Daj. Pozna, da vse me površi, in da sem tukaj in izgleda.  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.  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


Hi, beautiful, are you getting married or do you enjoy planning your wedding that may never happen? Me too. You may have asked yourself, well, how am I gonna wear my hair to my wedding? What's gonna look the most timeless and the most important question of all?  These are the three questions we all ask ourselves when picking our wedding hair. I've worked on countless brides in my lifetime. Let me just tell you, a lot of them get the wrong hairstyle. I do talk them out of it. However, a lot of them come to me wanting the ugliest I've ever seen. I don't like that. Let's fix that, shall we? Think modern but timeless. That's the goal we're trying to achieve with our wedding hair.  And I feel like we just have to remember where we came from and how much we've progressed. And then after that, we'll go over some don'ts of wedding hair. Some things that I really don't like that people ask for all the time. And it's just honestly, a little tasteless.  I also think it's important to preface this video by saying these are just my opinions. So don't get mad if you disagree with something I said, okay? I love it, okay. After you watch today's video, hopefully you'll be walking down the aisle living your extra, extra life. And that I think is really the goal of any bride. So with that all said, let's get it started. So let's begin checking out some iconic celebrity wedding hair from the 80s and also just a few randos that had some really interesting do's. Ah, here we are, the first image. If I were living in the 80s and getting married, I think that this is how my hair would be. Listen, she went for it and I really appreciate it. She said, tease it. And they said, is that enough? And she said, no, keep teasing. And they said, is that enough? And she said, no, keep teasing. I'll tell you when it's done. And it was never done until the actual minute her wedding started and she walked right down the aisle and that is how she ended up like this. That's what I think happened at least. These 80s bangs were just the most wild thing, just blown out in every direction possible. She was hit by like 80 mile per hour winds, it looks like, and her hair just went all back and she let it stay there. And then she hair sprayed it iconic. She's definitely living her extra life. Now we have Princess Diana. Yes, she had one of the most iconic 80s weddings. She went for a simple look to the point where it almost looked like she wasn't attending her own wedding. Looks like the hairdo took three minutes to do, which I bet it didn't. I'm sure it took like 10 hours to do. This haircut, the shaggy, I don't know. It's kind of like a weird bowl cut, like a mullet moment. I usually like mullets, but not on Princess Diana. It wasn't really flattering at all. I would have liked to see it out of her face. I feel like it would have really opened up her face. This is just giving me nothing. Oh, oh, now that's what I wanted. It looks like she has a faux-hawk going on. I don't think that was the idea behind this, I'm sure. But I mean, she said, teases the gods, I want height. And they said, yes ma'am, I'll do that for you. And she got that. She did that. She has the flower crown with the obnoxiously large veil. It's all just together making this wonderful 80s glam rock princess wedding hair. And I'm here for it. Now we have Cindy Crawford. Give me all the tool you have in this fabric store. And they said, but that's a lot of tool. And she said, I know, I want it all. And she got it, girl. She got the entire fabric store of tool. I mean, wow. That's a dress, but we're not talking about that. Then she has this crown that looks like it's growing, like crystallized salt formation. That's my best descriptor for that. And it's going across her forehead. It just looks uncomfortable, first of all. Then we have this high, high bun that looks like she threw her hair out in a hurry. It's just making this entire moment even that more spectacular because she has the most extra wedding dress with the most extra gloves and extra jewelry, extra veil. And then her hair is just thrown up into a bun. And she still looks gorgeous. So listen, if you're going to do it, might as well go all the way. This woman said, let's take athleisure wear and make it into wedding wear. This sweatband she has in her forehead might be useful because she might be a sweaty girl in her wedding day. I know it's a little stressful, I'm sure, I'm sure, you know. And she said, but I want a sweatband with a veil on it. And she got that exact thing that she was looking for. You go girl. And then we got this permed hair that was then teased out to then create this kind of nest, like a bird's nest, if you will, on the top of her head. I love it. Now we have Mariah Carey who did this down moment. Looks like she washed her hair, put a little curling serum in and then walked out of the door, which knowing Mariah Carey was not the case. I'm sure it took several hours to do this hair and it was a flop. I mean, it doesn't look good. It doesn't. I don't really have much to say about it because it's just, holy shit, that dress is crazy. Oh my God, that flower arrangement. Yep, that's Mariah Carey. That's what I expected from her. It looks like she's trying on the dress for the first time and she didn't do her hair and she's just trying it on, doing a fitting. And then she ended up walking down the aisle that day and getting married. I don't know. It just looks like they didn't do her hair. I don't know. This wedding party is looking like poodles. They look like poodles. I am not saying poodles are ugly by any means. I wouldn't ever. But let's focus on the bride. The bride has this updo with this snooki Jersey Shore poof with the bun. I actually enjoy that it's off her face. I think that it actually looks quite elegant in that sense.  Again, another bird's nest. I feel like on your wedding day, bird's nest should not be there. But I actually think she looks kind of like the most put together of everybody we've talked about so far. I like that little choker she has on with her beautiful decolletage out, quite elegant. And actually, what the, do I like this hair? I don't know what's going on right now. I guess she looks kind of cute for the 80s. Should we go with that? I don't know. Good job girl. You go girl. And lastly, we have Heather Locklear marrying Rod Stewart. Rod Stewart, holy, man, you went for it. Honestly would marry him. It's fine. And then Heather Locklear said, let's do the least with my hair. And I actually think she kind of killed it. Obviously her hair is a little bit shredded. She has a very, very short layering at the top. That is then blown out to perfection underneath this like cascading crystal waterfall of a tiara crown. I don't know what you call it. It's kind of chic. I don't know. She kind of looks dope. It's fun. I'll take it.  And those gloves. Now that I got that out of my system, let's do some don'ts of wedding. Don't do this with your hair when getting married. Let's do the don'ts. One of my favorite don'ts is braids. Especially fishtail braids. Let's keep the fish in the water, babe. Let's not bring him into her hair. It doesn't look right. I don't know what it is about fishtail braids. There are some things you can wear fishtail braids to. It's festivals.  For some reason, I'm judging you.  But fishtail braids, braids in general for weddings, it just looks like farmhouse vibes. I don't know. It's just weird to me. Here's a couple of images of braids that I hate. We have this crown, fishtail crown. It's kind of like acute when you first look at it, you know?   