 First of all, thank you very much for giving me the opportunity to give this course. I don't know the level of what you know and what you don't know. So I would appreciate if I say something you don't know to raise your hand and say, can you explain this? What do you mean by that? And the same way if I start becoming too boring, repeating things that you know except the other speakers, please raise your hand and say, well, you know, that's too much. We know how to add. Okay, I'm not going to start at that level. So and again, during the breaks, feel free to ask me any questions you have. So what I offer to present, it's something that goes back many years. And of course, things have been happening to it as we are moving on. But I will start with some things that happened long time ago, so long that when I tried to prepare the first lecture and I looked at some of my old papers, I had to think a little bit to figure out what the heck I had written down some time ago. And it was a challenge. Okay, so I'm going to start with the plan of the class is to discuss in some detail some definitions for generalized front evolution of fronts. And when I say evolution of fronts, typically I have in mind mean curvature plus something else. And I want to describe what you heard like the level set method and explain a little bit how it works. So the first thing will be level set. And I want to describe a little bit, very closely related approach using the distance function. I want to describe a more geometric notion. And I want to use all of them to study a little bit, discuss a little bit phase field theory, which is one of these heavy sounding words to describe to talk about, let me put some more words here, phenomenological, I don't know, maybe I put too many O's here, phenomenological approach to phase transitions, or in the words that everybody understands is scaling limits of reaction diffusion equations. And then I will sketch but I will not describe it because it's hard to do it on the black board. The opposite thing, which is the microscopic, so phenomenological is microscopic, microscopic limits of particle systems. And finally, I want to touch upon motions of interfaces with random velocity. I will not be able to do all this. And it depends on how much material I have to review. So aside from that, how many of you, because a lot of the tools, if not all of them, will come from the theory of viscosity solutions, how many of you know what that means? Feel free to say if you don't know, you don't know. So only four of you. Okay, so I will make the deal. I will use it now without saying, just saying what it is and state some theorems. And then, by my afternoon lecture, you go and read about it and you come back. No, I think what I will do is an aside if you are willing to do it, because if I put here a zero, something, I mean some, you will discuss the solutions, I'm afraid you're going to be stuck with me for a long time. So let's see how it goes. If I can do it, I can, or otherwise, for those of you who are interested, we can meet during a break or something for half an hour and I can give you the key elements of the theory. Okay, so let's start from here and there is a difference between the courses you saw last week in terms of mean curvature, motion and motion of interfaces, and the way, what I'm going to describe today. What I assume you heard last time, last week, was smooth things evolving and figuring out when they have singularities, what is the nature of the singularity. I would say that was basically, what, not basically, I mean that's a very important area. So the approach I'm going to take or the level set theory takes is we don't care necessarily about the level of the nature of the singularity, we want a notion of evolution that goes through singularities and characterizes everything in one way and of course, when you have something that goes through singularity and doesn't feel it, it's going to be weak, all right? And then you can add to that singularity, the theory, you can add to that regularity and try to say how much regularity, so it's the other way around. Now from the point of view of the geometers, this is, you may say it's below the dignity because they care really about the singularities and I have a story where with a very famous geometer, we was ages ago, we were in a restaurant eating and there was another geometer and something else and the famous person asked me, what are you working on? And I said, at that time I was working more on mean care about your motion. I said motion by mean care about your motion, he said, fantastic, let's talk about it. And the other in the younger geometers said, well, you know what he does? He does weak solutions and the famous geometers said, ah, okay, I forget it. There's no reason to talk. So having said that, let's start the level set. Now the level set is attributed to Osir and Sethian and if you ask Osir, it's attributed only to him and not to Sethian because he thinks that Sethian did not do much there but really goes back to previous observations. Neither of the two developed discover level sets. There was actually a paper by a guy at MIT before that we were discussing this and then a paper of Guy Barl who was looking at problems and combustion and he had come up with this. So