 It's really a pleasure to welcome you here. It's a meeting organized by friends of IHES and the Simons Foundation. So I would like to thank Jim and Maréline for making this event possible. And so in two words, IHES is a French institute, mainly in pure math and theoretical physics. And we have a small staff of very high level people. And we host every year more than 200 visitors from all over the world. And one third of them are American based. And here in the US, we have a sister association, friends of IHES, and Jim and Maréline are actually our coaches. And you just became coaches. And so it's another very good occasion to thank you again. And today we will have a lecture by Sylvia, which is a French mathematician now at NYU for some time. And she has been awarded a lot of very distinguished prizes. And the last time I saw Sylvia lecturing, it was at the International Congress in Rio. She did a wonderful keynote speech. And even if I was very far from her topic, I think it's one of the best talk I saw from all this week. So I'm sure you will enjoy it a lot. And thank you, Sylvia, for doing this talk for us. But thank you very much for the kind words. Now you're setting up high expectations for me. Thanks all for coming to support IHES and to listen to a little bit of science. I must say that it's the first for me. I've never given a breakfast talk before. So it's an interesting... Okay, so I want to tell you a short story about, of course, a story that starts from physics and that leads into mathematics and intriguing mathematics. And this is not uncommon, of course, as you know, and it fits well, I think, with the spirit of IHES and the Simons Foundation. And so let's start the story with the superconductivity, which, as you know, is a phenomenon that happens in particular metals and alloys. It was discovered in the early 20th century. And what happens is that in these materials, when temperature is small enough, if you cool them down enough, suddenly they lose their resistivity and they can flow superconducting currents. And they're explained by microscopic theory now via the appearance of cooper pairs. And one of the striking features is that the superconductor then starts to expel magnetic field. This is called the Meissner effect. So you see on this picture, a magnet is levitating above a superconductor thanks to the magnetic forces that are created around the superconductor. So what I'm going to be interested in here is a regime where you cross a certain critical field that's called HC1, the intensity of the first critical field. So it means you have enough magnetic field that's applied to your superconductor so that it cannot repel it completely. So this Meissner effect starts to break down and instead, you see how it's described, the magnetic field starts to penetrate a little bit and it penetrates through some sort of flux lines, through some special tubes that are called vortex tubes, vortices. And these vortices are surrounded by superconducting current loops here pictured in green. And what is found is that these vortices become more and more numerous as you increase the intensity of the external field and they repel each other. So they repel each other, however, they are also confined to stay together by the magnetic effect. So it's as if they are both confined but also pairwise repelling. And the result of that, these two competing constraints is that they tend to form these beautiful triangular patterns, these triangular lattices. So this is called an abricose of lattice and these are observations in superconductors. It was a beautiful success of theory when the physicist Abricose of, who got the Nobel Prize for this, first predicted from the basis of the model on superconductors that there should be these periodic patterns of vortices. And then they were looked for and observed. So it's one of great success where theory prediction precedes experiments. And so, you know, we're sort of intrigued by this phenomenon. Like when you repel each other, you have a crowd of vortices, but they have to stay together. Why do they choose to organize themselves in these triangular lattices? That's sort of the main mystery here, at least for me. So I told you that Abricose of predicted the vortex lattices from the model. The model is the one that was brought forward by famous physicist Landau together with Ginsburg. The both also got Nobel Prizes for this. And this is the functional, right? You don't have to read it all, but this is the sort of order, what's called the order parameter. This is the gauge of the magnetic field. And you have to minimize this to understand the ground states of your system. And well, from this, you're supposed to see that you're going to have triangular lattices of vortices. It's not very transparent there. But in fact, you can almost do it. So it takes some work. So let me again describe this model, right? So what are the vortices here? They're going to be the zeros of this function psi. And psi is a complex valued function, if that tells you anything. So zeros come with a winding number. And then you have all the sort of magnetic data. So the intensity of the applied field you see here is a parameter you can tune. A is the gauge like in Maxwell's equations, if you want. Epsilon is a material parameter. Usually it's denoted in physics. We talk about Kappa, but Epsilon is the inverse of Kappa. We're going to take it small. And here, this is a two-dimensional description, sort of matching a little bit this picture that I showed you here or this picture. Okay. So I told you there's