OK, so let me first define the words in the title, which also gives us a chance to introduce some notation. So that's the nonlinear Schrodinger equation. Defocusing with a plus sign, focusing with a minus sign, which will be morally irrelevant for the entire talk. Which one is which? But plus is defocusing, minus is focusing. OK, and this is the evolution equation for a field. At each time t, it maps rd into the complex numbers. One conserved quantity for this is the mass. So the mass of u integral squared, it's conserved for the flow. And what does mass subcritical mean? What it really means is that conservation of mass precludes interesting nonlinear behavior at small scales. So if u solves the equation above, then so does u lambda. But if you look at how the mass of u lambda varies, what's it going to be? There's a Jacobian factor, which means this. And then we have this factor. So if u lambda is going to have, right, u is some solution which has an interesting nonlinear behavior. If I want a solution that has the same interesting nonlinear behavior at small scales, then I'm going to need to take lambda to be big. But if I take lambda to be big, then the mass is also going to be big if p is smaller than 4 over d. So mass subcritical is a complex way of saying this. So that is the topic of the talk. These definitions will last throughout. Now maybe the most important announcement about the theory of the mass of critical NLS is that it's not dead. You don't hear many talks about it, but there's a lot of things that we don't understand. It turns out. And if you want to get a flavor for what's known and not known, a good place to start is not this talk, but a lot of the early work of Kenji Nakanishi, who probed it quite deeply, building on a lot of earlier work with whose references you can then find in the aforementioned work. OK, so it's not difficult. And this is perhaps one of the reasons it gets neglected is that it's absolutely trivial nowadays to prove that it's globally well posed in L2, or for that matter, in each one. It's a simple application of striccard systems. Before striccard system, it was torture, but now this is not. OK. On the other hand, one of the peculiarities, and if I can be a little bit poetical for a moment, in a sense, looking at the mass of critical, you feel a little bit like Alice through the looking glass. By the way, looking glass, old fashioned would be a mirror. You feel a little bit like Alice through the looking glass in that all the characters are the same, but they don't behave like they used to in the mass critical and supercritical case. In particular, this is ill-posed in the scaling critical sublase space. So it's ill-posed in HSC. Oh, by the way, what is HSC? HSC is the scaling critical regularity, which is d over 2 minus 2 over p, which mass of criticality is the statement this is negative. So rather than explain what I mean, let me prove, following an unpublished paper of Colleen de Christen, and tell the meaning of ill-posedness. Let me give you a quick sketch of what goes on. So imagine you take initial data, which will also introduce one of the key enemies in this regime. Imagine you take initial data, which has height about 1, but is very fat. I mean, it's living truly at scale L. It's not that it's flat in the middle. It's living truly at scale L. And if you calculate, so this is going to be my initial data, U0. And if you calculate U0 in the critical space, you'll find, well, just by dimension analysis, you can guess that this is going to be about L to the d over 2 minus SC. SC is negative. So this is a huge power of L. If L is big, this is big. Now, it's fat, so it's slow moving. And so not much is going to happen for times, so it won't disperse for times, say, so that means the nonlinear effect can have an effect over this time scale. And now, if I imagine, let's consider another initial data. I'd know U1. And what it's going to be is 1 plus L inverse with my final decision of notation here, L minus 2. So this is a number times U0. And you can see that U1 is relatively close to U0. Well, it's about L to the minus 2 times the norm of the initial data, which admittedly is a huge number. But now let's look at times near the end of this interval. So the solution is going to stay high to about 1. So one of the equations is going to have basically a 1 here to the power p, and the other one is going to have a 1 plus L to the minus 2 to the power p. Now, over a time interval of length L to the minus 2, that's going to add up to something. It's going to add up to a phase shift of about size 1. Agreed? The time derivative of an interval length L to the minus 2, they differ due to the phase factor. I mean, due to the size of this, 1 has size 1 plus 1 has size about 1. 1 has size about 1 plus L to the minus 2. L to the minus 2 is small, so we're taking a peak power. Still 1 plus L to the minus 2, or p times L to the minus 2. We care about that. OK, so over the course of this extremely long time, they're going to phase the here until 1, I mean, is more or less not parallel with the other. Maybe some parts of it are parallel, but some parts of it are anti-parallel. What L was big, so L to the minus 2 is kind of small. Ah, that's better. Yeah, OK. That explains at least some of the puzzle faces I have. Probably all the puzzle faces, I'm sure. OK, this. I mean, time has dimensions of length squared, so that's the only thing that makes sense. OK, so that means that at later times, so when T is around about L to the minus 2, these things, I mean, if I put a plus sign between them, at least I'm consistent. At this very much later time, I'd probably get more cancellations if I put a plus than if I put a minus. So at this time, OK, so this is what? Huge solution starts very close, but over an enormous amount of time isn't very close together. Of course, it doesn't smell like it'll pose this at all, right? Take for granted. That is the sort of thing you might take for granted. OK, we could rescale it. We could use critical rescaling, the one written over here. So you can make this time, the time where this divergence happens as small as you want. OK, that's not terribly dramatic. That's sort of scaling critical equations do that, right? Anything that can happen can happen arbitrarily quickly. But it's still a very large solution, OK? A very large solution which just phase decoheres. And now the thing which sort of differentiates the supercritical and the subcritical setting is the action of Galilei boosts, right? So for initial data, Galilei boost is simply a translation in Fourier variables. So I take my totally harmless function and I modulate it very highly. I'm multiplied by a highly oscillating character. The HSC norm, where it's AC is negative, fairly soon I'll start putting in absolute values and mindsets. AC is negative, but I highly modulate my initial data. The norm will go to zero, right? But of course, the nonlinear, I mean, the symmetry of the equation just corresponds to shifting to a moving reference frame. The nonlinear dynamics, the nonlinearness of the dynamics won't change. The same dynamics is just whizzing left to right, OK? So then you Galilei boost to get small data with this effect. So it's globally well-posed in the mass space, but it's not locally well-posed, at least not uniformly locally well-posed, even in the neighborhood of the origin in the critical subspace, OK? And sort of that's one of the enemies, OK? In the mass critical case, Galilei boosts a serious enemy. In the mass supercritical case, energy conservation basically kills it. It never shows up, OK? But in the mass critical, and I'm saying this persistence in the mass subcritical case, the Galilei boost has to be taken seriously. Ronald? Yes? But this only proves the absence of continuous dependence on the. I mean, that's why I put speech marks, because I know that someone's going to object that it's not a very serious one. What it does show you is that you can't expect to write down the same arguments that you do in week one of the, right? So something has to change. It's the looking glass phenomenon. In fact, I mean, let me continue my metaphor one step too far. There is this explicit looking glass here, which is the pseudo-conformal transformation which some sense intertwines the subcritical and the super critical by putting a coefficient in time. OK, what now? So I want to talk about two theorems, one of which well, let's talk about two theorems. So the first one, I want to make a little proposition. Let's take our standing equation. And now let's ask, so if u n, so sequence of functions u n, and u infinity are solutions to NLS, and they have finite mass, you can read the finite mass for the next sentence, the initial data of the sequence of solutions converges weakly to the initial data of this last solution. Then I want to tell you that u n at time t converges weakly to u infinity, which engenders a number of questions of the form y. Why is it true? Why would you care? And maybe y wasn't done before. In fact, it was done before. It's in a paper of Kenji. Not the statement, but the proof. I mean, when you see the proof, you'll see the proof was there before. And the real question, let me try and answer the question, why would you even ask this question? We know well-posedness in the normed topology. I mean, I should say, this statement is exactly the statement of well-posedness, not in the normed topology on L2. OK, now I wish. So we're going to end with this boat hook thing. This is well-posedness not in the normed topology, but in the weak topology. This is both statements here in the weak topology. So why? Well, the first thing is it's a little bit of a test of our understanding. Because well-posedness, I mean, for a linear map, sort of standard exercise in functional analysis, for a linear map, continuity from the weak topology to the weak topology is the same as boundedness, which of course is the same as continuity from the normed topology to the normed topology. But for a nonlinear map, they aren't the same. In particular, strong or norm continuity does not imply weak continuity. Let me squeeze a simple example over here. Well, let me let you think about a simple example while I erase. So an example of a map, which is norm continuous, but not weak continuous, would be something like this. F goes to the absolute value of F on something like L2 of the torus. Characters converge weekly to zero, but the images don't converge. Don't converge weekly to zero. OK, is the other direction true? I don't know. I mean, does weak to weak continuity guarantee strong to strong continuity? I don't know. Of course, in terms of saying this is an interesting statement, it's not relevant, but I don't know. For linear maps, again, it's trivial, but for nonlinear maps, the fat books that I own don't give me any hint. OK, so why am I thinking about this? Why am I telling you about this? Well, because it plays a role in some joint work with Monika Vishan and Xia Yi Jiang, Zhang, I guess I'm first, which is about non-squeezing. So Monika's talked in a couple of places here in the first conference, and also in Berkeley, about the fact that we've proved simple non-squeezing for the mass critical case, and she tried, to my mind she explained, but maybe she tried to explain why it's not a very easy problem. And part of