 Daj. Pozna, da vse me površi, in da sem tukaj in izgleda. Odličim, da se vse zelo, ki je zelo, na vsega vsega vsega, je to vsega vsega, ki je dve iz vsega vsega, Paweł Bjernat, ki je zelo vsega vsega in Maci Mariborski, ki je vsega vsega vsega vsega v Ejsternji instituciji. Tako, zelo je vsega vsega, za vse vsega superalizacija, tvoja je začelija vsega istrijev. Domajno vsega vsega je Minkowski spashtajm v d plus 1 dimensijnih, d je zrp. spesjelimi dimensijnih, in vsega je dimensijnja speshtajnja. Vsega je, da videlimo, izvedena v d plus 1 dimensijnih ukledenjo spashtaj. Tvoja vsega vsega sodate This semilinar wave equation with gradient non-linearities. So this is a very nice geometric wave equation, which is interesting for us. I mean, us physicists working in general relativity, because it shares many features isteni češnji vse je zelo stranje in je zelo, da vse nisi tukaj mislijo vse tešnje, kaj je to pričajne preferrede isteni češnji. Zelo se razglednjeni na to. Znamo, da vse zelo je vsezavljeni, ki je to vsezavljeni in tukaj nekaj zelo izvah vsezavljeni je to zelo, ki semilina vs. izgleda. V 1 plus 1 dimensijnega, in variabu R je, kaj radija, variabu, kaj je zelo pozitiv. Zelo bo se pričočite glasboj dinemijske vzivnih data za taj izgled, zornostanh data do zelo. Jedna z vseh vseh izgleda je, kaj soluzije potrebne vseh vseh in finajtih, in in finajtih. Ako izgleda vseh o finajtih vseh, nekaj trebamo vseh. Prvo, da je energija, kaj je vseh vseh, kaj je vseh vseh vseh. želite poslisto, and you see from this expression that for the energy to be finite, the solution u must go to a multiple of pi, both at zero and at infinity so and we take a convention that this function at zero is zero, so therefore the value k, ki je vsebejmer na infinitivne vsebejmer, je dobroga tukaj, dobroga tukaj dobroga tukaj, je to prizavljena na vsebejmer, tukaj je večem vsebejmer na tukaj tukaj, začna je, da je pristroga, nekaj najšelih sektor占, zelo jen zvršenja, zelo jen z k, Zelo, da sem tudi, da se je zelo, da se je zelo, da je vsega vzela. Zato, da sem vsega vzela, kaj pa je, zazam, pozitiv, konstant lambda, zato sem vsega vzela. In, da zelo, da se je zelo, da se je vzela, da se je vzela, bolj načo nači je podeloNetov nije skala najgratnije ali pomožete, ne zelo že nači je nači, nači ne zelo nači je podeloNetov nači je opravil a zači je nači da je napravil način. Vse tento iz pobroadi, nekaj je dobrov s fifteen z embarrassing shinji vskega koncentrativa v skrpisi tudi to opravljenje. Svačujem sve niferne valdovate vsevaj, če bo delavno, in popravljenje z Mikalš Strůve, načo zelo, da vsevenje vsevačne obostele spremidlje obostele bo države. Vsi je in opravljenje na czetko strano. In tudi pravda, ki prijeljali informacijo na priste, ki se zelo vse odpočili, začal se začal začal začal začal začal. V kontrasti, superkončnice dimensijcij, ne je nekaj nezvori. Pozolim, tudi se izgleda v 3 dimensijani, ki je, od vseh vseh, najšlič najlepših, prizaj, da v fyziknih vsečnih mavcij, kako je izgleda sigma modela. Tudi tudi izgleda je najmajno o izgledajsej delovini soličnih, ki je izgledajsej izgledajsi in zvršenje. but as far as I know there almost nothing has been known about higher dimensions. So to set the stage for blow-up analysis of blow-up I need to tell you first about self-similar solutions because you will see that they really govern singularity formation for this equation. So, self-similans solutions, by definition, are invariant underscaling. So, they are functions of one variable, which we call Y, which is R over capital T minus T. And capital T is just added for convenience, and this is allowed by translation invariants in time. If you plug these ansats into the wave map equation, you get an ODE of this form written here. Now this ODE has a singularity at the origin. It has a singularity at y equal 1, which corresponds geometrically to the past light cone of the singular point T0. So this is a point T0. This is the past light cone, which corresponds to y equal 1. This equation also has singularity at infinity. So by finite speed of propagation, what really matters is what happens inside the past light cone and on the past light cone. So analysis can be restricted to the interval from 0 to 1, closed interval from 0 to 1. And if we have a smooth solution of this equation on this interval, then this is an explicit example of singularity formation in finite time. Because for such a solution, the gradient of solution at the center blows up as 1 over capital T minus T as time goes to capital T. I'd like to make a remark that even though the analysis is restricted to the interval from 0 to 1, it is essential, at least if you want these solutions to participate in the dynamics, that these solutions remain smooth outside the light cone, all the way to infinity. Otherwise, they do not play a role in dynamics. And there are examples of such self-similar solutions, which are fine inside, but are singular outside the light cone for some subcritical wave equations, but not here. Ok, so we would like to find if there are such smooth solutions for this equation. And one way to approach this is by kind of shooting argument. So it means we start with solution, which is good, smooth at the origin. Such solutions form one parameter family, which is parameterized by the gradient of solution at the center, which I denote by C. One can easily show that these local solutions at the origin extend all the way to the open interval from 0 to 1, but they are in general not smooth at the light cone. There is one, before I discuss construction of such solutions, well, it's worth to see this explicit solution, which corresponds to a particular value of the parameter C, C0. So this is an explicit