Like it just doesn't look timeless or elegant in any which way. It just looks like you are stuck as a 12-year-old girl. Like you had this imagination of getting married when you were 12 and having a fishtail braid and like you stuck with that and did it. It wasn't good. Then we have this photo. It's just like chunky fishtail like laying on your shoulder. Like it's literally a dead salmon on your shoulder. Like I don't know, the braided crown area. It just screams childish and like daddy issues to me. I don't know if that's just me. It possibly may be, but I just don't love it. Next we have the overly textured hair. It's mostly just updoos that have way too much curl, texture, you'll see. Here it is. Okay, so this was a trend, you know, 10 years ago. It was cool at the time, but let's stop. She's like, I want to look like I didn't do a lot to my hair. So let's make sure it's really textured. But then they just overdue the texture and it ends up in this like weird overly textured moment and just has way too much going on. It's so busy.  Like these little pearl flowers in her hair and then she has the spun with the braids and the twisties and the texture. It's like, I can't, there's too much hair. Like that's what my reaction would be if I saw her hair in real life. Here's another prime example of too much texture.  This is some new stuff here. And then I'm just like, it's not, you're not doing anything. You're nothing about this is good. Then she has the braids going into the textured mass and I'm like, whoa, pick a style and stick with it. Up next we have the bump. The bump is something I see nearly the most often on every kind of ugly wedding hairstyle. The bump is always a driving force for those ugly wedding styles. This is what I mean by the bump. The crown region that is just so volumous. I get it if you have a bad head shaping or trying to correct something, but it's just so weird. I don't know why it drives me crazy. It's just such a, like it's like a, for lack of better words, sort of like a tumor on the back of their heads. I apologize, I don't want to offend anybody there, but that's the only way I can really explain it. It just looks like a growth out of the top of the head. And I don't know why it's there. Let's just flatten that out a little bit, shall we? Again, this style is semi-cute, but you lost me with the bump. The big bump on the back of her head. For some reason, every girl just goes to the bump when they're planning their wedding hair. They're like, I want the bump. I want more volume back here. What other day of your life have you ever wanted that? Why now?  You're gonna have to look at these pictures forever, girl. I'm trying to save your life here. Now we have the tight curls. The crunchy tight curls. This is the result of showing a old school hairstylist, a photo of beachy waves, and you walk out of the salon looking like this and they say, it'll drop. It'll drop later. If somebody tells you your waves or your curls will drop later, they probably don't know what they're doing because there's many ways you can make waves stay in the hair and not have them super tight at first in order for them to drop. I used to hear that all the time in the salon.  Like, what, are we timing it? Are the waves gonna know when to drop? Like the wedding bells start going off. They start tapping and the waves are like, oh, it's our time. And they just drop down like, no. There's waves to make hair stay in place. It's called hairspray. It's called the right products. You gotta have holding products, put clips in before she walks on the aisle. Like this is what happens also when you spray the hair with hairspray and curl it and it gives you crunchy curls. They're just not good.  First of all, we have the bump and the curls going at the same time and a little bit of like a twisted braid on the side. It's like, whoa. And like 10 pounds of hair extensions, but live your actual life, you know? I mean, it looks like she was going for a wave moment and she got curls and she said, well, I guess we're going with it. Now we have the overly perfected Sinyan.  Why is your hair quaffed to perfection around a semicircle sock bun?  It looks like a shrub growing out of the back of your head. Let the hair move, girl. It's so big. Like this isn't the worst photo actually. I get where they're going. I don't mind very quaffed hair. Listen, there's time and place for it. This just doesn't seem like the optimal look for anybody. The weaved pieces on top of the extremely large Sinyan is a moment. It's not a moment that I'm here for, I'm sorry. It's gonna be a no for me, darling. It's just a lot. Now we have the bad headband. This just feels like culture appropriation to me. This girl just went to an Indian wedding and said that is my look and then stole the headpiece right from the bride. A white person in one of these, it just seems a little culture appropriating. It's also just like not cute. Kim Kardashian even did it. It's a little bit better. It's not as a low. It doesn't feel as Indian wedding vibes to me, but it still doesn't need to be there for any real reason. Put a tiara on her, a crown. I think that would be better for you. And last but not least, one of the all time worst looks anybody can do themselves in their wedding day is the sock bun. Do you remember the sock buns? It was a thing. Everybody found out you can use a sock to make a bun and every girl came to school with that damn sock bun and some of them even did it to their wedding day. They even tried to dress it up with like a little bit of hair around the sides, just to like really make it wedding-y. And it's a flop, girl. It does look very yummy. Like I would eat her hair. It looks like a mix between chocolate and vanilla and like a caramel swirl going on in there. Delicious donut, I would eat it, but I would not put it on my head for my wedding. And that's all I have to say about the sock bun. I think that one kind of explains it so. So that wraps up my don'ts for wedding hair. Glad we got that out of the way. Now let's check out the do's for wedding hair. Let's see what we got. The headband, the simple stylish headband, if you will. It's dainty, it's small. It just adds a little bit of extra oomph to the style. It's beautiful, it's elegant, and just screams I'm living my extra life, but I also look like a bad who knows fashion. And then we have this headband with the pearls. That is a legit headband. It just screams timeless beauty to me. Just has a sort of elegance that says, again, I want you to look at me today because I'm gorgeous. And yes, I am that.  And that's what that gives me. I love a good dainty headband. It's beautiful. Now we have what I like to call the cool girl wave. Now this is a wave that is not overdone. She looks like she got her hair smoothed out. However, they didn't overly style it, overly curl it. She has the perfect amount of volume in the back to make it kind of look a little 70s-ish. Still timeless, beautiful. And also she has that gorgeous dainty headband on. And this whole look is just classic, timeless and effortless. And I love it. And now this is even a little bit more wavy, but still effortless. I think it is very reminiscent of the time we're in right now. I think this could possibly not age well. This is the one that I think maybe you might look back on and be like, oh, well, I look like I rolled out of bed. But it is very modern. And I think the less is more approach is elegant. The headband I think is important for this look because the look is so plain and simple. I think you need to dress up just a tiny bit with something you wouldn't regularly wear, which is the crystal headband or the pearl headband. Or a little tiara, you know, why not? The easy updo, the beautiful effortless updo. This is how you do texture without looking like you overdid it. You see that bit that is coming back from her face is textured, it's not perfectly smoothed out, but it's not like curls going on the side of her head. None of that happening. I love this sort of creation they made with that pearl, almost like a necklace going through her hair. She even has a little bit of a braid in there, but I think it works. And then the bun is just chill. It's not so crazy over styled, but it does look like wedding hair still. There is a fine line between a good low bun for a wedding and an overly styled one. This is how you do it right. Another beautiful updo. Not too much bump in the back. She still has that volume, but it's not like overly accentuated by this part and this part being pulled back tight with the really big bump in the back. It's kind of like an overall volume. And I love that she has a few pieces down the front. They're not perfect. They're very tousled.  