what is the idea? The idea is I have a set here, gamma t and any time I write this I will not be talking just about the boundary. For me everything is going to be three sets, the front, inside, the outside. And we are going to assume that it involves for now very concretely with the normal velocity, which is normal, a function of the, I want to use, I have three different notations. I need to remember which one I want to use now before I get it. Yeah, so with the prescribed normal velocity. So let's say this depends on the normal, the curvatures, the direction x and t. Okay, so n is the normal vector. I assume everything is smooth. And I want to describe the evolution of that. And the canonical examples we will be using will be one. Basically this will be trace of for today plus let's say a of x t. So that's mean curvature plus first order term. And I put the minus sign. We are interested in something that obeys the maximum principle and things like that. Those spheres, balls are shrinking, so I have to put the minus there. And then the observation of the level set theory is to say, okay, that's what you want to study. Let's assume that the gamma t is really the level set at time t, the zero level set of a function u. And then let's take omega t to be the place where u is positive. And of course, that becomes a set where x is, whatever it's, u is negative. So we assume this. Let's also assume that u is smooth. So all these things are assumptions. I mean, this is formal derivation, u is smooth. And let's also assume that du is not zero on gamma t. And then try to express this in terms of the function u. So you advise for the students, at some point in your career, introduce a notation and stick with it. If you don't, especially with signs, right now there's a big confusion in my mind what is plus and what is minus in terms of this, because I never pay any attention. You know, I wrote the paper and then I copied. And I never, so what is the notation? So the normal velocity. Okay, so what's the idea? The idea is that think of particles. You don't care what is happening on the surface. So if you take a point on the surface, it's moving around in space. You don't care what it does while you are on the surface. You only look at the normal direction. And so in that case, the evolution, the vt will be ut over gradient u. Okay, the du, like the normal vector will be this. And the gradient of the normal vector, the curvature is going to be, so you take all that stuff, you write it, you take this relationship and you find that ut has to be equal to du times v. And let me not rewrite everything that goes in there. Okay, so you just take this and you put it inside there. And this is going to involve second derivative, so u, first derivative, so u, space time if you like. So doing that, I'm a disaster with chalk. So construct an equation that looks like that. So what is f? I can write it down. f is f of xpxt will be what you get. It will be p. And then here you put whatever else you have. So here you're going to put identity minus p cross p. So p hat to simplify things would be p over length of p over p squared. Sorry for that. I'll get it right. So that's the f that you get from here. Now you make another assumption for the model that not only the zero level set of the function moves with this law, but you make the assumption that all level sets of u move with the same velocity, which means that u will satisfy everywhere this equation. And we'll start with some initial condition ux0 of x where u0, the place where u0 is zero is our initial front. And the omega zero is the place where u0 is positive. And of course that is the place where u0 is negative. So derivation is very simple. You have a particular motion and the example, let me write down what it is for that case. For that case, the PDE will look like trace identity minus du tensor du hat d squared u plus alpha xt a, I'm sorry, maybe minus the way I put it. So that's the equation in a concrete way. This equation up here gives me this level set. And as I said, I'm almost positive if it's minus, but if it turns out to be plus, it doesn't matter for the theory. Okay. And you may have seen the same equation for GTA. You may have seen it in the case that u is a zero. That's the same equation. So this is a nonlinear equation that we got it through this modeling and we made some more assumption. We came down to that. And there are several researches with it. Notice there is a singularity when du is zero. So you need to make out of that sense out of that. And this is not an equation in divergence form. So it's not something that you can solve by integrating by parts. And that's where you need the theory of Riscorsi solutions to make sense of that. And let's say the theorem you have here is, let's start with the comparison principle, which for now you take it for granted, that if u and v are uniformly continuous, so uc stands for uniformly continuous in Rd. And one data is less than the other. Then the solutions indeed are ordered. So for now you accept this and think of it as a maximal principle. And it is a maximal principle because the argument will be in the right way that you look at the maximum of u minus v and at the maximum you have an order for the derivatives and everything and it's going to