 Thank you very much. It's a tremendous honor to be here at this conference devoted to Frank Mel's birthday. You are very young. And I don't have to repeat what people before me so eloquently said how influential Frank was, but I do want to say a few personal things. I was lucky to join the University of Chicago in 2005 and this was around the time when I think you started coming. So let me try and borrow just for a minute or two a page out of Thierry's book, the Quantum Mechanical Talk, and so let Psy of TX be Frank's wave function. So we're all complicated people. We're complex. Of course, this is complex value. That is why we need a self-adjoint operator. And so we measure Frank as a function of time like this. So this is, let me abbreviate, Frank, and so how do we pick this self-adjoint operator? Well, let this be Eckhart Hall at the University of Chicago. So let's take a smooth bump function around Eckhart Hall. And then these are the observations I made about Frank's visits as function of time you would come. Then there was an exponential tail and at the beginning I thought this is a periodic function in time, but this is too easy. So I never figured out the law of your visits whether they were random or quasi-periodic. But interestingly enough, even when you weren't there, this was positive. So you tunneled from Paris and then it's a unique continuation principle and I don't know. Maybe we should have a chat as to sometimes you came in the spring, but then you come back in the fall sometimes, and then you disappeared for three weeks. So there was a double resonance. There was a conservation law, namely if you integrated over one year period and this was, I think, a conserved quantity. By the way, coming back to this, I have a little theory you might have noticed there's so much fake news in the world. Where does it come from? It comes from non-self-adjoint operators because if A star is minus A, then this is imaginary. So the real part is zero. So if the real part is zero, it's completely fake. All right, well, enough of this. It's hard to follow Simon in a geometry talk, especially for a non-geometer like myself. I'll still try. So I'll talk about the ancient relative of the Ricci flow that Simon presented. This is the gradient flow of the Dirichlet energy and I will only be in two dimensions. This is the critical dimension for the for the heat flow. Energy is conformally invariant in this case. The gradient flow was introduced a long time ago by Ilse and Samson for geometric reasons. I think they wanted to study harmonic maps in fixed-homotopy classes. And so here is the heat equation. UT is the projection of the Laplacian. We view U as a map into a Euclidean space, so it's a vector. The Laplacian is the standard Laplacian, the PD Laplacian. The nonlinear term you get by projecting onto the tangent space at U as usual. And for the purposes of this talk, it's enough if you consider U0 to be smooth initial data. The right-hand side of this heat flow, the harmonic map heat flow, is called the tension. And as is typical in heat type equations, you also see the same if you take, say, a damped dispersive equation. It's dissipation of energy. This will play a huge role. Years ago, Struve, 85, showed that this has an evolution in a certain sense. He identified the singularity, so more about this in a moment. The stationary time independent solutions that somehow we expect to converge to in infinite time, if exists in infinite time harmonic maps. So here is Struve's theorem. He stated this for general manifolds. We will stick to the R2 and S2, but it's stated here a bit more generally. So take initial data in H.1. N is viewed as an embedded manifold, so you don't have to worry here. We don't have an L infinity continuous embedding from H1 to L infinity. You don't need it. You can define H.1 as functions that almost everywhere take their values in the manifold, since we're viewing N as embedded. Then Struve noted that, say, for smooth data, you have a global smooth evolution, but up to finitely many singularities at which he identified loss of compactness and regularity, and that is through energy concentration. I should really point out here that the names will appear later on the slide, but it was Karen Ulmbeck who discovered this phenomenon of bubbling, and I recently watched at least a brief moment of a YouTube presentation that they had at the Institute for Advanced Study on Fluoromology, and the question was moderated. The talk was by Helmut Opfel, and then Karen spoke and she said, I discovered bubbling, and to this day it's hard. But for the young people, she said, they grow up with it, so it's like second nature, but this is really, she's right, this is an amazing phenomenon, and as we heard all week plays a huge role in the theory that Kallus and Frank developed. Singularities can indeed form. This is this well-known paper by Changding and Ye 92. They indeed constructed finite time blow-up. Later we will see also the work of Raphael and Schweher on blow-up. So an absolutely fundamental theorem here is this Ching bubbling theorem from, is it 95? What it does, it says that along a sequence of times, so this is the analog, if you want, of what Kallus explained in Sergei Pontoise on Wednesday, the sequential-solidon resolution. This is sequential-solidon resolution. So you go up to, say, the first singularity in time, it's located at x0, and then along a carefully chosen sequence of times, where does this come from? It comes from this energy dissipation inequality. The fact that your gradient is L2 in both space and time. So you then pick a sequence, we will have this on the