I think this is a very modern look. Well, it aged perfect. I don't know, but I do think is great for the time. Sometimes it's okay to dress for the times. I think we can all agree that sometimes when you look back at wedding photos, who the f*** cares if it's like ridiculous? At least, you know, it was of the time period. And I think that is what's going on in this look. I think it does look a little disheveled, but I appreciate it. And I think it's a well done, disheveled, low updo. Now we have the chignon. However, done well. And a modern type of chignon. This is still elegant. It's still very pulled back and tight. And there's not a lot of messiness going on, but it's modern. It feels new. It is still very wrapped.  The most beautiful turd I've ever seen, you know? I love the bit of criss-crossiness going on above the chignon. And I think it's so great how it's like laying on her neck. It's very low. It's a beautiful style. And it's quite timeless, in my opinion. This one is even more sleek. This one is a great size. It looks like a small, tiny little croissant you find at a French bakery. And they say, we, we, young lady. And you say, thank you. It's elegant. It's timeless. It's beautiful with her hair color. I love the low chignon. It's not too big. It's not overdone. They didn't try too hard. This stylist who ever did this is great.  The glamour wave, I think, is an amazing choice. It's like an old Hollywood look. I think it is sort of timeless. Some people go really overboard with this one too, but we won't talk about that today. The way it's done in this photo right here is perfect to me. I know some people really want that refined look and I can get with that, but do it right. Do a relaxed wave and have the wave formation all the same and have everything. Very uniform, sexy, smooth, silky, shiny. And that's how you do a wave correctly for a wedding. Here is a more tousled version of the glamour wave, but still very much a glamour wave. She has a beautiful amount of body in her hair. I really do like the one side tuck behind the ear. And then she has that really wavy look on both sides of her head. And I think this is just great. I love her hair length. It works perfectly with the dress and she can always put one side behind her shoulder and change up the look a little bit, but it's beautiful and I think it's perfect for the look that she's going for. Now we have the low ponytail. I actually really dig a ponytail. Ponytails are great too because you can really slick back your head and give yourself a facelift with them. High pony is probably not.  Add a little bit of texture and a little bit of wave to that. I love the way she has this jewelry in her hair with the little wrap around and it's a very thick ponytail, almost a little bit too thick. I think it's good for her. And I think it looks elegant and classy while still having a down, up do. I think it's great for the dress she picked. It's showing her decolletage, but maybe she didn't want all the hair off her shoulders. So she chose it in between and I think it's perfect. And this is how you do a great low ponytail. Another great low ponytail. This one's a tiny bit higher. Still think it's great. I like the way they overlapped the hair and a little crisscross pattern. Did a soft wave on the ponytail. There's not a heavy amount of extensions in her hair that makes her hair look heavy and lifeless, perfect amount of volume in her crown area. And lastly, let's talk about the easygoing bun. This looks like her hair was thrown up, but in the most gorgeous way.  Like, why can't I do that every single day of my life? Why is this the most perfect messy bun I've ever done to myself? That's this bun. Sitting at the perfect height where it isn't too low, it isn't too high. It's still very sleek, but messy. Really says I care, but not too much that I'm not a fun girl and I can't get down on the dance floor at my own wedding, you know? She looks like she can get down.  Those are my dudes. I compile that list to hopefully inspire you to make better choices on your wedding day. I'm just kidding. Those are just my opinions. Take it or leave it. You don't need to do everything I said. Those are my dudes and I love all of those looks. Now that we've discovered all these beautiful, fabulous looks, what's the right look for you on your wedding day? Consider the dresses detail. Is there a lot going on up here? Are there beautiful straps you wanna show off? Or do you just love your decolletage? Do you have the perfect shoulders? Do you wanna put a lot of shimmer on your shoulders? Do you wanna show that off? Do you have jewelry? Maybe do a little chignon. Maybe do something a little bit more edgy, like a messy bun moment. Do you wanna do a ponytail? But there's a lot of back detail. Maybe opt for a bun in that situation. Next, you wanna choose hair that's gonna fit the theme. Are you having a very ritzy, ritzy wedding? Are you spending big bucks? Are you doing a quick on the go wedding? You're gonna do your own hair for it. Well, these are all different things you should look at. There's different complexities to each look.  Those are pretty easy to do yourself at home and you can totally do it. I believe in you. Look up some tutorials. Or maybe you're doing top dollar spending all this money on your wedding and you have the best hairstyle as well, then see what they can do as for their opinion. Maybe do a look that's a little more complicated, like some glamour waves or the easy going bun. What I'm saying is dress for the occasion, even with your hair. If you wanna be glam, be glam. If you wanna be chill, be chill. My next tip is longer is not always better. And I'm talking about hair. Every girl says I need to grow my hair out for the next two years before my wedding. First of all, there's extensions. You can just tape them in for the night. They look very natural. It's easy. It's cute. But I think having shorter hair for your wedding day is severely underrated. I think a lot of the time, the hair is very overwhelming and heavy. It's all around just like a lot to look at and a lot to deal with and looks very uncomfortable. Shorter is sometimes better and less hair is sometimes better. Don't put too many extensions in your hair, please.  And one of my most important tips of all is to be you. Do what makes you happy. This is your day and your husband's day. Don't forget that. If one of the ugly hair cells would make you happy, go for it. This is not my wedding. These are just my opinions. Whatever makes you smile, what makes you glow on the outside, is all that really matters. So f**k their opinions. And I got your back and I might roast you for it, but at least you were happy. So if you're wondering now, Brad, what would you wear for your wedding day? How would you do your hair? Well, I'm glad you asked, I know. If I was a female with longer hair, I think I would just like bohemian beach waves. I want to look like I didn't try. That's the vibe I would go for. All right, that is all for it today, guys. Don't forget to follow me on all the social media networks, including Instagram, TikTok, and Twitter at BradMondoMYC. You can also find my Facebook page and my Snapchat show under BradMondo. And I mean, to have beautiful, amazing, fabulous hair on your wedding day, you need beautiful, fabulous, amazing products. And I got your back. You can shop X-Mondo hair linked below. Go for it, have some fun, do a little shopping. Let me know what you got. You can also follow X-Mondo hair on Instagram to stay up to date with new product launches happening all damn time. And that is all for today, guys. Thank you so much for watching.  And I'll see you next time. Bye, guys. This is Glitterati, our styling serum. Glitterati will deliver hold, shine, and lightweight hydration combined with the most beautiful touch of pink sparkle. Glitterati can be used on any hair type or texture. We've also added argan oil and chamomile extract to help strengthen your hair and just make it feel and look luxurious.


