 Okay. Welcome everybody. It's another edition of the basic nature seminar. We have one last one in December. I now forget the date, maybe it's December 9th. And it's my pleasure to introduce our colleague, whom I hope everybody knows, Claudio Rezzo, who will tell us about minimal surfaces. What? Problems and about minimal... Eyes of parametric problems first. So, well, welcome everybody. I mean, of course, this is going to be, I hope, a pleasant but quite elementary introduction to a piece of mathematics, one of the oldest pieces of mathematics. So, actually, in preparing these notes, I realized it was very difficult to put some technical stuff inside. So, I'll try to make comments here and there. But then, essentially, I hope you will follow the river of ideas, which actually start... I mean, I was hoping to talk more on minimal surface and eyes of parametric problems and then, actually, preparing these slides, I got fascinated myself. So, actually, it's probably going to be at least half and half on these two topics, which are clearly very, very closely related. So, unfortunately, I mean, there is something we have to do. Everybody who speaks about this story has to start with this picture. So, let's do it and let's do it quickly. So, the problem starts 800 years before, I mean, before the birth of Jesus. So, according to the poet Virgil, who was supposed to write a poem celebrating the greatness of Rome and the special of the Emperor Augustus. Of course, the greatest enemy of Rome was Tunisia for a few hundred years, so, Cartagena. So, he had to justify that Cartagena's people were smart, but they were also a little bit cheaters and they had to justify the fact that Rome was so angry with them. And he invented this story, which is now commonplace. But, of course, nothing is true. So, Dido, the Queen Dido, was escaping around 800 years before Christ. His brother, who wanted to kill her, nice family as usual in those times. So, he, she landed with few ships in the coast of Tunisia and she asked for hospitality to the local king. And she offered to pay a symbolic quantity of money to buy the piece of land that could be covered by a hawk's hide, okay, and compassed, actually, by the hawk's hide. The king, of course, had to say yes, because by that time, hospitality was the highest crime you could commit, worse than homicide, actually. So, he said yes, and then she played the trick. She knew mathematics, and so what has she done? She has taken the hawk's hide, she has turned into a long strip, and then she has taken a piece of land and compassed by a lot. So, and now the question is, what is the longest, the biggest area you can encompass with a long strip? Apparently, Dido knew the answer, quite unbelievably, okay, and now this is called the isoperimetic problem. So, I give you a wire, a perimeter, a positive number, and I ask you which is the biggest area you can actually encompass with a curve of this length, okay. So, the story, Virgil's story is essentially that Dido knew the answer. Actually, there is a slight technical problem because, of course, she wanted to have some piece of coast, so now we picture domains in R2, but now actually the real problem is when you fix, for example, a given boundary given by the coast, okay, and then you solve the problem. But actually, that's the same amount of difficulty once you think about this. Actually, and the answer, there is no mystery here, okay. So, this is the starting point of this seminar. So, we actually have an inequality which holds for every domain in R2 with regular boundary, so you see, I forgot to ask which is the, ah, okay. So, everything in R2 satisfies this inequality. So, the area inside the curve, suppose, of course, that the curve is simple, I mean, so no self-intersection, it's embedded, and so it's a nice smooth curve in R2. Then the area inside, actually, you should first prove that there is something called the inside, but okay, let's assume it. So, the area and the length are related by this inequality here. And actually, I forgot to write the most important part in some sense, which is when does the equality happens, and the equality happens if and only if the domain is a disk and the boundary is a circle, okay. That's okay. Now, actually, preparing this talk, I realized that, well, I knew this was almost every mathematician in the 19th and beginning of the 20th century gave a proof of this theorem. I never realized that actually now there are more than 50 proofs of this theorem. This is the simplest one that I know, but I mean, I'm sure, I'm ready to bet. It's a one-line proof, okay. So, how do you estimate the area up to 4 pi? Well, you make a stupid trick. You make one as i over i, and it's this integral, and then you just use Cauchy formula. So, Cauchy formula tells you that 2 over i, blah, blah, blah, is equal to the integral over the boundary, okay. Then you use Fubini's theorem. You switch integral. So, this is a double integral, one over omega, one over gamma. You switch them, and you are left with this. And then you realize that this is actually the differential of something. It's actually the differential of this. There's essentially nothing to be proved. And then you know what is the integral of this by brute force computation, and this is less than or equal to L squared. So, let's take a kind of a toy problem, which actually if you think five minutes is not really a toy problem, it's almost the problem. So, what are we doing? We want to maximize the area inside a curve. Now, of course, not every curve is a graph, but let's restrict ourselves to the case of a graph of a function over some interval of the real line, okay. That's a specific case, a special case, okay. And actually, this is the one which takes into account the cost problem, if you think, because the cost would be kind of the horizontal axis. But you have a constraint. You have the constraint that you want to fix the length of the curve. In this case, you have a function, you have the graph, and you want this graph to have fixed length, okay. Now, okay, you have to decide where you are putting this graph, and this is the typical, the prototype constraint problem. Actually, this is the problem that Euler and Lagrange studied when they tried to understand what was the Euler-Lagrange equation, okay. So, this is, well, it's one of the problems. At that time, they were studying a bunch of problems. So, now, so, this is where Lagrange multipliers are born. So, you have to find kind of the solution of the Euler-Lagrange equation for the constraint functional. The functional is this. This is the Euler-Lagrange equation. And then you have just to study the solution. These are ODEs in one variable. Everything is very simple. So, you can actually write down explicitly the solutions. And the solution looks like the graph of a quadratic, I mean, the