work. But of course what I didn't say, that it's a fundamental assumption here. So fundamental assumption is that you need to have ellipticity. So when you come down to this problem, you need to know that this problem is elliptic degenerate. We don't care whether it's degenerate and this is a degenerate elliptic problem. So I assume you saw at some point last week the degenerate the ellipticity. So we say that f of xp whatever is uniformly elliptic. If this matrix is, I'm writing it as if f were smooth but then you can write this thing in different ways. If x is less equal to y in a sense of symmetric matrices then f of x is less equal to f of y. It's greater or equal to f of y. It's less equal. It's like a little plush. Yes. Okay, you're going to have to be louder. Monotone in this, it has to be monotone in this argument. I will say something maybe in the next lecture that says that if you want to have a front evolution such that it has a comparison principle, in that sense a comparison from the front evolution means an inclusion principle. So it means that if you start with two sets, one inside the other and you want them to stay like that then you have to have a degenerate ellipticity. You have to have monotonicity to that. So I'm imposing here the notion, the idea that I will have to have the comparison. Say it again. It comes with a little bit gray hair difficulty. Yes, like a symmetric matrix. That's what I mean here. When I write that it's in the sense of symmetric matrices. Everybody's okay with this? So that's a big assumption. All right, so let's review what I did so far. I start with something smooth. I made some assumptions. I came down, I rewrote in terms of that smooth functions. Smooth function, what it means to move with this normal velocity. Then I made the extra assumption that not only that set, the zero level set, but any level set of you moves with the same velocity so you don't have this equation only when u equals, only on u equals zero, but on every set. So that means you're going to get a PDE in all of our d and that's the PDE. Now, what I didn't say is once you come down to that, you forget all your assumptions. Namely that u was a smooth, the level set was smooth and so on. And you say, okay, now I have an equation that I can describe globally in time. That's the viscosity theory that I mentioned. This equation now holds as a viscosity solution. This problem has a global in time solution. So now we look at one and I should have said here if u zero, so existence, u zero is uniformly continuous in RD, then there exists a unique u. And that's global in time in RD cross zero infinity. And we're going to say in view of that definition that the triple use omega zero, gamma zero, omega zero complement moves normal velocity v. If omega t is the set where u is positive, gamma t is the set where u is zero and gamma zero and omega, okay, it's clearly that. And u, Sol's star, I'm going to start abbreviating now this thing. I'm going to continue rewriting these three things. So that's our definition. So no regularity now, nothing. No assertion that the set omega, this set is a nice set. No assertion even that the set is a boundary of something, but it can be as complicated as a boundary of something. So what you get this way is that the set is the boundary of some open set, but has this property. So the problem, the theory is as general as this. Okay, now are there any flaws? And for that, we need to assume the uniform elliptist, I mean the degenerate elliptist, plus some other technical conditions on the way the f depends on x and t and so on. But let's stick with this. All right, do you see anything that looks weird in what I did and something that needs to be... Okay, so let me pause the following question. Is the evolving of sets unique? So if I'm interested in this property, if I'm interested in that, is this unique? What do you think? All right, the easy answer is when someone asks if this question is better be because otherwise I wouldn't be standing here to present it. So we know the answer is yes. But why am I making a point about that? Where is a flaw? Or what am I missing the way I'm describing things? So let me be a little bit more suggestive about the way we solve this problem. Will we given omega zero, gamma zero, omega zero closure complement? What did we do? We find a function u zero such that... Where did it go? We found the function u zero that had the property that omega zero is the positive level set. That's negative and that's zero. Okay, so this is omega zero. This is omega zero complement. This is a multi-d picture, but it's better to see like that. And that's a good u zero. This is an equally good u zero. Okay, I assume this thing is uniformly continuous. All right? Or for all I care, okay, this is not continuous, but the theory extends to that. That's another function that is zero and so on. So yes, the PDE, what I wrote here about the PDE is correct. An equation with this property has a unique solution. But as a motion here, in principle, for every u zero... Where did the equation go? For every u zero, I get a different ut, u, and therefore I get a different front. Yes, observation or another assumption. It's not an assumption when I get an f that is derived by a v that has this property. But if