In this session I will talk about an often neglected part of a tracker project that's monitoring and evaluation. What we want is to make sure that projects are running smoothly and that we learn from our mistakes. And it's really not rocket science but you need to be, it requires some conscious effort and you need to plan for it. In this session I hope that you will learn more on why monitoring evaluation of a tracker program is important. Some selected items that you can monitor for your tracker program. We'll also have a presentation of some practical tools that you can use for monitoring at a more technical level in the HR stew and my colleague Brian he will, he will show you some slides on that. You can go through some different types of evaluations that you can use to learn from your project. Can you tell us, I think I'm hearing your background. Yeah, the monitoring and evaluation. It can provide information on what an intervention is doing, how well it's performing, whether it's achieving its aims and objectives. You can also guide future interventions and activities. And it's really an important part of accountability to funding agencies and to the other stakeholders in the project. And plans for monitoring and evaluation, it should be done at the beginning of the project, beginning of the intervention, so you know what you want to monitor or evaluate that very often we're pretty bad at this in our fields. And why is this sort of the neglected or considered a stepchild of project planning. Maybe it can feel like it's not 100% necessary. It's not sort of the it's not the burning activity to be done it's more important to get things up and running and to require funding etc. So it can feel pretty abstract perhaps might not be high on the agenda of stakeholders. It's really important and I want you to take this away from the session that monitoring and evaluation can be done very extensively very structured very well plans, according to rigid protocols. You can have big teams working on monitoring and evaluation. But the monitoring and evaluation that is good is the monitoring and evaluation that you actually do. So it doesn't have to be that extensive. It doesn't have to be a huge big complex thing. It's just more important that you as a project manager or project owner consciously sort of pay attention to how your project is running. And what you can learn from it, or what you can learn from your own mistakes or other mistakes, or the factor surrounding your project that you can do something about or that you can adapt to. It's better to pick a few key things that you would like to pay attention to in your project and actually monitor them. Then to just close your eyes and think that things are going pretty well and dandy, but you don't really know because it's a bit uncomfortable and it's a bit of a struggle to find out, etc. So a little is a lot better than nothing. So there is a difference between monitoring and evaluation. So monitoring is sort of regular collection of information about all project activities. This is your day to day gathering our information or week to week or month to month, something you do at regular intervals to keep track of how your project is progressing to identify problems quickly. For example, I will assume that it is important that all nurses who are using your tuberculosis tracker program that they have a functioning device. But if you just close your eyes and you don't actually check whether they have a functioning device, you have no idea and you have no way to actually help them to fix that problem. So you need to decide what you want to keep look out for and decide how you're going to do it. And this is something that is being done at a regular intervals. And it's there to sort of inform your day to day decisions, what should we do today, what should we do tomorrow, what do we need to change straight away, what problems do we need to fix along the course of the project to make sure that things are running smoothly. And it serves as input to evaluations. So, once you through monitoring have discovered a systematic set of problems, for example, you can then evaluate. Why did we end up here or what are the factors contributing to this and what do we do about that going forward. So evaluation, it has more sort of a judgment or value perspective to it. You want to say why and you want to say is this good is it bad. How should it have been done. And it's sort of just sort of recording and accounting here you assess. And an evaluation is, it's more used to do major decisions, not just these day to day but like where do we want to be next year in two years time, what key areas do we need to acquire funding for. And it provides information for doing planning and something that you do more periodically not every day every month that maybe you do it at the start of your program. And again, please just post questions or raise your hand if you have any questions. I think what is important is to clearly define in the beginning what you want to monitor at the start of your projects. So and it should be closely linked to the goals of your tracker program I think we've been talking a lot about this and this academy that you need to be clear on on why you are doing a tracker program. Why do you even want to start this project, what you want to achieve, and then you need to figure out what to monitor and it needs to link back up to that goal. So go back to the first tab in your project planning template look, look at what you wrote there. And then you would need to sort of link your monitoring activities to to that goal. And you need to think through what success looks like in your project. What would make your project owners happy what would make you happy when would the stakeholders in this project sort of pat them on themselves on the back and say okay we did good with this project we achieved what we want to achieve. It can be different things it could be less children missing vaccine appointments, it could be more happy nurses. It could be that all health facilities in a region is using the system at a rather certain intervals. It could be that the system is always up. There is no gap in the data. This will vary a lot from project to project but you need to be have a conscious thought around what the success look like in your project. And then, once you've done that you can pick some key indicators that are important to your project so if you're, if you're concerned about the less of the children missing vaccines you can say that we consider our projects. Successful if less than 5% of the vaccinated children are missing their vaccine appointments for example. How many children are showing up how many children are getting their vaccines what do those numbers look like last month what is it now what is it only on a whole year. And you can also change what you monitor as the project