 I'm very happy to be here and so today I will speak about universal optimality of E8 and the leach lattice. And so today I will explain what universal optimality is, what is this property and during the next five lectures I will try to prove to prove this property. And so this this notion of universal optimality it comes from the field of energy minimization. So here is a very classical problem considered in mathematics and physics sometimes in computer science. So we have some metric space and in this metric space we want to choose a collection of points and we suppose that these points interact with each other and the energy of interaction it depends only on pairwise distances between points in the set. And then the game we are playing we are trying to minimize this energy. And so in this series of lectures we will not speak about general metric spaces but we will speak about Euclidean space. And so here is our and so let's I'll just formalize what I have just said. So what we do we fix some potential function or potential profile so it would be function from the interval 0 infinity which is somehow the values which distances between points can take and it takes values so our energy is always real positive or negative and so now if we have for example a finite set so we say that potential p energy of a finite set in the Euclidean space it will be the following quantity. So this will be the sum over all pairs of points and we assume points to be different so self-energy is not taken into account as in many physical problems. So then we would like to minimize a quantity like this and so here are straight brackets they denote the Euclidean norm and so but of course somehow because we are working in Euclidean space maybe being able to compute this energy only for finite sets is not very interesting for us because for example we think about possible for example physical applications and then we want to study something like a crystal so we would like to also to give a definition which would work for infinite configurations but again about doing giving a reasonable definition which works for all sets it's a bit too difficult so we will restrict ourselves to good subsets so namely we call them point configurations and so a point configuration in the Euclidean space it would be a non-empty discrete subset of the Euclidean space and so what it means it means that every ball every closed ball it contains only finitely many elements of this set and so now what we can do now we can define energy for so maybe just small piece of notation for the future so we will denote by b with lower index r of x still depend on d it will be the closed ball of radius r about x in the Euclidean space and so now how do we compute the energy of point configuration which might contain infinitely many points so we do it by limiting procedures so what we take we will take our our so we'll take all pairs of points which line the intersection of our configuration with an open ball of radius r about 0 and then the compute the energy of this subset but now what it will do I will take this r bigger and bigger and bigger and so now I will also need what I will need I will need some kind of a renormalization I will divide this energy by the number of points in the intersection and so now I would like to take a limit of as r goes to infinity and of course such a limit might not exist for some configurations and for some potentials p so I take limit inferior and so I will call this quantity this will be the lower p energy of a point configuration and so as we see in this definition this lower energy it is something like average energy of interaction of one point with all the others so for this we need this normalization factor here and so now in some cases just a limit like this would exist and then we say that this is not the lower energy but just the p energy of C so now if limit exists or the same expression as here number it's p energy and so just a small remark is that with this definition so our energy p energy it can be some real number or it can be minus or plus infinity and so now what we want to do we want to what's the name says energy minimization we want to find configurations with minimal possible energy but now to somehow to rule out trivial solutions we need to introduce another notion which is the notion of density of a configuration so we say that so that the point configuration it has density 0 if the following holds if the following limit exists and equals to row so the now so density it would be the average number of points per volume and we computed in an obvious way so we take our configuration again intersect it with a big ball we center it 0 and divide with all this by the volume of the ball and if this number exists then we say that it is a density of our configuration and so now when we try to so let's maybe before we come to minimization let's do a few examples which will illustrate these two concepts so one nice example of a configuration in Euclidean space it is a lattice suppose that lambda is a lattice and so in this case of the density of lettuce it is just the inverse of its co-volume so the density will be just one over the words co-volume and so also now we can apply our definition of p energy and then we see that the p energy of a lattice will be the following quantity so it will be this number assuming that the sum converges absolutely are you assuming any positivity properties not at the moment but this will come come in a bit later but actually in the for the energies we are interested in they will be positive so they will do consider repelling potentials and yeah also this assumption will be true so we'll consider energies whichever like fast decay of course our