some work, but you can do it. And the work we did, other people also participated, is that when Epsilon goes to zero, you can do a sort of asymptotic analysis. You do see that the points that are the zeros of psi have this repulsion. And this repulsion is a Coulomb repulsion or logarithmic repulsion because we're in 2D. It's as if these vortices were behaving like electrostatic charges, even though that's not where they are. But they're acting like charges that are quantized in the sense that they're charged has to be an integer. Okay. So you can derive this rigorously mathematically. And that actually leads us to studying a discrete problem. And I'm going to describe that in terms of a Coulomb gas. Coulomb, this is all going back to the 18th century, was the first to postulate the electrostatic interaction between point charges as being inverse of the distance. And then you have Poisson's equation that relates the potential generated by a charge density to the charge density. And you see it involves Laplace's operator. And what is the connection? Well, the connection is that the fundamental solution to the Laplacian is the Coulomb interaction. So it is this. If you want to solve for Laplace's equation, you might want to just solve for minus Laplacian equals Dirac. And then based on that, you can recover the solutions to Poisson's equation by convoling with this solution W. And W happens to be exactly the same as what you see in Coulomb's law of interaction. Let me write it here. Just more descriptive form. This is the W. This is the Coulomb interaction in dimension one minus the distance. Dimension two minus log of the distance. So I told you it's going to be logarithmic in 2D. Dimension three, that's what Coulomb had first postulated one over the distance. And dimension bigger than three, one over x to the D minus two in general. So with that, you solve Laplace's equation and you find the sort of fundamental interaction of nature. So after we've seen that these vortices interact logarithmically, well, you can ask yourself, let's look more generally now. I have an assembly of points x1, xn. For example, on my vortices, but maybe something else, it doesn't have to be in superconductivity anymore. And let's look at these points with these pair interactions. So you see the repulsion between xi and xj via this W that I showed you before. So for instance, minus log of the distance. And here we add just for convenience, we add some confining potential that's to keep the points together so that they don't fly off to infinity. So v is something smooth enough, growing. Okay, so this is always repulsive. I put myself in the simplest case, where I have only repulsion. So when xi becomes close to xj, you see minus log near zero, that's very large. So this thing blows up. It's very repulsive. It's singular. Every time two points want to get too close to each other, the energy becomes infinite. So they really don't want to be close to each other. Okay, so this is the sort of simplified model for the interaction of these vortices. But now we can consider it in a much more general setting. We can be in any dimension. We don't have to be in dimension two. And we could look at other possible interactions, maybe not one over xd minus two, but maybe general inverse powers of the distance. We call that Reese's interactions. Okay, let's understand. For instance, you might want to understand what our minimizer is looking like. So again, there's going to be this feature that they want to repel each other, but they have to stay together collectively. So the competition between these two should lead to these interesting patterns. What's going to happen in higher dimension? We've seen what happens in 2d, but so you can also generalize this to manifolds. If you're interested, people do that. They study Reese's energies on manifolds. That's nice in the way because you don't have to add the confinement. If you're in a closed manifold like the torus, you can just look at the interaction. This is called the Reese's s energy. And having this parameter s is kind of nice because now you can tune it and you can start to play with s. You see that when s goes to zero, formerly it's the same as the logarithmic case because you can expand near zero and see that it's like a logarithmic interaction. And when s goes to infinity, this is the best packing problem. The best packing problem is I give myself points around each point I put a hard sphere and the hard spheres cannot overlap. So it's like billiard balls. And then you ask to put the maximum density of balls per unit volume, like to pack the space as tightly as possible with those hard spheres. So this is 2d for now. I mean the picture here that I drew is 2d, but you could draw it in any dimension. But that's a 2d representation with hard disks. So you see in 2d, these hard disks here, they are drawn to form again a triangular lattice. Intuitively, you can guess, oh, that seems like the best. What else are you going to do? And of course, if you were in 3d, you sit on the on the market when the people pile up oranges, right? You want to pile up, like do layers of this, but then shifted. That also seems to be the best way to pack hard spheres. But you're going to see it's not an easy question. Continuing with manifolds, here is what you see if you're on a sphere, right? So on a soccer ball or on the earth. And here what's plotted is you maximize product of distances, which is the same as minimizing minus sum of log of distances. So again, the logarithmic interaction, these are called Ficcate