a tentative definition of what the slice should be so if say pi is a map from E r and t is an m-dimensional normal current the slice of t through pi at t is the current defined by the following formula so it's the boundary of the current t restricted to pi I think it's less than t because the boundary of t restricted to the set pi be your identity so remember we had the restriction operator defined for forms or functions so what I'm implying over here is that I'm restricting on the indicator function of this so when I actually write this t restricted to some set omega what I mean is what we actually defined with the following notation restricted to the zero-dimensional form which is the indicator function of the open set omega so this would simply mean if I want to evaluate t restricted to omega on little omega I'm actually going to do this now one of the things that you would like to imagine is that although I have my slicing function defined in the following way so there is a fubini type argument that I can use to define my slicing function so look for instance at the following situation so assume your curve is given by the graph of something so if the curve is defined by the graph of that of course it's pretty easy okay so now consider pi to be the projection on the first variable which is x1 and the map t pi x1 is nothing but the Dirac mass which is sitting at the point x1 f of x1 okay so now the question that you would like to ask is maybe the following I would like to reconstruct somehow what is my current over here this line by possibly integrating over the sections the intersections and of course one thing that I could for instance do is well I can test the deltas against the test function phi and then for every x1 I get a number so for instance assume that I'm integrating this over what actually happens and this is a simple remark in this case so assume you're actually making the following thing so assume you integrate d pi x1 so this you can actually test on a function g and then multiply by a test function psi x1 and then integrate by dx1 so test in this situation what you get okay so in this situation you just get the integral of psi and then you have g of x1 f of x1 write dx1 and you actually check right away this is nothing but the current I mean the integral over gamma of the form psi of x1 g of x1 sorry g of x1 x2 dx1 so and this is nothing but the action of the current gamma on the form psi of x1 dx1 g which actually is nothing but the restriction of gamma psi composed by dpi okay so and this actually we have done it in I mean for one-dimensional curve but it's not difficult to see that you can actually generalize it to n-dimensional graphs okay so one very nice formula that therefore you can imagine is what you get when you're slicing the current with your function pi is the following so if you want this is a proposition given the way I have defined you the current the slice of the current if t is a normal current then you actually have the following identity that in some sense the integral of t pi x acting on some form omega and multiplied by some function psi of x dx I mean to t restricted to psi composed by dpi computed on omega okay very good now you can take of course recursively slicing so say if I give you a two-dimensional surface right you can first slice in one coordinate to get one-dimensional curves and then you can slice it in the second coordinate to get actually Dirac mass so you can either take this definition which I gave you which is operative and sort of imply inductively over more and more projections so which would give you the slice with respect to a map pi which instead of being r-valued is going to be rk-valued or of course the other thing that you could try to prove is the existence of a map which simply makes this identity true but this times for a pi which is an rk-valued map okay and remember here dpi of course I'm meaning dpi1 wedge dpi k okay so both things are possible so let me summarize then here what is the main slicing theorem that you can prove in the theory I guess I must have messed up when I was preparing the lectures yes yes here it is okay so let t be a normal dimensional current pi from e into rm a leap sheets map with m less or equal than k okay then there exists a map x rk into dpi x normal m-k dimensional curve k-m dimensional current weakly measurable so it simply means that when I take this dpi x I test it against the form omega then I get a function which is measurable as a function of x the following properties so first of all dpi x they are boundary and its boundary are supported the counter image then d identity 1 holds then second d integral over rm of the mass of t pi x dx is equal and so this is an integration of measures so what I mean is that this is going to give you a measure if you want to understand how this measures x on a function of f you integrate with respect to this measure the function f and then you integrate with respect to dx and this is equal to the mass of the restriction of t on dpi if pi is a one-dimensional t pi x is given by the boundary of t restricted on pi bigger than x minus t restricted on minus the boundary of t restricted on pi and then finally fourth d integral of the mass of the boundary of t pi x is less or equal than the Lipschitz constant of pi times the mass of t plus the mass of the boundary so this is the slicing you have and somehow you can now make the following type of I mean you can think about carbons also in the following way you see one thing that I can do is the following I can take oh maybe I should actually point out something more over here there is the fifth important point if your carbon t is integractifiable also the slice is close to every x and maybe a little comment about this fifth property as you see integractifiable carbons have been defined in such a way that you're just pushing forward via Lipschitz maps okay so in a sense you can imagine that the only thing you have to care about is trying to prove this statement over here right the fact that your slice is integractifiable in the Euclidean setting and for top dimensional carbons and then you will simply see that essentially what you have stated over here is justico area formula so that's another way of actually slicing submanifolds or rectifiable objects in Rn in general but more than spending time on this I want to actually spend some time on the effect so now this point number five is simply telling you if you actually choose equal to k then your slice is going to be exactly a family of Dirac masses which is varying with x okay of course one