progresses so it's not like you decide on something at the start and then you're stuck with that. But maybe you figure out as the project progresses that you would like to pay attention to other things. I think this is a good consideration with tracker programs because the tracker programs you really are looking at data and looking at work from a different perspective. So this is a great opportunity to really kind of redefine traditionally what you're monitoring so. For like an API program you want to of course find the dropout rate of the children. However, there's this opportunity to kind of trim down all the fluff and reevaluate what you really want to monitor. And I think someone made a good comment earlier about. I think that they made a comment about how it's important to do quality over quantity. This is why it's the stepchild because I mean you have so much work to do. So, really focus on like really take some time with the groups to decide what you really want to monitor. And I think many of the things that you could monitor you will sort of figure out if you have a key indicator or a couple of key indicators that you do monitor all the time. And you see that we never get data in on time. So, and I'll get to that later but you should always do some investigations if the numbers or if the results from your monitoring isn't what you expect, you need to do some investigation and then maybe that will that again will highlight that. Well, the numbers aren't coming in because the nurses are lacking devices or because because then you make that phone call and you say to the tier district managers and say, Listen, we have this huge problem we never gets any numbers from this region. Why is that. Oh, it's because of blah blah blah or they feel that the system is always lagging. Okay, maybe that's a really an indication that you for a period of time you should pay close attention to your server performance for So you don't have to monitor 100 different things at all times but adjust as you go and if you identify problem problematic areas in your in your program then of course you monitor that more closely for a period of a period of time until you fix. Thanks for the good comments, Kim. Any other comments. Okay, is Brian are you on Brian. Can someone ping Brian to make sure he's ready for his slides later. If he doesn't answer now. Brian is here and now has co-hosts if you want to meet yourself Brian. Yeah, I sorry I wasn't I couldn't. I'm not sure if you're ready to answer this Brian but I mean you've been part of a pretty complex tracker program in Palestine, for example. Do you have any reflections around what sort of things you monitored in your in your tracker program with where you were following mothers and children. Quite detailed. I mean there's, there's different ways from approaching that as I'll be going into in a second slides but definitely monitoring of data quality to make sure that, you know that some important data points within the tracker program are actually being entered. So for example, that might be like a pregnancy identification stage that we had making sure that the very early entrance into the program that the key questions are being answered before we get into more clinical workflow. And also making sure that, you know, usage is pretty consistent across different users and organization units, so that you're getting for something like pregnancies it's not as seasonal something like malaria right so but you might still have. You might still expect to see consistent numbers coming in from all clinics instead of having, you know, rapid spikes. Another thing that we looked into was, since it was a point of care tool, we wanted to assess how frequently people are actually creating are creating events and enrollments at the point of care as in during work hours, when they would actually be seeing all kinds of clients. So developing a monitoring system to check when certain data points were actually being entered into the system by looking at like the attract entity value audit logs, and being able to assess. Oh actually like 15% of all the data points that are being entered in this tracker are happening after the clinic the clinics closed for the day at around like 46 o'clock. And that implies that you know maybe we should be encouraging more point of care use or we need to be talking to the clinicians about where some of the reasons or obstacles why you can't enter in all this data when the client is visiting the clinic. So those are some some things that we looked into. Yeah, that's great. And I think you also had a good, a good example that I've heard previously for example seeing that you're now collecting individualized data, you can also drill deeper so what I've heard for example in the case of this Palestine project is you wanted to monitor whether the women were getting a certain measurement I think it was hemoglobin or something that they should receive once or twice during their antinatal care. They should get a certain blood test done. You can monitor whether the women aren't getting this, this test and if it's routinely proving etc. And then you see that before you would just know that's okay you have 100% coverage for for this test in the population because you have 10 pregnant women and you can test conducted and use everybody should be happy. But if you drill down a bit more in the data you would perhaps see that you've conducted this test five times on woman a and five times on woman B. And then you have eight women that have never gotten this test, which is not the purpose of the program so I think there are many things that you can monitor and that tracker really allows you to monitor in terms of your service provision. Yeah, definitely. When it comes to the actual data outputs that we get not just for the, the implementation of the project but for monitoring of the services. There are so many different ways that you can actually use the tracker module modules to assess quality of care indicators. And, or effective coverness ratios so you can actually see are the timeliness of ANC visits at the right gestational age intervals actually occurring for individuals. And then also like are they linking at those different antinatal care visits to labs results in a timely manner, so they can actually get treatment if they need it at the right time. Rather than just waiting until the very end of the pregnancy and then adding up all of those numbers. So, you can really look at this two ways because you can think that okay you need to actually monitor your tracker project per se the IT and implementation project you need to monitor certain indicators on whether sort of introducing the system is working as you would like. But you can also use tracker data to monitor your health program and that's, I guess that's the main benefit of tracker right because you now get data that you can use to pay closer attention to how you actually provide the service itself so that's two different. Two different things to monitor. Thanks Brian. Any questions from the chat or comments. Just keep them coming if you have any.  But you we have some examples here of things to monitor and it's by no means exhaustive this is just a little list that I made when I was making the slides. You can monitor a whole lot of other things and we'll cover that in the mentee afterwards but here are some few things that you can monitor you can monitor. And users of the system how many users the percentage of users like usage analytics Brian will cover this a bit later on how you could practically do that. You could somehow measure user satisfaction how happy are you are the users who are using the system as it's, is it working for them. You could look at the tracker program itself you could look at the number or the proportion or somehow counting the attract entities you have in your system the mother is the children the stock the cases, whatever, to see if it goes up or if you have a big proportion or small proportion or coverage etc. You could pay attention to the completeness of your tracker data, for example, we know that. If you compare it to aggregate numbers if you're collecting the same type of information. If you're doing to different data collection activities. Sometimes you could find that it differs so example from doing COVID vaccinations now we see that in certain countries you might have recorded that you have administered. 