later side where the tension vanishes, and when the tension vanishes, that gives you elliptic control. It's a paleo-smale condition. So as it made a theorem, it says that you have harmonic spheres. These are harmonic maps from R2 to S2. But you can resolve the singularity at infinity. And then the energy splits. So you have equi-partition of the energy. U t0 would be the weak limit of the harmonic map heat flow into time t0 in the energy sense. And then you have these bubbles of energy. S2 here is special because the energy is quantized. So it will be 4 pi times the absolute value of the degree. Depending on the orientation, it can be positive or negative. And moreover, you have this typical description of what bubbling looks like. So the energy must concentrate in these bubbles. Those are the omega k. You subtract the value at infinity. And then you have an error in the energy sense. And this is a local, only a local statement. This is false globally. Globally, the harmonic map heat flow does not satisfy the paleo-smale condition, but locally. These scales lambda kn have to tend to zero. And the centers akn have to tend to this one singularity. And there is an orthogonality condition. And this orthogonality condition is the well-known one. That features exactly the same as, say, in Pira-Felstog yesterday, where he had the elliptic profile decomposition. So here is the orthogonality of the scales. It's either the scales lambda on the previous slide, so either the lambdas are radically different, or if the lambdas are comparable, then these centers have to be radic. They have to diverge from each other relative to these scales. And so these pictures are taken directly, cut and pasted out of Ching's 95 paper. Let me try and explain. They might look a bit counter-intuitive because our harmonic map heat flow takes its value in the sphere. Nevertheless, Ching drew kind of a big sphere with spheres attached. This is how this bubbling is visualized. It's not to be taken literally. And the key in Ching's theorem, which then was followed up by numerous papers, they will be listed in a moment, is to show that no energy can concentrate in the next. In the next, the energy has to evacuate. This particular picture from Ching shows comparable scales and separate centers. The next one shows the same center but radically different scales. This is bubble on bubble. So a word about harmonic maps, even though this has more information than I need in my work with Jander and Laurie, we don't really, at the moment, need the description by rational functions. But let's briefly review these remarkable properties. In two-dimension only, do you have weakly harmonic maps being equal to classically harmonic maps? Also very surprising to me at first was that critical points of the energy have to be minimized in the homotopy class. Moreover, from the harmonic map equation, which is a second-order equation, you can deduce, seems magical, first order Cauchy-Riemann system, the plus and minus, don't ask me which is which. One is conformal, the other one is anti-conformal. And, as I said here, they are the unique minimizers in the homotopy class of the energy. The energy's quantize is 4 pi times the absolute value of the degree. And moreover, you can say why, because you see Cauchy-Riemann, it means you're conformal from the sphere to the sphere. It's well known what that is. Those are the rational functions precisely. And so the degree is the maximum of the degrees of the reduced numerator denominator polynomial. And moreover, there is something remarkable. It's on the bottom of the page. Let me start with that. That's the Bogomolmi identity. It says that the axis of the energy over 4 pi times the absolute value of the degree. So this is written here, I suppose, for the positive orientations of 4 pi degree. The degree is exactly the L2 failure of Cauchy-Riemann. All right? So the first integral vanishes if and only if you satisfy Cauchy-Riemann. What's on the top of the page is how do you do this? Well, there is the hardest part turns out to be this regularity theorem. What you need to prove is that weekly harmonic maps are continuous. If you look in Helen's book, you will see that he then says that the rest is kind of standard elliptic regularity. This is quite interesting. And the way I understand it is that you have this basically have an equation Laplace F equals G. If G is in L1, just think of the fundamental solution, which is a delta. It's the limit of L1 approximate identity. So you can't hope to have L infinity control from L1. But if this is in the Hardy space, H1, then in two dimensions, then you have an L infinity bound. And that's not easy. This has a strong resonance here with France. This Coifman-Leon's Mayor Sem's paper, absolutely remarkable. The difficult structure and they discovered how the nonlinearity actually has this compensated compactness. Structure and allows you to prove that this actually is highly non-trivial in the Hardy space. As it may, there is a geometric quantity, the Hopf quadratic differential. The quadratic differential is only by design, do you add the dz squared so that it scales. By scaling, this is a quadratic differential. The function phi is holomorphic. And because of compactness, this Hopf differential has to vanish. That is where you get the Cauchy-Riemann from. So just like in both of your work, there's a Liouville theorem here that is absolutely crucial. That turns a second order equation into first order one. And so on this slide, what I try to do is to state kind of the state of the art, too many states. So that is the sequential soliton resolution theorem that we