the thing I want to say is that don't apply the methodology there. Here, there is a more direct argument. So I want to show, which sort of uses some of the ideas in that paper, but cuts out some of the horrors that come from criticality. OK, so what is non-squeezing? So I'm going to say, so let's take a ball that's in Hilbert space, center it at z, and have radius r. OK, and z is in L2. There's everything in this first part of the talk is about L2. OK, so I take a ball of radius r in L2 around some point, and this is where my initial data is going to belong. So my initial data is going to be in this set. And then you can ask at some later time, does the solution, u of t, lie in a cylinder? OK, now what do I mean by cylinder? I mean a cylinder with a cross-section is two-dimensional. How do I pluck out two real coordinates from a complex function? What I could do is take the inner product with a unit vector. So this is a unit vector. So that's plucking out two real coordinates and leaving the others alone. And I'm asking, are these two real coordinates in some ball centered at a complex number alpha of radius little r? So now the real question is, is it true that for every initial data in some ball, at this later time, every solution starting there is in the cylinder? And the answer is this cannot happen. So every is this in the cylinder. And the answer is no if little r is smaller than bigger. OK, so in the generality of general symplectic transformations, symplectomorphisms, in finite-dimensional Hilbert spaces, this is true, there's a famous theorem at Gromov. It's the Gromov non-squeezing theorem. But there's no sufficiently general analog in infinite-dimensional cases to cover the PDs of broad interest. But starting with Cookson, the question arose, can we prove this assertion not for general symplectomorphisms, but for the flows of our favorite Hamiltonian PDs? The flow map of a Hamiltonian PD is an example of a symplectomorphism. And the only example you need to care about while we're here. And the answer has been proven to be no, I mean, this thing doesn't work in various cases. I should say the fact that we're working on L2 is because this is the symplectic Hilbert space in which the dynamics is Hamiltonian. You don't get to change this in finite dimensions. This is a given. And for the PD case, it's also a given. In fact, Cookson has some examples where he shows that if you fiddle with that, you can get squeezing. If you don't work in the proper symplectic space, you work in some other Hilbert space. You can't get squeezing. So you have to work in L2. You can't work in H1. So let me quickly talk a little bit about the history without writing anything down. So as I said, the idea of studying this PD was initiated by Cookson, who's talked about, who's proved a relatively general theorem with strong compactance assumption, which covered various Hamiltonian PDs in the circle. And then Bourguin extended the Hamiltonian PDs on the circle on the torus to which this is applied, as well as proving it for the cubic NLS on the torus. The I team, Colleen D'Aquil, Staffelani, Takaoka, and Tao, proved non-squeezing for the Kordovic-Devries equation on the torus. In fact, there were a very long paper proving regular global well-posedness in the symplectic space. And then they had to write another long paper to prove non-squeezing. And that's one of the reasons. I mean, we list various scientific reasons in the papers. But maybe the better emotional reason is you don't understand local well-posedness fully if you can't prove non-squeezing. It's sort of a new level of challenge on the local well-posedness front. And so Hong and Kwan gave a simpler proof of non-squeezing for KDV on the torus. Rumugu, this is not chronological order. Just do what I wrote it down. Rumugu showed that the Benjamin Bahunian Benjamin Bahunian equation is non-squeezing on a torus. And Mendelssohn proved somewhat conditionally that the cubic Kline-Gordon on the three torus is non-squeezing. OK, now, if you listen to what just passed, the word torus appeared in all statements of results. And in fact, we were at Berkeley and listening to MSRI, listening to talk of Mendelssohn, and two people in the audience said, but what about RD? And so, well, what about RD with criticality, right? This is sort of part of the innovation of Mendelssohn. And so we wrote a paper about the cubic NLS in 2D, OK? And as Monika hopefully explained to you, it's long and complicated. And if we go to mass of criticality, we can avoid some of the complexities coming from criticality. And that's the sort of thing I wanted to sketch. Can I ask a question? Yes, please. Why do you need a cylinder whose, what do you say, a co-dimension two cylinder? Because it's hardest. If you can't squeeze into a co-dimension two cylinder, you sure as expletive can't squeeze into a co-dimension 20 cylinder. Fair enough? OK. OK. But it is important that this co-dimension two cylinder is a conjugate pair of coordinates. OK, so if we were on the torus, so what's the basic idea of all these things starting with coaxing?  You invoke Gromov, right? Or one of the proofs of Gromov by the original Gromov statement, or some specific capacity arguments. And you say, well, I have witnesses to non, actually that's not quite how it's done, rephrase in the language that works for us. You find witnesses to non-squeezing in those finite dimensional context, and you need to show that witness persists in the original PDE. OK, so that's our approach. It's not how these previous papers phrased it, it's just sort of estimates in ta-da, but for the whole Euclidean space setting, right? This is the better language, I claim. OK, so if we're on the finite dimensional, if we're on the torus, we could finite dimensionalize the problem purely by introducing a Fourier cut-off. On the Euclidean space, right? You need to first, well, you need to both truncate in frequency and in space, and it turns out that the best thing to do is to truncate in frequency, deal with that problem in and of itself, and then up posteriori truncate in space. But on the torus, of course, you don't need to truncate in frequency. So how would we truncate in frequency? We could replace it by the following Hamiltonian PDE. Let's write this with the L-key of U-N, plus some non-linearity. Let me save myself some bother later and call it F-N. What's this non-linearity? You take the function, you project it onto frequencies less than or equal to N-N. You take the non-linearity of the projected function, but that's not all. And then, right, the point has to be Hamiltonian. If you write down the Hamiltonian with a projected U, when you form the equation, you differentiate it to U, there's a P left over, right? And that P sits out here. So by Gromov's theorem, if we're on the torus, there exist solutions to that PDE which start in a ball and end up outside the cylinder. And what we'd like to do is take N-N and send it to infinity. Now, if we send N-N to infinity, the ball is weakly closed. Well, the closed ball is weakly closed. The limiting initial data, passing to a subsequent weak topology, the initial data will end up inside the ball. So we take our initial datas, and we can take a weak limit and make that our limiting initial data. It will be in the ball. Now, if you're outside the cylinder, I mean, the exterior of the cylinder, it's an open cylinder, the exterior of the cylinder, is also weakly closed. So if I take these things at later time and take a limit, these points outside the cylinder, if I take a weak limit, it's still going to be outside the cylinder. So U in at time t converges weakly to something which we pray is U infinity at time t. So the fact that it converges weakly is sort of relatively trivial. Well, that's the proposition. The question is, is this function actually this function? Is it the solution? And that's exactly the question weak will pose in this. And so on the tourist, this would be a complete solution. On the real line, this is sort of half of the solution. So in the mass critical case, you have a problem which arises from the mass criticality of this equation. It looks like cubic NLS with a low frequency projection, which should make it more harmless than mass critical NLS. But that you need, I mean, if you're going to prove that the sequence converges to anything, you better be able to prove it uniformly bounded, right? So if you're going to treat the mass critical, you need to prove that these sequence of solutions U N are uniformly bounded as N N runs off to infinity, OK? Now, this is a pretty non-trivial coupling constant. In particular, I mean, while as a Fourier multiplier, it has a sign as a convolution kernel, it doesn't even have a sign, OK? So it's not clear how to prove. OK, so you could say, I mean, the simplest approach would be, well, Dodson proved it without the P N, so it's just perturbative. And I'm trying to claim that it's not perturbative because putting a minus sign is not perturbative. Turning this plus to a minus sign is not perturbative. We're doing it for large data, right? And this thing has both sides. It's not clear you get upper or uniform bounds, OK? And a lot of the meat of the paper, where I'm not talking about it, I always do seem to be talking about it a lot, is precisely dealing with this equation, right? You could try and reprove. I mean, you could try and just copy out the Dodson paper, and every time you see the non-linearity, put this in. But then when you get to the fundamental monotonicity formula, it breaks, so you're back to nowhere, OK? So let's prove the proposition. Does the proposition still exist? Yes, OK. So u in at time t is by the Duhamel formula. OK. So what do I do now? I'm going to take weak limits. Well, I would like to take weak limits in everything inside. It's not quite possible. Here it is, right? This is a linear transformation. So weak convergence of the inputs guarantees weak convergence of the outputs. OK. Now what about this?  These would actually be equicontinuous functions with values in L2 in time. That makes sense, right? That would give you half of compactness, right? But at L2 regularity, you don't have that property, right? So the Galilei boosts show you can have no, I mean, right? Imagine a highly oscillatory wave packet whizzing by. At one moment, it's there, and the next moment, it's completely gone. OK. So you have no uniform continuity in L2 in time. You could try to go to a negative regularity, but it's ill-posed in negative regularity, so you're not actually doing something sensible. OK. But if you think about what I just said, these Galilei wave packets whizzing past, they don't hang around. They go very fast. How do we prove that they sort of don't hang around? Local smoothing. OK. So local smoothing is enough to show you, after passing to a subsequence, that this will converge to some function in spacetime, locally in L2, locally in spacetime. Once you have L2, you can get high. I mean, it's locally in strict-out spaces, right? So you can go from two, for example, in one dimension, you can go from two all the way up to six, but you can't get six, right? Because you don't have any compactness in L6, so it's scaling symmetry. OK. So again, it's crucial here that this is subcritical. OK. So by compactness here, you can extract the limit. Once you have the convergence of these in suitable Lp spacetime spaces, you can get that this thing, right, the P in ought to be harmless, and after a page it is, this turns into integral from 0 to T. I, T minus S, the class in F, V. OK. Now, are these two things equal? Well, maybe, maybe not. But at almost every time, right, convergence here and weak convergence here, I mean, at almost every time, sorry, the weak limit of this is going to be this. So this is true at least at almost every time. OK.  