general definition it allows also maybe to define number when this assumption is not true but we will not consider examples like like this or maybe we'll consider one of them but later in the series of lectures and so here is another example which will play a role a bit later so the lattices are very nice configurations but unfortunately there are too few of them so because being a lattice it's a very restrictive condition so we would like to relax it a bit a little bit and instead of considering lattices what we can do we can consider periodic configurations so from analytic point of view can periodic configurations are still nice they have usually well-defined density also it's easy to compute the energy for periodic configuration and at the same time we can say that almost any configuration can be somehow approximated by a periodic one at least if it's in some reasonable sense so so let's consider now next example would be periodic configurations so for this we fix a lattice lambda and then the configuration is lambda periodic if C plus L it will be again C for all elements in lambda and in this case it's also very easy to compute the density so density it will be the number of elements of C modulo lambda and because you know that configurations they are discrete sets so we know that this is some finite number again divided by the co-volume of lambda and so maybe what's a bit more involved computation it is how to compute the energy so here again we assume that our potential decays fast enough so what we can do we can take some over all elements of C mod lambda and we will consider separately x and y which are different modulo lambda and then we have a sum like this some over all elements in the lattice and in the case if x and y coincide modulo lambda then we just have to sum over the lattice without zero element we will get a sum like this and so now let's define so now what does it mean to minimize p energy so maybe maybe at this point you have some questions to the examples oh yes yes you're right thank you and so now what we would like to do we would like to fix our potential and to find the configuration which minimizes the p energy of course here one obvious problem is that for example if our potential is repelling then the points will just like go away to infinity we'll get something like that empty configuration is the best or that point should be as far away from each other as possible and so one way to overcome this is what is sometimes done in physics to make to do somehow change the shape of a potential which somehow makes this trivial solution impossible and another way the way which we will as which we will choose we will fix the density we will say that we are searching for a minimum only among configurations with some fixed density definition so we say again so let see be a point configuration configuration suppose that it has this density row and we assume that row is a positive integer and also let p not integer sorry positive real number be any function then we say that c minimizes p energy if it's p energy exists and every other configuration so every configuration in the same Euclidean space of the same density as of this time has lower p energy at least equal to this number and so another terminology for this is to say that c would be a ground state for p terminology stolen from physics and so now what we in general expect we expect that of course this ground states they will know what what happens if we for example fix density and then change our energy profiles somehow we expect that of course the set of ground states it will also change for together with a energy profile and it's quite easy to come up with easy examples like that but maybe maybe there is a nice family of p energies such that several energies have the same ground state and so these questions they bring us closer to the notion of universal optimality so how the ground state do the ground states depend on the energy profile p and so first idea is of course that they can depend a lot and so here's one example so let's consider a Gaussian core model and let's look at it in our favorite gledian space R3 and so Gaussian core model it's quite often used in physics and chemistry so first it is quite simple on one hand and on the other hand it is also rather realistic and useful so what we do here we fix some positive number alpha and we define our potential function to be the Gaussian with this exponent and so now here one observation it is that because of the symmetry Gaussian has it is what we can do we can fix a density and change exponent alpha and study ground states or on the other hand what we can do we can fix alpha for example to be pi and then change densities and search for ground states for different densities and this will be essentially the same problem so here's so what we can do so fix and change alpha and see how ground states will depend on alpha and this is actually equivalent to the fixing alpha which I like to be pi and then change density and because this is somehow used more often maybe let me just use this terminology for for a moment and so if we have fixed alpha oops sorry and therefore our energy becomes this function so what experiments tell us about the ground states for different densities the first is if density is smaller than one or much smaller than one then we know that face centered cubic lattice it seems to be the winner in this case and so what is a fade center cubic lattice it's a lattice like this which for example gives us the best packing it would take a usual cubic lattice Z3 and then in the center of each face of a cube switchers which give us a tiling of a Euclidean space we choose one more one more point and now if rho is bigger