use. Notice that it improves upon, by work of the authors listed on the bottom, the energy also to L infinity. So remarkably, you can show that this resolution that u n is what, the omega 0, that's a harmonic map. It's the weak limit by a suitable rescaling by rho n plus the other bubbles. This converges to 0 locally on disks, but also uniformly in L infinity. So, and notice the condition on the top of the page. What is t u n? That is the tension, that is the right hand side of the harmonic map. Heat flow is Laplacian u plus u squared u, the projection of the Laplacian onto the tension plane. And most important is this rho n that sits in front of it. That is the scaling. And this scaling is, you can view it in space and time. But because of the parabolic nature, of course, you can move from one to the other, from time to space. And what you see here is that you get this bubbling decomposition on disks of size rho n. Needless to say, the top condition, the tension condition, you can multiply by a sequence, r intending to infinity, it will still hold along a subsequence, everything up to subsequences. And so that is where the big r intending to infinity comes from. It gives you some leeway, you need this flexibility. And so you have these harmonic maps, they are non-constant of course. You have orthogonality of scales as before, either the mu j n have to diverge from each other, or the centers have to diverge relative to the mu j n. Notice that the top condition is vacuously satisfied if the u n happen to be harmonic maps to begin with. And the tension vanishes, that means that theorem will hold for any rho n. And in fact, bubbling for harmonic maps was done, this name is perhaps missing down there, by Parker in the early 90s. He proceeded, Ching was simultaneous with Ching and he did this for bubbling of harmonic maps themselves. The third bullet point down there says that of course you want to stay away from the boundary, you finagle always in such a way that you don't hit the boundary of the disk where you're actually working. The last bullet point is fundamentally important, that's the property of the sphere, that we can talk about the number of bubbles K, just by dividing the energy by 4 pi. That is fundamentally important. Okay, so this is sequential soliton resolution, both in H1 and L infinity. Very noteworthy I already mentioned, Ching on the bottom, but then Ding Tian, no, Ching Tian revisited the energy and redid a certain vanishing of energies in the neck by different methods. Very important but not surprising to the expert, there's some Poesive identity actually, you integrate by parts by extra grad in many of you. So in this community you speak of this as virial, we speak of this as virial, the elliptic is Poesive. Ching Tian were the ones to the best of my knowledge who added the point-wise, the necklessness, which geometrically is very important because it says that the image is connected, you have a connected image in the limit. So here is a childish rendition of what the geometry looks like, you have the big disk, this is Aaron Rowan, and inside you would have the scales of the harmonic bubbles. So this is a picture in R2, this is not in the target, this is in the pre-image, and you can have bubbles inside of bubbles, it's complicated, just like we're used to also from dispersive equations. So what this slide attempts to explain is how to use two, which is this all-important dissipation of the energy, it's this, this is the key, just like if you have a damped dispersive equation, you will have this property, but for you index T. So this is absolutely crucial, and you see now where the sequences of times comes from in Ching, in the previous slide, the bubbling slide, well you, if you have two and time is infinite, then you just pick a sequence of times such that that root Tn times L2 norm tends to zero, if it's finite then you have this other limit, then you apply the sequential soliton resolution, and you see that you can do this at these scales, globally in time you get the parabolic scale root Tn, or you get the reverse parabolic scale root T minus Tn, and there you have the palace male condition, and you can bubble both in energy and uniformly along those sequences of times at these scales. So here are some open problems, so the best of my knowledge open to this day, I mentioned in these fundamental 90s papers, is the body map in finite time which is a weak limit continuous, in fact topping proved that if you pick this very sensitive to the target, if you pick a non-analytic target it fails, you also lose the uniqueness if you mess with the target, then there is the question of concentration of energy, are these points unique or do they depend on the sequence of times? You see this is really the underlying issue here, is that the 90s literature only controls this process by a sequence of times, I showed you how to select them from energy dissipation, that is how you get your palace male, and then everything depends on the sequence of times you chose, and you immediately have these uniqueness problems. Topping in Warwick made fundamental contributions, I tried to sketch here in that theorem some of his results, for example he showed that if all the bubbles have the same orientation, then you have a uniqueness of these points. Okay, so now what we really after is not imposing any conditions, but before I do that let me state this very recent theorem of Yadzik Yende and Andrew Lurie about equivariant harmonic map heat flow. I should be quick to add that these results by Yadzik and Andy are rooted in their work on wave maps. So even though one might think one should start