But the right-hand side is continuous, right, by subcriticality and stricats. Actually, just by stricats. So if I change V on a null set, I wouldn't change the right-hand side, given how it appears, and I'd make it continuous. So I can delete this just by changing V on a null set, and now V solves the Duhamel equation that U infinity solves. OK. Now, just because it's an L2 solution, it's a continuous L2 solution, in the mass subcritical case is not enough to say it's the solution. You need stricats bounds, right? Basically, because you can't make sense in the nonlinearity for an L2 function. OK. But of course, U in abates stricats bounds, because, unlike the mass critical case, you can prove that solutions of this goofy equation abates stricats bounds by the exact same proof for using the subcritical, in the, when the prediction is missing. OK. So these U ins obey stricats bounds. So V obeys stricats bounds. So V is truly the solution which makes it U infinity. OK. With this stepped out with, I think we'd agree, much more swiftly than the mass critical case, dealing with the space truncation, we basically parrot the mass critical case. There are some simplifications, but it's very similar. OK. So that was topic one. I hope, I sort of hope you're not desperate for a change, but you're going to get one. Maybe you are. OK. What's topic number two? And where do I write it? Wanting to future speakers. This is a disaster. So the second thing I want to talk about is scattering at the critical regularity. OK. Large data scattering at the critical regularity. Let me state a theorem. Me first, I guess. Don't work with Satoshi Masaki. Now in Osaka. Jason Murphy. Currently in Berkeley. Monica Bichon. Currently in the audience. Now, there are a couple of conditions. One is mass of critical, because that's what I'm talking about. The second is I said the word scattering. And unless your non-linearity has a big enough power, that's simply not going to happen. You need a little bit of non-linearity. I mean you need a little bit of power in the non-linearity, otherwise there's no way scattering can happen. And then there's a technical condition, which is not the thing we most eager to change in the theorem. So it's going to stay for the foreseeable future. Well, S is negative, because that's what we're talking about. And it has to be more than minus one. OK. Now, suppose I have a solution to NLS with the interaction variables. I guess you could call them beginning quantum mechanics. The interaction variable has the following property. So if u was a linear solution, this wouldn't change. So we're going to ask that this interaction variable obeys the following bound, because I put a super together for everything. So I have a maximal lifespan solution with the property. I mean, if you can't extend it, you have to stop. But if you can, you should. That's how you define the u. If I have a maximal lifespan solution with this peculiar property, let's call it star, because we're going to talk about it a lot, then u scatters, OK, in the topology sort of written here. So what does u scatters means? It means it looks like a linear evolution.  So scatters in the sense that f of t times x to the sc converges to x to the sc times some function of x, but not of t. We'll call f infinity. OK. So the first thing, we're going to devote a fair amount of time to some remarks. Zero's remark. Star, I mean, scattering does imply star, right? If this dude is going to converge, as t goes to plus infinity, say, and really, that's all that matters here, right? This thing happens near infinity, plus or minus. OK. If this thing, if converges in this topology, well, then certainly is bounded in this topology. So scattering clearly implies, I mean, scattering implies star. That's my first actual remark. For small data, if I take my initial data at time zero, and for various reasons in the paper, that's actually a bad idea. But for conversation, we'll do it. If I take my data at time zero, it's in some weighted space. And so in particular, it seems like I'm saying such solutions exist. I seem to be taking it for granted. OK. So it's been known for a while. In particular, Nakaneshi Nozawa proved that this problem has global solutions, is locally well posed for data. Sorry. The problem is OK for data U0 obeying that condition. So x to the sc times U0 at x. OK. This is a scaling critical space. Many of the predecessors had done analysis with an epsilon of room. It's the scaling critical space. And you can see that if I write it correctly. OK. Right. Taking a derivative is reciprocal length. So derivative dsc has the same dimensions as length to the absolute sc. And of course, I made this mistake up here as well. OK. So this is a critical hypothesis. And solutions exist. And what do I mean by this? Local solutions exist for big data. So locally for large data and globally for small data. In particular, small data solutions are base data. OK. So these global solutions for large data. Right. It's only local. So compacting tools. Interesting. But for small data and these solutions, so these are base data. And it's not a very strong assertion. But at least we have no reason to doubt that star holds for all solutions in the defocusing case. Right. If you have a reason to doubt, then please tell me because I have no reason to doubt. OK. In the focusing case, OK, of course, it doesn't hold for all solutions. So star doesn't always hold. For example, you have solitons. Of course, as I remarked number zero, if your solution happens to scatter, then star will hold. But it doesn't hold for all initial data because of solitons. How do we see that solitons break it? Well, for a soliton, right, this has a limiting profile near enough. This thing is wiggling around in a compact set, which means in function space, which means when I propagate it by a large amount, it's going to get spread out astronomically, right? It's going to lose compactness. Basically, it's going to spread out. And if it spreads out a lot, then this thing is going to go to infinity. So star certainly doesn't hold for solitons. Therefore, it doesn't hold for all data in the focusing case, but not a scattering. So one peculiarity again about the mass of critical case is that now solitons are stable, right? At least ground state solitons are stable. Therefore, ground state solitons cannot be minimal obstructions to scattering in any topology, right? Because they have a neighborhood in short space that also don't scatter. Now you've made me not have to do it. Future speakers, it's actually pretty easy. There's ample room. So now, one more sort of general remark. Why are we making such an assumption? Well, if you know about the study of critical equations at non-conservative critical regularity, at the moment we don't really know what to do. Except to assume such uproar rebounds. And there is no conservation law at the critical regularity. So listen. There is something that smells like a conservation law at the critical regularity, which is the pseudo-conformal energy. So the pseudo-conformal energy, which is the energy of the solution of the mass critical, I mean, comes from the mass critical equation is pseudo-conformal invariance. If I take a solution of the mass critical equation and I do a pseudo-conformal transform, I get another solution which has an energy. That's a number. That energy is the pseudo-conformal energy of the first solution. So the pseudo-conformal energy does give something, but you need more on the initial data. You need that a full weight is in L2. And in general, it will only give you something at the critical level if P is big enough. So, you know, for the pseudo-conformal energy to lead to something useful, which I mean critical bounds, it requires that the critical regularity of the pseudo-conformal thingy has to be smaller than the critical regularity.  You do hold, okay, with conservation of mass and bring this up towards zero. So it being too negative is a good thing. It being not negative enough is the bad thing. And, well, if you look at that, that gives you a quadratic equation for P and the solution is commonly called the stress exponent. So, if P isn't too negative, as defined by a quadratic irrational, then everything's, then you can get uproaried bounds from the conservation, from the pseudo-conformal energy at the entity for this smaller class of initial value. This is a much narrower set, well, this is a strictly narrower set than discussed in the theorem. Okay, now, dirty secret. My hypothesis actually embodies uproaried decay. Okay, so if we define this function J, which is the solution of the Heisenberg equation for the position operator, we get the signs right. They're easy if you think about it, but I'm not thinking at this particular moment. Okay, so that's the solution of the Heisenberg equation. If you differentiate it and the following claim for the answer, you'll see that it is the answer. They differentiate both things in time, and you'll see they're the same, and you'll get the signs right as, now I can get the signs wrong again, again, if you think for a moment, you'll get it right, and particularly you'll think for a moment you'll see if I'm right, just by differentiating. Okay, so this is the center of mass intercept operator, I don't know. If you look, I mean this is a conserved quantity for example for the linear Schrodinger equation, it's sort of total logical, and what does it represent? Well, momentum is also conserved, the center of mass goes in a straight line, and at time zero where is it? J tells you. The center of mass at time zero is a conserved quantity, sort of a time-dependent conserved quantity. Okay, and this operator is uniterally conjugate, these are unitaries, these are unitaries, it's uniterally conjugate to your favorite operator which is either multiplication by X or differentiation. So in particular if I want to take functions of this operator, which I do, I want to take an SC to the power of this operator, so I'm going to take a star, it is equivalent to the statement that J, this operator, vector operator to the power of C, I explained how to take functions of this, this thing times U of t in L2 is bounded. Okay, now if you use say this representation and you sub-leave embedding, this will actually tell you that you have some upper e decay which is starting to make the theorem sound less interesting. I'll try and counter that shortly. SC derivative isn't L2, that's sub-leave embedding, right?  