than one then here it seems like that there is another obvious winner which would be a body center cubic lattice and this lattice is they actually do all to each other after right scaling and the body center cubic lattice it looks like this so again we have we take a usual cubic lattice Z3 and now we add one more ball in a center of each of these cubes which we can imagine they are tiling our whole space so this will be a geometrically different lattice and for big values of rho this one seems to be the best so we see that there is certainly a dependence of ground states on the energy even for this very nice family of potentials difficult to come up with a nicer family but now something quite interesting happens around density which equals to one and so this was observed in the 70s by Stillinger so what he have found that if density is about is close to one then actually face coexistence of these two lattices it will have better energy than each of them separately so and so how it looks like so we divide our Euclidean space into two parts and here we consider so our total density has to be rho but we make for example half of them to be cro one and another half or two such that their average is rho and on this side we for example arrange our points in a maybe FCC lattice of density also we have to rescale it and on this side we will make it as a business lattice of another density and because of this play with different densities it turns out that we can achieve some improvement sorry but boundary does not play that much role right because bug like the dimension of the boundary is small right and like in that list in terms of our definition somehow the density of all thing is still well defined and the energy is still well defined right exactly that's why I don't understand how it can help what he proposed yeah I think it happens because because he has he has two different densities on two different sides and again so it has to do with the convexity of this you can draw a graph of like how how for example the energy of such a lattice depends on density and how let energy of this one depends on it and then around one there is something like a kink and he uses this kink to create an improvement so in the case after case of maybe two doesn't it's not something very strict right now but it looks like would look something like this no maybe not like this maybe maybe okay maybe I should draw it but he does use the somehow convexity properties of these two graphs of the energy of this lattice how it depends on density and energy of this lattice how it depends on density so we see that the ground state and it can depend in some rather complicated way is just to make sure that I'm not totally mad is it true that raw is equal to one plus two over two yes I suppose so yes and so how however to universal but universal optimality it is exactly this phenomenon that in some metric spaces we can find the very like a nice family natural family of potentials for example like Gauchan's and the same configuration will be a winner for all elements in this family and also to make it a bit more formal we will need one definition so we say that the function g again from the interval zero infinity to the real numbers is completely monotonic if it is infinitely differentiable and satisfies the following condition so it has to be like a perfect repelling potential so it is positive it is decaying and it is convex so and this pattern goes with all the derivatives so it's derivative derivatives they have alternating signs and so this will be true for all positive k for all non-negative k and here is one example so what kind of function is this for example an exponential function with a negative exponent will satisfy this condition so and so here is the definition of universal optimality which was given by Henry Kohn and Abinav Kumar so they say that the configuration is universally optimal if it is a minimizes energy for all functions p which are completely monotonic functions of a squared distance so it is a slightly stronger condition than just being completely monotonic so we say that so now a point integration c with density with some positive density rho is universally optimal if it minimizes p energy p so whenever p is a completely monotonic distance of a completely monotonic function of squared distance so p it is g of a squared and g is completely monotonic yeah but I think it is actually necessary so at least at least there are configurations which do do do this so I mean I mean we do to have like less less conditions on on on g power laws okay the power laws are also also fall into this category yes and you cover potentials sorry you cover potentials oh it's expanded one hour okay so this I'm not I'm not sure about maybe maybe mix of power and exponential it's a better function you know that much anyway yeah but maybe okay maybe maybe maybe not because the best of functions sounds like it's something oscillating right so this okay maybe yeah but this I have to check check the condition and here are the small remarks on how to be if you want our function to be sorry configuration to be universally optimal it actually it's sufficient that it minimizes energy for all Gaussians and this happens because of the following theorem which is quite old and classical probably it's theorem by Bernstein at least in some of its formulations and it says that every completely monotonic function can be written as a convergent integral of the following shape for some measure on measure mu on the interval zero infinity I was surprised by this not patients should not use our meaning our square it's kind of confusing but it is like this is to use the R squared it is somehow stronger condition and because