points, the things that maximize these products of distances. And people are interested in that for all sorts of reasons. One of them is computational, like you want to put points to interpolate on a sphere, like let's say you wanted to compute things on the surface of the earth, you want to have well distributed points. How are you going to put your points as uniformly distributed as you can? That's one solution. They're also interested in that for carbon molecules, which have these soccer balls structures. Again, you see here that there's obviously the points are uniformly distributed on the sphere. That seems natural. But they also form interesting patterns when you look closely. Microscopically, so there's a sort of macro structure and the micro structure. You do see something that's reminiscent of the triangular lattice, except now you have this sort of topological problem, which you can't really, you can't really tile your sphere with a very nice lattice. So you have to have this location, like these things that are called scars. So again, something quite intriguing to try to describe. On the torus, that's kind of fun. Here it's the inverse power S, optimal distribution of points. And you see on top here, the points prefer to go on the outside of this torus because that you have larger distances collectively, right? Whereas if you go to the inside here, these guys here and those guys there would be closer. So it arranges itself like this. There is a uniform distribution, but only on a sub region of the torus. And then there's a critical S above which suddenly the distribution becomes uniform. So the existence of that critical S is called the poppy seed bagel theorem for obvious reasons. But again, if you look microscopically, maybe after all, it's the same. It's still these triangular patterns. Again, something not completely clear, but to be understood here. So maybe it doesn't depend on S here. So now we have this interaction and we've tried to minimize it. We can try to do something else, which is we can add temperature. That's what statistical physics will tell you to do. Let's now look at probability of seeing the particles at x1, xn, so probability density, which is now like the Gibbs measure. So it's exponential minus beta times the energy. So what this does is it says, okay, you don't want to see just minimizers. You want to see configurations with sort of higher probability if their energy is lower. But you also want to see other possibilities. And this is going to be mediated by this beta, which is the inverse of the temperature. So when the temperature is effectively zero, and beta is infinity, then you're really weighing this thing very much towards the minimizers. But when beta is very small, you essentially have no interaction. Okay, so there's going to be a whole range in between depending on temperature, where you expect to see what was called temperature effects. So I said you can do this. You can put this into this Gibbs measure. Why not? But in fact, this is an important model in physics. These types of things arise in quantum mechanics. If you want to describe the fractional quantum Hall effect, you end up looking at things like this with logarithmic interaction. In 2D, it's a toy model for Plasma. Plasma are considered as point charges with coolant interaction. It's also quite important to look at this when you let the charges be possibly positive or negative, where now it's not just repulsion, but the charges with same sign do repel, but the charges with opposite signs attract. And that's the sort of other side of the story, which is also quite important in theoretical physics. There is in particular, it's related to this special type of phase transition that happens in two dimensions that was predicted by Kosterlitz and Taules. Again, some recent noble price happened related to this. So there's actually many reasons to look at things like that. And another reason I'll describe now here concerns particularly the logarithmic interaction that I mentioned, which you can think of as an inverse power. I told you log belongs to this family of inverse powers up to a sort of low asymptotic expansion. So it's eigenvalues of random matrices. So what are random matrices? Physicists started in the 70s, I think 60s, 70s, under the impulsion of Dyson and Wigner to try to understand the spectra of very large atoms. What is the sort of typical spectrum of a large matrix? If you understand what happens for a large symmetric matrix, Hermitian, then that should be a good model for a large atom. So they started with the simplest thing you can do, which is draw the entries of your matrix at random, make them IID. That's the simplest possible thing and make them even IID Gaussian, if you want. First case. Okay, so let's say you take a large matrix, make the entries IID Gaussian and impose like they were doing some symmetry. So let's say the matrices are Hermitian. So it's not quite that it's IID. You only draw half of the triangle at random. Well, then you find that the law of the eigenvalues, which you can compute algebraically, is exactly of the form that I described here, which means that's the partition function, that's the Gibbs measure of a Coulomb gas, we call that a Coulomb gas or a Ritz gas. So it's exactly like a Coulomb gas or more precisely a log gas in 1D. It's not quite Coulomb but log. And if you do with real symmetric instead of Hermitian, you get a slightly different Coulomb gas or log gas. So here it's for beta equals 2 that you should find. And here it's for beta equals 1, always have a