Okay, and, okay, so that actually tells you your solution is in this space. What's this horror? This is Lorentz space notation, this is the weak LP space where this is P and the infinity is weak, right? Lorentz space, L infinity. And that's a scaling critical norm, okay? And in some sense, the theorem says it implies scattering, but you know, if I told you I had a scaling and critical norm that looked like a strict arts norm, right, you'd expect just to prove scattering in a page, it doesn't work. And the problem ultimately comes from this infinity, okay? It's the standard problem. Knowing that an L infinity norm, I mean, knowing that a weak LP norm is small, doesn't mean if you sort of go far enough in time, sorry, it's finite, it doesn't mean if you go far enough in time, that same norm is small, right? This is the classic well-posedness argument for critical equations, right? If an LP norm is finite, then on a small enough set it's small, that's montane convergence, okay? But it doesn't work in weak spaces, okay? And this, of course, this problem reappears everywhere, right? Trying to solve, I mean, local well-posedness for data in critical weak spaces doesn't really work because you can't find anything that's small, right? And we have the same problem here, okay? Now, small data solutions obey this. Why do we know small data solutions scatter? Well, one way of saying why we know small data solutions scatter is for small data, the fact it's actually in this space. Again, the two is a little rince, two, right? I mean, it takes almost nothing to win, okay? But it does take something, okay? So, right, again, this is not, I mean, we're through the looking glass, right? This is mass subcritical, everything's a bit funny, right? For mass supercritical equations, you know, you have critical, upper-ary bounds in L infinity, and you want to get to bounds that aren't in L infinity in time, then there's 50 pages in between, okay? Turning an infinity into a not-infinity can be complicated, okay? So, how do we do it? Well, in some sense, I've given away the model, which is, right, all the papers that have been written on conditional well-posedness, right? Assume upper-ary L infinity bounds prove scattering, right, for a mass supercritical equation, or a conformally supercritical equation. Okay, do you want to talk about wave? Okay, so it's three steps, which again, right? I mean, so existence of minimal criminal, right?  Step two, I promise I will not leave this board or use the eraser. Okay, so existence of minimal criminals in their almost periodicity. Number two is we're going to reduce to self-similar, and number three is we're going to show that they don't exist, okay? Now, many papers could be summarized with the following three sentences, but of course everyone is different, and again, right, as I've tried to assert, we're through the looking glass, so everything smells the same, but when you talk to it, it's different. Okay, I mean linear profile decomposition in this peculiar space, okay? What are your defects of compactness? They're a little unusual, they're Galilei and Scali. Translation is broken, and time translation is broken, okay? We'll talk about that later if you care why. What do I mean by self-similar? So normally self-similar means it's a function, if I take u, I can write it as a function of t to the minus one over p, some function of x over root t. So that's what self-similar, without speech marks, quotation marks, means. What do I mean by self-similar with quotation marks? I mean the function depends on time as well, but as you vary this parameter, this function varies over a compact set, okay? So instead of being a single function for the truly self-similar, it varies over a compact set, so it behaves to some degree, like a truly self-similar, except of course you can't use any elliptic theory because it's wiggling in time, okay? And then you want to prove it doesn't exist. So why is there only one type of N and V? Why is there only one enemy? In a sense, this is the only type of enemy that's consistent with what's written over there, right? If you look at the scale of that function in order to be in that weak space, it has to spread out at exactly this rate, okay? Of course, takes easy to trivialize, but not quite so easy to prove. And then why doesn't it exist? It doesn't exist because we can prove that it has finite mass. And of course this solution grows up at zero, but finite mass solutions don't blow up at all, okay? That's the end. Yeah, that's the end. Do you have any guess as to what might happen if the data does not satisfy the weighted or too normal bound? Okay, so... What happens if you take data and do not satisfy this weighted norm? Well, I mean, this is the property of the solution, right? So, okay, so... Let's start with data that does not satisfy that, and... At times zero? At times zero, okay? Yes. Do you have a guess as to how the solution might reflect? Well, okay, there are many ways to not obey this, right? One is we do not decay at all, right? Now, I mean, if you have decayed infinity, which is slower than critical norms, that means the decayed infinity can continuously mess with you forever, okay? And so, probably you can't have scattering, right? If you want a more severe singularity at the origin, well, then it's probably opposed. There's two answers. There's probably more. So, in the sense that you have to do a lot of work because you have this norm, this infinity, right? Yes. So, what is the advantage of your condition instead of just assuming that you have a... The U of t is in the right space. Okay. So, one... I mean, this is a space for which you have local opposites, okay? I mean, it's a condition at a single time, so I can sample the solution at a single time, mess with it a little bit, and then it will construct a solution into the future, right? This, at a single time, doesn't mean much of anything. And if I do somehow choose a time where it means the most thing it can mean, if I make even the slightest modification for the solution, it might not exist at all, right? This is not...  You need an inductive hypothesis that actually leads to solutions, right? And that's why sort of taking critical LP bounds doesn't seem to go anywhere, right? In the mass supercritical setting. Yeah, and there's a... Can you say anything about p2 over ea's? It's modified.  Because, I mean, small data, the literature is enormous. Large data, the literature is quite scarce. At this moment, I mean, I have nothing to say. Maybe if I think about it, I will have something to say, but at this moment, I have absolutely nothing to say. I haven't thought about it really.


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Thanks for joining us at Think Take Hawaii. Today's topic is midterms. When we were much younger, that referred to days of high school and college. Unfortunately, now it refers to both national and state and local elections. And I think for the first time in quite a while, we've got a lot of close races and a very, very close margin in the US House and Senate and control in a number of state legislatures and in election control offices that are up for grabs. So we're fortunate to have with us today, Tina Patterson in Germantown, Maryland, mediator, arbitrator, business coach and consultant Louise Ng, partner at Denton's Law Firm, one of the largest in the world and one leading women's rights lawyers here, as well as very, very active in community service and the mom of a putting theatrical performer as we understand it. So David Larson also of Mitchell Hamlin School of Law and the immediate past chair of the American Bar Association section of dispute resolution, leading dispute rugs over and. So today's topic, midterms. What do you think folks, where are we headed? Louise, you wanna start us off? Oh, I was hoping you were gonna start off somebody else since I started off the last time and didn't have much to say.  Okay, well, I'll start it off and then I'm sure we'll have some great insights as we go down the road. But I have to say they are, the midterms are uncomfortably unpredictable. And it's, you know, candidates that I just think are, and I'm looking at the national scene that I just think are very well qualified are running very tight races. And I can never tell from day to day based on the numerous campaign ads we get, you know, who, which one is ahead, but it just seems neck and neck. I hope, of course, that the Democrats can maintain a lead. But of course the economy looks like it's gonna be a big issue. I saw something disturbing an NPR headline, I think, or was it a newspaper headline about how many voters when faced with the choice of economic security and, you know, democratic principles would rather vote for economic security. And now those are who are of us who are comfortably economically secure, you know, can be worried about those views. But, you know, you can understand why people are looking for economic security. I just, we just see too many instances internationally where people are pretty much done deals with the devil, so to speak, in order to have an economically secure life. So that's my thoughts out there. Okay, and we have a pretty good historical idea over the last 40 plus years of what the Republican economic policy is like and what trickle down really means for most of us. And we started to see what at least the current democratic economic policy looks like. And they're vastly different. Tina, any thoughts on how much people really understand of the economy and the policies that are in conflict here? Chuck, thank you. And I'm going to say this for the average person, they don't know. And I think part of what's underlying this is because in some ways the policy seemed to be very close. And as Louise mentioned and I echo the sentiment, people are not looking at the overall policy that they are thinking about the day to day and they're looking at things such as the challenges in supply chain, whether it's getting groceries and the cost of groceries going up if their heating bills have risen astronomically. So they're not thinking about the differences in the economic policies. They're really looking at, we believe that there's a recession coming and how do I best safeguard myself against that recession? Well, let's ask the question, David. You've done some work and some research on this. What do you think the Republican policies really offer? The populace. Well, I think that we just saw from the UK that the policies don't offer very much. And we had a prime minister who went kind of to be extreme of cutting taxes and corporations and wealthy and the pound dropped precipitously, created a huge economic crisis and after a matter of days, her term is over. So, I mean, that's what I think the Republican policy represents. That's, it's just an ineffective political mantra. You know, you had asked what kind of get back to the first question about what we're thinking about with elections. Now, one thing I'm thinking about right now is that the road to authoritarianism is to disqualify or destroy elections. If people begin to believe that elections aren't fair, that's the gateway to authoritarianism. And for increasing number of people, they seem receptive to the idea that maybe that's not such a bad thing. And what's really distressing to me is that even if you buy in the short term to the idea that if you do get an authoritarian leader, you believe that their values and interests will line up with yours, that's just one generation. You know, what happens to that second leader and the third leader? And if you think they're gonna continue to be responsive and attentive to the constituents when they have complete power, that's just a fantasy. So this idea that maybe this is really a viable alternative is very disturbing to me. And this idea that, eh, maybe we don't have to worry about elections because elections don't create those good results anyway. Look at the chaos where elections don't work. That is frightening to me. Well, and you look at some of the individuals that are running, whether it be governor's race in Pennsylvania or Senate in Georgia or in Arizona, and kind of get a, what can people be thinking? What's going on that gives these people such a substantial support base? Well, picking up on something Tina said, I think that economics are complicated.  