 Thank you very much to all the organizers for the invitation. Is my mic working? Someone tell me if it's working. It's working? OK, great. So today, I'll be telling you about soliton quantization and random partitions. And just to begin, I want to emphasize that the theory of soliton quantization has a vast history in theoretical physics. And in fact, during the lunch break, I went to the personal library of Abdus Salam. And you can see here right next to his desk, there is a big book on solitons. Right next to it, many books on string theory. And what I'll be doing today is I won't be doing anything non-rigorous. Everything will be totally rigorous. But I'll be focusing on the underlying classical dynamics of solitons. So let me write down the equation that I'll be studying and presenting some recent results for. So I'll be studying these classical Benjamin-Ohno equation. Let me write the equation down. So the time derivative of v plus v times the spatial derivative of v is equal to some constant epsilon bar times j, the Hilbert transform of the second derivative of v. So this is the equation that I'll be looking at. So epsilon bar throughout is a positive constant. And j is the Hilbert transform, which I can write in the following way. I just have to tell you what this thing does to a plane wave. And what it does, either it multiplies by minus i times the sine of the plane wave. And that would be of case positive, or it multiplies by 0. So this is the Fourier multiplier of j. You could write it as an integral operator, which is singular. And yes, so without this j, this equation would be the viscous Berger's equation in one dimension. It would be a diffusive equation. However, due to this non-local operator j, this is actually not diffusive. It's conservative. And so the goal, in general, so given v of x0, possibly depending on epsilon, you want to find and study the solutions of this equation. So there are many well-posedness results for this equation in literature, which I'll mention if you want to know. And a special example is this special initial condition, which is 2 cosine x. So we'll be revisiting this initial value problem with this initial data throughout the talk. Now that we have that on the board, let's look at the outline of the talk. So first, I'll tell you about some special solutions to that equation. These are the periodic analogs of multisyltons, also known as multi-phase solutions. Then I'll tell you about the problem of studying this equation with arbitrary initial data. After that, I'll tell you the quantum versions of each. And the main goal of the talk, which is part 3, let's see if the pointer works. Oh, yeah. The main goal of the talk is to arrive actually at a non-random partition. After that, I can state some results for random partitions. So this material all appears in these three works of mine on the archive. So what is a soliton? Well, here I'll start and present these solitons. So a soliton is just an example of a traveling wave. So a traveling wave is any v of xt that has this kind of form. It has a permanent form eta. And it has a speed, a wave speed c, in some direction. And so throughout the talk, wave speeds is going to appear in red. It's going to be very important for us. And if this eta is localized in x, then we say it's a solitary traveling wave or a one soliton. If you look at the survey by Terry Tao, why are solitons stable? There's a very important footnote at the very beginning where he says that some people actually require stability in its definition of a soliton. But here I won't be addressing questions of stability. The term soliton was introduced by Zabowski-Cruzgl, where they studied not this initial value problem on the board, but they studied the same initial data for the KDV equation. And this is a plot of the trajectory of the initial data to cosine x for the KDV equation after some finite amount of time. And what they point out in this figure from their paper is that there's a bunch of bumps, eight of them, and they call them solitons. And these are interacting. So technically speaking, these are not solitons. These are periodic solitons because they studied periodic boundary conditions. So just be clear and define what is periodic soliton. Okay, a periodic soliton more precisely is a periodic traveling wave. So a traveling wave is any shape eta that just moves in some direction with some fixed speed. And we say this is periodic if the wave form eta, this permanent form is actually periodic in some direction. And we call these for short of one phase wave. So here is actually a photograph of a football stadium sized wave pool where from the paper of Benjamin where he studied these periodic traveling waves in two plus one dimensions. And these are very importantly not stable. In the same year, Benjamin discovered this Benjamin-Ono equation. So what I wanna do now is present to you what are the one soliton and one phase solutions of the Benjamin-Ono equation. So here's a Benjamin-Ono equation. I've already written it on the board. And the Benjamin-Ono equation originally arises as a one plus one dimensional model at the interface of two fluids. Okay, this is the original formal applied math derivation by