 This morning is by Sylvia Serfati in search of the abricus of lattice Thank you very much. So first I would like to of course thank the scientific committee for inviting me to give a lecture here. It's a great honor and I will start also by apologizing because I am not going to talk about something that Poincaré worked on However, you know if you think of something that Could have been in the spirit of Poincaré you have to find You think that maybe it has to have some connection some strong for connection with the physical phenomenon and some interesting mathematical content, so hopefully this is this is fulfilling this Fitting the bill And so it starts from a physical model Which is related to superconductivity? So in fact superconductivity was discovered one year before Poincaré's death in 1911 by somehow by accident by Camely owns and So what what turns out is that if you take these particular materials or alloys and cool them Below a certain critical temperature. They lose all resistivity And it explained it's it was explained later by the presence of superconducting electrons core called Cooper pairs Which form superconducting currents? and One of the most striking features of superconductors is this Meissner effect that you see here on this picture Which says that the superconductor? Repairs and applied magnetic field And by repelling a magnetic field it can levitate above a magnet and so here you see a nice picture of a superconductor in levitation The other thing that happens is that there are some phase transitions depending on the external Applied magnetic field so there is a what's called a first critical field hc1 At which in fact the magnetic field does start to penetrate in the material And it penetrates via what are? vortices So these vortices are a bit like the vortices that fluid mechanics Talks about and that that point I had been interested in Except that they are quantized so they have an integer charge and they correspond to if you want like a Place where there is a normal phase So the superconductivity is lost in the center of the vortex and they are surrounded by like superconducting currents little loops of superconducting currents and What you see is that the higher the applied field the more vortices Will be present in the sample and in fact they arrange themselves in these things that are called abricos of lattices So this is the term from the from physics It's named after abricos of physicists who got the Nobel Prize for this in 2003 and who had predicted that such patterns should be observed and later after Several years after his prediction they were indeed observed and you see that they they for there are perfect triangular lattices and they are understood they can be understood in a very used realistic manner by the following reason so you have a You have these vortices and they repel each other At the same time they are sort of confined to stay in the material by the external field So there is a sort of competition between confinement and repulsion and in order to Arrange themselves in the most favorable way. They they find these triangular lattices and this is in fact something quite difficult to Explain and to understand mathematically as I'm going to try to to show you so the abricos of prediction was based on the model by Ginsburg and Landau that was introduced in the 50s on purely phenomenological basis, so later it was explained by a quantum theory But but at the time it was it was just phenological and it it proved to be very very successful in describing in Repredicting the phenomena that happened in superconductors and so again these Earned them noble prizes. I mean Landau got the Nobel Prize for many other things But Ginsburg certainly got it for this model and so here here is the The model it says basically that the states of the sample are the things that minimize This energy functional, so just a few things here For simplicity I will restrict to a two-dimensional domain and the vortices if you want to understand them from the from the model they correspond to the zeros of this complex valued function psi that's here and The they have they come with a charge or degree Which is the winding number of the zeros in the complex plane and here? There are two parameters the intensity of the external field here and epsilon which is a material parameter and Corresponds to the size the characteristic size of the vortex core and it will be taken to go to zero and A here is the gauge of the magnetic field, so I'm not going to give too many mathematical descriptions of this but This is this is a model that With with my collaborator it and Sunday we have been working on for for many years and one of the sort of eventual conclusions of our work was that if you want to understand the minimization of G epsilon in the end you find that Vortices essentially interact indeed according to a Coulomb type of interactions as if they were electric charges In in the plane interacting with the with simple electrostatic Behavior and so that that that justifies this explanation that I was giving you before It's as if you had charges that were repelling each other and that were confined together by an external potential and in fact it turns out that there is a much simpler model that Contains these ingredients, which is a discrete model, which is also Also a sort of classic standard model, which is the model of a Coulomb gas So let's focus on this one for a moment And in this