There's lots of things happening in the world that are entirely outside of our country's control. And I, what I'm hearing a lot is inflation, inflation, economics, economics, I'm not hearing. What I would do differently right now to change that in the immediate future. I don't hear any of that. All I hear is that inflation is terrible, so vote out the incumbent. I don't hear how I would resolve it. And again, people are, apparently that's resonating. I wish people would ask the second question, what will you do to solve it that hasn't been done already? No, and that's a great point. That's the big question mark to which no answer has been provided except a few fringe people saying, well, if we can get rid of social security and Medicare, that's gonna free up a lot of money. But it also liberates money from exactly that sector of the population that most needs that help. So, Tina, what's your thought on where we might be able to go? What would really help this economy the most? If a policymaker came to you and said, what two or three things would you most wanna see happen? That's a good question. I think we'd have to really examine and I was gonna touch upon some of our foreign policy positions have impacted our economy. How we, what is it called? QE2, which is the adjustment to our currency in light of other currencies and how we've tried to balance that and we haven't been successful but literally looking at the supply chain imports, exports but also what are we doing in terms of living wage? I think the corporate taxes need to be revisited as well. And that isn't necessarily economic but that is more a policy. And when we think about the supply of goods and services or sorry, the availability of goods and services, it's due in large part when we have a lot of our work that has gone offshore. And I think some of the people that we hear running for office bring back jobs to America. Well, folks, let's talk about if you bring back jobs to America as you're stating what does that cost and what does the living wage really mean? And there have been studies that show what a living wage actually means and most businesses, the majority of small to medium sized businesses can't afford that. So where's the middle ground? Do you have intervention at the state level, the federal level or at another level? And I'm not saying that economic policies can't help. They certainly can but I think we have to have a real realistic discussion about it and what that means. It also means sometimes the policies that we have interstate policies that we have sometimes actually do more harm than good or for the states and both states involved. How we tax. I know I have friends who live in Texas right now and the supply of housing, affordable housing is becoming an issue to the point where people are now thinking, do I stay where I am or do I move elsewhere? And again, this is back to the economics of what can you afford when the cost of living has continued to go up? And for those who are, let's say, at 20%, 30% of the AMI for a particular area, those are the individuals that need the transportation most that may need to have access to good paying jobs and are we really willing to do that? And I think if we look at our larger corporations it's having that conversation of, can you really do what we're hearing from these politicians? And if you can't, why not? That's a great question. Hey, and affordable housing is an issue. A lot of people outside of Hawaii aren't aware of it, but Louise, that's been a major, major issue for us here for decades. Hey, do you see anything happening that's gonna bring about favorable change? Chuck, I'm not sure I have the best handle on that. I think what he has always been economically challenged, but it seems to me that part of the key to affordable housing is one of the intractable problems we have had about the very slow and Byzantine permitting process we have that just adds cost and unpredictability to the ability, it makes it an economic disincentive for developers to build housing. And of course another issue we have is of course the most economically attractive housing to be built for developers is luxury and housing for second home buyers. And of course we have a lot of that. We have a few developers who are really trying to address the rental and the workforce housing market, but we need to create more incentives or maybe just get more developers that are looking to those markets as well. And then helping them, incentivizing them by just making the whole development process easier. And I'm sure that's just scratching the surface, but just my anecdotal observations. Good insight. David is affordable housing a major economic issue in Twin Cities and Minnesota as well. Yeah, it is. We have a major development going on. We had a board motor company production plant in St. Paul, huge area on the Mississippi River. And they decided to stop building ranger small pickup trucks and decided to close the factory, which opened up a huge acreage right on the shores of the banks of the Mississippi River. And they've decided to make it a kind of a mini city, a highland bridge. And they're building a tremendous number of thousands of apartments and buildings. And of course, as you go through there, there's a lot of nice apartments that are kind of on the luxury end. And initially there were proposals to make certain that they're gonna have a certain percentage of affordable housing. And as the project has been going on, that has become less of a priority. And there's a college in town that needs some athletic facilities. And there's a lot of acreage there. And they went to the developer and kind of pitched the idea of putting their collegiate softball stadium, baseball stadium, maybe their hockey right there. And they got a receptive audience. And that was where the low income housing was going to be on that part of the parcel. And the assurance was, well, if we do that, we'll move it somewhere else, but it's not clear where that would be because it's pretty much all planned out already. So that's kind of just an example of even when we go into a project with good intention sometimes, things change. And when opportunities come that sound attractive for any variety of reasons, it seems that the affordable housing maybe gets pushed to the backside. That's a great point, David, because that reminds me, Chuck and I know well that we have a very similar issue here with a stadium that has been rusting for a very long time and had great ambitions as a sports stadium to attract people. And now they're looking at redeveloping it. And the idea was to develop a sports complex with residential work play style around. But there is a group that's trying to rethink that. Do we even need a stadium? Shouldn't we be using that for affordable housing? And that certainly is an important issue that needs to be discussed and looked into more. I mean, do you need that piece, that entertainment piece to attract people or should you be looking to reallocate the usage of your land? There's other places that can be used for recreation, but here we need the housing. But yeah, it's a good point. And I think we also need developers who have a commitment to the community they are in. Oftentimes when we get outside financing, there are folks that's gonna be on making, getting an exit strategy and making profit as opposed to thinking about the communities in which they are putting their money. We're seeing that in Honolulu and I'm sure many other cities see that too. Yeah, it is a difficult issue. One thing, St. Paul, both St. Paul and Minneapolis passed enacted rent caps, rent limitations. And Minneapolis has it kind of indexed with some inflation criteria, but St. Paul said hard, fast, 3%, that's it. You know, there's only no way, no, maybe some extreme exceptions, but it's 3%. And the problem is that people that were started to build, stopped building and they stopped building and parting in this Highland Bridge development too, but other parts of the city. And then some of the people were lower income builders. And now the problem is that if you're not gonna build any apartments at all, is that going to help with the shortage of housing? So sometimes, you know, you got a paradox of trying to do the right thing and then suddenly the response is maybe unanticipated and makes the situation even worse. So now they're backing off and they're giving all kinds of exceptions. They're gonna rate the rent increases for all kinds of things. And you've identified some key areas, not only the government bureaucracy, the permitting process, things that need to happen to make government more collaborative with affordable housing development, but maybe then things like changing the economic system, the tax system to incentivize that rather than the short-term high-level profit margin. And we see that not just in housing, but in energy and virtually all industries. And we're here from the current president and from some of the Democratic Party people that change in the tax system to reallocate away from profits and into incentivizing infrastructure development and energy development. But on the other side, we saw with the former president that major tax cuts at the highest level didn't result in more jobs or more infrastructure building and development. They actually resulted in buyback of stocks and a higher degree of concentration of wealth at higher levels. How is it that people don't get that? Are our candidates getting the right message across rather than the easy sound bite? Because that's one issue. Well, yeah, check it.  