Benjamin and by Ono. But I also wanna point out in the physics literature recently this same equation was derived by Vigman and Bogotsky where they studied a large number of point matrices. So Vigman and collaborators have been looking at both classical and quantum Benjamin-Ono over the years, also 2D random matrix models. And what's very important for them in their argument is that the, sorry, the vortices actually accumulate in a bit more densely towards the edge. So in a paper with Kahn, Forrester and Telez, Vigman studied for random matrix models in 2D this kind of bump that occurs towards the edge. So that bump is the reason that they think that at the edge of this large number of vortices you should see the Benjamin-Ono equation. So in a sense, the Benjamin-Ono equation is predicted to be the universal dynamics at the edge of a large number of vortices. But for now I won't be addressing any of those universality questions. We'll just start with the equation. Okay, and Benjamin and Ono in both of their original papers they each found both a one soliton solution and a one phase solution. So I'll present those now. Okay, so how many one phase solutions periodic traveling waves do you have? We have a four parameter family of them given by three variables labeled S and one variable labeled chi. And it's written here. So it's very important so I'm gonna write it on the board. One phase solutions. Okay, so you give in S1 up, S1 down, S0 up. Three real variables, these are open conditions open inequalities on these variables. And some chi one in R. I can build for you some solution of that equation. Okay, and what is this solution? Well, I need to tell you what a wave speed, what the wave speed is and what the permanent form is. The wave speed is this average value of S1 down and S0 up. And the permanent form is written here. It may look like a horrendous formula but it's definitely not if you compare it to even the KDV equation where you need elliptic functions to write down the periodic traveling wave. Okay, so there's the permanent form. It's written through this nice cosine. And you can see what's very important is that the wavelength is gonna be, oops, the wavelength inversely proportional to this, sorry, the wavelength is proportional to epsilon bar. So at low epsilon, this solution doesn't actually exist anymore. Okay, what we're gonna do, I wanna encode the conserved quantities, these S's by a picture. Yep. As epsilon bar goes to zero, this, you can see that this thing becomes effectively constant or doesn't have a limit. It's rapidly oscillating. Right, yes, thank you. So the limit does not exist point-wise. Right, so you have rapid oscillations at small dispersion. And so what I wanna do for these solutions, it has a certain, what I'm gonna call the Dover called of Critchford Profiles. And what this is, I'm just gonna draw here, write my three variables, S1 up, S1 down. Oops, let me draw first a picture. Okay, at these lower regions, these local minima, I'll put these variables up. And at this local max, I'll put S1 down. And then the rule is that where I have slope negative one, I'll shade it in. That goes down to negative infinity. And here I'm gonna have another shaded region. Shaded region I'll call bands, B-A-N-D, band. Here, the variable here is C on the real line. C is for wave speed. And I'll write F as a function of C, which depends on these S's, okay? And the reason we use variable C for wave speed is that in this formula, you can just see that the midpoint of the band is the wave speed. So this midpoint is the wave speed. Okay, so the claim is that you write this function down, v of xt, it does depend on epsilon bar. You plug it into this equation, you get a solution. And it is a periodic traveling wave with this permanent wave form here on the left, and it moves at some speed. Okay, now notice that in order to have this permanent form be two pi periodic in space, I have a condition on the band length, okay? So you can read this off from this cosine term pretty easily, but it is so important for us that I'm gonna write it down on a big board, two pi spatial conditions, okay? So this is the one phase solution. And the statement is that vS chi B, xt epsilon bar is two pi periodic in X, if and only if the band length S one down minus S, oh sorry, S zero up, minus S one down is equal to epsilon bar times N one for sum N one in Z plus. I'm defining F to be, I give in my S's, I draw these variables here. So this, I'm calling it a Dobro-Cott of Critchiver profile, I'll see you'll see in a couple slides, but these S's variables are appearing in their paper, their description of these one phase solutions, okay? And so there I had no constraints on any of my S's, but now I have a constraint on the band length. So if I'm gonna draw this picture again, right, there's the big band that goes off to infinity, here's this first band. And here I might take like N one is equal to three here. So this thing is gonna have length three epsilon bar, this direction here. And what that will do is it will chop up this profile, this reach underneath into three pieces, okay? And those three pieces correspond to the three bumps on that plot there. All right, so more can be read