model you just have n particles or n points sitting in the plane Or on the real line. That's a variant. That's also interesting So here they sit in RD with D equals one or two and you see that we We have in this model a sum of logarithmic repulsions between the points so this thing will become very very large if the points get close to each other and we have a sum of We have a the effect of a confining potential capital V So think of capital V as something that grows at infinity say for example that grows quadratically at infinity and The the parameters here are cooked up in such a way that the two effects balance each other the effect of the repulsion and the effect of the confinement So this is Again, what's the Hamiltonian of a Coulomb gas and if it's in one dimension if the points are Constrained to live on the real line It's called a log gas Because then the the logarithm is no longer the Coulomb kernel in in one dimension So a motivation for studying This model for studying minimizers of this There are several motivations actually but one of them is that minimizers of this also happen to be maximizers of this type of pairwise interaction where you have here the product of the distance Times the product of the exponential. So to see this just take the exponential of minus W and These things arise in a completely different context in the context of interpolation and are called weighted fakety sets. I Will come back to that a bit later So this is this is no longer about physics. This is about Trying to find best interpolants for numerical computations, for example And you can also know that you can reduce yourself to products of such distances on Manifolds of for example on the sphere by choosing a potential V Appropriately and using stereographic projection. So really a motivation for this can be to study the maximization of such products and On any general manifold two-dimensional surface Okay, so the question in in this in this Setting is to understand now the limit as and the number of points Tends to infinity Okay, how do the points, you know, how do the points arrange themselves in order to minimize This total interaction. So here is a numerical simulation for quadratic potential for 29 Points and you see they tend to arrange themselves relatively uniformly In something that looks like a disc But and here is not yet very very large. So we'll see what happens for a larger end This thing is in fact more interesting even if you add a Temperature to the system and you start to do statistical mechanics So the the Coulomb gas is really a statistical mechanics model where you're looking at a probability measure on the space of configurations of endpoints X1 Xn again that live either in the plane or on the real line and Here you see you put in the standard Gibbs factor So you put it exponential minus beta times the Hamiltonian where beta represents the inverse of a temperature and Z is the normalization factor that makes this whole thing a probability And W n is as before so this thing is is Interesting in physics, but it turns out that it also has some close connection With something that arises from probability theory, which is a which is random matrix models So If you take these particular very particular values of beta Which are one and two and if you take the points to live in dimension one or two Then it turns out that this probability Here is nothing else then the law of Eigen values for some random matrix models that are like the most most classical random matrix models So what I mean by that is the following you take an end-by-end matrix and You draw the entries at random according to a Gaussian law and you draw them in an independent manner, so it's iid random variables Okay, so Gaussian entries and then you compute the law of the Eigen values and What you find is exactly this You fight exactly this law of the Eigen values So then it depends on the symmetry that you impose on the matrix So if you don't impose any symmetry The Eigen values are going to live on the complex plane And it corresponds to this case what's called the Gini Bronson ball For the case of v quadratic and beta equals 2 If you impose the matrix to be real symmetric or complex her mission Then you're going to have Eigen values that live naturally on the real line By symmetry and again the law of the Eigen values is going to follow some formula of this type with beta equals either one or two and v quadratic and these are called The Gaussian unitary ensemble and the Gaussian orthogonal ensemble So there is a very very large literature on random matrices It takes off this form and most of it exploits the fact that There is a matrix model behind which means there is some sort of algebra That allows you to make computations and to understand What the Eigen values can do But at least what What you can see from this Coulomb analogy from the analogy between Eigen values of the random matrices and the Coulomb gas model Which was first noticed by Wigner by the way and exploited by Dyson What you find is that the Eigen values of these random matrices tend to repel each other In a Coulomb manner. In fact the Eigen values interact exactly like Coulomb particles in a confined potential and Here is a plot here is a numerical simulation for the Eigen values of matrices without symmetry so Matrices with just iid Gaussian entries and what you see is indeed that the The Eigen values there are many many of them They tend to distribute