I think Koleas is right. And I think David mentioned this as well. We are in the era of sound bite. Can you say it in 30 seconds or 240 characters? And this is the opportunity to put the zinger out about the other candidate. And I have said this, what Louise said earlier is that's great, you're telling me how bad the other candidate is, but tell me what you plan to do when we've got this list of issues, when we've got limited resources and we have communities that need answers to these resources and we have to do long range planning. Where are you in this? I think some of the candidates that we're hearing from are feeding into or responding to this sense of scarcity and fear. I'm going to lose something. I'm gonna miss out on something. Something is being taken away from me. And this is how I can hold on to it because this candidate is saying, we'll go back to how it was in 1970. We'll go back to how it was in 1980. And the truth of the matter is we can't go back. This is the reality that we're in. Chuck, that question just asked, I think links up with a question you asked earlier. Bob, what would you like to see a elected representative do? What policy changes you like? And I was thinking about that and thinking, well, what I would like is that somebody to focus on the proliferation of social media sites and websites and misinformation that's getting out there and try to make some effort to control the conversation so that there's not so much misinformation, ricocheting around in virtual space. So you ask, how can people not get it? Well, one reason people don't get it is because there are this myriad of websites, you got Rumble and Perler and GAV and TruthSocial that are putting out a fair amount of misinformation and some people are only paying attention to those. And they don't see what the actual consequences will be because they're not being told that, they're being told other things. So I think it's understandable why people aren't getting it anymore because they're getting so misdirected that it's difficult to figure out what's accurate and what isn't. No, and that's a great insight. And you think of the old days of radio and at least there was and still is a national public radio, it may not draw a great audience but at least it offers a much more independent, less biased sharing of information and gathering of information than certainly Fox News or those on the other side of it. There's also public TV but we don't have public social media. We don't have an alternative to Facebook, Twitter, Instagram, TikTok and the rest of those. Tina brings a smile. Hey, can we put you in charge of engineering, public social media, PSM? Is this gonna be a cabinet level position? I chuckled because you mentioned TikTok and I hadn't really thought of TikTok as being a reliable news source. Not that the others that you mentioned were that much more reliable but TikTok definitely wasn't on my list. But then again, I don't know if I'm their target demograph. But if you can make the social media position a cabinet level, I'm in and I know who's gonna be on my cabinet team with me. I'm looking at three of those individuals right now. I saw a recent survey and you probably find it that said that millennials get most of their news information from TikTok. That has become a new source and people are believing whatever they're seeing on TikTok. So I'm definitely not that demograph. Well, and it's a little scary because my daughter's kids are teenagers and during the pandemic, Grandparent's Day was done on Zoom. So could actually participate even though it was in Boston. And one of the first questions that the teacher who was conducting it asked was, oh, what are your news sources, your information sources? And the parents, a lot of them came out with New York Times, Washington Post, Wall Street Journal, ouch. But the kids unilaterally and unanimously all came out with social media as their sources. And we laugh about TikTok but it may not be an information source but it certainly can be a propaganda source and it certainly can be a message source for the kids who spend a lot of time on that. Yeah, I made reference to millennials. It's really younger than that. The generations that are following that group that are really paying the attention that the platforms like TikTok. And just to get back to one thing because I know we're gonna run out of time that I think is noteworthy is in the last few days we saw what's happening down in Texas and Harris County and predominantly Democratic County in Texas. And the state said, okay, we're gonna pick up this county or we're gonna send inspectors there to watch the election. And they announced it right before the election in response then Harris County says, let's have the Justice Department come and we need that election monitors coming to see what these inspectors are gonna do and whether they're gonna be intimidating voters. So as we move in the midterms, I think there also is a concern about voter intimidation and on the one hand, and also inconveniencing voters taking away drive through voting, taking away 24 hour voting. Those things did exist in Texas, they're gone now. And a lot of the things that were done during the pandemic to make it easier for people to cast their vote in a valid way have been taken away. So as we go to the midterms, it'll be important to see what an impact that has on voter turnout. Well, that's a good point that we've been talking on the national level but certainly our local and state races are important too and especially the races for secretary of state in a number of areas, it's not to mention legislators who are gonna be voting on voting rights issues. That's a great point, David, that we need to be mindful of, well, getting people out there to vote but also making sure that they have the ability to exercise their right, not just this election, but in future ones. Yeah, we've got here in Minnesota, we have secretary of state election and this is another one that's kind of heated. The Republican challenger is an election denier who does not believe that Trump lost the election. That's the secretary of state, that's the person that's gonna be overseeing her elections. Doesn't believe their last election in Minnesota is kind of famous for having clean elections and great voter turnout led the nation and voter turnout. So we've had really a good reputation for elections and nonetheless, we've got somebody running for that position who doesn't believe in elections. How strong candidate are they? Well, fortunately right now, they're not leading, but they're not, it's also not a landslide. Of our contested races, I think that's probably the biggest gap, but it's not so great that you can say that it's a guaranteed win. It's a solid win for Democrats, you just can't say that anymore. Well, and we know that we have quite a number of people running that are not only past election deniers, but who have already come out and said, they're not necessarily agreeing to accept the results of this coming midterm election, if they are not successful. Somehow they've all overlooked the fact that apparently the presidential election was corrupt, but not the Senate races in 2020. How did that happen? Well, probably the same way that most of the self-contradictory positions happened, right? Which is, Republicans all voted against, unanimously voted against, the passage of Biden's economic relief bills and stimulus bills. And yet all of them are grabbing for that money and taking credit for getting that money. So... Of the 12 states getting the most money, eight of them are red states. And we've got all the local officials, you've got the national officials, the Senate representatives of Congressional who voted against it are all now celebrating all the things that that money is bringing to their states. So if they didn't support it, they're getting, but they're getting in twice as much money as the blue states and they're celebrating the money, even though they tried to prevent the passage of the bill. Okay, so in our last minute or so, for each of you, one thought that you would love to see voters put high in mind as they go to the polls or cast their votes for these midterms. Tina. Go to a reliable source for your voting information. Please. I had the same thought that, you know, people should be paying attention to the news, maybe read the news for a change, reliable sources as you say, go to NPR, think critically, ask questions of your candidate, make sure that they are supporting things like voting rights and they have solutions that make sense. Don't just go for the sound bite. David. Yeah, and that's what I would say. You know, to agree that you're in a lot of criticism, you know, ask the second question, what would you have done differently and what will you do if you're elected that you believe is gonna change things significantly? And we want that second answer. You know, and I think that points us to a really good place to wrap up today, which is that maybe what's missing the most is objectively reliable information about the choices and consequences that we're facing. Thank you all for joining us. Thanks for asking the hard questions and giving us some thought-provoking perspectives and insights on all of those. Come back and join us in a couple of weeks. Thanks for joining us at Tink-Tak, Hawaii. Take care. Thanks as always, Chuck. Thank you so much for watching Tink-Tak, Hawaii. If you like what we do, please like us and click the subscribe button on YouTube and the follow button on Vimeo. You can also follow us on Facebook, Instagram, Twitter and LinkedIn and donate to us at tinktakhawaii.com. Mahalo.


My name is Paul Withington, I'm co-owner of the Coconut Kitchen. The Coconut Kitchen started nearly 10 years ago now.  Pechea always makes all her own sauces, all her own everything really in the restaurant. So things were going so well in the restaurant, we thought let's try and do some retail products as well. So it was 2013 we set up CK Food and Drinks to take the Coconut Kitchen brand into the retail space. The sauce really is a reflection of the restaurant so because we like to do everything packed full of flavour in the restaurant, freshers can be, you know, as naturals can be, we reflect that in the sauces and the idea really behind the sauces is that the customer can replicate the dish that they would have had in the restaurant at home. The local shop in Pechealy Spa was our first customer. It was Conrad really at Spa Pechealy that spurred us on to take it further and to expand the range and to take it into a professional retail packaging. We probably have around 400 independent stores which we sell through distributors that we use in the UK and then we deal directly with the local stores. We're also exporting into Germany, France and Scandinavia as well. I doubt we would have got this far without the support of the Welsh Government really. Mentoring has been a big part and also the support of trade shows has been key I think really. We've had exposure at the Specialty Food Fair, Food and Drink Expo in the UK and also at Nugra in Germany. The prices that we pay through the Welsh Government have made it financially viable to actually go to these shows. We started off producing all the sauces here and bottling them and packing them here. Last year it got to a point where it was just too big for us to cope in the restaurant so we've now with the help of the mentor from the Welsh Government she's helped us transfer all our products to a great company called Shoda Sources in South Wales and we can now touch a book and we can kind of increase production when we need to and the fact that they're BRC registered means we can access new markets in the UK and abroad. They use all fresh ingredients the same as they have access to actually fresher ingredients than we would have done. So we're more than happy now that the result we've got with Shoda Sources is as good a quality and not better than what we have. On our sources we say inspired by Thailand made in Wales and it's really key to one of our USPs really is that a lot of the competitor Thai sauces are all made in Thailand, shipped over here, packaged up in containers where obviously some of our ingredients have to come from Thailand but it's all made here in Wales and that gives us I think a USP in terms of keeping close to the product, being able to control its quality and also obviously keeping the local economy here in Wales.