someone had given you an f, that's how you'd find the f. I walk in and say I want to describe the motion and I give you an f here. So if I ask the following question, I give you an f in such that the problem has a comparison principle. Then define this for the initial data and this for the solution is this unique. For example, that's an elliptic problem. I don't come from there. The PDE question. And the PDE question is I solve this problem, degenerate elliptic, and I am asking the question, are the level sets uniquely determined? In other words, are these functions going to give me different solutions? And if I do it for the heat equation, the answer is yes, different things. So it doesn't hold for our best possible PDE. And it's good that it doesn't hold because otherwise there was going to be something too weird. So the extra assumption we need, the extra assumption is that f has to be geometric, what we call geometric, which is a code word for the following property. And I'm not assuming. This is now a property of a PDE, which is the following. If I solve this problem with initial data U0 and then the sets where U is positive and so on, depend, I will use the word strongly, although I'm not going to quantify that strongly on U0. That's the property I'm stating. And that property is not going to be true with the heat equation. So let me write down what geometric means. It means that if I put a P here and XT, so if I dilate in the gradient vector and the matrix of the second derivatives, I get degree one homogeneity, positive homogeneity, but there is an additional property that the equation doesn't see this tensor in the conormal direction. So it's independent of that. And that tells you that basically you are degenerate on the surface and you are not degenerate when you move in the normal direction. Now let's check that for the heat equation. So what is the claim? If this holds, if we have this assumption, which we will call geometric, then the evolution omega 0, gamma 0, omega 0 complement is unique. So I will show you that. Watch the case. So meaning this question, this thing is unique. So it depends. It's independent of how you mark the initial set and you only do that. Okay, so a simple calculation. Why isn't it this geometric? Because the geometric property will be that lambda X plus mu P tensor P is independent of mu and positive homogenous of degree one in capital X. And you see that once you have this thing here, this cannot be like that no matter what. This has to be true for all X and P. And on the other hand, if this was the mean curvature motion, so if this was trace, it was, what did I do here? Correct. So if this was a trace, if our F of X and P was the trace of the identity minus P tensor P p square X, it's a simple exercise to check that F of X plus mu P cross P of P is the same as F of X and P. So exercise. It just kills this thing. And of course the positive homogeneity is automatic there. Okay, so now the real question is, why does this imply this uniqueness? So I like to show you that because it's different. It's the, when I teach, usually I go like that because I lose the origin. So I wanted to make sure there's one here. So that's what we're going to do. Notice I have hidden everything about, I'm not talking about regular solutions. These are just uniformly continuous solutions. So the equation here is described in a very weak sense. Not even almost everywhere. It's a very weak sense definition. But here is a lemma that F is geometric, implies that if U is a solution, as a matter of fact, it's equivalent. If U is a solution of star and you have a function phi from R into R, non-decreasing, then phi of U also solves star with initial data. So the result is that any increasing function of a solution remains a solution. And that follows its equivalent to geometric. And I will use this, which is a triviality to prove because assuming everything is smooth. Let's assume I'm dealing with things that are everywhere, they are smooth. They are not and one has to justify that in the viscosity sense and so on. Why is that? So again, for those who don't know this solution, which seems to be the majority of the people, viscosity solution theory allows me to do anything I can do when the thing is smooth, allows me to do it when it's not smooth. And so proof of that is let's call this, let me use the inverse, let's call U psi of E. It's easier to do the computation like that. Then I plug it in and I get psi prime of V VT equals F of psi prime of V D squared V. I'm just writing down the equation, so I'm writing the equation for U like that. Okay, from now I'm going to start forgetting the X and T. So this is the equation. So if U is some function psi of V, psi and U solves the star, V has to solve this equation. And now you see that things are simple. If it is geometric, this is not there, this is just a straight characterization you do when you compute the Hessian of the function, of the function of a function. And then you are left with psi prime, psi prime, psi prime, but the homogeneity, the positive homogeneity allows me now to take off this term and I got the same equation. And if you like to check it in... Now the statement that the heat equation is not geometric is very simple because if you do the same computation for