profile of the function which solves this problem is exactly the circle, half of a circle. And actually, you get also for free the condition that the intersection with the horizontal axis has to be orthogonal. So, not every circle, not every piece of a circle is a solution, but actually only those which cut the horizontal axis orthogonally. This is for free from the data, okay. Well, so this was a critical historical proof in, but now, oops, okay. But now you see, well, even staying here. I decided right from the beginning to look at domains in R2, that means connected things, okay. Connected objects bounded by some curve, okay, with boundary given curve. So, actually for me in this, I'm also one of the reasons why I like this topic is because I think it's a nice way to learn how to make a good problem out of a known one. Okay, so let's suppose this is well known. Up to what I said up to now is really very well known. How can you complicate a little bit the problem in an interesting way and make it non-trivial even by modern standards? Well, one way is to drop the assumption of your set being a domain. So, domain means connected and bounded, okay. So, what about disconnected regions? So, suppose now you're asking the same question so you want to maximize the area of two regions at the same time having a given perimeter. There is a guess by symmetry, by symmetrization arguments you can make a guess, but it's highly non-trivial to prove that actually this guess is the solution. This is now called the double bubble. You see the configuration you have to construct. You see the two regions are those inside that picture. So, first, of course you'll notice that a priori if the two regions could be completely disconnected actually the solution puts them close to each other. They are touching each other and this is forced by the solution. And also, the boundary curve, now of course it's not one circle but it's made of three pieces of circles intersecting, every time they intersect they need to intersect with three arcs creating an angle of 120 degrees. And actually not every configuration like this you have kind of a condition about the mutual radii of the circles. So, for example, if you are taking two regions of equal area the double bubble would have, that's the only degenerate case so the intermediate arc would be actually a segment. It really looks like a figure eight. So, if you are prescribing the two regions to have the same area so if this is equal to this then actually this circle becomes a straight line, a piece of a straight line. That's for example the... And this was actually proved only in 1993 but still with, I would say, reasonably elementary arguments. I mean, it's kind of a very ingenious proof by... What about three domains? Well, three domains, that's getting much more complicated but we know the solution. So, now you give me three positive numbers and you are telling me, okay, A1 is area 1, A2 is area 2, A3 is area 3 find the least perimeter way to enclose three regions with the area, the corresponding areas. And this is the solution. We know that this is the solution. Again, this was an old guess but this has actually been proved only in 2004 and now proofs are becoming definitely non-trivial. So, what about more domains? Well, we don't know. There is a beautiful group of people doing also computer... I mean, mathematics with computer, graphics and actually numerical computations who actually try to draw all the possible solutions by computer evidence, okay? And this is what... So, first one is the DDoS problem. Double bubble, triple bubble. Actually, so this picture is about all regions with the same area. The two things here have the same area. The three things here have the same area and so on. So, the interface is actually a segment here. So, what about four? Four is the first one that we don't know. So, this is an open problem. Is this one the best possible configuration? And actually, from four onwards, we don't know. Four, five, six, up to ten. Different colors actually here they like going and check how many other regions each region touch. So, that's why you see blue, green, blue, white, red, yellow and so on, okay? Because you will see now I give you from 11 to 20, from 21 to 30. So, you see strange color starts disappearing. I mean, gets rarer and rarer. And from 31 to 40. And I think even computers at that point got bored, okay? So, that's all we know. But you see what is the pattern here? Okay, so one problem would be prove it. Okay, take the 23rd of these and prove that this is actually the optimal configuration. Okay, this is one important problem. But actually, there is one which is probably more important looking at this picture, which is kind of the asymptotic behavior as the number of cells increase, okay? Because you see, look, always look in the heart, art of the picture, of each of them. So, what's going on? Well, around the center of this, whatever the center means, you see that starts looking like a very regular configuration made of hexagons, okay? And this goes more and more, it's more and more clear, okay? So, in some sense, irregular things are thrown to the boundary and inside they look like an hexagon tiling of the plane, okay? Well, so is this just an accident or not? Mind you, I mean, this seminar is completely non-historical, okay, because this, I guess, is a question which actually is older than the isoperimetric question, which is this one. So, now, suppose that instead of taking a finite number of regions of equal length, for example, if you want to have this intuition, okay, now you take the whole plane and you want to divide the whole plane with infinitely many regions now of area one by spending the least possible perimeter. So, perimeter is cost, okay? And actually, B is new, this, okay? So, building a perimeter is costly for you, okay? So, you want to minimize this, this problem, okay? So, this is now automatically a global problem. Of course, we have never seen an infinite honeycomb, but suppose for a moment that such thing, and this is the problem which was, I mean, the fact that actually the solution, you see, when you take these very old problems, the moment they were actually asked this question, they also gave you a solution, the problem is that the solution was never right, okay? So, when in the antiquity, like 2,000 years ago and even before, when they mentioned this problem, they also told you, well, of course, B's know the answer and the answer is the perfect hexagon tiling, regular tiling of the plane, okay? Of course, the B, I mean, maybe the B's know, but no human beings knew, okay? It's quite easy to prove it if you assume that the cells are regular polygons and that's probably what they really knew 2,000 and something years ago, okay? But then, if you drop this, actually, it's not too complicated to prove if