300,000 COVID vaccine doses, but in tracker you've only registered 200,000 of them so you could pay attention on whether your tracker numbers are matching the aggregate numbers. It should be interesting makes checking it with population figures or what you expect the, the, the, the number of tracked entities to be. You can pay attention to timeliness. So this could be this data coming in on time like Brian mentioned here for AFI adverse events following immunization if you get a reaction after getting your vaccine you could perhaps monitor how do people register the AFI after the vaccine is given or after the adverse event was reported for example how long did it take to file the official report. How many reports were actually investigated or checked out. And other data quality indicators. So through technical performance monitoring of your program you can check monitor through various different technical tools I'm not the expert on this but we have people that know how to do this but you can check the uptime of your server make sure it's not down and response times, how long does the user have to wait before the page loads etc etc. So you have to pay attention to these things and monitor them regularly and really keep tabs on that. I mean I've done evaluations and visited. We don't have to mention countries but where you see that people are getting up at four in the morning because that's when the server is most responsive and they'll climb to the top of a hill to enter their data. Data hasn't been entered for months because the server doesn't reply or it can't handle the loads or it doesn't function well in the afternoons when people are actually doing the work, etc etc. But if you don't pay attention to this both looking at the technical performance of your program but also going out there and checking with users, interviewing them, visiting the field, hearing how is this working for you in practice then you won't know. And it might be a huge problem for a long time and then your project is a bit of a waste. And you can support you can do you can monitor your support team for example saying that you have a call center that will support your series you could check the call center performance how many phone calls are they able to pick up on time how satisfied our people with their performance. This is just a very short list of a million things that you can monitor and I think if we now move to the men see. And when you are monitoring whatever you choose to monitor in your project, it should be, it should lead to a root cause analysis so not just seeing the results but then asking the question why are we getting this results. What can you do about it, and then of course implement the changes and continue monitoring it's pretty sort of an obvious message but there's a lot of things that we like to monitor but then again same as the same as the first mentee is being done with the results that's being put in the door it's put in a report or it's it's put on a PowerPoints but then are you actually taking action based on the things that you do monitor that maybe it's better to monitor fewer things and focus on doing something about the results. Then monitoring 500 things and using all your project resources on that and then no one has time to fix the issues. Now we are at the famous word of the day capacity and competence. It's the word of the day. I'll leave that up for 30 seconds capacity and competence. Okay, and next now we will I'll give the word to Brian he will talk about more technical solutions you can use to do monitoring through the tries to. I will give the fortune just say next slide Brian when I should click your. I was going to share slides but that's fine we can use this. Yeah. Yeah, that's fine. Um, great. So, um, we've talked to a bit. I think most importantly to understand the types of data that you want to collect for monitoring. But often those data that you want to monitor for your tracker program to actually measure it's it's coverage and the the rollout. So maybe there that's not what you can actually develop in DHS to. So, here are a list of potential tools that you might want to consider, and I've listed them in sort of increasing order of customization and complexity. So to fit your own use case for monitoring for monitoring tools. I think that the first one which I'm sure everyone on this call is familiar with is dashboards. There are a lot of indicators that you could build about your tracker program in DHS to that you could then utilize as a type of admin level monitoring dashboard. It might be just the number of of new enrollments or events for your tracker program. The number of org units that have reported successfully on a given month, or it might be some data completeness indicators as well so if you have a form where you know that, you know, every 10, every one of the 10 questions actually need to be fill in, then you could build a percentage to assess what is the average percentage of indicators that are actually being filled in for this event in the tracker program. And another thing that you might consider of course then is validation rule analysis. So, with that same idea of building on program indicators and indicators for your for your program monitoring. At an administrative level, you could also run some routine validation rule checks for to say, if you are rolling out this program incrementally. And you know that you should be adding on more users and more tracker events every month, then maybe if one organization unit drops off and stops using the system you could be alerted through validation rule analysis that this month, this clinic using the system less than it did the previous month, and that might be some nudge for you to get sent via an email or DHIS to report of a list of all of the clinics that are not using the system as average. So that's a good question from so digits. And we'll get to that in a in a second. So there are also a variety of tools that are available in the DHIS to app hub. These are not core DHIS to apps, however, they have been developed for other purposes and our public use. So you can look into the DHIS to app hub and explore two separate apps which I which I'll go into just briefly here. You can look at usage analytics which is the app that I'm presenting here. And from this you can get a list of the number of favorite views and dashboard views. So you can see which dashboard is the most frequently visited I think. Yeah, or the number of dashboards that have been viewed. And then this within this usage analytics you can also see the number of active users in your system month over month. So there's a lot of interesting things that you can drill down with this usage analytics app. It's exploiting this within DHIS to every time that a favorite is opened from a dashboard. And you get like a data statistics event that's a that's generated and in the back end of the system and so you can actually analyze that and see which users are most active on your dashboards and most active with your analytics apps. There's also the user extended app which I'll go into just briefly, but for all these other types of tools here. I'll go through just briefly but at the most complex that you might consider are doing something