with harmonic maps and then go to wave maps, it was the exact opposite, that the harder problem was addressed first, and the ideas they developed there are absolutely essential for this work that I'm describing for harmonic map heat flow. So what did they prove? They proved that if you have an equivariant solution that means the usual thing, you have the sphere and you introduce this azimuth angle, I believe, that is called psi, and the equivariant condition simply means is that the action of SO2 on the domain commutes through into the target. And the commute is not completely correct, that's for k equals 1, if you have higher k then you wrap around k times, that's the equivariant k case. And before I had mentioned that harmonic maps have rational functions, so if they're equivariant rational functions, there can only be two kinds, this and z bar to the k, orientation preserving, orientation reversing, and if you write them in this angle, then they take the form 2 arg 10 r to the k, and so this is what they proved. They proved that you have uniformly in time, in continuous time, you have these scales and your solution to the harmonic map heat field decomposes, this is for infinite time, into a sum of bubbles. If you do the calculation just with the trigonometric identities here, you'll see that irrespective of the plus or minus doesn't change the degree. Think about that because if you change the sign of theta in the first line, you reverse the sign of psi in the first line, you reverse the sign of the first two components, so the determinant is 1, minus 1, minus 1, 1. If you look at this more carefully with the subsequent bubbles, you'll see that they all have opposite orientation, so any two subsequent equivariant bubbles have opposite orientation, so this is the opposite of topping. Topping wanted equal orientation. All right. A bit of the history, important papers. So van der Hout did the case of finite time and you don't have bubble towers, so one bubble. Del Piniu-Mousseau-Vey looked at the energy critical heat equation power nonlinearity. More relevant for this talk is the work of Davila Del Piniu-Vey, who constructed, of course, non-equivariantly on a bounded domain with parabolic Dirichlet conditions. They constructed blow-up solutions with several degree one maps glued together. And let me start with Gustavson-Nakanishi. Tsai, they showed some years ago stability of the harmonic maps for k greater equal 3, so in particular no blow-up, k is the equivalence class, k equals 2, they showed the existence of what they called eternally oscillating solutions. k equals one they left open that was filled in by Raphael and Schreyer, who constructed by QB plus Epsilon techniques blow-up solutions, stable, in a certain sense, stable blow-up solutions. So now what we're after, and this will take some work, is to describe this new work. It's been an archive for a couple of weeks, so with Jacek, who's here, and Rulori, fundamentally builds on their wave map paper using the idea of a minimal collision energy which will come in a moment. But in order to kind of set the stage, we need to get a handle on these complicated harmonic maps. We will not use the structure of rational functions, at least not yet. I should emphasize we don't do modulation theory. We tried, that is hard. So we're building that wave function. That's a hard one. You'll see what the theorem says in a moment. So let's describe the scale associated with the harmonic map. Not so surprising you go to such a big disk with a suitable center that you take 99.9% of the energy say, or 90% of the energy of the whole thing. That will define a natural scale associated with the harmonic map. It's a bit delicate, be careful, because you might have a Degree 2 map that has two Degree 1s sitting far apart. Then the scale would be this. All right? But you will see later in the talk that we can also look at these individually. Because we will look at what we call multi-bubble configurations. Now clearly, so center and scale, all right, associated with the harmonic map. And an m-bubble configuration is what I wrote there. Our notation is script Q here, omega x. And omega 0 is a constant. The omega j have non-vanishing energy, and we subtract as the authors in the 90s showed you have to do the value at infinity. And the constant maps here have to be allowed. This is the case of m equals 0 when the sum vanishes. So what we're going to try and do, you'll see the theorem in a moment, is to say that continuously in time the harmonic map heat flow remains close to multi-bubble configuration. Approach is a multi-bubble configuration for all times, not for a well-chosen sequence of time. And what does it mean to be close to multi-bubble configuration? So we have to introduce a distance, D, that measures the distance. And as you saw on the sequential solid and resolution slide, we have two senses in which we do this, energy and L infinity. L infinity is, of course, the harder one. So let me unravel this for you. I tried hard here. We tried hard with Jatzig and Andy to present this in a way that is least repulsive. So the first bullet point is not a surprise. You want on, of course, on the scale rho. Everything's local. The energy of u to be close to multi-bubble configuration. First bullet point, not a surprise. Then let me show you this very simple geometry. It's kind of a trivial picture. So the red one is dy rho. That is where I just showed you we want the energy of u to be globally on the disk close to the multi-bubble configuration. Then you need to give yourself wiggle room. In fact, you give yourself far away this blue disk, the outer blue