it to distribute themselves sort of uniformly and you will observe that they distribute themselves in a disk and They repel each other At least you can see that they tend to be not too close to each other on this picture So one of the things that we would like to understand is something more precise about this distribution and In particular is it true that if you decrease the temperature Which in physics corresponds to? Decreasing the disorder Will you see those points? Crystallize and organize themselves according to abricots of lattices All right, so you see here There is a certain Still a certain amount of disorder, but this we know corresponds to the value of temperature beta equals 2 So what happens as beta gets larger? We can expect that it's somehow related to what happens in vortices and superconductors So this is this is really the punchline of the story already So another model just to give you another example that's of interest to The physics community is a model of Di-blog co-polymers, which is called the otaku wa-zaki model and just to show you very briefly It's a model where you have a it's a phase field model You have a function u which takes values plus and minus one and its average is prescribed You have a greens kernel or greens function here g and you see that these This this u thing can be seen as a set of positive and negative charges that interact With itself in this coulombic manner except that here you add another term that essentially penalizes the perimeter of the interface Between the plus one and the minus one phase. So this is essentially what's written here. You have two phases interacting Via a screen who long colonel and this is the perimeter of the face and the there is this Competition because the perimeter term of course wants everything to be relatively round and in one piece and the coolant Kernel interaction here Prefers some rapid oscillations between plus and minus one And so again, you have a competition between two effects. So it turns out that in this model There are many many possible regimes, but people expect that Minimizers of such a thing are periodic Exhibit periodic patterns and numerically you see periodic patterns however mathematically it seems Almost impossible to prove very few things have been proven So this is a numerical plot of what happens in this model You see you have these two phases plus and minus one represented on the picture by black and white And so essentially the the whites the white repels each other and the black repels itself and The perimeter of the interface is penalized and you see this is the effect of trying to optimize these two competing of constraints these two contradictory constraints and We'll be interested in the situation where the white phase becomes very very small compared to the black phase You can prescribe this The volume of each phase and so in that case you will see you see immediately that essentially the white things are going to become points and we are back to the same setting where you have points interacting in the plane and In fact, we will have similar results on this model as well Okay, so in in three models. I have mentioned against Bolando Coulomb gas or fakete sets and otaka was a key we have these Same ingredients in the end. We should have a 2d Coulomb interaction with large large number of points And we expect to see some sort of pattern formation in in minimizers some periodicity and even To see this abricus of lattice showing up Okay, so let's see what is known What is known and I will do it only on the second the simplest model the one that's discreet the Coulomb gas where I have endpoints x1 xn and Assume they minimize w n Then you can form the probability Distribution which is the sum of Dirac masses at the points normalized by the number of points Okay, so this is a probability Distribution and it's known that it will converge to a certain probability mu zero Which is the unit minimizer of this Interaction energy now, which is simply the continuum version of the previous discrete interaction So the this thing has a unique minimum among probability measures Because it's convex It's strictly convict and in fact all this has been understood since the 50s This is the basis of what's called potential theory Which goes back to actually gauze And which consists in understanding things that minimize such energies All right, so there is a there is going to be this mu zero which is the equilibrium measure And for example if you take v quadratic, then you compute you find that mu zero is essentially the characteristic function of the unit ball Properly normalized to be a probability measure And so what this corresponds to is exactly this Picture because you see here that the probability of finding a point is converging to the constant density on the unit ball The constant density Supported in a disc okay, so The point after that is to try to say something more precise so beyond This mean field limits if you sort of expand around this behavior of n mu zero when you you look at The error that you make compared to that You are able to find next-order terms Which should tell you something about? The microscopic distribution of the point so here is the sort of rough picture You have many many points There are going to be distributed in a set Which is the support of the equilibrium measure? we know what's the Global density of the points what's the sort of macroscopic density but as you zoom at