the heat equation, you're going to get phi double prime DU squared, and there's nothing you can do with that. Okay, so again, if you compute for the heat equation, the same computation is going to give me psi, psi prime V VT equals psi prime V plus psi double prime DV squared. And now you see that you have a problem. If anybody can eliminate that, tell us because then those of us who give the lectures would not earn our money not to come here only but so far in our careers. So this doesn't go away, right? And that's the difference, but that's at the technical level. Now let me describe to you... Now let me give you a rigorous proof based again on comparison principles or whatever, why this property... Not that one, that is almost rigorous. Why that property implies that the level sets are independent of the initial data. Okay, so here's a proposition. So we assume always now... I'm not going to keep you writing. From now on we assume degenerative electricity geometric. If V and V solve star and U0 positive is the same as V0 positive, U00 is the same as V and therefore the third would be true, then UT positive is the same as V positive and so why is that true? We devise... We need to come up with a function that, if you like, takes positive sets, positive level sets of U0 to positive level sets of V0. So a map that takes the sets... Not the set, yeah, it takes the sets with certain level sets, two of U to level sets of V and if we manage to make such a change with an increasing function, so we let it go because now the U and V being solutions, if this increasing function is phi, then phi of V will be a solution but that means that phi of V will be positive when U is positive because this thing just measures the level sets. All right, so let me write it down and let me see whether I remember it. Let's call phi of T the infimum of V0 of Y of when U0 of Y is greater than or equal to T and since basically I'm going to prove the equality, I have to prove two inclusions, I have to do it the other way and this thing is the soup of U0 Y when V0 Y is less equal. Okay, so this is U0 and the positive, which of course is the same and this is also where V0 is positive. So what I do is I take a T, let's say T is positive, I take a T level set of U0 and I assign to that the smallest that phi of V0 can be there. So again, this is a T level set of U0 so here is U0 is bigger than T and I assign, I define as my function f of T to be the infimum, minimum whatever of the U0 of the V0 here and the same thing but the opposite for the supremum and why do I put infimum soup? Because if I'm outside the set, the sets will be unbounded and therefore the omega T or the omega 0 complement, the omega 0 bar complement will be unbounded so I have to put infimum soup. Now it's a simple exercise that phi and psi are non-decreasing. When I state this theorem here, I should have said non-decreasing continues. Non-decreasing but they are not continuous. In principle, they are upper and lower semi-continuous functions but let's forget that. Okay, I do that. Now where do I use the assumption that the level sets are the same? This comes from here. Phi of 0 is equal to psi of 0. This is a consequence because the 0, if I look at a place where U0 is strictly positive and I take the infimum of the V0 there, it's going to be 0 because V0 is greater than or equal to 0 in the same set. Okay, so then that implies, let me put it down the right way, implies that by comparison, this will imply that phi of V will be less equal to U less equal to psi of V Okay, there are two increasing functions. V is less equal to U initially because it's the infimum of the values there so I get this inequality and this gives us the answer. Okay, so check that. Comparison means we need to show that phi V0 is less equal to U0 is less equal to psi V0. Change that. So that's part of the definition. And then you plug in the comparison, this is true for T positive and then that implies the claim. Okay, I'll let you check that. But to do that was very important that these functions we built had that property. Okay, so far so good. We have a definition, we have a unique evolution in time of these things and it's normal to ask here what kind of sets are these omega t's? This set I call here, I said they are not manifolds, there is nothing in principle but what kind of sets are there? And so to do that, let's test some examples. So I assume that you heard about Gratian's theorem last time. Okay, so that if I have a nice convex curve or whatever and I let it move by mean curvature, that remains smooth till it disappears. So let me say like that, goes to a point that becomes empty. So you learned things like that last week. So right, so curve, moves by that omega t, boundary of omega t smooth and it goes eventually reduces to a point and disappears. So that's the good case. So I'm not going to draw pictures like that. I'll draw some but let's go now to the next case which again is like that. I have a ball, I let it move by mean curvature. This is a special case of that. It disappears and as a matter of fact for here the motion if I have a ball of radius at zero moving by mean curvature means that rt is minus n minus one over r, d minus one over r. If you didn't see that last week, compute it. Okay, so things are