disk, and then you zoom into the green disk. All the fine structure will be in the green disk. And this annulus, this very fat annulus between green and blue, we demand, and we will propagate this, of course, there will be propagation estimates in a moment, that you are close to a constant. So all the action in the green disk and on this annulus, you're close to a constant in which sense, that's the second bullet point, that in the L infinity sense, you're close to a constant and the energy is almost zero on that annular region. And the third bullet point just says in formulas what I said in words, that green is tiny and blue is enormous and the red one, which is the actual scale, rho is in the middle. Then as before, you have orthogonality of the bubbles, scales, lambda and centers are diverging. Notice there is no n. So what I'm describing to you is what does it mean for the distance to be very small? Well, it means that these orthogonality conditions give you very big numbers, right? So lambda, like you both do in your in your invention is in octopip and so on, and how the elliptic people do this. Separation from exterior neck, well, of course, you want your scale, that's the natural scale of the harmonic map to be far away from these, the boundaries, no surprises there. Then that's a bit delicate. How do we handle in our theorem about to be stated the uniformity? We cannot uniformly handle uniformity. We only handle it on a Swiss cheese region. So we have a bit of a complicated geometry. We then have to solve the heat equation on Swiss cheese region. So what's a Swiss cheese region? Here I tried showing this to you. So notice the green, this the tiny one in the middle is where all the action is in terms of bubbling. So I zoomed into the green. Then inside of the green you have more bubbles and the red ones are the actual harmonic maps drawn in their position and scale and you surround them by disks of equal radius. Congruent disks is perhaps what I was looking for. So disks of equal radius and it doesn't matter that the actual scale of the harmonic maps inside is perhaps much smaller. And notice here I allow collisions of these disks. Later you remove them by a covering argument. Of course you would want everything to be well separated but you can do that. So in terms of formulas, the DJ star is you take the disk of the harmonic map omega j, you remove the inner life coming from the inner bubbles but you don't remove it at the scale of the inner bubbles. So you get at the, not concentric, but congruent scale psi j. The picture showed you what I mean from this. Again you want separation from the boundaries. You never want to collide with the boundaries. And then finally if you put all these energy pieces, L infinity pieces together, L infinity only on Swiss cheese, do you get the delta? Delta is the infimum of all possible multi-bubble configurations. And then the theorem that we posted a few weeks ago says this, that if you take as smooth solutions of harmonic map heat flow up to the maximum time, that's where you cease to be smooth, that's where the first Stuve singularity sits, then that delta will tend to zero uniformly in time. And if the blow-up time is finite then you do this on the inverted parabolic scale. Here is the time t plus, time infinite you do it on the actual parabolic scale. That's root t. Moreover this is very important. You can localize in the disk. So that's what the yn is. So you have translation invariance so you can pick your centers to be arbitrary. And so what's the meaning of this? The meaning is that uniformly in time you remain close to multi-bubble configuration. And on the suitable scales, on these parabolic scales, but we do not control, we don't solve the uniqueness problem, we don't solve the dynamics of the bubbles. That we do not control, that requires some form of modulation. So as I said there is an analogous result in the infinite time. We do not solve the uniqueness. What does the theorem says? It says really that there are no destructive bubble collisions. We will say more about this later. And in particular you can, as a corollary, this is our theorem two in the paper. If you open it you'll see it's preceded by theorem one which says from every sequence of times you can pick a subsequence along which you bubble. There's no qualifier here. No mention of a palace male condition. So as Pierre Raphael so eloquently put yesterday, every proof starts with assume it's false, right? How else could we do this? This is a large data result. It's not perturbative. So you prove by contradiction, assume that you move away from these MBCs. These are multi-bubble configurations, but infinitely many times. So this way of thinking was so important for us. It really comes from dynamical systems, not to take too much time here, but if you look at the dynamical systems literature, in particular also from the 90s, something like Brunowski-Polochic, preceded by Chen Heal and Tan. What they look at is invariant manifold theory called the eases, so flows and maps, but in infinite dimensions there is a linearized operator with a gap condition and so on and so forth, leading to a decomposition into stable, unstable and center manifold. And then this is an equilibrium, obviously. And then they say, imagine you have a trajectory gamma which contains a point P, the P being the equilibrium, and here is your trajectory. Then they want to know, this is called a convergence theorem in the dynamics literature. When is omega gamma the singleton consisting only of P? When does that happen? This is a fundamental question, and as you can imagine, often you just have to answer