you assume that each cell is convex, convex part of the plane, it's extremely complicated to drop the convexity assumption in this problem, okay? And this has been done only in 2001 with actually, this is a piece of virtuoso for amen. You have to admire both the technical power but also the infinite patient of this man, Thomas Hales, so he ruled out all possible other solutions, know that, of course, this is only an asymptotic statement. So, how about filling a whole plane? So, of course, if you change something in a finite piece of the plane, this would still be a solution. So, of course, there is no uniqueness statement here because, of course, if you take one hexagon and you replace it with four triangles, only one, I mean, that would still be a solution because the limit as n goes to infinity of the ratio between area and perimeter stays the same, okay? But, of course, here, of course, the whole point is about this constant and this constant is the one obtained by the perfect regular hexagon tiling, okay? The two main difficulties, as I said, one is about convexity and the other one actually is very hard to exclude the possibility that actually you could decide in this problem to waste some space somewhere. I mean, why all these things should actually fill everything? Maybe it's more economic to waste epsilon areas here and epsilon area there, okay? So, that's actually the critical thing. Because actually there is a nice, simple proof if you actually assume that everything is covered. Well, so, how to make out of this a nice research project? Well, add one dimension. So, this was a problem about area and perimeters in R2. What about volumes and areas? So, everything goes up by one. So, the regions starts having volumes and the boundaries have an area and you ask exactly the same question. For historical reason, this is not the honeycomb, but, of course, has a different name. It's called the Kelvin problem, because Kelvin studied this problem for thermo-dynamical, I mean. And actually, so, exactly the same question. He suggested the solution. These are openly taken from Wikipedia. So, I'm just... And the solution suggested by Kelvin is already very strange. I mean, it's a very irregular thing, you see. Now, you would have expected, I mean, dodecahedra or, I mean, something like that. No, actually, there are better ways to fill three-dimensional spaces. And he proposed this, where actually it's not even true anymore that the boundaries are flat. In Kelvin's original proposals, there are some faces which are squares and some faces which are hexagons. But the hexagons are curved. So, he actually checked that this gives you a better way. I mean, you save area of the boundaries covering the same amount of volume. So, nobody was able to prove this, and in fact, ten years ago, two mathematicians found a better way, which is, so, this is the where-fellan partition, which I don't understand. I confess openly. I can just show you many pictures of this. Here you can see a description. Anyway, so their contribution is to produce something which has better density than the original Kelvin proposal of this ratio volume divided by area. But nobody knows if this one now is the solution. Okay, so this is a completely open problem. Okay, so this is another direction you can take to... But now, let me go back to the isoperimetic problem in the plane. I gave you a proof, which is absolutely perfect, but as I said, doesn't seem to add much to our understanding of the problem. Now, actually, historically, the v-proofs, the most of the proofs of this type of theorem, symmetry arguments, after all, and you end up proving that something is a ball, something is a disc. So why not trying to prove that if you cut... I mean, you immediately seek, right in the proof, the symmetries through planes, for example, reflections. Okay, so that's one technique that I won't really mention now. But let me give you what I think is the proof that I consider most complicated but more interesting. Okay, so I give you the two extremes. Very simple, useless, very complicated, but actually I think it contains everything about the story that we are telling. So for a moment, this is, as I said, it's a constrained problem. We have a perimeter, we are fixing the area inside, we want to keep the perimeter fixed. But so suppose for a moment, you forget about the constraint. And I give you a curve and I ask you, well, which is the most... for a simple closed curve in the plane to collapse. Of course, the plane is simply connected. So you know that sometime, I mean, in some way you should be able to deform it to a point. Okay, now the question is, what is the quickest way to do that? Meaning the length of the curve gives you a functional over the space of curves. Okay, you can measure, I mean, this is a space where you can actually speak about the gradient flow. Okay, so at every point, you move by the gradient of the functional that you want to minimize. Okay? Well, this is actually the answer. So I just remind you, I just thought of the second line in class, but maybe... So for a curve in a plane, actually the only geometric... I mean, metric geometric invariant is the curvature of the curve, which is denoted by k there, which is actually the rate of change. Once you parameterize the curve by arc length, you go and check the derivative of the tangent vector. This gives a vector, and you compute the norm of this vector. So the derivative of the tangent vector, so the way the tangent vector changes, gives you a special direction, which is called nu, but it also gives you some kind of modulus, which is called k, the curvature. And this is essentially the only invariant, I mean, metric invariant of a curve. So now, what is this equation here? As I said, now you are introducing a time parameter, so you start with a curve, which is usually parameterized by a parameter called s, typically its arc length, and now you move it in time. So that's a time zero, and time t, this is evolving via this slope here. So every point is pushed in the direction of k nu at that speed, infinitesimal. This is called the curve shortening flow, because I said it's not difficult to check that this is actually the gradient flow of the length functional. Very interesting is this observation here. So you see, this looks like, well, instead of writing k and nu, I just told you the definition of k and nu. So k and nu, try to write everything in terms of the original map x, which is actually the unknown of the problem. So the tangent vector is, of course, dx in ds. The secondary, k nu is the secondary, but even with respect to the arc length parameter. So if you write everything in terms of x, this becomes the equation, the parabolic equation that you are studying. And this looks like a beautiful heat equation. You see, that's the key observation. So heat