great. Now, let's say I put two balls arbitrary close to each other. I mean first I put them very close to each other. Very, very cool. Epsilon distance away from each other. This will shrink, that will shrink. So this will go to a smaller ball, a smaller ball and then if they have the same radii like that. Okay, because each one of the two balls moves independently of the other. Let me take the first example and I'm going to draw something that doesn't look like convex but let's assume I have this picture. So these are, these still are balls. Okay, but they're very close to each other. So, and I let it move by mean curvature so what is going to happen? This is going to pop out and then we'll continue moving. We know what happens from there. So what if we are in a critical situation where this thing is zero and that thing is zero? So what happens if I am on that situation? So I have this figure eight and I let it move by mean curvature. Okay, and let's assume that moving by mean curvature has a mind can think. So if you are here and you want to move by mean curvature, you are confused. You don't know which one is the normal. How to go. Do you think of this as being like that or like that? And so it's like an intersection. You don't know which way to go. Some of you go that way. Some of you go the other way. So that means that if we let this... So if I take now my function u0 to be positive here all I will find is a picture like that. u will be zero here, u will be positive here there and u will be negative there. So what we have here is interior. And of course that is not very promising because you're going to ask, okay, but what happens everywhere here is it turns out that any set you can put here also moves by mean curvature, but I had to use a different definition. So let's... Okay, so you get something that is... I won't call it... It's not satisfactory. You say something like that and a lot of work has been... was done to figure out when things have interior or they don't and I will give you some examples in the next lecture and I think the most general result is by Sigurd and if you want to explain that to you... Okay, so the question is do you have interior? Is it 55 minutes or an hour or what? Is this clear? The issue of interior? And you can think of other pathological situations like two crossing lines where again you have the same problem. You don't know what happens and as a matter of fact if you let this thing by move by mean curvature you are going to see this phenomenon and the explanation is the same you can think of this as being two half lines I mean two intersecting lines like that or two intersecting lines like that or everything I'm saying here there are results, okay? This is not just drawing pictures and notice I'm describing mean curvature I keep saying mean curvature if I put a velocity if I put a normal velocity if I put an extra term if I put mean curvature plus something you don't have this picture because the plus something knows how to go, right? I promise to say some things that are newer so let me do something very dramatic now suppose and of course you don't know what that means at all but suppose I have mean curvature I said if I put a velocity I know how to go if I put a velocity I will put a small velocity here epsilon but I will put here Brownian motion in time okay so Brownian motion how many people know what the Brownian motion is? okay so that's better so it is from the analysis point of view is the prototypical example of a function which is continuous and nowhere differentiable from then of course it has some probabilistic properties that are important and so on now even how to make sense out of this it's problematic because that will amount to taking the equation the level set PDE that corresponds to that will be and this is now a stochastic PDE and things of that let's not worry about that there is a theory for this okay there is a viscosity theory for that but that's the equation okay so what happens to this motion if I do that what is the effect of putting here a Brownian motion? it's the effect of putting something that changes signs plus minus so it gives a direction but also does it in a very dramatic way because basically what I'm doing is I'm adding to the velocity plus minus infinity right Brownian motion is nowhere differentiable so the b dot is either plus or minus infinity at every point so I'm putting here something extreme where is the picture so I take these two balls and I put something very extreme at that point like that or like that but huge so what happens here and I'm willing to put a small huge thing epsilon is any epsilon so what happens is you let this move by mean curvature and things are good there is no interior I put it epsilon because it depends on epsilon the epsilon is because of this epsilon that's very dramatic I have something that looks okay I mean we know that it's pathological the two balls touching move by mean curvature I add an epsilon I add something that is really huge and somehow it selects one of the two I would call that a stochastic selection principle and you see that in many applications in analysis when you have diffusion so let's say you have a potential with a minimum and you perturb the motion by a small Brownian motion this thing is going to start