this case by case. This is exactly what it means to go from sequential solid and resolution to continuous in time solid and resolution. It's the exact analog of this. And so what Brunowski-Polochic say, he started out going into too many details if the flow restricted to the center manifold is stable, orbitally stable, and if the unstable manifold has some compactness, for example, it's finite dimensional, then if this fails, there has to be another point in the omega limit set on the unstable manifold. This is quite natural because so this is stable, unstable. If this is orbitally stable, then your trajectory should come infinitely often close to a point on the unstable manifold. And if you happen to know that every limit point, every point in the omega limit set has to be an equilibrium, well then you're done because you can't have an equilibrium on the unstable manifold. Arguments of this type, this is truly the Brunowski-Polochic theorem from 92 that I just described to you. It's a very short paper, but it builds on, you know, the construction of Chen Hel-Tan that actually do have these manifolds. They build them by Lapunov-Perron method. All right, so this is of no help to us. We don't have such an invariant manifold, so just in the back of our minds we had these concepts floating around. All right, so if the theorem fails, this is the fourth bullet point, then infinitely often you have to stay away from zero from this delta, this infimum of all possible configurations by an amount eta. And then the key is that by energy dissipation and the sequential soliton resolution you could then find a sequence of times which is not too far apart. That's the Tn minus sigma n much less than rho n squared, parabolic scaling, so that along the sigma n you come arbitrary close to the multi-soliton, multi-bubble configurations. That's how you can think of it as a manifold, multi-bubble configuration. So we approach this manifold arbitrary closely infinitely many times, and then you have this kind of situation which you have to lead to a contradiction. So no infinitely many trips. And absolutely crucial to now lead this to a contradiction, which takes some time, is this notion of collision interval and minimal collision energy, which is exactly what Jenda and Laurie had in their WaveMap's paper. So where is where the estimates? You need some kind of quantitative control. Well, I will be very brief here. To all of you, this is kind of obvious. Do you integrate the heat equation by parts against ut phi squared? What could be easier? Ut is a tangent field. Then the non-linear term, is it somewhere no, drops out. And you get an energy stability type estimate. Somewhat surprising is, that you can also go backward in time, but you're not solving backward in time which you cannot do for heat equation. But the second inequality which seems to go in the wrong way that says that at a previous time your energy is controlled by the later time. So that's backwards. But you pay a price of course, which is the difference in energies which drops out in the first line which is decreasing in time. So this controls the energy on parabolic regions and it also tells you, this feature prominently in previous talks, in the dispersive setting, you cannot have concentration in the self-similar region. This is a universal property that one checks case by case of critical equations. This is the source for example in, I shouldn't be repeating these things, log-log blow-up of Fong and Pierre Fehl. You have a root T minus little T while Schödinger and heat have the same parabolic scaling, but then you have to have a log factor somewhere because you can't have the quadratic self-similar, parabolic self-similar type concentration. Somewhat surprising is that we rely on parabolic Strichert's estimates. What on earth is that? I told you that this heat flow paper draws substantially on the wave map literature and this is no exception. So where does this come from? This comes from Terry Tal's analysis of his own caloric gauge. We do not use the caloric gauge, we have no need for a gauge, but we use this piece and the proof is so simple, this is our proof, it avoids dyadic decomposition, it's so simple that I'll walk you through it. So it's called Strichert's because you do it the same way you do TT star, so look at the second displayed, let's look at the first, what does the lemma says? It says that if you take L2 data and right-hand side L1 in time, L2 in space, then you get L2 in time, L infinity in space control of the solution. So it's Strichert's. Where does this come from? Well do TT star, you get the formula there e to the t plus s Laplace because you're in two dimensions, you put L infinity inside, then of course you get from the fundamental solution of the heat flow, you get t plus s to the minus one and you immediately recognize this is a hunk of transform, this is a nice exercise for students, they go crazy with it because if you substitute s equals t u, then the t goes into the forcing f and then use Minkowski and you get something integrable in u because you get 1 plus u inverse then u minus a half and the students, it drives them crazy if you don't know this trick how would you think of it and so this is trivial, you put this L2 in time and then you use duality, you're done. So this is Tao's parabolic Strichert's and for us it's a little bit more difficult because we need this on a Swiss cheese region, we lose control in point wise sense in terms of the inner life of the harmonic bubbles, they might have bubbling inside, we eliminate that with the Swiss cheese region so here is the statement, I hope it's not too