off from these profiles. The average height of the wave is given by what would happen if you went down here and put this band right next to it and drew the center, or equivalently, you can draw like this. This picks out A, the average height. Next, the wavelength, like I said, it's proportional to epsilon bar, but it is also inversely proportional to the length of the band. That means that you have these open conditions on S's as if you shrink the band, then the one phase solution, the periodic traveling wave, becomes a one soliton. Okay, so in that limit, the second limit down here, if you shrink this band, the wavelength diverges and you get a one soliton, the one soliton's written there. So this one soliton, you can also see that as epsilon bar goes to zero, it becomes constant. In the other limit, if instead of shrinking this band, I allow the bands to merge, then I approach a constant. So what's next? So next, there actually, there exists kind of super positions of these one phase waves or periodic solitons. These are called multi-solitons. So let me define that for you now. So what is a multi-soliton? A multi-soliton is by definition, or n-soliton wave, it's just any function of x and t so that in some asymptotic regime, it becomes the literal additive sum of one soliton. Here is a picture of an actual solution to the Benjamin-Ohno equation. This is the two soliton of Benjamin-Ohno from work of Christie and the Atmospherical Sciences. And in this case, for the case of solitons, the asymptotic regime of interest is long time. So at long time, these solitons will get spatially separated and then the function will be pretty close to the actual sum. Similarly, an n-phase wave or the periodic multi-solitons, this is the same thing. It's just some v of x, t so that in some asymptotic regime, it's actually a sum. So the regime of interest is not the scattering regime of long time, but instead when we have well-separated wave speeds. I'll show you on that on the next slide. For the Benjamin-Ohno equation, the n-soliton n-phase waves are discovered respectively by Matsuno and Satsuma Ishimori in 1979. What I'm gonna work with is a presentation of these n-phase solutions by Doberquot of Critchiver. So, all right, so here is the formula. The Doberquot of Critchiver formula, this appeared in their work on the modulation theory when they show that the Wittem equations were completely decoupled, but at the very beginning of the paper, they did something remarkable. They gave a completely new formula for these n-phase solutions. And here's the formula. It's some constant times the logarithmic derivative of a determinant of some matrix. Again, you might think this looks horrendous, but it is simpler than the inexpensive formula for the n-phase solutions of KDV. And indeed, you can see that these simple plane waves here, just like we had a cosine from before. Okay? And as promised, I needed an asymptotic regime where this becomes a bunch of periodic solitons. The asymptotic regime is when all of the gaps diverge. Okay, so I'm calling gaps the reasons between SI up and SI down. Okay? And again, the wave speeds will be the midpoints of the bands and the wavelengths will be inversely proportional to the bands. Okay? So here is this formula for these solutions. Okay? So let's look at a couple of examples. What about an n-equals two-phase solution? Well, in order to construct the n-equals two-phase solution, I need three n plus one, which is seven parameters. That's gonna be five s's and two chi's. So I can collect the s's and draw this picture just as before. And the chi variables are phases, determine where I am. Okay? And in this example, you see three bumps, but I thought I was telling you about a two-phase solution, and the reason you see this is that actually this first band has length epsilon and this band has length two epsilon. Okay? And that's why it's three, three bumps. If you allow these two bands to merge, you would get back the periodic traveling wave from before. Okay? And here, next slide, here's another example. Here's a 26-phase solution. So I need to specify for you 79 parameters. I put these parameters down, all these s variables, just write them down, and I choose some chi's and then I plot the function. Okay? Now, some of you may have seen this picture before. There's absolutely nothing random so far in the talk. The claim on this slide is that unlike the previous slide, when I chose some s variables like this and some chi variables and I plugged it into the formula and I got this picture, instead, if you choose s's that are fairly close to this curve, here and you choose certain chi's, you will get in fact some function v of x that is pretty close to cosine x. Okay? And what's important is that this is non-random but there's an important point process we can even see right now which is the collection of the wave speeds. So the wave speeds are a bunch of points and they're drawn in red. Okay? So what I've done here is I've taken some s's, I've taken some chi's, I plug it into the formula which is very crazy and then I get some v. The question now is can you go in the opposite direction? If I give you a v like some initial data, is that initial data actually