self-similar solution. This was first found in three dimensions, well, first proved to exist by Shatah, and then found in explicit form by Turok and Sperger, and last year it was found, well, generalized to higher dimensions. We believe that this solution is only self-similar solutions for dimensions 7 and higher, so I'll come back to this. And this should be contrasted with the harmonic map flow, for which in dimension 7 higher done no self-similar solutions. So let me just one slide how this solution F0 was actually found. So this was inspired by Herbert Koch's talk in Banff two years ago when he considered generalized KDV in powers with slightly supercritical powers, p equal 4 plus epsilon, and similarly here we can think about dimension as a continuous parameter and introduce positive parameter epsilon, which is d minus 2, and change variables, and we change variables and rewrite our ODE. We obtain this ODE in new variable F tilde and X, and on the left-hand side this is exactly the static equation for the harmonic map in two dimensions. And on the right-hand side you have a perturbation. So the idea was to start a perturbation analysis and construct this solution perturbatively in epsilon, but at the first step it turned out that the right-hand side is zero. So therefore this is an exact solution. So you can think about this as an educated guess. OK, so how about other solutions? To understand other solutions one has to understand analyticity properties of solutions at y equal 1. And this very much depends on the dimension. And actually in each dimension it's different. So if you take this equation and multiply it by y squared and 1 minus y squared and differentiate, and keep differentiating, you will get, here are here all equations, and I wrote this first two over there, which have to be satisfied for the solution to be smooth. And from the first equation you see that dimension 3 is distinguished, because when d is 3, then f at 1 must be a multiple of pi, so actually which must be pi. Now, so this is what is written there, that in dimension 3 the solution which is smooth at 1 is parameterized just by gradient of at 1. In dimension 5 there are two possibilities. Either in the second equation, either both terms are zero, well the first term is obviously zero in dimension 5, but the second term could be zero, but or, and this is this case, and maybe, or we have to solve this system of equations and we get this behavior. And actually this explicit solution, which I showed you before, has this exactly this expansion at 1. In even dimensions there are no restrictions like that, in even dimensions the solutions are parameterized by the value at 1. So once we understand these analyticity properties, so we have one parameter family at the origin, one parameter family at the light cone, and we have to match them in a smooth manner. So, and here is the theorem, that in each dimension from 3 to 6 there is an infinite sequence of these shooting parameters such that the solutions are smooth on the whole interval from 0 to 1. And this is a pretty standard shooting argument in odd dimensions, in even dimensions it's slightly more complicated but still pretty simple. Now, there is another method of constructing self-similar solutions or more generally is solving, well, solving solutions for ODEs, which is a variational method. And this is a functional for which self-similar solutions are critical points but you see here in order for this to be finite you have to renormalize it, so you have to subtract the value of the solution at 1. But if you don't know this value a priori, this is a problem. So for this reason these solutions have not been found before because, well, there is a variational proof of existence of one of the solutions of this infinite family in dimension 5 by Shatah, Kazunava Shatah and Tachvildar Zadeh but in this case F1 is just pi half which by the way is convexity radius of the sphere so it's an equator. So these solutions probably, well, it would be hard to find them by these variational techniques. OK. Now, once we have some self-similar solutions the next step is to analyze the spectral stability so the spectrum of small perturbations because this is essential in understanding the role they play in dynamics. So it's standard to introduce so-called slow time so the blow-up happens when s, the slow time is infinite and change variables so these s and y are similarity variables and in these similarity variables the wave map equation takes this form. So self-similar solutions are just solutions which do not depend on time so there are stationary solutions of these for which the right-hand side is zero. So the standard procedure is we linearize around these self-similar solutions a convention is such that eigenvalues lambda which are positive and unstable so if we linearize around them we get a quadratic eigenvalue problem quadratic because it involves lambda and lambda squared and the quantization condition so we want solutions of this linear or the e which are smooths on the close interval from zero to one. This is a condition for quantization of eigenvalues and this is the conjecture form of the spectrum I will give evidence for that in a moment so first of all there is a so-called gauge mode which is always present in similarity variables which is due to the fact that we don't know this time capital T so by shifting this time by time translation we generate a mode which is explicit and it has eigenvalue one so this is even though this eigenvalue is positive this is not a real instability so that's why it's called a gauge mode then the conjecture is that there are exactly unstable eigenvalues so positive eigenvalues bigger strictly bigger than