quadratic confinement. In the case with no symmetry, you just draw your entries uniformly at random to be complex numbers. And so now your eigenvalues, instead of being in the line, so here you see you get a law on the line, here a law on the line, now here you get a law in the plane, a law for the eigenvalues. And this thing is called the Gini-Brand-Sambol and it's exactly a 2D log gas at temperature beta equals 2, inverse temperature beta equals 2 with quadratic confinement. Okay, and now here you see a plot of a typical set of eigenvalues of a complex random matrix like this. And you see it's very clear. They go uniformly in a disk, right? It's a uniform distribution in the disk and now microscopically it's less clear. There's a sort of shaky distribution of points. What you should be able to see is that they repel each other. They don't like to be too close to each other. There is a logarithmic repulsion, which is seen in the Gibbs measure, which is called repulsion of eigenvalues. You see it whether it's 1D or 2D, so the eigenvalues typically don't like to be too close to each other. So let's continue and see that more systematically. So here we're plotting a two-dimensional log gas or Coulomb gas with quadratic confinement. So because of this sort of rotational symmetry of the confinement, you expect things to be kind of rotationally symmetric, right? So you are not going to be surprised to see that these points, they always lie sort of in a ball. That's the effect of the confinement, right? They have to sort of stay together, and it has to be rotationally invariant. But now let's play with the temperature as a parameter beta. So we're going to start from high temperature, which means relatively low beta, and we're going to decrease the temperature. And you're going to see the effect it has on the point patterns, always kind of uniform, vaguely uniform in the ball. Maybe you believe me for that. But you see, as temperature gets decreased, the shaky nature of the patterns diminishes and becomes kind of less and less shaky. And now you see it towards the end, the temperature gets very small. It actually becomes quite regular. And well, eventually you would believe that you have sufficiently many points. And if beta really is infinity, you should see a triangular lattice. Okay. So what's happening? In order to describe that, we're going to do a sort of a zoom. And because we want to understand these patterns, right, at very small scales, I told you about the micro scale, the macro scale. What you want to do is you want to zoom your point configuration. So your points are going to align some set. In general, it doesn't have to be a ball. It can be another shape. Then you zoom and you see these points that are now well separated. And you imagine that n goes to infinity, n is the number of points. So as n goes to infinity, your configurations, your blue, your red, your green, are going to fill up the whole space. And once they've become infinite configurations, we define a sort of infinite volume energy for them. So I won't give you a formula yet. But let's say there's some W, some energy on infinite point configurations, which are to be understood as a helium, if you know physics. So it's infinite number of positive charges with a uniform negative background charge. And so we prove that you can do this as n goes to infinity. You can make this limit rigorous that you have this energy. And what is the end result is that with temperature, there's going to be a limit, what's called a point process, which means it's a point configuration, but it's random. Also, because you remember, we're drawing things according to a probability law. And that limiting point process, if it exists, or maybe there might be several, but must minimize the sum of two things. So this Coulomb interaction energy plus one over beta times a certain relative entropy. So here, you don't need to have the definitions, but physically, this is not surprising. An energy plus one over beta times an entropy, this is always the kind of structure you expect in a statistical physics problem. It's a free energy, if you want. And what you can think of is that there's a competition. If you want to make the sum of two terms smaller, you have to see what they want. The first term, W, my interpretation I'll give you is that it prefers order. We think it prefers order like it would be really more maybe at this triangular lattice in 2D. The relative entropy here, that's relative entropy with respect to the Poisson point process. That thing prefers the Poissonian distribution. What is the Poissonian distribution is when you throw your points uniformly at random independently of each other. So you just have a rain of points and they don't care about each other. Okay, so that's kind of more disordered and I'll show you a picture maybe right now. So you see on the left, that's the Poisson point process. The points get to fall wherever they are. Uniformly, they don't feel each other. On the right hand side is plotted the Gini point process. That's the one you get as the limit of this planar log gas or Coulomb gas at precise temperature beta equals 2. What happens that when beta is equal to 2, we know how to compute everything. We have a little chance in a glimpse into the system when beta is 2 thanks to a particular algebraic structure. And that's why we can plot this. Unfortunately, for other values of beta, we don't have access to such things. So this what was presented before is a characterization of the point processes that value for any beta. And that's what we, that's