scale root n so Because of course you're in dimension two So if you have n points in dimension two in a box of fixed size the typical distance should be one over root n So if you blow up at this scale around a point you will see a configuration of points now Which are well separated from each other and which should fill the whole plane as n goes to infinity Okay, so if you're on the real line is the same situation you have a point living on a certain segment with a certain density and If you zoom You're gonna have points that now live on the line And so you would like to understand the microscopic Arrangement of the points by understanding The sort of energy that governs The distribution of this these points on the real line or these points Sorry these ones in the plane or these points on the real line, which are the blown up configurations Okay, so there is some computations involved in doing this. This is a hard part of the work But what we find in the end is that these points After blow up they interact according to Again a logarithmic interaction, but you can see them as screened by a fixed background charge So it's what physics physics the physics literature calls sometimes a gelium So it's not it's not very important But it means that there is something that makes them hold together even though they repel each other logarithmically It's as if there was a background charge. That's Exactly uniform and negative So it's opposite and it holds them together because it has the opposite sign to the positive charges So mathematically We find ourselves with with this equation where you see here you have H Which is like the electrostatic potential Generated by a sum of positive charges Dirac's at the point and this negative background charge That I was mentioning and so long down now at the limit when n goes to infinity lambda is just this infinite discrete set of points filling up the whole plane So the point is we are able to define a total interaction energy of this set of points So I will give you formulas here, but you don't have to really follow them So essentially the the energy that we define call it w is defined based on H and It's defined essentially like this so it's essentially consistency taking the limit over larger and larger boxes of the Square norm of the gradient of H except that it has to be computed in In our renormalized way because this thing that these things don't converge So here is the proper way to do it if you're interested. Okay, so we have a definition Which works for arbitrary configuration of points in the plane and this is the point So you have an infinite number of points They fill up the whole plane instead of summing their pairwise Coulomb interactions with some sort of infinite sum you replace this Difficulty by by computing something which is local and which is more like amenable So there was this case of the line. So here I described the situation in the plane So in the line, it's in fact, it's the same. So what you find if you blow up is you find a sum of Dirac masses and What you can do is you can embed the real line into the plane and solve For this equation which means you can you can view the positive charges on the real line as living in the plane and you can compute the Potential H that they generate on the whole plane Provided that instead of having this constant one as a background charge now you put a constant Density on the real line as the background charge So you you make a sort of singular charge on the real axis which is negative And it's going to sort of balance these positive charges positive discrete charges on the real line and if you compute You can then compute the energy of this configuration in the same way as if it were in 2d And so this gives an energy and the point is that we can derive it as a The limit of the original problem. That's the main point Okay, so this is the way to compute So you have this configuration of points in the plane, which is infinite you take balls of radius r and you average of a larger and larger balls this energy and If you're in the real line you make you embed this real axis into the plane and You compute over a larger and larger strips the electrostatic energy generated by these guys, so The point is that this thing can be derived rigorously from the previous models, okay, so what we show is that after blow-up Around almost any blow-up center in the support of the equilibrium measure You converge to minimizers of this limit energy w So we did it for ginsbolando. We did it for Coulomb gases and for the otaka wasaki model That I mentioned before so so this is this is really the picture We have this picture. We know the sort of average distribution and when we blow up these guys have to minimize a certain interaction energy and Hopefully this is going to start to explain why they arrange themselves Triangularly, okay, so that's the next question. What are the minima of? This w so first thing is you might want to have a more Tractable formula for this w. I gave you a definition which is valid for arbitrary configurations in the plane Here is a computation when you assume something more which is when you assume that the Distribution of points lambda the discrete set of points has some periodicity so if you assume that it's a Pattern of points that gets like repeated Definitly the same pattern Then you can compute the w in a much more explicit fashion and you find a gain What you would expect which means you