equations, every time you have a function of a given variable, y, you can actually deform this function via the heat equation by introducing a t parameter and then requiring that d in dt of the map x, y, t is actually equal to x, y, y, the Laplacian in general. This is the heat equation for the map. So this is nothing, so it's kind of interesting. So the length, the gradient flow for the length function is exactly the heat equation. This is a prototype, this is kind of a meta-theon. I'll comment that. So again, which are the solutions to this problem? That's a good way. And also, actually first, because once you know that something is the heat equation, the first thing you want, you are very happy usually because the heat equation is a linear equation. So then you take every book on partial differential equations and at the beginning the first three or four chapters are good for you. Okay, for this type of equation, this is a heat equation, you can throw the three chapters away. Why? It looks like a linear equation, a linear PDE, but it's not. Because every time the curve changes, the arc length changes. So here, s depends on x. So this is just a trick, it's an illusion. X is the map. I mean, I'm thinking of x as a map, for example, from s1 to r2. No, no, no, no, no, it's the whole map. Sorry, no, no, it looks like the first coordinate of something. No, no, it's the whole map. But now you see every time, so for every t, it's a family, so I'm actually looking at x as a family from some interval, okay, to r2. Now, every time I fix t, I need to compute the arc length, and the arc length is nonlinear. Okay? So here, this is just, as I said, it's an illusion of the way you write the equation, the linearity. So how bad really this equation is? Well, again, let's take the toy problem, which is, again, not exactly a toy problem. So suppose, actually, you are taking a graph, I mean, your initial curve is a graph of a function, and you want to know how does this graph evolve under curve-shortening flow? Well, this is the answer. So here, you see very well why this is a very nonlinear problem. Okay, actually, here, I tried to argue, so I mean, I gave you some indication of the fact why this should be the length, I mean, the gradient flow for the length. But then, so, of course, what are we asking now? So is it true that this flow actually exists? So typical thing in parabolic equation, this is a system, actually, it's a system of two equations because I've done, of course, these are two components, each of them must satisfy an equation. So this is a parabolic system, you know, what would you like to do? Well, of course, you always try to prove short-time existence, long-time existence, convergence, these are the three steps in every... So short-time existence is okay, no problem. Long-time existence fails, and this is easy to see because if you take a circle of a given radius, this shrinks to a point in finite time. Okay? So there is a kind of a maximal time of existence, but still, you would like to know what does it happen to this curve when you are approaching the maximal time of existence for the flow? And this is actually extremely hard. The way people understood something about this is actually they noticed that there is a monotone quantity in this flow which is actually given exactly, you see, by the length square of the curve at time t minus 4 pi the area inside. Okay? So if you compute the derivative of this quantity along the flow, you get that this is non-positive. This is actually easy, but then... Well, easy, easy but cute, I mean. So then, out of this, so what is it really meaning? I mean, and then, actually, there was the hard part of the story which was proved in 1987. Really, at our time. And actually, this quantity is not... I mean, on one side, you know it's monotone, but on the other side, you know that it goes to zero as t approaches the maximal time. So now you need a little bit of geometric intuition. What does it mean that this quantity is going to zero? Well, you see, the length square minus the area inside is going to zero. So for example, a situation like this where it might happen. You see, you cannot collapse the area inside by keeping some kind of perimeter. So if the area inside goes to zero, also the perimeter is going to... must go to zero. This is why this kind of... this statement is amazing. Okay? I don't know, is it clear? I mean, I'm happy to... You see, the geometric interpretation of this limit. Okay? It cannot happen. So your curve cannot shrink to a segment, for example. So where does it go? This is actually what is usually called in these things, ruling out singularities of the flow. So then out of this, Gage Hamilton and Grayson, I mean, one for convex initial data and then Grayson in general, they managed to prove that actually this flow exists, of course, up to maximal time and suitably rescaled the limit of the curve when you approach maximal time becomes a circle. So of course, curves disappear. I mean, we constructed this flow to make curves disappear to somewhere. Okay? The length must go to zero. So eventually you will get a point. The interesting question is how you get a point. So you need to... Once the curve is around this point doing some mess, you have to take a length, rescale, and see how is it going to a point. Okay? If you do it properly, what they proved is that actually this becomes more and more the round circle. Okay? Already this one. Now, this is fantastic. If you like differential geometry and geometric analysis, this is a fantastic theorem. It's kind of curious the fact that you had to wait 1990 to see this, but actually why you had to wait 1990, because of course Hamilton was so frustrated by the Ricci flow. Let's say, look, let's play the easiest game. So the Ricci flow for whoever knows what I'm talking about is exactly... I mean, it's a very close variation of this idea for curvature in any dimension. Okay? You could not prove any theorem. In fact, he could not rule out singularities of the Ricci flow. He said, but how is it possible? Let's go back to the simplest case. And so in this sense, this is the simplest case of the Ricci flow, geometric evolution. Or the mean curvature flow, which was actually already existing. Okay? As for the isoperimetric problem, you can extract out of this theorem immediately also the isoperimetric. You get another proof of the isoperimetric inequality because, you see, this problem forgets about the area inside. Okay? Actually it's changing the perimeter, it's changing the area, everything. But of course you can rescale every time. For example, you can prescribe the length of the curve to stay fixed. So you are doing something funny, because you are taking a curve shortening flow, rescaled in order to keep it of given length. Okay? In this way, Grayson theorem becomes the isoperimetric