one of the things we know. So you see the competition when beta is very small, that means very large temperature, just Poisson, when beta is intermediate, some competition between wanting to be ordered and wanting to look like this. And when beta is infinite minimizers, we expect them to crystallize. We expect them to finally find their periodic minima. Let's look a little more at that question. So what we would want to do is understand the minimizers of this W-Coulomb interaction energy or RIS interaction energy. You want to work with RIS. So I have a formula for you. The formula is when I restrict myself to periodic point configurations with little end points per period. So imagine you're on a torus or imagine you have a square, a box. You're going to have end points that are going to be free in that box. And then you're going to repeat that pattern periodically. Just copy paste it. So that's periodic, but it's not necessarily a lattice that already allows you for some freedom because you can move those end points the way you want. So that's the interaction of these end points or the interaction of the total system copy pasted from these end points. It has a sum of interactions g of pair interactions plus some constant. And this guy g, not too surprisingly, ends up being the green's function of the torus. So it's exactly like our Coulomb-RIS interaction except it's periodized. It's made periodic. And now it's much more complicated. You can find a very simple form for it. You can write it in the form of an Eisenstein series. And now you get yourself into the world of number theory release. So we're still with that question what the minimizers look like and can we say that there are lattices like the triangular lattices I showed you for superconductors. So now let's sidestep a little bit and let's ask ourselves how often is it that we can answer a question like this? The question is you have some interaction say big U. Pair interaction potential x i minus x j. You have an infinite number of points. So maybe you do it like this. You look at the points that fall in the ball of radius r. You sum all the pair interactions you divide by the volume. Something like that. How often do we know that the minimum is achieved at the periodic configuration or even better at a lattice? So the answer is in 1d we can do it usually. That's okay. In higher d there's very few instances where we can prove something like that. It's actually an important question. The crystalline structure of matter depends on things like that. Not exactly this type of interaction but we don't really understand it from a mathematical point of view. We can't prove it. So I told you about sphere packing earlier. The hard spheres in the plane. In the plane for this hard sphere thing yes we can prove it. Triangular lattice is the minimizer. There was some sort of if you want generalization of this that was obtained by Radin more on the physics side and by Tile on the maths side. That shows that you can have the same result for a sort of caricature short range Lennard-Jones potential. Some interaction like this you see where you have a very steep decay a very steep well here. So it prefers a certain distance then goes to zero. So it's actually quite close to hard sphere interaction because hard sphere is just saying the interaction is infinite until distance one you know until distance the size of the ball and then it becomes zero. So they were able to extend this crystallization result to that. There's the honeycomb conjecture that was solved. That's a story about tiling planes by hexagons and minimizing total perimeter. There's the 3D sphere packing that's the oranges on the market that was solved actually not so long ago by health. And then we get to this really nice stuff which is the cone and q-mark conjecture. So that dates from 2009. What they say is there are special lattices that happen only in dimensions 2, 8 and 24. Very special things. The sort of mystery of numbers if you want. So in dimension 2 the special lattice is the triangular lattice or friend that we've seen before. In dimension 8 there is some guy that's called the E8 lattice. Dimension 24 the leach lattice. I won't give you the exact definitions. And these lattices are universally minimizing. That means if you take an interaction as long as it's completely monotone essentially then this lattice is the optimal. What completely monotone means okay so it means you have to be a function of the distance square and this function has to be decreasing. The Coulomb interaction satisfies that. All these risk interactions satisfy that. It has to be decreasing and it's derivative. So derivative is negative right and then second derivative has to be positive. Third derivative has to be negative etc. You alternate. So really like these inverse powers that's the type of thing you want to think of. Okay so they conjecture that and that includes sphere packing by the way. So suddenly all the sphere packing problem maybe in dimensions 2, 8 and 24 can be solved. When 2 it was already done 8 and 24 that's new. So what they proved was for regular interactions. So you might wonder well these risk interactions Coulomb they're singular at the origin. Maybe it doesn't fall in that category. So we proved a little you know lemma if you want that if you can do it for smooth interactions you can do it for Coulomb risk. It implies the same result. So it's not restrictive. Okay and so then there was a breakthrough that happened three years ago now. Beautiful work of Marina Wiesowska. She was able via sort of