find again a sum of pair-wise interactions So here g can be computed explicitly It's the green's function of the underlying torus on which these points live This thing is a constant, which is also explicit So you find yourself with trying to minimize this sum of g of a j minus a k Over a set of say n points, which are again repeated periodically Okay, so there is a formula so even though we gave a complicated definition in At least in some simpler settings It amounts again to something you would expect and to something much more explicit a Discrete energy a discrete sort of coolant interaction So you see the effect of the background is that instead of having a log here You have rather the green's function of a torus, which is something slightly different So it's a solution to this by the way Okay, so this g if you want you can express it Even more explicitly you can solve for the green's function of a torus and you find some Eisenstein series So so g can be expressed as an Eisenstein series And it turns out that I found that such quantities also arise You can even write them on general Riemann surfaces. They arise in in Arakulov theory. They are called heights so the question here is to identify the configuration of points a1 to an which minimize such a thing such a height and Well, maybe if there are a number of theorists in the room that have ideas we would be interested I'm sorry advertising for the problem Okay, so there is one thing we can do Which is that we can minimize this formula Among configurations which are already Themselves a perfect lattice Okay, so a simple lattice if you want, right, so you take configuration of points Which are just du plus dv with u and v two vectors Such that the unit cell has volume one, so you normalize the volume Okay, so then you only have two parameters of minimization. It's actually the angle between the two vectors and the size of one of them and Then using the explicit formulas that I showed you before which means using the formulas in terms of Eisenstein series you find that This guy this function w is minimized among the lattices by the triangular lattice So this is the place where you're like because We derived this limiting interaction energy and at least we can see That it makes the difference between different kinds of lattices, so it sees a microscopy a Microscopic behavior and it can tell you that the triangular lattice is the best among all lattices So triangular lattice is better than say the square lattice So this is the one thing we can prove the triangular lattice is better than all the other lattices However, we don't know to prove that it's better than anyone else Which wouldn't be a lattice Okay, but at least it's consistent with the observations that I showed you at the beginning on superconductors where you see these triangular lattice this abricus of lattice All right, so the proof of this in fact boils down to using theorems from the fifties that were already known on Number theoretic Quantities So here you state the zeta function So you found the zeta function of a lattice so here now lambda is a lattice, right? So you some one over p to the power s Over points in the lattice So when s is equal to 2 this is critical this some would not converge And it turns out that when s is strictly bigger than 2 this can be viewed as one way of regularizing the sum In the Eisenstein series thing that we have to compute here is a sort of another way of regularizing the same sum some of one over p squared and what we can prove is that they're essentially equivalent so you can you can Transform one minimization problem into the other and then fortunately the other minimization problem had been solved So the question of minimizing this thing among lattices Had been solved and it was known that the best is the triangular lattice Okay, so here Modular functions and number theory are the things that give you the answer a little Digression which is there is another proof of this fact about the zeta function due to Montgomery So what he observes is that the zeta function is the Mellon transform of the theta function and now the theta function is this thing you sum exponentially decaying terms over the lattice And you ask yourself Again the same thing. What is the lattice that minimizes this quantity? And so Montgomery proves that again among lattices. It's the triangular Which is the best and then by using the fact that the zeta function is the Mellon transform you obtain the result for the zeta function and The reason why I'm mentioning this is that it turns out that in superconductivity again in the Guinness-Bolando model But in a different regime of external field than the one I was studying So there's a sort of two different regimes of external field Then you find this problem so you you also have an abricus of lattice and What you find is that if you try to explain why this abricus of lattice you find to a minimization problem You find a minimization problem, which once restricted to lattices gives you the question of minimizing this theta function So both the theta function and the sort of zeta function Which are Mellon transforms of each other arise in the same superconductivity model and why? Why that is so far. I can't say Okay, so this is a remark and so we are led to this question, which is just what I said before We can conjecture now That this triangular lattice is not only the best among all pure lattices, but it's the best Among all