custom whether that's a an SQL view for certain tables in the back end that are not exposed to other DHIS to apps but with a SQL view can actually extract that information for for routine monitoring. One example that may be useful is to consider the program temp ownership audit table. We remembered earlier in our discussions about security and privacy this notion of breaking the glass or accessing a record that's outside that was initially enrolled outside of one's organization unit where they can actually capture data. And when that happens and you have to write in a short note about why you're accessing this this this record. If you go into the program temp ownership audit table, then you can find a list of all of those breaking the glass accesses and then you can assess sort of how frequently that breaking the glass feature is being used on your program. Other things that you might consider for free for SQL views might just be getting the number of new events by user that have been entered or as I as I described to Anna earlier, looking at the time when events are being created as well, so that you can assess some data use in hours or after work hours. There's also ways for you can extract data from the DHIS to API for similar types of information. One way that we use those custom scripts were to generate reports on scheduled messages that have been delivered from the system which messages were being the most frequently delivered to to the to the patients in e registries. And then you could also as I'll go into a bit, even exploit this the the platform of DHIS to to develop your own app that might assist your, your program monitoring, and I'll discuss that just briefly There is a question in the chapter Brian, can we check on the dashboard if someone is struggling with sinking or if someone is getting error messages. Yeah, so with with Android monitoring that will be one of the upcoming slides. I'm not sure if we can see that per user, maybe hi may could chime in but we could also. It is possible to see the the error logs that you can get. Yeah. And so this is the an example of the user extended app. Oh, it's a slightly different from what I had earlier, but if in the user extended app here. You don't just see the information that you get from the user app. This is a way to display all of the information about users that is exposed through the API. So now you cannot just see here the, the name and surname of the user, but also their, their roles, and their groups, and then also the last login that a user had in the date that that was last updated. And then so if you want to filter and then export this data you could actually make a table routinely of the number of active users and by group of users, which ones have not been logging in recently. So that might be something to look into. Okay, so this was. So this is a an example of what you can get from the Google, Google Play console. So, and we use this on our programs as well for you registries we had a, we had a custom DHS to Android app that was based on the DHS to SDK. And so we added a few more features into this and what we could actually do with that when we had our own fork of this Android app is that we created our own at Android APK, and then loaded it as its own listening into the Google Play Store. And what that allowed us to do was to actually use good Google Play and Google Analytics features to assess things like crash reports for certain endpoints of our of our app. So you can see that we had like a very serious problem with this, a number format exception I think it was like an integer versus a floating point, and that helped us debug the, the custom app that we released. And the next one, yeah, you can go to the next screen after that. One of the other features of the Google console was that it also allowed us to see the number of installs that were being made, and the number of uninstalled that were being made as well. So now we can actually see this is it's not exactly like an MDM but we can assess like how many of this, how many times that this app has been installed, and how many times it's been uninstalled by accident, for example. And so we can actually track the increase of app downloads over time. And same with the crash reports down there. So this one has been has been useful for us as well. I think I added another image here earlier but I know it was in the previous one I think you need to refresh on it's okay. I also use Google console analytics to track the number of which version of Android your devices are running on. Right, because maybe that there's a problem, not with the app itself. Some users are are using a earlier version of Android that may have compatibility issues with the Android app. And so that is a useful piece of information to know the problem with using the Google console though is that you need to have your own listing on the Google Play Store. In order to do that, you need to sign the app yourself you need to compile it yourself, and you probably need to have a developer who's familiar with the Google Play Store to SDK, in order to make your own listing. So, the, the Android team at DHS to is working on solutions that you could actually get similar types of analytics that you would from Google analytics, but in a, in a way that is hosted on your server and could compile those Android statistics and deliver them to a software called Matomo so if you can go to the next slide on it. So Matomo is also a an open source tool that would deliver you statistics on visits to your, to your URL, how many of them were from Android devices, how many of them were from the DHS to app. Where were those visitors coming from, et cetera, the duration of visits things like that. There's a lot of information that could be gleaned from these types of monitoring tools. And I think that in the coming weeks, the DHS to team will be posting information about integration with Matomo tools in the community of practice. So be on the lookout for some updates on how to use Matomo for your Android implementation. So this, speaking of like more in depth. I saw some people mentioned earlier in the in the mentor meter about server status and sort of performance of the server. There's a number of different tools which we don't really have the time or scope of this, this workshop to go into. But one of the, the favored ones and spin suggested, I, I linked to it in the presentation to a community of practice link. So here's discussed how you can use Prometheus to collect server statistics, and then Grafana, which is a visualization library to actually analyze your performance statistics. So you can see things here on, you know, memory usage over time server requests. So one big piece of, you might notice is that you get a high number of server requests at the beginning of the day as people sort of log into their tracker for the first time, or when they return from holiday and then it sort of falls back over time. And so now you can actually plan your server resources according to those thresholds of maximal requests during the day. Right. So you can make sure that you have server capacity to meet that demand. But there are more technical documents and how to do that, which I can send and link to. Next. And this was one of the last ones that I had in my list of tools, which is building your own custom app. And in our implementation in Bangladesh. In the DDRB. One of the analysts was was very gifted with with JavaScript and building his own custom apps. So he actually built a custom program monitoring app, just specifically for implementation with the DHS to apps framework. So here is an example of how he has a listing by user and then the number of, yes, and then the number of monitor the number of mch enrollments that were made for that user in the past 15 days, the number of mch enrollments