technical so assume you take data u and 0 that are close to harmonic map omega on a Swiss cheese and you can take big L equals 0 then you remove no holes, it's also allowed you might have zero holes then there's this dichotomy here if you have no holes then your parabolic scaling works, your time tau n over which you have control goes all the way up to the spatial scale Rn of the big disk squared otherwise you can only go up to time epsilon n squared where epsilon n is the inner life that's the micro scale of the bubbles how do you prove this by a three step argument you first use the contraction in infinity then to control the right hand side nonlinearities you use tau's parabolic strichels and then you use stuwe non-concentration energy L4 regularity improvement alright, so this is small energy h2 control at the end so here is this central notion that's due to my two young collaborators the notion of minimal collision energy, this is a key definition, okay let me try and work you through it so this is about two sequences of times, the sequences of times are called sigma n tau n they approach if you want infinity if t plus is infinite if t plus is finite say 1 then they approach 1 and you have this infinite sequence of them, we call the smaller one sigma n the bubbling time because that's when you're close to the multi-soliton manifold and tau n we call the ejection time because that's where you ejected from, this is only heuristic we don't have invariant manifolds but you ejected from the multi bubble configuration, bubbling and ejection, that's the first and second property, so delta small delta separated from zero but still small and then you want that interval ion in time to be on a parabolic scale small compared to rho and squared, what is rho and that's the physical scale where you're actually working so it makes no sense to go beyond that because you're physical, then you lose total control, so you have to stay within the largest possible stability scale, that's the third bullet point and the fourth says that at bubbling time you have quantization of course which you have for free because delta small means you're close to multi bubble configuration, you have quantization of energy so you converge to some 4k pi and if k is zero then you actually cannot have any multi bubble so we take k equals one and the lemma is that if bubbling fails, so again argument by contradiction then you have such a minimal k and it's at least one and the proof is based on both the energy dissipation which gives you the bubbling along a sequence of times, that's where you get the bubbling times from, always always this vehicle, alright always the sequential soliton resolution and you of course need propagation estimates here so what's the key lemma? the key lemma is truly the key, that's why it's called key lemma, I suppose and it says that if you have so what did the previous slide say? if the theorem fails, what's the theorem? it says that delta, what did it say? it said that in one version delta of the solution and then here you have say in infinite time you pick a y and then you do root t, this tends to zero as time tends to infinity and in finite time you have this and then there is the caveat that you can allow this to move also the center can move but so if this fails that means you move away from the multi bubble configuration infinitely many times and then we have these minimal collision intervals, these sigma and tau n with the positive k, the quantization of energy, everything after selecting a suitable subsequence so the key is that this bubbling time, tau n minus sigma n, that time between bubbling and ejection rather has to be large compared to the largest harmonic map scale squared that's lambda max and squared alright, so look at this picture unfortunately it's a bit coveted here but look at this picture what is lambda max? lambda max comes from any multi bubble configuration which need not be unique that does this four that means you have this, so again very heuristic you have a multi bubble configuration, all my manifolds look the same, they always look like this don't ask me why, well I can't draw any others so you find a multi bubble configuration that's our q of omega and you come close to this alright but you might also have another one, so you come close to this that's written as a sequence of finitely many of course k is the number so harmonic maps and then you take the the largest natural scale associated with the harmonic map notice this picture in this picture you might have picked a degree two harmonic map then this is lambda max or you might have picked two degree one harmonic maps then this would be lambda max now you will say this is very strange because no it's not strange because it would then take lambda max in time squared to create a collision between these two because they are separated by lambda max alright that is the logic behind this ambiguity of such a picture so this is absolutely key and the proof sketch takes a while but if again everything by contradiction if not then you could find these bubbling and ejection times that become arbitrarily small relative to the lambda max and squared scale say this one here and such that you have bubbling times sigma and twiddle and ejection times talian and here is an attempt a picture that shows you what's going on so lambda max is the big green one then the pink one is what it is the root of tau in minus sigma and tilde it is essentially the root of this which is allowed to be much smaller than that alright but notice that this means that the pink one will also hardly move over that time scale because that's the stability