possible configurations or at least that it achieves the global minimum of This function w that we define okay, and Maybe a reason supporting the conjecture is simply the experiment, right? Since you see that in superconductors there are triangular lattices since we derived The fact that rigorously from the Guinness-Bolando equation from the Guinness-Bolando model We should see a minimum of this w While nature gives it to you nature's nature tells you well The minimum should be the triangular lattice now go prove it We have no idea how to prove it at least what I can say is that w Can be expected to be a sort of quantitative measure of disorder of a configuration of points in the plane if you think that The the most ordered configurations are the triangular lattice The most order is the triangular lattice and things that have a higher w then Would be more disordered if you want So now there is the one-dimensional case, which is particular So I told you there was a one-dimensional version of w and then you can again compute everything explicitly And then you can completely solve the question so in one dimension It's not too too difficult to show that the regular configuration, which is z putting the points Putting the points equally spaced Is the minimum over all possible? Configurations achieves the minimum of course 1d is much easier than 2d right you have much much less geometry in one all right, so What do we do once we have proven this we have derived this w as A sort of rigorous limit from these problems with large number of particles So just a few things that we do we expand The partition function We get a finer asymptotic expansion of the partition function, which was not known before We get a large deviations type result, which tells you that Essentially if you have finite temperature there is going to be a threshold Below which all the configurations live except with very small probability So w has to be smaller than a certain threshold As n goes to infinity and this threshold decreases as the temperature is decreased The threshold converges to the minimum of w which means that as you let the temperature decrease You should indeed crystallize To the minimizing configurations for this w and if you believe that the minimum is the triangular lattice this Conjecture that we don't know how to prove then it gives you a proof of Crystallization in the Coulomb gas Okay, and in 1d of course you have a complete proof because in 1d the minimum is identified and Two things I want to point out which are two things that sort of follow from our analysis Is that if you want to solve the conjectures there are two different things that would suffice but that both seem very difficult So the first thing is it would suffice To compute the minimum over periodic configurations But periodic with respect to a larger and larger box Okay, so that means if I give you again this formula All right, which is the formula when I have like just n points in a torus repeated periodically if you want if you can identify the minimum of this and let the size of the box go to infinity or Let n go to infinity then It's it would suffice to to identify the minimum Right, so it suffices to identify the minimum over periodic configurations as long as you can take the period arbitrary large But still this we don't know how to do Okay, second option is compute this integral This integral is the partition function of the Coulomb gas Right, it's an explicit integral, but with a number of variables that goes to infinity So if you can compute this integral and take the limit beta goes to infinity In the order n term you would find the minimum of W The problem is that as far as I know nobody knows how to compute this integral Except if beta equals 2 So if beta equals 2 You can recognize your square of a van der non determinants and using the fact that the square of a determinant You can expand and there's some algebraic manipulations that you can do And eventually you completely compute exactly zn But when beta is not equal to 2 it seems you cannot do the first trick and You're sort of blocked here Except in dimension 1 if you replace R2 with R1 Then these integrals are computed. They are called Selberg integrals and this thing is completely not Okay, so again two open problems So now we'd like to Maybe extend the picture a little bit So what we what we find ourselves with is in fact a crystallization problem and There are many such things in In nature and it's a more general family of problems You know you give yourself some interaction potential V And you look at the sum of pairwise interactions V of X i minus X j Over a family of points in the plane and maybe some kind of boundary conditions. I don't know and you ask ourself we ask ourselves When can we say that the minimum of this total interaction is? To put the points on a lattice To put the points on a perfect say triangular lattice so really the importance of this question I think cannot be understated because This is really the reason why matter Organizes itself in crystals right you minimize some of total interactions between Between atoms say in three dimensions typically and you want to understand Why things organize themselves periodically and what I would like to tell you is that there's very very few instances Where we can explain this rigorously Not another area where this arises there's this thing called the cone and q mark conjecture where They take a potential like this