new transform the introduction of new transform on modular forms very deep stuff to prove the conjecture for the sphere packing problem in dimension 8. Then quickly it was extended to the dimension 24 again sphere packing by work with her and Mila Rochenko. And then they attacked the full concomar conjecture which is about all interactions not necessarily sphere packing and they solved it in dimensions 8 and 24. So that's nice that's the first time that other than dimension 2 we can say something about minimizers being periodic. Okay it is in these sort of odd dimensions right odd in the sense they're not they're even but they're odd they're they're weird right why 8 and 24 but now it says that our W or Coulomb or Ries you know interaction for infinite configurations now we have an answer in dimensions 8 and 24 it does crystallize now we know things crystallize and you can plug it with temperature you can let beta tend to infinity and show it crystallizes by the way dimension 1 you can also solve right. Now let's not be carried away in some sense into thinking that in any dimension we should see lattices at zero temperature because another mystery but in high enough dimension minimizers are not lattices so it's like when you when you pass dimensions around 11 you can start to see counter examples there's better things to do than doing a lattice when dimension is large except you come back to 24 and suddenly it's better again but okay so it's actually much more subtle than that I can point out also that if you're interested in doing dimension very large it's not just a curiosity because it's not physical but it's actually a much interest to theoretical computer science and coding theory to understand for example sphere packing or interaction energies in very large dimension okay so we're left with a still a conjecture dimension two we still don't because they're conjectured they didn't prove it in dimension two which with one could think would have been easier but no it's harder so we still don't know but we believe that the triangular lattice is the global minimizer of these Coulomb-Ries interactions and of course that's in a way supported by experiment because we saw the triangular lattices in the experiments on superconductors we observed them and we proved remember we derived rigorously this w from the Ginz-Bolando energy so here we have a path straight to that function so if the Coulomb-Cumar conjecture is true that gives you a proof of crystallization in superconductors in this triangle in these abricots of lattices okay so what there is the only thing there is so far is a restricted result that says the triangular lattice is the best among lattices just among lattices so say you look at all lattices of a fixed volume you have to normalize things you compare them so is the triangle better than the square for instance and the answer is yes the triangular lattice is the best and that's related very directly to a result from number theory from the fifties and then revisited in the eighties by Montgomery the famous result that says if you look at this thing that's called the Epstein-Zeta function of the lattice you see some here over the lattice 1 over p to the s this thing is minimal at the triangular lattice into the end so here you see the series converges so s is bigger than 2 so it's not completely obvious that it has anything to do with the original problem so you have a little transform that you have to make but you can derive from that the question about the w within the class of lattices and now to show you the extent of our ignorance we don't even know how to prove something like this in 3d like we don't even know how to prove that a certain lattice is the best among lattices in dimension three there's two there's actually well there's two or three candidates to be the best lattice there's the bcc there's the fcc lattice and there is not a universally minimizing lattice in 3d because you can at least numerically check that bcc is better for long range interactions and fcc is better for short range so there's a transition there's there cannot be an universally minimizing lattice even for these monotone interactions and we can't even prove or find the best among lattices that's a question for you know people doing number theory all right so right on time to conclude well i i hope i've shown you that if you're interested in coulomb and rith interactions there's many motivations for for those important physical systems superconductors random matrices quantum mechanics plasmas and that in the end they involve an interplay of analysis probability geometry number theory so i've i've sort of swept under the rug we do a lot of that analysis and a lot of that probability i haven't told you how we do it but there is there is some and then we can rigorously extract from these mathematical problems some difficult mathematical crystallization questions or which we have very few answers and many open questions so i mentioned is the triangular lattice really universally optimal into these that's the quantum conjecture the part that's remaining open how much of this behavior is really specific to coulomb and rith interaction is the monotonicity really the question or not there's also questions i haven't talked about but is there a finite temperature phase transition for crystallization in a 2d or 3d coulomb gas so when you really go back to the case with temperature physicists say mostly on numerical basis that they see a critical beta above which you have a solid and below which you have a liquid and we see that