somehow if you have a because I think if we go because it's not a very difficult computation but if we had something which was for example here function p it also will be completely monotonic if it's but somehow just completely monotonic is not enough for us for our needs and so for example like a conceit corollary for for this would be so that that C is universally optimal it is equivalent to see minimizes the energy for p equals to the E minus alpha R squared and here for all positive alpha so now I have a question how do I bring back the upper blackboard and so now we gave this very nice definition so not we but Henry Cohen and Abinav Kumar gave this definition but now of course the question is is there at least one object which satisfies this definition and as we have seen in for example in dimension 3 we don't hope for an object like this so in dimension 3 it will not exist and so Cohen and Kumar they actually studied the same question also in a compact metric spaces like on spheres and projective spaces and they found many particular examples of this universally optimal configurations and they also found a few in Euclidean space the first easy rather easy example it is that the lattice of integer numbers in dimension one it is universally optimal which is of course maybe not so surprising because what can be better than the lattice of integers so integer numbers inside R it is universally optimal and they've also conjectured the universal optimality for the few more lattices so they conjectured that the A2 lattice which is a hexagonal usual hexagonal lattice in R2 and also E8 lattice which leaves in eight dimensional Euclidean space and the leach lattice which leaves inside 24 dimensional Euclidean space that these three lattices are also universally optimal and so now the theorem which I would like to present in this series of lectures it's a joint result obtained together with Henry Cohen Abinav Kumar Steven Miller and Danila Ratchenko and so it says that the lattices E8 the leach lattice they are universally optimal one case is open so Z is now A2 yes A2 yes A2 is still open and some power it turns to be more difficult than E8 and leach lattice yeah but maybe I will come to this little bit of the in the last lecture so I'll see why things which works and big dimensions why they don't work in small dimensions and so maybe now what I don't know what would you like to make a small break so maybe maybe we can make a small five minute break and then I will explain tell a bit more about the methods of the proof and so at the second part of the lecture I would like to speak about methods of the proof of this theorem and so the proof it in print yeah I guess so it does follow from this result after after some limiting procedure but it's not it's not difficult because packing problem it also can be formulated as this energy minimization problem for some suitable like one way for this potential it's that it for example energy is infinitive two points come too close together and just zero if they are if the distance is enough of course it's somehow it's contradicts a little bit our rules for the potential but but it's somehow but it's very close to and so let me tell you know about like our sex our strategy of the proof and so the key ingredient of the proof are the linear programming bounds and I will introduce this technique in a moment and so essentially what this method is it means that we have this very difficult nonlinear problem about configure about point configurations but we replace our difficult problem by an easier one we relax it so to say by a different a question which we now ask not about points in our space but rather about functions on this space and so the problem which we will study it will be actually now it will be certain convex problem in the space of functions and because it's a convex problem here this word linear programming comes from which is a bit no strange and maybe also old-fashioned terminology but at least it works very well in some nice cases as I will tell show you in a minute and as we have to work to do this convex problem in the space of functions so it's convex it's nice but what now what is not so nice that our problem becomes infinite dimensional and for this we will need some trivial facts from Fourier analysis and in particular we will need to establish new Fourier interpolation formula and now the last piece of our strategy it is after we first relax our problem to some convex problem which magically turns out that even after relaxation we still get a precise result for this very nice e8 and leash lattices then we do manage to solve this problem by introducing new Fourier techniques what we were able to do we will reduce our question about of universal optimality to some to the proof of a positivity of some function of two variables and unfortunately this step we could not find a better way to solve it other than check it numerically so the last would be the numerical verification positivity of a certain function and for example after some transformations we can think of it as a function for example on this unit square and so let me start with the first point which would be the linear programming bounds so here we'll need two technical definitions so one of them we will recall the definition of a Schwartz function so the Schwartz function on the f-credian space it is a function which is infinitely differentiable and also the function itself and all its derivative they decay faster than any inverse power so and also we know that and this will be true for all positive numbers alpha for example we can take them to be integers and for all multi indices beta and here also one piece of notation so as we have to use a Fourier