inequality. So I think this is actually the best proof of the isoperimetric inequality. Actually, I should have said already, I mean, I'm borrowing pictures and text from everywhere. Okay? So sometimes I remember to say where, but... Okay, now I forgot... Oh, no, no, it's more than a book. It's an amazing proof. It's an extremely complicated proof. No, no, no, no, no. This is actually a 400 pages book, which is called The Curve Shortening Flow, which contains only one theorem. No. It's extremely sophisticated proof. Because actually, you see, the problem you have to imagine is you have to rule out all possible ways singularities can occur. So if you think in a higher dimension, you can manage to save some time, I will say something. Actually, in higher dimensions, singularities do occur. So it's quite amazing that actually they don't here. Anyway, I prepared also another thing. So this link I put there is kind of fun, but now maybe I'm getting a bit slow. There is actually a mathematician who had fun creating a beautiful computer simulation of The Curve Shortening Flow, where you can actually go there and with your mouse draw the curve you like. And actually it's amazing. And you will see the evolution. The evolution in time. So actually, I suggest to you, which is people, experts in the field know that this is the nightmare of The Flow. So if you want to know, if you want to check whether you are, if you want to check whether this guy is good and the program is good, draw this. Actually maybe one just one comment. Why this should be the nightmare? I mean, this is not just fun. You see, evolving by curvature means that this point will start extremely fast in this direction. Boom. Evolving by curvature will tell you that this point will go extremely slow in this direction. This is almost flat. This is extremely curved. Okay. So who wins? Who does it win? Who does win? Or whatever. The amazing curvature here or the empty curvature here. Actually this is the hardest part of the theory. They even, nobody wins. So this just becomes they flow. So you will see the evolution of this curve is that this snake enrolls and becomes more and more round and round and round and then eventually it will be round. Okay. So there is kind of a compensation phenomenon between these two things which is amazing. And in fact actually now, look, I think I will start improvise because I think I've prepared a three hour seminar. So let me show you and in fact you are a bit too silent. You should be free to ask things. Let me show you why people have been you see? So this was about, so this would be a second hour of the seminar. So let's play the two-dimensional game. So let's flow by curvature a surface now. Instead of flowing a curve in the plane let's flow a surface by curvature. Now unfortunately this, I mean we're not spending three or four slides to explain what it means by curvature for a surface because a surface now has many curvatures. It has the Gauss curvature, the mean curvature and so on. The right thing to do is the mean curvature whatever it means. And actually these are for example computer simulations of mean curvature flow. This is exactly the same thing. It's the heat equation we played with. But now unfortunately it's interesting because it's difficult. Every type of singularity is allowed. Things can collapse to a segment. So a surface could become a segment. Necks can disappear, can pinch. So this was exactly what I was trying to convince you that here was not happening. So a neck a one-dimensional neck does not disappear. A two-dimensional neck disappears. So a connected surface could become connected just by pinching in between. Excellent question because back to... So the question is even since I was I hope I said I'm starting from an embedded curve so nice without self-intersection. So one crucial property that you have to prove is that actually this embedded property stays along the flow. And this is highly non-trivial. So in fact you can prove even more. The curve-shortening flow is this other nice thing. If you start with two curves that don't intersect at time zero and you flow them they will never touch. I don't know where they will go but they will never there will never be a time where okay. But actually answering your question because as I said I mean one of the hidden messages behind this is how to choose a good problem well how to choose a good problem unfortunately if this is a really good problem if I knew how any idea of how to solve it I would do it myself. But what about the curve-shortening flow for non-embedded curves so what about if you start from something like this I mean it makes perfectly sense to flow the gradient flow by length the length gradient flow of this curve. Nobody knows what happens in general so that one we can play some game and we will find out. But there is no general theorem in principle this can also disconnect and even more difficult is what happens in R3. So suppose you take a closed loop in R3 and flow by curvature well maybe there the guess is still that it becomes kind of a planner now it's non-planner maybe in time it becomes a planner circle nobody knows how to prove it. Because actually they have examples for both of these things where singularities can happen. Now just another important historical comment but not really not just historical there are very simple proofs and historical these are all the proofs that were known for almost 2000 years about the isoperimetic problem which actually the main argument is like the one I outlined there. So if you start from something which is not the circle then there is actually a procedure which makes it for which you can the output of the procedure is to enclose more area than before. So now the question is, is this a proof? See we are trying to prove, so the statement is the circle is the one which has the total area inside. For 2000 years there were about 30 proofs of this type of statements. So actually this hits one of the most delicate points in the calculus of variations. Unfortunately for everybody but actually Perron a very important mathematician of the beginning of the 20th century when actually criticizing a colleague for giving yet another proof of this type since the colleague was a geometry he was an analyst of course he was claiming that geometers are not able to write down proofs, correct proofs and he said, well the argument you gave, the poor guy was called Steiner which actually was a very smart guy but he passed into history for this mistake. So what Perron wrote, actually this is taken, I mean translated from the German but it's actually taken from Perron's paper. It's not very nice to have a comment written on paper. Anyway, so theorem, among of curves of agreement let the circle encloses the greatest area, proof by Steiner. For any curve that is not the circle there is a