so they take this sort of minimization problem and They say if it's of the form f of x squared with f completely monotonic So what completely monotonic means is this so the each derivative each case derivative has the sign of minus 1 to the k Okay, then they conjecture that the minimum of this is achieved By the triangular lattice always So that's in dimension two and they also have a conjecture in dimension eight There is something that's the ea mood lattice that would do it in dimension 24 the leach lattice so that means that the abricus of lattice is expected to have some sort of Universal minimizing property is going to minimize a large class a large class of starch Such problems and and so you can ask the problem differently What is the class of potentials V for which the minimum is the triangular lattice? Again We don't know and so as I said there are very if you even enlarge the problem and you ask yourself Given a minimization problem minimize an energy and interaction blah When do we know that the minimizer is periodic? When can we prove that the minimizer has some periodicity? Either then in dimension one which as I showed you I showed you one example either and other than in dimension one there is Very few instances. I mean, I don't know many so it's something that's sort of Very intriguing and at the same time we don't really have any Tool to attack such a problem. So let me give you a few positive answers Few positive answers. The first positive answer is the sphere packing question so in two dimensions you take You take disks of fixed size and you want to arrange themselves In such a way that maximizes the number of disks that you can put in a box Or if you want you minimize the interaction with the first hard sphere potential Then it was proven By writing in the 80s that the triangular lattice is the minimizer, which is of course what you would intuitively expect but You know to go and prove it is something else. So This is the proof of this Now this was slightly extended by Florian tile in 2006 where he looks at the Again this sort of interaction where V is a slight perturbation of The hard spheres potential So he takes a potential of this this form which is sort of caricature of a lino drone potential Or you see there is a fixed distance Which is slightly which is very highly favored So it's very highly favored to put the potential the points at distance say one here from their nearest neighbor and For such for such a potential he manages to give a rigorous proof of Crystallization and this is almost the only proof These two are almost the only proofs that are known I would say on This on crystallization from so now if you compare to our problem We were very far off because in our problem. We have a very long-range interaction the interaction is logarithmic Whereas this thing is very short range. So We really not in good shape to apply any of their Of their results Now there is something a little Surprising even which is that there is another problem, which is soft and which doesn't look so far off from ours so I Would like to rephrase our question in the following way so you can sort of heuristically wave your hands and see that the minima the mini equation of minimization of w is the same essentially as the question of Minimizing a sort of h1 star norm of Sum of Dirac minus one so h1 star denotes the dual of h1 whatever that means some functional space Right, so what you're trying to do is you're trying to minimize a certain way of measuring The difference between the sum of Dirac's and the constant one Okay And so now there is this problem which is to measure this distance in the dual of the lip sheets functions and This thing is it is in fact the same as measuring the Vasselstein distance in optimal transportation Between the sum of the Dirac masses and the constant charge one So Concretely what this means is the following problem You're given you give yourself points in the plane you imagine it fills up the whole plane and You want to attribute to each point a certain domain of the plane of volume one and All the mass that's in this domain you attribute it to the point and you compute the integral of the on the cell See I of simply the distance between each running point and the point Xi the black point Right, so you take all the mass on the cell and you transport it According to the March counter of each transport problem on to the point Right and the cost that you pay Is essentially each unit of mass that you transport into the point you pay the cost equal to the distance to the point Okay, so this is this is the and then you integrate right so this is the total cost for each unit of mass You pay X minus Xi and you integrate over the cell and so the question is what is the best? Partition of cells and what is the best distribution of the points in order to minimize this whole thing? right, so What is the way that if you want if you want you you you can transport the mass to these points in the most efficient way Well, it is the triangular lattice and this is a There is a proof in preparation by born policy and tile which hopefully will appear soon So they managed to really prove it the best distribution is the triangular lattice and you see This is for a different way of measuring this distance, so it's very strange that for this way in fact The proof is not very very difficult. You can do it. I mean it's smart, but it doesn't seem Extremely extremely hard and when