analysis we have to introduce Fourier transform and so here I just because there are several different conventions about how to write down the Fourier transform I will write my favorite way of introducing it and the following and so here by the dot I will denote the usual scalar product so standard inner product gradient space and so now when we have this notations we can formulate what a linear programming is and so this method is used quite often in metric geometry and then analyzing optimal configurations and so usually for each space for each kind of problem there is a separate adaptation and so Henry Cohen and Abinav Kumar they developed a nice adaptation of linear programming to this energy minimization in Euclidean space again let P be any function any energy profile so but this time we do want it to be a repelling energy profile so we don't allow P to take negative values and we suppose that F is a we were able to construct a Schwartz function and so we suppose that this Schwartz function satisfies several nice conditions so first we suppose that F is bounded above by P to all points except for zero and also that the Fourier transform of F is non-negative then we know that then every configuration of Rd with density rho it will have lower P energy which is at least this number which will be rho times the value of Fourier transform of F at point zero minus the value of F at point zero so we have obtained an estimate for possible P energy and so now let me give you a short proof of this statement and so this is what I say that we have somehow replaced our problem about finding good content about configurations of points by convex problem for functions because now our problem becomes just finding such suitable F and the conditions which we have one F they are convex conditions so F has to satisfy these two inequalities and also this value has to be optimal for us and in general as we will see now in a proof that this is a sort of clearly a relaxation of our problem so in general what we obtained this the question about F it is a weaker statement it will just give us the lower bound in most of the cases but of course we will not replace our initial problem and so let's do the proof and so here now I will just to make my life a bit easier I will not do the proof for general configuration but I will prove this statement for periodic configurations so suppose that C be a periodic the lambda periodic configuration and lambda is some lattice so then we have the following we can write the P energy of this configuration on the following quays so and so now at this point we just used the computations which we have done before when we considered periodic configurations so now what we do we can estimate this number from below using the proof in the fact that F does not exceed P so we can write in the following quay and now because F is already defined at zero where P was not we can rewrite this in the following slightly nicer way maybe more symmetric so we will write the sum over all x and y in C modulo lambda not excluding the case where x is equal to y and now the only term which we will have to subtract would be F of 0 and so now what we do at this point we use the Poisson summation so we write our sum like this and again so here lambda star is a dual lattice so it would be the set of all mu such that mu times L is an integer for all elements of the initial lattice L and so now we do maybe the somehow the most non trivial part of the of our proof so what we will do we will replace this double sum by absolute value squared of a single sum so we leave our summation over dual lattice and so now here the trick comes so we have written a sum like this and so now okay okay sure it should be F and so I think now it's not correct and so now a good thing which car which we see here is that this the absolute number squared of course it's a non negative number so here we can use our conditions from the theorem that the Fourier transform of F was a non non negative so we can again estimate this number maybe I'll do it in this black part so now we throw away all the terms in our sum over the dual lattice except of the the one where mu equals to zero and so now this the the sum of exponentials if mu equals to zero it's very easy to compute it it will be just the number of elements of C modulo lambda and we have to take it squared and now this is if we see that we can cancel something cancels out here and what we will get we'll get the number of elements in our configuration modulo lambda divided by the co-volume of lambda and this is just the density of the configuration so so this proves our statement and so the general case where we consider not only periodic configurations it was actually so general case was proven by Korn and Corsi Ireland and so now we can look at the proof and we what we try to see is in which case we can hope for a sharp bound so in the proof we have seen that we when we prove so we could get a sharp bound only if all the inequalities which we have written in our proof they are actually equalities and when by each of these estimates we essentially did not lose anything and again some by considering different case and doing numerical experiments we do know that actually sharp obtaining sharp bound by this method it is a very rare occasion so usually we get just some estimate but this estimate is not good enough to completely resolve the problem but still in some cases it does happen we do get sharp bounds so when can it happen so again suppose that so we make the following assumption suppose that lattice suppose that our optimal configuration that is actually a lattice and that it minimizes p energy for some p and we suppose that this can be proven by