one and there are infinitely many negative eigenvalues which correspond to stable directions so what is the evidence for this conjecture so first let me discuss the solution this explicit solution f0 for which this conjecture applies as well so in this case we have a luxury that we know this solution in closed form so and we can change variables like this and then this equation we had here becomes so called Hoine equation which is a generalization of a hyper geometric equation which it has four singularities at 0, 1, d minus 1 and infinity but this is not very helpful because a so called connection problem for Hoine equation is unsolved so in contrast to hyper geometric equation I mean it's not helpful in terms of proving this conjecture it is very helpful actually in terms of computing the eigenvalues because for example Maple knows Hoine equation and can compute the Vroidskan of two solutions so this mean that you have no integral representation of solution to d3e that's what behind this and this is the reason why the connection problem is this the connection problem means that the connection problem is this that if I have a singular boundary value problem and I take a solution which is good at one end this solution will be a super position of good and bad solution at the other end and connection problem is well for hyper geometric equation the coefficients in front of of bad and good are explicit not in this case so they are explicit in terms of some hyper geometric functions but still it is okay but there is a way around it so and actually we can take this solution which is good at the origin this and take a power series expansion this is a standard fuxian analysis we take this good solution in terms of a power series around this would be actually a Hoine function but we don't well this power series is guaranteed to have a radius of convergence one because this is the nearest singularity but in general it is not smooth at one and the way to look at smoothness of this power series at one is to look at the asymptotics of the coefficients in the in the expansion and this coefficient satisfy a three term recurrence relation which can be solved and these are asymptotic solutions of these recurrence relations and this is a bad solution for which this if this coefficient c1 is ton 0 then this won't be smooth but and this is a good solution so the quantization condition is actually that this coefficient is 0 and this can be used to compute eigenvalues with great precision and confirming this conjecture I showed you and quite recently Kost in Donning and Glogic actually showed rigorously that this does not have positive roots but as I said this technique very much depends on explicit form of solution f0 so we don't know other solutions in closed form actually so so what we do for other solutions first of all we can rewrite this problem in a self-adjoint form so this is just by change of variables we can bring this quadratic eigenvalue problem into a standard sturmluvil problem with operator an which is self-adjoint in this Hilbert space actually this is very much related to the Laplacian on a hyperbolic space now the point however is that eigenvalues of this problem which I call mu and eigenvalues lambda of my original problem these are two different things so because in one case I want solutions which belong to these functions space x in the other case I want smooth solutions and if you analyze this you can see that these eigenvalues actually coincide but only if the eigenvalue lambda is bigger than d-1 over 2 and in this case so by analyzing this problem we can learn something about eigenvalues of our problem but only in this range and it's not difficult to see that if we take the gauge mode which I mentioned before and transform it to this function psi we get function like that which is an exact solution of that with eigenvalue d-2 now from the shooting construction of these self-similar solutions we actually have a complete control of the number of oscillations of these self-similar solutions therefore we know the number of zeros of this function so we can apply Sturm oscillation theorem on this it follows so this is different in dimensions 3 and 4 and 5 and 6 so in dimensions 3 and 4 this function has exactly n zeros so there are exactly n eigenvalues below and in this case there are n-1 eigenvalues so this gives us the exact number of eigenvalues for lambda bigger than d-2 but this leaves open the interval from 0 to d-2 so there is a gap and actually maybe I should mention so today there was a preprint by Kullow Raphael and Scheftel in which they for a very similar problem harmonic heat flow problem sorry the heat flow for semilinar wave equation for heat flow for power load nonlinearities show that there are no gap eigenvalues and it's very likely that this proof can be extended to this setting as well now so there is this gap so in this sense I don't have a proof that's why I call it conjecture however numerical calculations show that in fact there are no eigenvalues in this gap actually there is one surprising thing that in dimension 6 there is exactly one eigenvalue of our problem which is not an eigenvalue of this self-adjoint problem so so here is a table computed numerically these numbers we are physicists so we want to have quantitative results so these numbers will play a role in a moment so you see that these are different dimensions there is this gauge eigenvalue 1 always and all eigenvalues are negative this is the least dumped eigenvalue which will play a role and this is for solution with one instability which will play a role in dynamics because this is for co-dimension one dynamic it has one positive eigenvalue the gauge mode and this is