Good morning. Welcome to CMC Markets on Friday the 26th of February and this quick look at the week ahead beginning the 1st of March Before we get started on that just get through various risk warnings before we get on to the discussion I'm not so much a discussion, but my look back at the events of the past few days because I think if anything This week can be characterized by I think a single question Whether an increase in rates or yields is good for stock market or bad for stock markets Certainly, I think in the context of the discussions that central bankers have been having this week They would argue that higher rates are a broad reflection of an improving economic outlook and thus should actually be Very positive for stock markets, but as we well know and have known for the past 10 years or so The economy is not stock markets and the stock markets isn't the economy So just because the stock market is doing well doesn't necessarily mean that the economy is doing well And if the economy is doing well, you could conceivably argue that the stock market might do Not as well as maybe it should have been doing say for example five or ten years ago simply because central bankers might be inclined to pull away Some of the fists of some of the monetary policy support That they've been putting in over the course of the past 12 years So this is where we currently are With respect to stock markets. I think it's important to remember that even though We've seen an awful lot of choppiness this week and Stock markets have tried to go up. They've tried to go down and Really this week haven't ended up going really anywhere I think there is a concern about over frothy valuations at a time when particularly US Treasury yields and UK guilt yields I've come an awful long way in a very short space of time And I think that more than anything is what is spooking the markets If you look at where UK guilt yields were for example at the beginning of this year They're around about 0.2 percent They're now point they're now 0.75 so that's at 55 basis points Above where they started the year if you look at US Treasuries Treasury yields It's been a similar sort of story. They were around about 0.91 percent at the beginning of 2021 and they're now one and a half percent so 60 basis points higher With most of that Rising yields coming in this month of February which we are now Coming to the back end off with the 1st of March beginning on Monday So I think more than anything it's not so much the fact that rates are going up But it's the speed that they're going up and you've got a stimulus plan a US stimulus plan That's due to be pushed through The US Congress over the course of the next week or so an extra 1.9 trillion dollars on top of the $900 billion that we saw pushed through at the beginning of January and I think there's an there's a perception or a fear that maybe A fiscal side is probably doing too much Lifting and you could see an economic surge over the course of the next six months because we're certainly not going to get it over the next three because Restrictions will still be in place And I think one thing that we can well, I think one thing that we can Acknowledge over the course of the past few days is the massive rise that we've seen in copper prices from Where we were at the beginning of this year now that does appear to be some evidence potentially The copper may have topped out now if copper has topped out That could potentially be bad news for stock markets because if we look at the way stock markets have gone this year And we are still up for February So despite all the volatility that we've seen this week February is still set to be a positive month for the footsie The DAX and the S&P despite all the hysteria about the big declines that we've seen This week We're still up on the month nonetheless Canary in the coal mine could be a potential reversal in copper and a key day reversal on the daily chart So I'll be paying particular attention to the copper price in the context of whether or not we can go higher in stock markets now there could be a Divergence starting to take place, but we've seen big gains also in other base metal prices on the basis that We're going to see a significant shortfall in supply relative to demand for all those renewables That the copper that copper is going to be used in terms of batteries Electric cars solar panels Wind turbines anything that conduct an electric current Also seen Brent prices Move higher over the course of the past Few weeks. You can certainly see that here just shy of $70 a barrel $66 a barrel. I think it's been the high. We've got an OPEC meeting coming up over the course of the next few days in fact, I think it's next Thursday and This is something again that we need to be aware of At the moment OPEC plus has been broadly in agreement about keeping the production curbs exactly where they are To basically try and pinch off Too much supply now that prices are about $60 a barrel You could see some crack starts to open up between some of the smaller countries who want to take advantage of higher prices By pumping out more oil into the market and that could well Act as a headwind the crude oil prices over the course of the next few days and weeks That would also be welcome for inflation expectations If we start to see evidence of a topping out in commodity prices Which have been which are back at levels last seen in 2011 2012 particularly where copper is concerned so those are the two things to Two key commodities that I'll be paying attention to over the course of the next week or so It's also a big week for macro We've got non farm payrolls coming up before I come on to that. Let's have a quick look at the FTSE 100 as we can see Despite the choppiness of the past few days. We haven't really broken out of the range that we've been in Since the beginning of February We're still above the lows and even though we've come down from the highs in the middle of a month We have the middle of a month We actually haven't come down too far and actually we found support and every day this week around about 65 40 so based on that The downside for the FTSE 100 at the moment Looks fairly limited. Why do I say that given that I'm concerned about highs in copper and Oil prices well, ultimately while the FTSE 100 is very Susceptible to moves in copper and crude oil prices. It's also not a complete proxy for them One of the one of the key one of the key Beneficiaries of the past week or so has been banks From a much steeper yield curve. We've also seen travel and leisure Pick up quite significantly over the course of the past few days And we've also seen retailers pick up as well and the FTSE 100 has an awful lot of them. So I Think overall The FTSE 100 up my bias for the FTSE 100 still remains more positive than negative And I don't expect crude oil prices or copper prices to come crashing off But what I do expect to see is a little bit of consolidation at these higher levels Which in the short to medium term should well be positive for the FTSE 100 It's certainly I certainly don't expect to see it below this trend line that I've drawn in through here From these lows here the 6540 in the short term still expecting to see a retest of potentially these These series of peaks all the way through here. So let's draw in a little trend line for here Go through there. So that's my next trend line resistance on The FTSE 100 looking for a retest of that over the course of the next few days Similar sort of story for the German DAX again here daily chart Yes, we have made marginal new lows But what I think is significant about this particular chart here is that every time we've gone lower These long lower shadows would appear to suggest that there's fairly decent demand Anywhere near 13600 in a similar way there was fairly decent demand all the way over here When we dipped back below to this level here and even though we spiked below that And the 50-day moving average on these three or four days here That move low approved to be fairly short lived simply because it wasn't followed through by similar down moves and equity markets S&P 500 same thing here very much in the uptrend that we've been in over the course of the past Few days and weeks if we look at the weekly chart to see whether or not the candles are telling us anything there Not so much. Certainly a little bit of a rollover may well see a retest of 3600 but I think if we are going to see a correction in equity markets Then I think the the bigger correction is going to come in US markets rather than it rather than European markets more broadly the NASDAQ I think is going to be very important in the overall scheme of things this 12700 level here which thus far has held but is Acting as a very very case to pull and we have broken below the 50-day moving average So the NASDAQ is flashing red. It's flashing some warning signs about Further upside going forward and certainly here the weekly charts are suggesting that we could see a little bit of a correction lower Particularly if the economic recovery story Continues to gain traction because I think some of the key beneficiaries of the pandemic have been stocks Ultimately and companies are ultimately that don't generate an awful lot of profits, but obviously do and have Been very very good in terms of growth over the course of the last few months companies like Peloton and zoom And who who I will be basically looking at as part of my look at the week ahead Other companies like net like Netflix as well Could well suffer as a result as specifically as we head into the summer and the weather improves They may well find their subscriber numbers drop off simply because people won't Be staying in they'll be going out and obviously lucid restrictions will mean less eyes on Streaming platforms. So again, you could see a little bit of softness there so Looking ahead to the week ahead. We've got us payrolls, which is you out on the Friday We've got services PMIs. They're going to be very important in the context of How The the economy is doing and it's not going to be any surprise To know that there is going to be a continued weakness in Europe Got the US base book. We've got the UK budget now I'm going to devote a little bit of time to the UK budget simply because of how the pound has performed Over the course of the past few weeks We are starting to see some signs of a little bit of a reversal in the pound from The levels that we saw all the way back at the beginning of the year but also from what we saw back in September and I have published a piece on the budget which can be found in the insight section and use an analysis section Where I talk about what Rishi Sunak may be looking to do as we look ahead to Next Wednesday, but I think more importantly. I think the pressure will be on For him to do no harm. What he's going to be doing looking to do is plot the road to Recovery so while we've seen a little bit of a reversal in the pound at the moment, we could see further declines Going forward. We could see a move back to 138 20 and this 50 day moving average through here and 137 12 these series of highs through here which prompted the break higher I think we could well see a retest of that, but it doesn't change my overall view That we will see the pound continue to push higher over the course of the next few months UK still leads Europe and the US in its vaccination program and assuming That the Chancellor of the Exchequer doesn't do anything stupid then week We will probably see a furlough extension In the coming week and that's one area that's proving to be extremely expensive and it's due to end in April We could well see the cut in VAT to 5% Extended that is due to expire at the end of March. I think it's I think it would be fallhardy in the extreme if he were to raise VAT all the way back to That 20% level at a time when aviation Restaurants bars and other leisure venues have yet to reopen and I think that is a worry going forward I think the support measures for those particular sectors will continue to have to stay in place For the foreseeable future either that or he could well come up with a whole new tax regime for a sector That is likely to take a very long time to recover Because something like that just can't be dreamed up on the hoof So these are the areas that I think are going to need an awful lot of Tweaking shall we say and I think that's and they could well get extended into the middle of the summer And we could have another budget in November when