constructing appropriate function f your programming method first function f so it means that in all the our inequalities on that blackboard we we don't have any losses so and so this give us quite restrictive conditions on the function f so first we know that our function of x it has to be equal to p of x for all x which are in in our latest lambda maybe accept for zero and also we know that the value of the Fourier transform of our function it has to be exactly zero for all y which are in the dual lattice again without zero and so now the question is so something that we observed numerically that actually in in some cases these conditions knowing these two conditions it is enough to identify the Schwarz function f uniquely so only by giving these two kinds of restrictions we can then reconstruct f at least in the space of Schwarz functions and so here is another result which I would like to present during these six lectures and hopefully also to prove it so it is the following interpolation formula here maybe before I write the formulas so one more remark so we see that these two conditions these two equalities have to hold but also they have to hold to the second order because of the inequalities which we have so if our function p is nice and smooth then it means that f has to coincide with p at all lattice points but it also it has to satisfy up to also the first derivative has to satisfy to coincide and the same happens for the Fourier transform because we know that f at point y is zero but also I've had it is a non-negative so it means that it is the derivative of f hat also has to vanish at this point and so now the question is whether we can reconstruct function by knowing its values and values of its Fourier transform and their first derivatives and some discrete set of points and so suppose that d and and 0 be either 8 and 1 or 24 and 2 then every f in the space of radial Schwartz function so here we consider not any Schwartz functions but only Schwartz functions which are invariant under rotations around 0 so any radial Schwartz function is uniquely determined by the following values so we have to compute f of square root of 2n its derivative at this point Fourier transform and the derivative so it's Fourier transform and this should be you know you should know these numbers for all integers and which are bigger or equal than to number and 0 and and 0 it depends on a dimension so if dimension is 8 then and 0 is 1 and if dimension is 24 then and 0 is 2 and so more precisely we can write in the following way we know that there exists an interpolation basis consisting of functions a n b n a n tilde and p n tilde and they are all radial Schwartz functions and again so we have a such four sequences of numbers with index n which starts from n 0 and so this basis is defined by the following property so such that for every function f we have the following formula so equation by i f for interpolation formula so we know that f of x equal to the following sum so n goes from n 0 to infinity and here we have so f of square root of 2n times a n of x plus its derivative value of Fourier transform at this point and so also what is important is that this series it will converge absolutely and of course the convergence is not very difficult because we started with our function f which was a Schwartz function and it means that these values over which we are interpolating they are already decaying very fast so the only thing we have to ensure that the values of these functions a n and b n and a n tilde and b n tilde they don't grow too fast for example that they grow at most polynomially in index n for fixed x and then we'll know that this serious converges okay so actually we proved that they can be arbitrary sequences yes so in any dimension if we to take for example Schwartz functions and consider these its values such for example all square roots of integers and its values and the same Fourier side we know that there will be some relations between these sequences but there will be only finitely many of f of not finitely many but it will be finite dimensional space of relations and then it will be actually described by some space of I would say modular forms or modular forms like with general sense but but I will come how probably explain this on our next lecture a bit more and so and so let me write a little bit about the properties of these functions a n and b n so so it's actually it follows from our claim that this formula it is an interpolation formula for example it should works for functions a n and b n themselves and so what does it mean that they are interpolating basis so they have the following properties so so it means that for integers any pair of integers m and n which are bigger or equal to n 0 we know that for example functions a n if you evaluate them at points square root of 2m so they will be it will be this value will be equal to 1 if n is equal to m and it will be 0 otherwise and at the same time the derivatives of this function they will vanish and the value of its Fourier transform at such points they will also vanish and the derivatives of the Fourier transform is also vanish and for the function bn we will have a slightly different story so here are the values of the function bn they will vanish at all our chosen points on the other hand the derivatives they will vanish everywhere except for one point where m is equal to n and the values of bn and the values of the derivatives they will both vanish on the Fourier side and another useful symmetry here it is that if the Fourier transform of a n it will be actually equal to the function at a n tilde and the Fourier transform of b n it will be equal to b n tilde so the Fourier transform in all these