a superposition of interacting periodic solitons? That's the question. That's the question I'll address now. So before I go on, are there any questions about this review of these multi-soliton solutions? Okay, please interrupt for any time for questions. Okay, so now I'll tell you about the problem of addressing this equation with arbitrary initial data and what can we even say about this? Okay, so the results that I'll present from here are from this paper that I've written on finite gap conditions and small dispersion asymptotics for this problem. Okay, right, that's a typical problem and figuring out how to solve that is part of how to set up rigorously the complete integrability of the problem. Right, so okay, so let's see how do we get started? Well, the main goal in this next part, just like I wrote down these periodic multi-solitons encoded the conserved quantities through a profile, I wanna do the same thing now for an arbitrary initial condition. Okay, so let's start. So let's look now at some simulations of what this equation looks like for generic kind of arbitrary initial data. Here's the equation again. Now I consider an initial data here on the left, which is independent of epsilon bar. Okay, if you choose a large impulse initial data, we have hope of studying the small dispersion asymptotics of this problem. Here's a simulation from Miller and Shoe, who in what you can see here, you plug in some initial data and then over time it starts to shoot out these bumps. A remarkable thing is that if the height here is two and the height of these bumps is about eight and so you expect that an initial impulse of height H should become this should spit out these bumps or solitons that are about height four times H, okay? And moreover, you expect that as epsilon bar goes to zero, so here it's about point four, you expect to see universal behavior in these solutions that is independent of the initial data. Okay, so I just wanna mention that there are analogs of a Dubrovian universality that have been formulated by Masuaro-Romainzo-Antunes for this equation. Okay, so let's look at one more simulation here. This is from a recent survey by Jean-Claude So, which includes a definitive survey of pure and applied research on Benjamin Ono and the intermediate long wave equation. And here's a plot of what another epsilon independent impulse looks like. It shoots out those bumps before. These bumps are moving through time, which is this axis here, the bumps move, but something you can make out that's really interesting is that you see these faint ripples on the side. So you see these low amplitude waves on the left. Okay, and so for this equation, Peter Miller and collaborators have been developing non-local Riemann-Hilbert techniques for this problem. So I'll point you there if you're a Riemann-Hilbert person. And also if you wanna see a definitive survey of the KDV case and the analogies with random matrix theory, please look at Professor Grava's critical asymptotic behavior for KDV and random matrix theory. Okay, but so for now I won't be addressing these difficult questions about the fine local details of these patterns that form from a solution. What I wanna do is just locate some conserved quantities and then figure out how to formulate the quantum version of this problem. So how do we get started? How can we possibly compute anything for this model? Well, there's one structure that will invoke now, which is it has a lax pair. So, namely, I start with this equation. It's a remarkable fact at the bottom of the slide that the eigenvalues of something should all be conserved. So let's work through that for a quick second. Here's the equation again. Now I ask that I have a periodic initial data and it's, let's say, smooth and it doesn't have to be epsilon independent anymore. And let's say vk are the Fourier coefficients, okay? Then let's consider the Hardy space in the circle and use this Hardy space with this basis e to the ihx to write down this specific operator. It's not a tophalus operator. It's a perturbation of a tophalus operator by epsilon bar times derivative. You write down this matrix here. The claim is you can check that it actually is essentially self-adjoint and that it has a unique self-adjoint extension. And moreover, that the spectrum is actually simple. It's got no accumulation points and it only goes to negative infinity and it's bounded above, okay? So this, sorry? Any epsilon, any epsilon, yep. So this is a remarkable fact here that formally, if you just work with the equation and rewrite the equation as a lax pair using this lax matrix and some other ingredients, then you think formally that everything should be conserved, the spectrum should be conserved. Okay, so you would expect that these eigenvalues would be conserved. But technically speaking, when you work with a lax pair, you've only rewritten the equation. You have not incorporated the boundary conditions and the regularity of the data, okay? So at first glance, this equation doesn't make analytic sense, but figuring out how to extract from this spectrum something that a bunch of conserved quantities that are actually meaningful was discovered recently by several authors. So let me