hopefully the economic outlook will be that much clearer I think we do need to look at long-term solutions. It is a time for long-term solutions And I think while it's entirely right to be concerned about the level of the current deficit Borrowing costs are still fairly low I mean the government can afford to be creative when it comes to time frames in terms of Closing the gap when it comes to plugging the tax and spend The the the tax and spend gap He could look at new to he could look at new long-term funding or savings vehicles That will be one area 50 year bonds infrastructure bonds could be another area the government could look at Extending the debt profile of UK borrowing UK already has the longest remate payment profile in its terms of its peers at over 10 years Other countries are looking at 50 year bonds Austria Belgium and Ireland have issued hundred year bonds so-called century bonds You could encourage People with some savings to put them in national savings and investments They cut the interest rate on that which at a time when government needs money and excess savings Seems a remarkably short-sighted thing to do So there's any number of things that the Chancellor could do so ultimately I think when we look at the pound The outlook will continue to be fairly positive We look at this candle here and it does appear to suggest That we could well be overdue a little bit of a pullback in the short to medium term and As such, I'd be very very cautious of being about overly long of sterling as we head into the budget similar sort of story when it comes to Euro sterling if we look at this euro sterling chart here There's a potential hammer on the daily chart there with this very strong upper thrust candle here that suggests That we are probably overdue a little bit of a short squeeze back to 88 Maybe even the 50-day moving average all the way up there around about 88 60 88 70 But I'm still of the opinion that euro sterling is likely to go lower But we could well be due a little bit of a pullback in the short to medium term Okay, so I think that's about as much time as I'm going to devote to the UK budget It's it's going to be very very important But in the overall scheme of things I think more important than anything else is that the hope is the Chancellor won't do any harm He might Announce a rise in corporation tax That's currently at 19% you could push it up to 20 21% But that may take effect From next year. It could be that he will sign post a rise in corporation tax Rather than bringing it in straight away so In politics Sometimes optics is everything And while he can afford to be generous now he can start sign posting Tax increases Further down the line and certainly corporation tax is a tax on profits. Do you not make in any profits? You don't pay any tax I mean amazon plays that game very very well Anyway, because it ciphers its profits out of the UK so Digressing ever so slightly there So that's the budget on the third of march We've also got us payrolls and the dollar the performance of the dollar In recent days has been Fairly interesting. What's been notable is that we actually haven't taken out these previous lows through here So the dollar has consistently Tried to push lower Or what it hasn't been able to do Is to take out the previous lows that we've seen in the cmc dollar index here I think that's the very very I think that's the key thing to keep an eye out for People talk about dollar weakness An awful lot And it's very keen and very very keen to ride off the dollar But every time We threaten to break lower. We get a nice little pull higher And we've got what it looks like a bit of a bullish reversal here Which suggests again That we could see further dollar upside If that's the case Then the euro May well find it's very very difficult to go higher And if we look at euro dollar, we had a little bit of a false break on the daily chart here Above the 50 day moving average long upper shadow there Suggests the error above 122 is very very thin As such we're probably going to get another downward thrust perhaps towards The 120 60 area Which again is very fairly key support going forward So if we get a retest of 120 60 70 We could we could well get a retest Of 119 and a half Certainly, I think a stronger currency Is not necessarily a bad thing And one thing I think that And the resilience of the euro has done Is it will limit Inflation cpi inflation And while that's probably a bad thing for europe because their rates are negative It's probably not such a bad thing to the uk Because it certainly makes import inflation that much more difficult to filter through so Certainly in terms of the overall inflation outlook Higher pound is probably going to take some of the edge off Any move higher in headline cpi so That's euro dollar for you So looking ahead to us payrolls. I'll be hosting a webinar usual time Next week 115 Onwards for half an hour to 45 minutes And the thing about this payrolls number is it's going to be important In the wider scheme of things in terms of How well the us jobs market is starting to recover Now just as a quick recap on that what we've seen is up until The december number We saw seven consecutive months of job games job job job games And they came to a shuttering halt in december The us economy shed 227,000 jobs at the end of last year And the adp report was also negative now since then in january We've seen And a decent a fairly decent recovery Now the prospects of further fiscal support on top of the 900 billion past the beginning of january also bailed as well We saw january payrolls rebound With 49,000 new jobs added the unemployment rate fell quite sharply to 6.3 in december I'm not really paying too much attention to that Because of the width because of the low participation rate nonetheless You would hope That as the vaccination program rolls out across the us US businesses become much more optimistic about the outlook going forward and that's where the beige book Will come in and that's obviously coming out on the 3rd of march as well So it's going to be quite a busy day Look for look for look for signs of positivity in the beige book But more importantly, I think Look at the under employment rate at the moment That's currently sitting at 11.1 percent. It's still well below the april peak last year of 22.8 And it's still well above the low which we saw at the end of 2019 when it was at 6.7 so As I say virus cases are falling hospitalizations are falling deaths are falling mercifully across the us And I think that's playing a part in the rise in optimism now expectations are for 133,000 new jobs to be added in february And weekly jobless claims are still trending between 700 or 800 000 So they're they're down from the peak that we saw Around about 950 at the beginning of january So the hope is that december is the low point or an aberration And subsequent months will see a continuation of the gains that we saw in january We're still well below the levels of unemployment And the number of jobs that we lost in the months of march and april last year is 21 and a half million We've only added back around about half that number so While the headline unemployment number looks Failing low at 6.3 6.4 relative to where it was You have to remember That that does not include people who've given up Looking for work and have dropped out of the workforce has also been a more than 2 drop In the participation rate as well But nonetheless it's about the direction of travel and the hope is that we'll get an upward revision to january But we'll also see a fairly positive February number given some of the data that's been coming out on the isms manufacturing services isms and What have you So keep an eye on the services pmi's or the services isms the us also on the third look for a pickup in uk services pmi's And as well because we saw a nice little pickup in the flash number When it came out a couple of weeks ago Fact is uk is already talking about a roadmap out of lockdown Unlike france and germany and that will I think continue to help underpin the pound going forward okay, so that's um That's the major Currencies that we've looked at let's now have a quick look at some of the companies that are reporting their four-year numbers next weekend housing market very much in focus certainly percept certainly potential for a stamp duty the stamp duty Stamp duty divot the stamp duty extension to be extended The threshold to be extended beyond the end of march that certainly helps The likes of taylor wimpy and persimmon and look at the share prices. Obviously, there's the original there's the original Korean post lockdown plunge Since then we've come back Round about halfway in the case of taylor wimpy And yet when you actually look At the total completions. It's not hard to see why they were down 39 percent They're down they're probably going to be down 39 percent year on year which is not surprising when you saw The suspensions that we saw in march and april We can expect to see a recommencement of the dividend starting with the 2020 final dividend Management haven't completely committed to that and I think it will really depend whether or not they can actually Match their expectations for 2021 but certainly I think in terms of the rebound taylor wimpy's share price Rebound has been a little bit Underwhelming if we look at say for example persimmon and look at their Share price rebound. It's probably been slightly more positive. It's certainly Rebounded an awful lot more than say for example taylor wimpy And that's because they paid a second interim dividend to 70p in december. So The payout is obviously encouraging a few more people to buy into the shares forward sales are looking fairly decent So that will I think will be a key bellwether for both persimmon and taylor wimpy going forward as I say my all of the There'll be various comments about all of these in my week ahead Which you can find on the news and analysis section of the website. We've also got earnings from prudential and aviva four-year numbers from those two And in the u.s. We've got the likes of zoom Which has been a big big winner from the pandemic, but which is now starting to show Signs of a little bit of a slowdown as the 200 day moving average We're currently bouncing off it. We're well down from the highs that we saw in october And I think there are there's big questions being asked about the fact that weather evaluation of around about 120 billion dollars is really Is really credible for a for a company that Doesn't even turn over a billion dollars a year so um, sorry, sorry a billion dollars a quarter so I think that really is the key question when you're turning over less than a billion dollars a quarter Is it really credible to have a market cap of around about 120 billion? Yes, the company's profitable But is it worth what the market's got it valued at and I think this is the this is the question that's being asked by an awful lot of investors about some of these Pandemic stocks as I like to call them similar sort of question with respect of room which basically allows you to buy used cars online that has fairly eye-watering valuation as well I mean, we've seen it here in the uk with kazoo And the kazoo is talking about going down the ipo route Spending lots of money on shirt sponsorship in the premier league is kazoo And looking to take advantage of the ipo boom that we saw in the us last year and With room the big difference with room is it's not making any money. It's losing money as is expected to appear Carvana so this market is a very very challenging one. Ultimately if I'm going to buy a car I want to get in it Find out if the gearbox is any good find out if the brakes are any good I'm certainly not going to buy it online even if it is delivered and I've got a seven-day money back guarantee on it I want to drive it first But call me old fashioned. That's probably because I'd prefer to actually sit in a car before I think about driving in a way anyway, um, that's um, that's pretty much it for this week Ladies and gentlemen, as I say, um, please tune in for non-farm payrolls next week on On the friday for between one and two Um, we've also got the budget Obviously on the third Of march as well and those will be the key but those were to those will be the but the two key items on the agenda along with opec obviously On thursday, otherwise, thanks very much for listening ladies and gentlemen. Have a great weekend and speak to you all Same time same place next week. Thanks very much