going like that and eventually will go over it, over the hump and it will concentrate on the minimum so I'm trying to say suppose you have a potential like that and you move then you have a particle that's moving then you can get stuck here or there but even if you are here and you start putting something around then eventually it will go over there and you will go down to here and somehow this is the picture here so why is this happening this happening for the following reason the two balls think that they either like that and they start coming inside but they are coming inside like crazy because the equation there will be this plus epsilon db so you have to solve a stochastic ODE something will oscillate they will think they are like that so they will keep going up and down and then a miracle happens which has to do with the scaling properties of the Brownian motion so there has to be some probability in it that by the time these things have come in and come out again because as I said they go in and out the time they come back and they want to close they get stuck by this because this takes longer takes longer to close so it goes up and it comes down again and it's open here but the two balls these things start to come in and they cannot go through, there is a barrier so they are stuck and therefore they keep moving this way and this has to do with the scaling properties it has to do with something that goes like that and they are scaling the factor Brownian motion scales like t square root so it has to do with this property and so it never closes and that's why we get this and now if we let epsilon go to zero this will converge to the gamma t that was here the maximum so everything all these things have proofs but this is going to be, this is a more complicated, I don't even know where I am this is a more complicated thing to prove but at least I'm trying to give you the idea and okay so there is this issue of interior and of course we are not going to exclude it by putting the epsilon here so the theory at least from in curvature was like we know there are some pathological configurations like two touching balls or two crossing planes and if we let them move by mean curvature we know there is interior and then for a while there was this classical question is if you start with smooth but the complicated smooth not the convex or something can you generate interior and so I will give you a sufficient condition not to generate interior that goes back to one of these in quotes conjectures of the Georgie and then of course the final result was proved by Sigurd and Tom Ilmanen and that was that indeed you can start with a smooth surface that comes in finite time very close to a minimal cone which is unstable and then it comes very close to something like that and then this takes over and creates interior so I mentioned there were conjectures of the Georgie so that's something that you may remember there was a meeting in Trento with the Georgie who is a god needily and everywhere else and the Georgie did not speak English but I have the feeling he understood English and he had two interpreters at that point it was Ambrosio Ambrosio and Giovanni they were his interpreters so when the Georgie went to say something he would call them nearby talk to them in Italian and then either Luigi or Giovanni would get up and say what professor did Georgie say and would go like that so for some reason towards that time in his career he had become interested in again motion by mean curvature and in this phenomenological description and the connections with minimal surfaces and so on and so he had posed a number of questions and for those of us who were younger at that time we wanted to work on this the Georgie conjectures so we are sitting in a big table at Trento and that's Villa we were up there and someone I don't remember who but there was me there used the word the Georgie conjecture one of the Georgie conjectures so one of the Georgie conjectures was this thing about the interior I will show you another one and the Georgie points to I don't remember whether it was you or Giovanni or Luigi they come nearby tells them blah blah blah and so then one of the two says professor Georgie says that he does not like to call these conjectures he likes to call them questions and if you want him to call them conjectures you have to ask him what the probability he assigns on them be true or not and of course all of us decided not to ask him because we wanted to work on the the Georgie conjectures I think one of them was this famous the Georgie conjecture which we don't even know what number the one about Laplacian equals f of u and we don't even know what number the Georgie assigned to it but like them everybody else forget it so this generation of interior and so one question of the Georgie was the following can you create interior if your surface if your initial surface is invariant in quotes under all possible geometric transformations so if you have any if you have a set I'm talking about mean curvature rotate it if you have a surf initial function u0 a set which had the property that if you rotate it translate it and dilate it does not hit nearby sets so if I have a u0 it's hard to to draw something like that but think of having some these level sets that are