method, symmetrization, reflections, blah, blah, blah by one finds a curve that encloses greater area. Therefore it encloses the greatest area. So he said look, geometers don't know really logic because in the same way I can prove that among all positive integers the integer one is the largest proof. There is a method that if you give me any number different from one I create a bigger number square. So one is the biggest number. Okay. That's one of those, I mean it's fun now after one, actually it's exactly 100 years old. It must be very tough to receive this kind of comment when you are alive because actually it's true. I mean he did this mistake. I mean actually many people did this mistake for 2,000 years. So what's the problem here? The problem is that nobody realized that before proving that the solution is the circle you need to prove that your problem has a solution. Okay. You see that's what is missing here. I mean this proof is correct. Except the problem has no solution. Okay. So that's the moral that Perron was trying to advertise to everybody and actually there is another famous example of this occurrence and now I fight back I feel like I am a geometer so let me mention a mistake done by an analyst of the same type which was another of this very famous you see we have lost a little bit touch with practical problems at least but that's what I feel. I mean in the 18th century mathematics was really much more related to everyday life and one of actually very popular problem was the problem of describing the shape of the perfect column of the most solid column. Okay. So for thousands of years columns have been designed to form this is actually taken from a 16th century book but this is based on Apollonius rules which were driven only by aesthetics. Okay. So the point here is that if you are more or less tall like this point and you stand below the column you will see the column will look for you like a cylinder. It's an optical thing and was just an aesthetic problem. Of course the Bernoulli's, Euler, Lagrange I mean okay but now you know some elasticity theory something you can speak about solidity of a column. So what is the most solid column that you can and then Lagrange wrote a famous paper where he proved that the most solid column is the cylinder and this is not true because he assumed that the solution existed in his space of functions so he restricted himself to study this variation of the problem. Now I don't write for you the law of solidity of a column but I mean he assumed the existence of a solution among smooth functions. It's actually easy to prove that it has to be rotational invariant. So it's really a one-dimensional problem it's a problem about the curve. So he assumed the existence of a smooth solution and then among smooth solutions the horizontal line that you rotate becomes a cylinder is actually the solution. Then people realized 20 years ago that this was actually the mistake done by Lagrange and actually for your curiosity if you want to know which is the most solid column it is this one. And you see why the point is that this solution must have singularities. If you continue this column a little bit I mean here they have put something in between the whole point of the picture but I mean the whole point of the picture is that the solution will look like this. It has a singularity there. This was just a curiosity but then unfortunately I cannot give you any idea of how to solve. Now you know there is a problem. You give me a variational problem geometrical or not. First you need to prove that the solution exists and then you can start analyzing the possible solution. There are more than in all this type of things. Geometrical problem. Unfortunately it's too complicated to tell you in a basic notion how people have circumvent, I mean solve this problem. So basically it was the birth of a subject which is huge. It's beautiful. It's called geometric measure theory. And basically the idea is you enlarge so much the space of competitors. You define something that they are usually called integral currents or rectifiable currents and so on. You prove existence in this space. Actually the three giants are here, Armgren, Federer and the Georgians in the 50s and 60s essentially. They developed all this theory. And then the problem becomes so you see in order to get the solution you have decided to take into account very rarified things. So curves will not be curves anymore in your intuitive sense. Surfaces will not be surfaces and so on. So then once you have such a solution you need to prove regularity theorems. So you need to go back from your monsters to real life. But this is one advantage is that usually these existence theorems for these kind of monsters are kind of typically I mean I know very few places where the dimension of the space that you are studying is actually mean important. But actually from there to the regularity the dimension is crucial. So here I was trying to describe to you what is the mean curvature of a surface what is actually the curvature of a surface. Now it's not that obvious like the derivative of the tangent vector. Now you would like to speak about the curvature as the derivative of the tangent plane but what does it mean to take the derivative of a plane? So you pick essentially what you can do is that you pick two vectors in this plane and for each of them you measure the individual curvature of the corresponding curves but then you have this in the terminology about how did you choose these two vectors. So actually none of this curvature is interesting. The two interesting things that you can do is actually taking the product of these two or the average of these two. So if you take the product of these two you get the Gauss curvature if you take the average of these two you get the mean curvature. And actually in another way you can also say you have a two by two so the transformation which sends infinitesimally one tangent plane to the next tangent plane nearby is an endomorphism of two planes so essentially if you are more algebraically oriented well if I give you an endomorphism between two dimensional vector spaces there are only two interesting numbers determinants and trace so Gauss and mean curvature that's exactly how the story goes and then what are areas of surfaces? Well area of surfaces again you should convince yourself but it's easy because surface is actually controlled by the determinant of these two numbers that I mentioned you see if you take two vectors think of what is the area of the parallelogram generated by these two vectors well this is actually the determinant okay square root okay now now suppose that you take now we are looking for surfaces which are critical with respect to some variations so you take a family of surfaces and we are imposing for example isopereometric problems and so on so