the eigenvalue I mentioned which is somehow surprising it is not an eigenvalue of the corresponding self-adjoint problem so we have self-similar solutions and which are spectrally stable they have no growing mode so this gives rise to conjecture because they are natural candidates for attractors and the conjecture is that actually this explicit solution f0 is a universal attractor in a sense that if I take generic in a sense in a vague sense so I mean we take some initial data and if they are large enough they will blow up and they will always blow up along along this solution f0 in this sense so they approach locally near the center this solution this is a non-perturbative result so we are saying this is universal for all initial data in the case of data which are close to this f0 actually there is a proof of non-linear stability of this blow up by due to Donninger in three dimensions and it is very likely that his technique can be extended to higher dimensions so there is numerical evidence for this behavior which I am going to show you now so and this tells a little bit more because it says that the dynamical solution approaches this self-similar profile exactly with the rate dictated by the least dumped mode which is here well the coefficient of course depends on initial data and the time of blow up depends on initial data so this is a numerical evidence for that so we see what we see here so these are snapshots these are initial data shown by this black curve this is in similarity variables and the dotted red line is the self-similar profile and you see as time goes on the solution approaches this self-similar profile and actually you see this approaches it so you see this is the light cone one so it approaches it also outside the light cone this is a more quantitative evidence for the same result so what we show here is the gradient of the dynamical solution minus the gradient of the attractor in a logarithmic linear scale and this shows that this is decreasing in time with this rate when lambda minus 1 is exactly the eigenvalue which was computed independently by this perturbative method and this shows this is a snapshot of solution at some late time and we compare the solution with a self-similar solution and the difference is the profile of this lowest mode so this is the evidence and this is just shown here in 4 dimensions but the same is true in dimensions 5 and 6 so this means that the solution is behaving like that and of course we could add more there are infinitely many stable modes so this was about what we call generic blowup or stable blowup but we know that for this equation sufficiently small solutions remain globally regular in time so there is a question what is the borderline between solutions which blow up and solutions which don't and there is a very very straightforward strategy to analyze threshold namely you take one parameter family of initial data which interpolate between small and large data and then you try using bisection of the parameter along this family could be a Gaussian with an amplitude you can find you into the critical value and then look how what is evolution for solution which has this nearly critical amplitude so in this case we have so this is a dynamical solution and the candidate for the critical solution which somehow sits at the borderline is obviously the self-similar solution which has exactly one unstable mode which is the first solution in our family which is not known in closed form so we have this critical solution self-similar solution it has exactly one unstable direction with eigenvalue lambda 1 and all other modes are decaying so by fine tuning we can make this coefficient as small as we wish and if this coefficient is very small then for long time we don't see this instability it is a standard settled point time behavior and from this there follows very scaling laws for example if you look at solutions which do not blow up but they are marginally subcritical so they have a slightly less than the critical value the gradient of these solutions in the center can grow to very large can become very large and how large it is it will scale with this so this gives rise to conjecture that the self-similar solution with one unstable mode is a critical solution whose co-dimension one stable manifold separates blow up from this person so here is a schematic picture of this behavior so here we have this co-dimension one stable manifold which we imagine as a smooth hypersurface in our infinite dimensional phase space and here is this self-similar solution which has exactly one unstable direction which is transversal to this curve and all other directions are stable and here is a curve of initial data which intersects this manifold and the data corresponding to the point of intersection will flow to this and the data which slightly miss it will approach it and then will go either towards blow up or global regularity surprisingly so I mentioned that this helped to understand critical behavior of Einstein equations because actually this picture which actually is very well known in physics of phase transitions because this is nothing else this picture exactly explains the universality of second order phase transitions like in ferromagnets so this was observed for Einstein equations and in this case the analog of blow up is black hole formation so this is numerical simulation illustrating this behavior I guess this is in six dimensions so this dotted line is this self-similar solution of co-dimension one and this is a dynamical solution the blue line which is fine-tuned to the threshold with this precision and actually this is a pair of initial data on both sides of a star so they evolve together because they are