close to each other so these are level sets of the same function and the georgies one of the questions was if these things are such that even if when you rotate them squeeze them dilate them or move them in time if they don't touch each other so there is no interior so if gamma zero is invariant under rotations dilations and translations in space time so no interior for me I didn't give a formal definition means that the set where u is zero doesn't have it's not an open set I'm sure it cannot be open set no interior means that the boundary of the set where u is greater equals zero is the same as the boundary of the set where u is positive and the boundary of the set where u is equal to zero is equal to the boundary of the set where u is negative that's the mathematical definition of no interior and it turns out that if you had under good conditions that's equivalent to saying that the set x t x t equals zero has no interior if that's the case this is for sure so let me show you actually I will start with this the next hour I want to draw one more picture so another of the questions I think was a question of the Georgia about this yes here so it's hard to draw it but it says that if you when you take a gamma zero and these are okay so this is u zero and this is u minus epsilon u epsilon nearby sets and if you take gamma zero and you rotate it it doesn't hit any other level set if you rotate it so there's some order on the way they are rotating one rotates everything and if if you dilate one let's say they're star shaped and you dilate one you don't hit another one and okay and all these things I will tell you what they mean analytically it's not an issue one thing that maybe you heard about it has to do with a set of if you have a set with positive in curvature initially it remains nice maybe someone mentioned that and that's the same as saying that in translation in time you don't have this property so it's like assuming that ut equals zero is positive and that takes you because it stays positive and then you have so I will state this before if I went to do at the first order velocity if I had this picture so if I'm moving by velocity a of x only and t so my geometric PD is this plus minus now let's not worry about that so now this is the case of a first order motion there's no there's a similar result there is a result that says that if a has a fixed sign there is no interior because if it has a fixed sign this will keep increasing or decreasing or and so there are two results a of x t fixed sign or a of x or no assumption so if a of x independent of time and of t there is no interior and so that's for the first order motion but there is a counter example if you look at this motion in this equation x minus t ux now notice there is a coefficient here that changes sign and there is interior so I will start with these things in the afternoon lecture so to conclude let me draw do another thing with a picture and the fascination of this interior led to the following question again and I think that was a conjection of the Georgian maybe interior is generated if you have a motion that has a completely different topological structure in one regime and the other and maybe when things come together the interior has to happen there and this was a little bit motivated by the this idea of the figure 8 of the two balls come together so here was another example that suppose you have a torus but it's a very thin torus thin means this is not thin thin means like that and you let this thing move by mean curvature so these are dotted lines there was a time where I knew what these things had to be called but let's say this moves by mean curvature so what happens this circle wants to close this circle wants to close but this is much bigger than that so this is going to close before the other so if you let this thing moving by mean curvature you are going to get an even thinner thing and at some point you are going to get a circle so of course it will disappear and everything is nice now suppose you have a very fat torus these are the bagels you like to buy if you are in the US they give you more food so you have a very fat torus so this circle is very small compared to the other then you are going to have the opposite behavior the one in the middle will close before everything else so this fat torus will become basically a pink sphere at some point and then it will open up, it will become something like a sphere and it will keep moving and so the whole idea was and this depends on the ratio of the radii and so the whole idea was that at that point at this critical radii you are going to have to see interior because somehow the idea was that one wants to go this way the other wants to go that way and unfortunately this was not the case because what happens is that these two phenomena the pink sphere or the circle happen exactly at the same time so there is some kind of solution that comes in and so this didn't work as an example and that's why one needed this very nice example of cigarette with Ilmenin and I think David Kroep so I went further I am going to start and then I will do the distance function and so I will try to show you a asymptotic reaction diffusion example and then go again back to the definition any questions? ok you want to thank me first