that were made in the last 30 days, total enrollments and then the number of pregnancy identification stages that were actually made after the fact. And this is different from other DHS analytics because you may have multiple users reporting to a single organization unit. And so you might want to actually drill down to the user level to see how are there any users which have been particularly inactive in the last 30 days. Maybe they're they've they've gone on leave, or they've retired and then you can deactivate their accounts, but it's a way to drill down into the data. And the, and also it's filtering by a user group up there so you can get a breakdown by user group. Next. And then here's another in another tab within the same app. He also built a platform to do a monitoring log of the call center. So it's a bit small and I apologize for that. But essentially this is a table of the of the issues which have been reported from the field, and then who logged them, what the issue was, and then the status whether this issue is still pending or it's been resolved. So as an example it might be that someone's device doesn't work, or they lost their password, or they've found a duplicate record. These reports can be from the can be delivered from the field. And so you might want to keep it in a log whether that's in DHIs to or it's an Excel spreadsheet where it's a Google doc whatever it happens to be, but you should keep a close eye of the this log somewhere. And then at the end of the day you can analyze and say of the, you know 100 issues that we've had we've managed to resolve 85 of them. And that's a really strong performance for your, for your system. That's all that I had, I think. And now, if there are any questions about those tools before we go on to evaluation. I just have a quick question, Brian. How hard is that actually use these tools out. I mean, if you just download the app, it's pretty self sufficient. It's not very hard. Yeah, so the first couple of apps that I shared our DHIs to apps that are found in the app hub. So it's it's pretty straightforward to use the user standard app or the usage analytics app. You could you could go to a link to the place where we can explore a bit yourself and what it looks like. So the ones that get more advanced you might need someone who has an experience with sequel to to query the back end of the database. And you may also have some might want to have a developer on hand to use this Android monitoring or server monitoring tools as well.  Yeah, there's a question from the audience. Yeah, I just progressed because I'm going to talk about that in the next. Yeah. Okay. Yeah. So hopefully everyone says that. Yeah. So I think I'll, I'll go through I have another four slides and 10 minutes. Thank you so much, Brian. It was super useful and you can reach out to Brian as well Brian at details to.org. If you have specific questions on his presentation or just posted in the slack. Thanks. So we've been talking a lot on monitoring up to now. But we also have evaluation and evaluation is, as I said in the beginning more concerned with assessing whether things are progressing well. Whether you're achieving what you're set out to do and if you're making the difference that you want to do. If it's happening the evaluation would seek to understand how and why the intervention has worked well if the project is unsuccessful questions can be raised to what could have been done differently so while monitoring is more just recording what is happening and an evaluation you would put more judgment to it and and try to understand why are we getting this result can we do something different the next time and very high level you could sort of think of evaluation to happen. Pre project or more sort of readiness assessments are you ready to start this is part of the part of this project planning template is part of this. You can do evaluation work in the middle of your project, should we change course, are we going in the right, are we going in the right direction, or end of the project what should we, or others do differently next time. And you can approach evaluation in different ways. I highlighted two of them here because I think this is what we do the most feasibility studies. Like, are we ready to start. Is this even possible to do. And we also do quite a lot of this implementation research where we assess the uptake the institutionalization the sustainability of the system. In any given context, like are the policies practices supporting your, your project, but I would welcome and I think many people would do that as well welcome more sort of evaluation on health outcomes, whether the digital health intervention actually achieves the intended results, both in a controlled more research type of setting but also in the uncontrolled setting in a non research setting. So really looking into your service deliver indicators, and for some projects, but then you would need maybe more sort of medical research competence, going in and seeing well are if you have a health intervention digital health intervention where your aim is to make sure that babies are born, not premature but later because you're doing something then are you actually achieving this medical results does it help. And I think, at least from from the his perspective that these two types of evaluations are the feasibility and the more research on the implementation itself is what we're doing more on. It's a key point that evaluation work should always result in some action points, again, linking back to where did the evaluation results end up is it in a drawer or is it as in a report management so that any evaluation should result in action points. It should be to organize your technical team differently, more devices, fix the design of the program, increase awareness in the user community, I mean it can, the list can be long. I added here, one example of a type of very sort of high level assessment work that we do quite a lot with the countries that we work with in the haste network, doing sort of doing looking at core areas for funding because quite often assessments can lead to future needs of funding, for example. So sometimes we do high level assessments in collaboration with or on behalf of for example global fund, so that they also have more knowledge to whether approve requests for funding for certain areas of the health system. So this could be to then go in and via various methods identify different challenges, you could identify that the there is an understaffed national team, they are the team doesn't have sufficient skills and teachers to maintenance for example and then it ends up in some the targeted activities. So, and of course again then you can link it back to the funding, the funding session that we had earlier or the budgeting. Where do you find budgets for this what do you prioritize first. Yeah, there is a question here are there any specific conceptual frameworks for the implementation research option which are particularly suited for or designed for it projects. I can come up with anyone right now but yes there are different types of what you call it. Yeah conceptual frameworks where you have a theory of change for example where you explain that you believe that these are the factors that impact that impact the progress or the successfulness of a certain intervention and then you try to monitor these things. I'll pick up some and share, share later. I think that was all I had I had a short exercise in the end but I don't think we have time for that. Any last final comments. I think my key takeaway is to remember that doing a little bit of monitoring and a little bit of evaluation is better than doing something designing something super comprehensive that you're not actually going to do or look at. That's with you out of the session I will be happy.