highlight this now. So let me define for you a dispersive action profile. What I do, I take the eigenvalues of that matrix, which is, sorry, not a matrix, it's an operator. I take the eigenvalues and I put a delta function there and that's gonna define my second derivative. But I'm subtracting off a shift of the eigenvalues, okay? So what I do is I try to put, so given v, v of x, and here, two pi periodic on the circle, and epsilon bar positive, then I plot this shape above ch up v where ch up is the h, h eigenvalue, eigenvalue of the lax operator, okay? And you plot this picture above the shift. And if you have them both at the same time, then you draw nothing, okay? So there's a lot of cancellation that will occur, okay? So the dispersive action profile is defined here and we have these conditions at infinity. It agrees with absolute value of c minus a where a is the mean, okay? And the theorem is that this is actually conserved. So a quick remark about this theorem. This result that this profile encodes all conserved quantities. So for any c, this is conserved. This was discovered is a degeneration of a result in the quantum setting by Mazarovsk-Leonid and also Sergei Veselov, okay? So I work to just figure out how the remarkable works in the quantum case imply this fact. And then this result was actually independently discovered by Gerard Kapler, very recently in April, who showed that not only are these conserved, these are actually accompanied by action very angle variables that evolve and they establish the global action angle variables for the periodic problem. So the fact that the periodic problem is actually integrable in the strictest sense in L2, which is the subcritical regularity where well-poseness was already known, was established definitively by Gerard Kapler. So a bit earlier than that, I showed that actually this construction of the dispersive action profile, which is true for, it's available for any V, if you plug in the multi-phase V, then you get back the Doberpov-Krichever profile. Okay, so it's consistency check. All right, so just to review, if you look at this picture here, then over time, this V is evolving in a complicated way, but at each time, if you take the V and you compute this F from V, you will get the same thing. Okay, so this entire picture is conserved as you evolve for this equation, okay? And we can see here, again, if I draw the midpoints of these bands, all of these wave speeds will be conserved as well. Okay, so next I wanna say, well, what does this look like for small epsilon bar? This will be the second result. So just let's quickly review, what is the epsilon bar equals zero version of this equation, so the dispersionless version? Okay, well, here is the equation, sorry, yep, in this one. So, yeah, in this one, this is from Sol's paper. I don't remember which was the specific function. Oh, it's just a heuristic sketch. Yeah, thank you. Yeah, any other questions? Right, so now let's look at the simpler equation when I just completely destroy this non-local term. This is the inviscid burgers equation, but in our problem, it's the dispersionless Benjamin Ono equation. I set epsilon bar equals to zero. Well, there's a similar conserved quantity you can write. It's the convex action profile. Okay, so this is a function, f of c, which is defined for any v, and it's defined by the same asymptotic condition we saw before, but now I just ask that it's first derivative up to a shift and code the relative volume on the circle that is where v is less than or equal to c. And then you can just check that for short time that this is conserved. Okay, so for any c, I have conserved quantities for the dispersionless problem. All right, so how do you prove this? It's like one integration by parts or you can use a method of characteristics. And indeed, if you think about what the method of characteristics is telling you, it says that this conserved guy, capital F, is just the relative mass of a uniform continuum of free classical particles on the circle with wave speeds that are less than or equal to c. Now, for this much simpler problem, we know that at finite time we have a gradient catastrophe, but there are all these conserved quantities encoded by this convex action profile, and these have appeared before, all as limit shapes. Okay, so the very special case of my favorite initial data, two cosine x, appeared in this, the Vergey-Karov-Loganchev Limit Shape. Okay, so you take the simplest nonlinear equation with initial data, two cosine x, it has infinitely many conserved quantities encoded by the Vergey-Karov curve. For general v, this is discovered by Limit Shape for sure measures by Akun-Gov in 2003 in the uses of random partitions. And more generally, these dispersive action profiles have appeared as limit shapes as well in the work of Nekrasov-Preston and Shaddaf's feeling. Okay, so all of these are limit shapes for models of random partitions, but here I'm arriving at them from the point of view of the limit just writing down this classical problem, thinking about the conserved quantities. Okay, and so as promised, I wanna tell you what are the small epsilon bar asymptotics of these dispersive action profiles? Okay, so here's the