mathematically okay so with all these questions i give you and thank you none none okay because the the temperature so how what you do is you you draw your eigen your entries to be Gaussian okay then you compute the law of the eigen values and it's going to come as there's going to be a Vandermonde determinant product of xi minus xj put that in the exponential you just put it as exponential minus sum of log you see it selects beta equals two because uh because the the the the Vandermonde is for i strictly less than j so if you want to write it as sum of i different from j you have to put a one half so naturally the temperature is two if you want and then people can extend in 1d for the for the laws of eigen values on the real line you can find matrix models for one two four and then there's people that have really worked tweaking the thing it's a little bit far fetched but you can find a matrix model for any beta on the real line but uh naturally these Gaussian models they sort of select the temperature which is two let's say and so you that's why you you that's why i told you when beta is two we can compute we have this matrix model you have it and when beta is not too we sort of leaving the world of random matrix i think you have to press on your mic i'm supposed to tell you this first question i mean in the matrix case uh so you show that you have some repulsion between the eigen values do you also have the same phenomenon of repulsion between the the eigen spaces that they correspond to the eigen direction so the eigen vectors and the second question is when you go i mean for instance in 3d either any cases where like you know penrose quasi periodic lattices can go can be one of the solutions or they are still far away from the from the regular lattices uh so about the second question i think in theory it's plausible we don't know we don't know why not i don't think that's what people believe from the observations and simulations and yeah that's true about the first question so um what people um show is this notion of delocalization so you look at the eigen spaces and you check that the eigen vectors distribute themselves uniformly that's related to quantum chaos also so they do study these eigen spaces so it's not a repulsion but it's a question of localization versus delocalization pressure well great talk thank you very much um so you mentioned about uh some applications of sphere packing to theoretical computer science and coding theory is it something related to computational complexity or error correcting code do you think you can explain a bit more i think so but yeah i'm getting into dangerous territory because i'm really not a specialist of this but i think yes i i think one of the points is uh for instance it's it's simple right you have a very large dimensional signal and you want to represent it by only a small subset of points and so that's why you want to you want to pack the space with balls that are quite small such that if you if you replace the the signal in any ball by the center of the ball you're not very far off so uh i think it's uh i think it's related to that the expert if you want to investigate more is is henry corn okay thank you very much i have a question coming uh online from denise it's about so it's about uh one of your slides if you can just come back i think it's uh it's the slides that precedes just your conclusion yes and he he's asking is the volume of the lattice prescribed yes i said yes i have fixed volume okay denise denise technical remark is that this beta ensembles sometimes it's better to think of them not in terms of matrices because it's for beta not equal to two is kind of artificial it's not very natural but you can always reinterpret them as random partitions models that's true so and then that's has a meaning for any value of beta and some relations to gauge theory in four dimensions yes yes yes and and the kind of a fun remark uh so these dimensions two eight and 24 in which you have the special this sphere packing problem is solved these are minus two the critical dimension of bazonic string super string and n equals two string uh so there is some some relation to string theory there right even more entry yeah thank you great talk so i think my question was mostly about uh the dynamic with the temperature as you mentioned a little bit with the the beta factor you said that for some transposition it might be possible that we can have a threshold for for a value where we can have crystallization i was just wanting to know in terms of the dynamic of the evolution of if you were to change the temperature do you see some property in terms of behavior of how the temperature uh you show some how the temperature evolve and what's the link between the that evolution and the structure when you try to optimize do you see some discontent yes so uh to repeat the question it's uh the question about crystallization in terms of beta what is the evolution that you see when you change beta and so the the the sort of conjecture i would say numerical context what you should see is a transition from a exponential decay of correlations to algebraic decay of correlations so you you you have to look at it in terms of the of the correlations between four particles and that's how it should manifest itself so it's a slightly different notion but it's a different notion from what the crystallization i described okay do you have a question here oh okay i thought you took the mic foot any more questions okay well let's all thanks Sylvie for a wonderful talk thank you very much for coming