surfaces which has maximal area, minimal area with respect to some problem okay so the idea is take a family take the derivative of the area and impose that this is equal to zero okay this is what I try to recollect here and then this is just a metaphysical proof but this is since the area is governed by the determinant of this thing it's not surprising that the variation is controlled by the trace okay because the derivative of family of matrices of the determinant of a family of matrices is actually the trace once okay of course this is not the proof but it's really what's behind everything so what does this mean this means that if a critical surface for an area problem is something which will have mean curvature zero or in general constant because for example if you add to this so okay here I wrote it in words so the same computation tells you also this if you add the constraint which is the volume inside then actually that formula proves for you that the smooth part of a solution to the isopereometric problem in higher dimension is a sum manifold of constant mean curvature okay so now the question is about surfaces in R3 and the question is are there constant mean curvature surfaces in R3 different from the sphere and so on so actually Alexanderov so now you see the isopereometric very popular way to describe the isopereometric problem in is the following so now instead of length, perimeter and area we have surface and volume so for example I'm fixing a given amount of volume and I'm asking you which is the surface with least area which contains that amount of volume okay so this is exactly the soap bubble problem okay so the literature is filled with pictures of soap bubbles and language by soap bubbles because that's exactly the problem that the soap is trying to solve every time you create a bubble you blow some fixed amount of air this is your volume and then the soap is trying to minimize the energy to cover it to become its boundary okay okay so there is about this soap are there which are the surfaces of constant mean curvature in Rn and here the critical theorem is this theorem by Alexanderov which says that the only embedded constant mean hyper surface in Rn is the sphere okay so in some sense this is also the solution to the isopereometric problem in R3 because this is exactly the boundary of your domain and this is the sphere okay so this explains to you why we don't see so this is actually the theorem that tells you that there are no soap bubbles different from this except if you assume that the soap bubble is actually smooth okay what about if you drop the smoothness can we see something else well and amazingly the answer is yes in 1978 Vente found a torus of course it's very difficult to imagine it exactly because you see it has self intersections so you have to imagine the torus like three spheres connected with three necks okay and these are the three spheres somehow and inside there are the necks which connect the sphere so this is a picture which tries to tell you what's happening inside okay if you remove this so this is an historically very important surface it's called Vente's tori on the other hand so let me just mention the last few minutes that remember I only give you the first derivative of the area okay so we just constant curvature arises only as derivative equal to zero for your problem but actually we are trying to minimize or maximize a problem so more actually minimize in this case the area among everything which encloses the same thing so we have an extra information about the second derivative so if it's a minimum not just the first derivative is zero but the second derivative is non-negative okay now if you go and check for this Vente's tori and everything now there are millions of these surfaces actually you will find out that these surfaces are unstable critical points and this also explains why it's very hard to see them you see in every natural problem unstable critical points are very hard to detect because as soon as you move epsilon away from it nature will go somewhere else okay the principle I mean a stable critical point it's easy to detect because everywhere you start you will flow there think to the curve shortening flow unstable point it's very hard to detect okay because it depends where you start and in fact in some cases wherever you start you will go somewhere else okay if you drop the compactness condition it's very easy to construct many constant complete constant curvature surfaces I will show you some look I don't know I think I should stop in a couple of minutes let me just show you some pictures okay remember the logic of my talk was it's easy for the the isoparematic problem for one connected domain what about two what about three four five six twenty forty honeycomb okay so what do we know for the two for the one connected domain okay does it and many other proofs what about this two domains well actually this is just it has been proved ten years ago this is the solution this is just another competitor just to give you the idea that adding one dimension makes the problem extremely more complicated and actually the fact that this is called the standard double bubble that this is the solution in this case has been given on Annals of maths 2008 okay here we are in now 2002 okay so we know that this is the solution for two domains and nothing is known remember for the planar case we knew also n equal to three for surfaces and volumes we don't know that even the three that the case three the dynamical proof I've already gave some comment and then okay then we started with minimal surfaces but well actually this is a good thing for a coffee for a coffee seminar minimal I mean soap bubbles in some sense are also a very special case of the constant mean curvature equation is when h is actually zero so the sphere has constant mean curvature proportional to the radius I mean inverse radius it's particularly interesting the case where actually you fix h equal to zero and you ask yourself where surfaces with h equal to zero because those are exactly the ones that the problem in the solutions to the soap problem when you fix the wire okay so if you take a piece of wire you put into soap and you pull it out you will see a minimal surface a surface with zero mean curvature so you can actually ask I mean do they exist who have studied them what are the open problems this is another story okay but now it took me too much time to find them so I have to tell you in 30 seconds because again these are unstable points so in fact if I show you pictures like this and say well look you were studying you were studying a very natural problem why we never saw them why we never saw a soap bubble like this I mean this is a very unstable point actually you need to prove it it's a very hard and beautiful theorem in geometric analysis the fact that