theorem, second result. As epsilon bar goes to zero, the dispersive action profile concentrates weekly on the convex action profile, as you might expect. So that's the result, yeah, that's the result. And what does this result actually tell us about the solution? Well, technically speaking, nothing, because I'm only studying the conserved quantities. However, because the equation is dispersive, unlike the ordinary wave equation, this is a dispersive equation. What that means is that we expect that at small time, this will form some dispersive shock wave, and then the anatomy of this shock wave will be built up out of low amplitude waves, modulated one phase waves, and separated one solitons, whose wave speeds should be pretty much captured by the wave speeds you see here in this picture. Okay, so that's a heuristic motivation for looking at the small epsilon bar asymptotics of these conserved quantities. Okay, but the result is only about the conserved quantities. Okay, so now I wanna move on to the quantum setting, and I'll give you some ideas to what's happening in the quantum setting. All right, okay, so the quantum version, this is from the second paper. Okay, so first I wanna now invoke another structure of the equation. I already told you that it has a structure of a lax equation, and now it actually has a structure of a Hamiltonian equation. So let's consider the Hamiltonian structure of this problem, okay? Here I'm gonna again use VK for the Fourier modes. Well, first of all, energy is conserved. If you form this big cubic expression of the Fourier modes, this is conserved over time. Next, more is true, and action is conserved, okay? So in phase space, we have well-posedness of the flow. Any loop that you draw allow the loop to evolve in phase space. The area enclosed by this loop will not change. That's what it means to be Hamiltonian. So here you have a bunch of different kinds of conservation laws given by the conservation of the action. And so technically speaking, what this action is with respect to the little one form of the first bracket of KDV, all right? So what I wanna do now is to tell you, compute the actions and the energies of these multi-phase solutions from beginning. Those will be the next results. So what does that mean? For any V, I have an energy. For any V, I have an actions that I can compute. Now let's just plug in those complicated solutions from before and see what we get. So, first of all, the classical energy. This is a corollary of the earlier results from the first paper. The classical energy of the periodic multisolatons will be given by this alternating sum of the cubes of those Doverquad of Critchiver variables, okay? This nice formula here can be more compactly written as the integral of C cubed against the second derivative of the profile, okay? So the energy depends globally on the profile. But next we can ask about the actions. The theorem is that take again this one form here. It's just an explicit thing. And now which loops are we gonna take? We take the loops which live on the Liouville-Torai defined by the multi-phase solutions. That means more precisely, I just go to these Doverquad of Critchiver formula and I just vary one of the phases. And the result is that the action is just two pi epsilon bar times the length of the gap. You remember I was very excited about these conditions we had earlier on the lengths of the bands, which was the spatial periodicity condition. Now I'm telling you that the gaps, which I didn't even talk about before, are actually the actions, okay? And these depend locally on the profile. All right, so right now this is actually a theorem about quantization, staring at us. If we take the first form of quantization ever formulated, which is the Bohr-Sommerfeld quantization conditions, the Bohr-Sommerfeld quantization conditions say that the action should be quantized because it's an adiabatic invariant. Okay, so now let's see what this looks like in a theorem. So let's recall that if you have any loop of any type, gamma i, in some symplectic phase space with an exact symplectic form, then by definition, the loop is h-bar Bohr-Sommerfeld if the integral of this one form around the loop is two pi h-bar times some integer. Why do we like the Bohr-Sommerfeld conditions on the actions is because it gives us an approximation to the energy. Okay, so here you can see that for the classical energies of these guys will approximate the quantum spectrum. This only works for integrable systems, but as you all know in random matrix theory, the relationship to quantum chaos relies on the chaotic version of this, the Good-Swiller trace formula, where all the predictions about GOE come from. But okay, so here is this result. Now I can restate it. These actions are given by two pi epsilon bar times the gap length, and that means that in order to be Bohr-Sommerfeld, the gap lengths have to be given by h-bar divided by epsilon bar. So let me write this down here. So what are the conditions now? VS Chi XT epsilon bar is h-bar Bohr-Sommerfeld. I'll do this for the one phase, if and only if the gap length given by h-bar divided by epsilon bar times n1 prime for some n1 prime in Z plus. Okay, so now let's take again