time scale that you get from both the stuve energy stability and the you know let's call it tau L infinity bound pointwise stability they're all on this parabolic scale and so if the length of the interval you're working with between bubbling and ejection is that small then the physical scale that corresponds quadratically to that will also be stable but notice what's going on inside what this shows is that you have two bubbles inside at a much smaller physical scale so then you can do violence to those because you are now evolving the heat flow on a time scale that's much bigger than the natural time scale of those bubbles and so on the right this heart is not a heart it's the merging of the two bubbles so that's where you have a collision and why is this a contradiction because then you could have from the beginning lowered your energy and that was not minimal k was minimal that's the key so if you had this type of phenomenon you could eliminate the larger scale and just work on the smaller scale and you would have a smaller k contradiction and so now just to conclude the proof in a minute so use the key lemma imagine the theorem fails then you have these collision intervals and you have that tau in minus to be at least lambda max squared that's the square of the natural bubble of the harmonic map and you are ejected at time on this whole interval so there's slightly different what I said before but you can make your interval smaller so that your distance then remains uniformly above epsilon right because if you do this eventually you have to go far away by Adas so then you go until you remain uniformly above epsilon on that interval jn and so now the claim is that lambda max n times the altunum of the tension has to be bounded below if not then you would use sequential soliton resolution you would use your bubbling lemma to find to find bubbling times within jn but that contradicts the first inequality this one this contradicts that you remain separated by epsilon for all these times and why is that finally a contradiction well because if you have this inequality on the whole time interval then look at this integral this is finite then you break it up into these intervals you pass the subsequence you can assume the disjoint everywhere on this you have to be bigger than lambda max minus 2 but that length is at least this so then you're summing up constants infinitely many say of a contradiction all right so happy birthday frank as final speaker I have to profusely thank the organizers the scientific committee so you know you know who they are just for sake of completeness I listed them and then the the tireless selfless guardian angel Elizabeth we cannot forget her she's been amazing okay thank you very much are there any questions I have a question is there any chance even at a conjecture level to say something about the dynamics of the states and the centers do you spend a lot of time on these system technologies that is modulation theory and that is hard and maybe we can have a chat in a few months and I'll have something to say at the moment I have no comment okay if I don't know are there any questions do I have one question or even two questions one about the jasek law there was a M in the energy we should expect the energy of the profile so this theorem in the engineering this is my M M pi this just comes from counting degrees so there exists such an energy it is related to the energy I suppose remember that for harmonic maps energy and degree are linked so then we can count M also from the energy if you want but asymptotically only at infinity but before you yes only for the limit yes yes yes another question perhaps you have your ultimate energy contribution which in the 80 k's you can say that you don't interact at the W1 or H1 level you need to reduce the energy and then drop the slow energy in your do you have any of them what is your role apart from the energy separation of bubbles when you do when you consider the equivalent the time or yeah we need this also in the L infinity theory so for our Swiss cheese we want the centers to be separated when they collide but that would be likely to be the case but then it's kind of a time rule no because we work on they don't move so we work on the feature at the bubbling times once you have ejection times all of this is lost you can only work indirectly there is no more structure but that structure is important at one time but infinitely often that's the bubbling time and then you propagate but only on the parabolic time scale where they at least that's why we pass to the bigger ones we do not control the collision indirect arguments we know that they have to happen but how they happen we don't control are there other questions we talked about wave maps and normal maps so is there a way to understand the class every question that you could have write by this method so I can't think of an abstract theorem that would say the class of PDEs to which this would apply or for instance are you thinking of Kinswagland it's not more like an answer yeah so yeah NLS is an interesting topic maybe you're thinking of Schrodinger maps because that's a geometric equation related to this so maybe the ingredients you need to have are purely to start well on an abstract level this does split into some kind of modular structure so you need the sequential bubbling we have this from the 90s and then you need the stability which is also a module and then you combine it with these dynamical ideas involving minimal collision intervals of Jacek and Laurie yeah so it decomposes into modules if you give me a PDE that has these modules then it should work