close but here you see that the unstable mode has grown to finite size so they separate and one will blow up and another one will disperse so the blue one will blow up and this one will disperse and this is again this illustration of this phenomenon now we look at the gradient of the dynamical solution at the center the gradient should approach the gradient self-similar tractor plus this saddle point unstable mode with very small amplitude like 10 to minus 26 plus stable mode and again this solution will blow up and it will disperse so this was in dimensions from 3 to 6 in dimensions 7 and higher I said before self-similar solutions except this explicit solution f0 so there is no natural candidate for this co-dimension 1 critical solution and the reason actually this self-similar solution sees to exist in dimension 7 is the spectrum of the equator map so the wave map equation has a trivial constant solution pi over 2 which is map to the equator this solution is singular at this origin but if you look at the spectrum of this solution then eigenfunctions are oscillating below dimension 7 but above dimension 7 the number of eigenvalues around this solution is finite and this phenomenon is exactly responsible that you lose these self-similar solutions but at the same time you gain this solution in a sense that this solution has finite co-dimension and actually it has co-dimension 1 if you neglect the gauge mode and the numerology here is actually you have the same phenomenon for all different kinds of equations so harmonic map flow and this is just everything is about in this case whether this 10 minus 8d plus 8 is positive or negative and this changes sign somewhere between dimensions 6 and 7 now so what we can do so so here I want to describe a completely formal construction of the threshold which is actually due for harmonic map flow asca is unpublished paper many many years ago so we take inner and outer solution so the outer solution is just this singular solution plus perturbations such that we tune away the single unstable direction and the inner solution this solution is singular at the center so it's not good so near the center we attach actually a static solution which is rescaled by some a priori unknown functional of t this solution is not good at infinity because it has infinite energy and the whole point is that the asymptotics of this outer solution near the center and the asymptotics of this static solution near infinity so they have the same they match so they can be matched this is called match asymptotics this is what herrer overlaskers is and by this formal analysis we predict the rate of blow up which is given by that and this is type 2 blow up in the sense that this functional of t goes to 0 faster than linearly so this is completely formal argument actually this argument doesn't work in dimension 7 because in dimension 7 this eigenvalue lambda2 happens to be 0 so maybe there are some logarithmic corrections this we don't know but actually this probably is not so much worth pursuing because now there is a new approach to type 2 blow up which was developed by Raphael Odnijanski for NLS which is applied to other equations in particular to the super critical wave equation by Kolo and it's very likely this approach which does not use any matching can be adapted to this case but still I believe with the same result ok, so let me finish by mentioning some open problems so I said before that the critical dimension is well understood but there is one exception namely we don't know what is the threshold of blow up in two dimensions in the following sense so it is known well this was this result by Strava that if there is blow up in critical dimension it must have a form of a harmonic map shrinking asymptotically to zero so this is the profile of harmonic map in our case so this function alpha of t the speed of blow up must go to zero for stable blow up well this function alpha has this form so this is a type 2 blow up which is derived formally by Ofchennikov and Siegel and then rigorously by Rafa and Rodnijajski however as far as I know it is not known what is this function alpha at the threshold so it is not known what is the speed of blow up at the threshold as far as I know so for this problem even if this is a power law does it have a logarithmic correction so it certainly is faster than t minus t squared so this is clearly seen in numerics but without analytic prediction it's impossible to detect some corrections numerically the second problem which I wanted to mention is the continuation beyond blow up so whenever you have a solution which form a singularity in finite time there is a question can you continue beyond and in this case well you can always because these equations are variant under time reflection so when you have a backwards self-simular solution you can attach it to a forward self-simular solution trivially and then you would get a weak solution which is singular at just a single point and this well such phenomena of the solutions immediately recover smoothness are well known mainly for parabolic equations and for a corresponding harmonic map heat flow so and actually we have numerical evidence that this happens and there is probably an interesting pattern of blow up times the sequence of blow up times the solution blows up and then blows up again and again another problem I wanted to mention is when we change the domain if we change the domain of the wave map I considered to a domain which is either compact or bounded or effectively bounded so we lose this person now the blow up does not really see the geometry of the domain because the blow up is a completely local phenomenon however whether blow up occurs or not does depend