as before, n1 is equal to three and n1 prime is equal to nine. That means I will have in this direction three, and then I'll have one, two, three, four, five, six, seven, eight. Okay, so what does the theorem say in general that actually what does it mean to be a quantum multi-salaton? It just means that you're a classical one, but the profile you draw has to be of a partition. That's it. So now we've completed what I promised, which is arriving at a young diagram, a non-random partition from dynamical first principles. So what can I show you here? What's nice is that this has to be actually volume proportional to h-bar, yep, and what next? Well, more is true, but actually in light of the fact that the, not an approximation of the spectrum, but the exact spectrum is known by works of Sergei Veselov and Nazarov-Sklyan and others, that actually if you write down this operator, which is you choose to be your quantum version of the Hamiltonian, the eigenfunctions are jack functions. You can regard that actually as a definition of a jack function. What this means is that if this is another result that the Borsamer-Pell approximation is exact if you just choose this renormalization of the coefficient of dispersion, okay? And this is discovered by Abana Vigman in the Lagrangian picture. And indeed here, now all you have to do to find the exact spectrum, you just draw these shifted young diagrams here and draw the same picture. So definitively jack functions are actually the states of the quantum version of the multi-phase solutions, okay? Last but not least, I should quickly say that it is a result of Dubrovian that if you look at the dispersionless problem and you try to quantize this, you arrive at sure functions. This gives a new interpretation of that result, which is that, well, at epsilon bar equals zero, the renormalized coefficient is still present. And that means sure functions themselves are actually also Benjamin Ono multi-soluton solutions. So this is a new characterization of sure functions. Okay, so with the last couple of minutes, I wanna quickly tell you how these ingredients of the dispersive action profile and the convex action profile have appeared before in the theory of random partitions. This is a quick statement of like one result of mine for random partitions and then I'll be in. Okay, so what are the random partitions? Well, start with a Fox base, which you'll see more in the next talk. I just take this Hilbert space and I have this basis and I declare it to be orthogonal with this inner product. That's what a Fox base is. It's just polynomials with an inner product. Then I have some ladder operators. So I can write multiplication by VK and H bar times K times dvk and I can define this Hamiltonian here. It's just a self-adjoint operator in Fox base in the completion. And Jack functions are by definition the eigenfunctions of this thing. Okay, and to connect to the conventions, here I'm using the cross-off variables, but to connect to the usual conventions, the power sums are a rescaling of VK and the Jack parameter is the ratio of these epsilons. Okay, and so you can define a natural model of random partitions where you just pick some specializations of VK, which are the Fourier modes of some of V, and you choose a random partition proportional to this law. Okay, so at first glance, this is a very confusing model of random partitions because it's defined through these transcendental special functions. But they have appeared before in a variety of contexts. So these are generalizations of sure measures. And these are also the generalization of the Poissonized Jack-Plancherale measures for my favorite initial data to cosine x. And there are special case of the McDonald measures introduced by Boarding Coral. Okay, so for these, this model of random partitions here, which has been introduced before, not first by me, they're in the, you can study, I can present now some limit shape results. So here's the result. Draw this profile of the partition. Sample from a Jack measure. Then there's two asymptotic regimes. I can set h bar goes to zero. And if I recall, h bar is minus epsilon one, epsilon two. And epsilon bar is epsilon one plus epsilon two. Look at this regime where h bar goes to zero. Then, oops. Then for fixed epsilon positive, the random partition concentrates in the dispersive action profile that I spent the whole talking about. Also, as h bar goes to zero, which is quantum to classical, and you also send dispersion to zero at a comparable rate, given through some beta, which is indeed our beta, then you get the concentration on the convex action profile, okay? And the last but not least, I'll just quickly say that results of Professor Guignet and Huang showed that for these models, these Poissonized Jack-Plancherale measures, at the local scale, you see the same universality as you do in random matrix theory. And so even though the results I've presented are only at the global scale, what I've shown is that if you orient yourself from the point of view of the limit where nothing is random, then the point process for which we see universality is in fact the point process of the wave speeds of the interacting solitons. So thank you very much.