the only stable minimal surfaces in R3 are planes so now we have an amazing theory existence theorem of many minimal surfaces which should be soap bubble or means solution to a very natural thing and we never see them so these are all soap bubbles if you put the boundary of the picture in the soap and pull it up you could get this okay this is a critical problem and then 1970 came and this guy Alan Schoen who is actually a crystallographer found minimal surfaces in crystallographic problems he was studying he actually took pictures and then so you see now this was another seminar but and now a few years ago this was to me an amazing discovery so don't you see a minimal surface there actually I should tell you I should give you a hint these things contain this subject which is actually this subject here so this is called the gyroid and this subject was known to exist in nature in crystals but then 10 years ago and now it's amazing I mean there are dozens of people who are studying this kind of problem I also talked with Antonino Skardicki about why and actually now I understand why because to me sound a bit strange every basically there is a theorem here every animal which looks lucent green is made like this okay so and now this has been here has been checked on these poor guys and it's true here so you see many papers even on nature and so on I mean so they take the wing of these animals they enlarge it sufficiently much and by sufficiently much I mean this so they took this butterfly and if you go to a scale of 3 microns 1 micron 0.5 1 micron you see that there is a network like this actually the fact that this actually is this I was not very convinced to be honest when I saw it but then they made a computation about optical photonic optical properties of this thing so if you build a network like this you throw light in here actually and only because it's because to me it sounds a bit suspicious when you go to that scale how do you recognize this surface from epsilon this surface okay so only this surface has the properties of capturing all light except the green the lucid green frequency so of course these animals presumably have developed this to hide green leaves and so on so and now the problem is are humans able to construct the same thing I mean wires at this scale in order to have the same kind of photonic ability and apparently this would be a fantastic advancement in optical fibers and nanostructures and so on because this would create ways to propagate to send information at the speed of light actually literally I mean if the problem of course information is a frequency is able to choose one frequency and apparently this is the hope okay sorry too long but I hope you had fun I did the principle is what I hinted with this sentence about the heat equation so the curve-shortening flow is the heat equation for a curve in a plane now in general this is not a theorem it's just a logical pattern I mean so the idea of emails actually and the Richard Hamilton was to take the Ricci curvature of a manifold and treat it as heat so you write down the evolution of a Riemannian metric via Ricci curvature the Ricci curvature if you write it down looks in normal coordinates at a given point like the Laplacian of the components well we can I mean because it's kind of technical but it's philosophically exactly the same thing so the Ricci flow is the heat equation using curvature okay which is exactly this one in fact for a surface since I told you I convinced you probably there are two interesting curvatures gauss and mean here for this type of problem the mean one is the interesting one but why don't play the same game for gauss curvature well people have done it and it's interesting what happens that's another story the area inside for a given perimeter yes you know already that your domain is a ball well is homeomorphic to a ball or I mean because if it's a ball I just compute the area the area of the boundary of the sphere and the volume of the ball and that's it I mean if you give me one specific thing I just make the computation in R2 yeah well it's happening that actually the length squared is actually equal to 4 pi the area I hope or the other way round but I mean I'm not sure I understand so in the plane which surface are you looking in the plane you take a disc you take a disc you have the circle which is the boundary so now if you already know that this is the ball and this is the circle there is no variational problem I mean this is one configuration so what which problem would you like to solve I'll offer you a coffee you asked me you explained me better what you have in mind any other you're right and I don't know I don't know it's a critical point but whether it's a local minimum I don't know it's certainly critical because it has all the properties I mean each smooth part is made of constant mean curvatures that's okay and then you also have some kind of condition on the way the tangent planes touch when the two things intersect okay to be a critical point so not if you take everything made of constant mean curvature thing you put them together you get a critical point the critical point equation also gives you a specific way in which they need to touch but if I'm correct this is also this property so we could check don't you feel embarrassed that bees and snakes and beetles have so much more than plumbers a long time looking poor butterflies I mean I cannot imagine what they looked for before any other remarks Anton by mean curvature flow you take a torus like the standard torus in R3 I'm sure that for the standard torus people know in general there is no understanding so of course here in this subject the key name is who is Ken Brown White there are two or three of this name so who is Ken for example has proved that if you start from a convex hyper surface in Rn the evolution by mean curvature flow has the same property as the curve shortening flow but convexity is crucial so it will approach the round sphere that's what I mean, I mean you take any convex hyper surface then everything goes well and since then this was the end the beginning of the 90's this is a great theorem of course and since then people have tried to remove the convexity assumption what happens in the non-convex case I'm getting a bit don't take it too serious but I'm pretty sure this will converge to a circle in the mean curvature flow nothing prevents surfaces to become curves unfortunately so I think if you start from around torus and you flow by mean curvature I think it just becomes I don't know I don't know but what I'm telling you is that there is no general I mean if you are able to do this in this specific case it's only by your ingenuity not because there is a theorem that is going to tell you anything they don't know further comments, questions? alright let's thank Taurio one more time