on geometry and we have some preliminary result for wave maps where the domain is so called anti the seater spacetime so I don't have time to go into what this time is or maybe I have so this is so if you take a unit sphere in in euclidean space and restrict to the upper hemisphere so this would be z and you take a round metric on this and then say I take a line of which goes to the equator and parameterize it by latitude theta which is changes from zero to pi half at the pole then the ads metric is the following metric where h is the round metric on this sphere this would be an ads metric in in two plus one dimensions but if I change it to four dimensions sphere this would be ads metric in four dimensions which we have here well as you can see from and actually the constant time slices are just hyperbolic spaces so the distance from any point here to the boundary is infinite the boundary is just an ideal boundary of the hyperbolic space this is like a disk model for hyperbolic space however null geodesics travel in finite time so this is from any point here to the boundary so this space time which is of great interest in string theory is effectively compact or effectively bounded from the point of view of light rays so if you take this as a domain and consider the same equation as I did we find that evidence that arbitrarily small initial data blow up and the time of blow up scales with the size of initial data as one over epsilon squared so this the conjecture is that there is no threshold of blow up in this scale so presumably the Einstein equation itself would blow up the Einstein equation itself perturbations of this perturbations of this of this yes, actually this project is actually is motivated by trying to understand the stability of this space time as a solution of Einstein equations exactly the conjecture is the same that arbitrarily small generic initial data lead in the case of Einstein equations to black hole formation actually there is a corresponding problem for the critical case from ADS3 to S2 for which it is known that there is a threshold for blow up for the same reason there is a threshold of blow up here so to have blow up in critical dimension you must concentrate energy at least the energy of the harmonic map so it means if we take wave maps in this case in the critical case we cannot have blow up for arbitrarily small initial data but it does not exclude the fact that solutions remain smooth but the radius of analyticity shrinks to zero in infinite time and this is the conjecture in the critical case the radius of analyticity shrinks to zero in infinite time both for this model and for the full Einstein equations and finally I'd like to mention my favorite model which is a supercritical Einstein wave map system this is a rather complicated system with very rich phenomenology this system has a dimension as parameter and actually that's why wave maps are such a good model for Einstein namely for Einstein there is a coupling constant which is Newton's constant and for wave maps there is a coupling constant we are called beta squared and because this they have the same scaling properties these coupling constants have the same dimension so one is the inverse of the other so therefore this is dimensionless and for this system there is everything is numeric except for existence of self-seminar solutions there is numerical simulations indicates that if this parameter is small enough in particular when this is zero we are there is no gravity when this parameter is small enough we have self-seminar blowup very similar to what I described but this blowup disappears when the parameter is large enough and this is an example of gravitational desingularization so this is for the cosmic censorship to start working the parameter must be large enough this is not a counter example to cosmic censorship because the wave maps have singularities already without gravity so gravity is not expected to help but actually it does if the coupling constant is large enough and what's even more interesting for sufficient large coupling constant we lose self-seminar solutions but we gain so-called discreetly self-seminar solutions which which only I think recently have been studied I think by Terence Tau for systems without gravity Thank you So you talked about the divine case or about the general case and you didn't say much about the expectations I don't know so this is well I didn't talk anything about well you know all the results are heavily heavily rely on numerical simulations and simulating singularity formation is numerically non-trivial so we don't know how to do it without this assumption so that's why all our results are restricted to the equivalent case sorry Any other question? I have a question so do you think that say in dimension greater than 8 so we know that there is this new type 2 type of singularity formation there are still self-simular solutions sitting there do you think that they don't exist in this case or did you try to Well I would bet they don't exist but I have no proof so it's one of the sizes so no for this is an ODE after all for this ODE this is an absolutely numerical evidence there are no self-simular solutions so you try it it's not just the proof of existence breaks down for the power nonlinearity no it's more complicated it depends on the experiment it's more complicated than that for the power nonlinearity the model is slightly richer because you can play it with dimension and the power of the nonlinearity so here I have just one parameter which is dimension but of course I could generalize this to higher equivalence and then I would have the second parameter and then I would have a similar picture let's send this picture