you change it into this we have no idea Okay, so here these are a few a few positive results and now Another positive results by the way is this honeycomb conjecture by solved by hails in 2001 Which tell you that if you want to minimize the perimeter Among tilings by region of equal area The honeycomb lattice does the best So now just one minute. I will take one minute to say to come back to these fakete points So you remember there was this question on minimizing fakete distribution of fakete points on a manifold so the product of Xi minus xj or more generally Sting with risk risk kernels the sum of one of our Xi minus xj to the power s so in fact this thing comes up a lot in this question of interpolation because the Optimal points give you the best way of interpolating a function and of computing numerically integrals There is a whole very interesting article by saff and Kuhler's In in mathematical intelligence sir and they explain why these things are interesting One of the reasons they give is the understanding of the carbon molecules that live on sphere You know these soccer ball types molecules, but also interpolation, etc and so here is a picture that you find here for Numerical computation of fakete points on the torus and you see You do like to believe that if you zoom around the point you see a triangular lattice picture seems very very telling in that case and so What what was conjectured in fact is that if you take this energy and you expand in powers of n You have an expansion all the way to all of one And people ask, you know to prove this thing so what we should be able to prove now Is that the order n term this constant here? Is the minimum of w so we should be able to identify this constant and relate it to this other minimization problem and maybe to prove to you that this thing is of interest to people just want to finish by saying that this was asked as Something like this was asked by smell as a seventh problem for the next century. So what happened is at the beginning of the 21st century. So this one this century Mathematicians congregated and they try to reproduce what Hilbert had done in 1900, which is to phrase problems for the next century And so this thing was recorded as smells sevens problem. So it's again is to find it's to find the minimization problem this minimization problem on the sphere, but to provide an algorithm That will give you a minimizer up to an error of log n Okay, so this thing is considered very very difficult. Of course, you want to do it in polynomial time And the reason why it's difficult is because if you look at the local minima of this interaction energy There the number of local minima is expected to grow exponentially in n and so Algorithmically, you know if you compute you have a very large chance of being stuck in a local minima not knowing how to Extract yourself from there. And so whether this is solvable in polynomial time or not Is not Okay, so I hope I've tried to Convince you that this abricus of lattice is in superconductivity, but is also sort of everywhere and That there are some interesting math problems Related to that. Thank you Thank you very much Like I reattached a great importance to the least action principle. So I'm sure he would have liked your lecture other questions again Did you make some kind of numerical simulation or computation on the quasi periodic crystals like Penrose dialing or something? We you know, some of them have high degree of quasi symmetries. So maybe some of them are just better than the equilateral lattice Did you maybe maybe you can make some computation? Yeah, I mean There are some first of all is difficult to do computations with this because again you get stuck in look You know, you cannot prove that you have a global minimum, right? It's very difficult to show that you're not in a local Sure sure sure sure, but so so far no one has found anything better So all the numerical computation seem to go in the direction of the triangular lattice Not me not myself, but some people have done yeah, and they seem to as far as you can see You don't see any quasi crystals. That's all I can say who knows Remarkless Sorry, you mentioned you you mentioned beta equals one and beta equals two and This corresponding to real and complex case But beta equals four wouldn't it correspond to the quarter and some kind of quaternionic case and can't it be for instance? simulated with or Understood more efficiently analytically So yes, be type was for it the quad quaternionic case That that's just more difficult to to analyze because the computations are more complicated from the algebraic point of view the you know probabilists work on this But in fact the point is not that the point is that you only have one to four for which you can say anything Using these and then in dimension one there are these trigonal Matrices which can work for any beta etc. But yeah, it does it Be the temperature of one half seems higher than temperature of one four Yes, would be one more So what you can do is you can see as you increase beta in these models whether you indeed see more order In the local statistics and you do you do so we did some computations which compute W for these Processes that are obtained as locals, you know local limits of matrix models and you observe that W decreases as beta increases, but You cannot make a whole curve, but it seems to fit like everything seems to fit quite well Thank you Let's talk it's up to 30