thing that you can ask yourself is what happens if instead of giving you the current I just give you the slices with respect to some particular direction in general of course you don't expect to reconstruct the current by this information because for instance if I have a vertical line and I'm slicing with vertical directions so I will not see this line and there's no wonder that I will not see this line because the identity is just telling you the identity which I cancelled is telling you that what you're looking by slicing is just the restriction of t to dpi on the other hand say that for instance you are on Rn so if you are on Rn you can take any form and write it as a linear combination of the forms which are of the type and these are finitely many and now you know that when you're actually evaluating t on any of these dxi you can reconstruct what this is by integrating the slices so if I give you a curve essentially this is telling you that if I give you the slice in the horizontal and in the vertical direction you should be completely able to reconstruct what the current is initially so that introduces a point of view which is very powerful you can actually think about currents as a bunch of Dirac masses which are varying with your parameter x on the other hand there is something more to that if I give you a generic map which to every point x is assigning Dirac masses it's not clear at all that this gives you a current which is nice so for instance think about having this Dirac masses think about having a Dirac mass which is depending on x1 and it's going to be a Dirac mass on the point x1 f of x1 but the function f is a completely crazy function which is varying a lot of course you will have something which is a current but you will not have something which looks like a nice current so what is decisive here is the following fact normal currents have slices such that this map has some regularity in x actually some hopefully some form of derivative in x which might be taken so in particular make the following observation if I know that the current I mean if I know that the curve has finite length and I'm slicing over here and over there and I know that the curve has finite length and no boundary what is the distance between these two Dirac masses? it's at most the length of the curve which is sitting on this slice of course the length of the curve maybe in there might be very large but it cannot be everywhere large if the length on a slab is very large everywhere then you have a huge length so this is for instance kind of telling you that maybe if I control the length of the curve I'm able for instance to control the w1 norm of the function f on which my Dirac mass is sitting this is true if it were always being just one slice this would literally be true you could really prove it because you could actually just say if this is x f of x and this is x g of x then sorry x2 y f of y you could actually say f of y minus f of x is obviously less or equal than the length of the curve intersecting this slab so x less than x1 less than y right and now you notice immediately this thing over here gives you a measure so this you could define as the measure of the internal of the interval x y where the measure mu is simply defined so this is simply defined the length of gamma intersected with the set x1, x2 such that x1 belongs to so you realize immediately that this is just a Radon measure so what you have discovered is that actually f of y minus f of x is less or equal than the Radon measure evaluated on the interval and that tells you that the function is a function of bounded variation so this suggests you that maybe being this slice if I interpret it in the correct way I actually get that this mapping is a function but of course you have to interpret it in a correct way because you don't have a graph of a function one possibility is that for instance my map does something like this I don't know so here it could be for instance two-valued curve and here one-valued curve and here another one-valued curve so now all of a sudden the slice is made of two points well as long as I'm following actually I do discover that I have a bv function but over here I have two bv functions but then when I join them all together here I have one single bv function with two values so it does not restrict to functions on the other hand what we are going to see actually is that there is a way of encoding this idea that since you're actually slicing a normal curve you actually have a nice dependence on x so this we will see at the beginning of the next lecture which is on Thursday so in which sense I can actually say that this is a bv map now what the only thing I want to say is that once you get this information you can imagine that then you can complete the proof of the compactness theorem or of the closure theorem because it would look a lot like when you're proving Ascoli-Arzela for instance you have the equicontinuity and then you know that you're taking values on a compact set when you're taking values on the compact set this is telling you if I look at the values of the functions at a given point I get the sequence of points in the compact set I can extract the subsequence and I can pass into the limit and for this I can do this on a dense set but then that's all I can do I can do it on a dense set and then I would have point wise convergence on a dense set of sequence of functions well the equicontinuity is helping you on the nearby points so once you have actually extract the sequence which is converging somewhere at a given point the equicontinuity tells you in the points nearby you cannot actually go too far off so this is in a nutshell the idea which is behind the modern proof once you have actually proved that this is a bv map you have enough equicontinuity in the x variable of course you will not really have equicontinuity because you know a bv map is not a continuous map but still you have some regularity in the x variable to be able to use the zero-dimensional compactness case to actually conclude that you're compact so we're not going to see this part but what you're going to see next time is how you encode or how you can define bv maps taking values in this space which is of course an infinite-dimensional space and how you can use the structure of a current to prove a nice simple estimate for the bv norm of this map and so that's going to take 10 minutes and then we move on really to the regularity theory so the first thing that we will see is how you can try to get hold of regularity by approximating with harmonic functions in some points