is taken in one dimension r squared or in rd? no no it stays taken in rd but we work with radial functions so it's in a sense it is one can think of it that because all our functions they will think of them as a d-dimensional function to take Fourier transform but essentially they are one-dimensional so this Fourier transform it could be replaced by some Bessel transform for one-dimensional functions and the derivative is the derivative of the function of r squared okay yes so here the derivative it's actually the derivative of one-dimensional function so it's somehow radial derivative in radial direction thank you and so here maybe I'll use some notation so we'll denote by lambda 8 this would be the E8 lattice lambda 24 this sub-index 24 this would be the leach lattice so these are very famous mathematical objects and maybe they don't need too much introduction but I'll just remind that these both lattices they are even and unimodular lattices and so even it means that if we take any vector L in this lattice then its Euclidean norm will be a square root of even integer and unimodular it means that the co-volume of the lattice is one and also one important fact it is that the so the shortest vector of course shortest non-zero vector in lambda d it will has length or has Euclidean norm square root of 2 and 0 where in 0 is like in the formulation of our interpolation formula and so now when we have an interpolation formula and we know this properties of the optimal axillary function what we can do we can combine this together and construct our axillary function and so by combining what I have just said so now we see that the only possible axillary function that could prove a sharp bound for for this fun for the lattice lambda d where d is either 8 or 24 at least the auxiliary function among among Schwarz functions would be the following one so this would be the function f of x which is the following sum because we know that such a function that will coincide with p up to second order in all vectors of of lambda and also it will vanish at all the vectors of as a Fourier transform of f it will vanish at all the non-zero vectors of the dual lattice which in our case is actually the lattice itself and so now but now of course we have extracted this information only from knowing the equalities but in in the proof of Conan Elkis about linear programming we know that knowing equalities is not enough we also have to prove the inequalities and this is what will prove the universal optimal the universal optimality for us so so we also need to know what happens to the function f in all other points now it suffices to check f of x actually does not exceed p except for zero and also that the Fourier transform of f is non-negative for all points y in the dimensional Euclidean space so we need to have a picture which looks something like this so so if we have our potential which looks like something like this then our function f has to be like this and Fourier transform has to be something like this yes so this we will do this for the gaussians right and so again so this is what you said so we will remember that lambda d it is universal optimal if and only if it is a ground state for all gaussians and so now a few words about how to how to approach this problem so now we have to prove some inequalities for now we do see where the proving positivity of a function of two variables comes from now we have somehow two variables here one of them is alpha the exponent of a gaussian and another is x which is a length of a vector and we need to prove some inequalities on our function for all parameters x and alpha and so come out to do this what will be very useful is to have an explicit formula for our function f which somehow would be we also get it if you could have explicit formula for example for this for interpolating basis functions and so now I will finish but now okay there is a trick which actually allows to do it only on one side I will also show it to you now in a moment but so let's let's first I'll write on how to first how to make these functions explicit so that we can work with them and prove inequalities and second part is also how to like instead of proving two inequalities how to prove only one so now we consider the the following generating functions right by f capital of tau and x the following functions so first function the function like this and second one the function like this the same only with still does and so here in these functions here x it will be a variable on space and tau parameter tau it's an element in the upper half plane so it's a complex number with a positive imaginary part and so now what we see that if we substitute tau to be i times alpha the purely imaginary number then what we will get here we'll get the function which we had before this will be this auxiliary function for the Gaussian with parameter alpha so now the inequalities now the inequalities capital f of i alpha x smaller than equal e to the minus pi alpha x squared for positive real alpha and all x in the Euclidean space and now we can take the Fourier transform of this function but we take a Fourier transform with respect to the second argument in the function so we consider alpha just as a parameter and compute the Fourier transform again for all alpha and for all y in the Euclidean space and so now these inequalities they will imply the universal optimality of the lattice lambda d all the letters were interested in and so and so here now I will show you a trick which helps us to compute this function capital f of tau and x explicitly and therefore it will also help us to find for example the interpolating basis and the auxiliary functions we are interested in and so the trick is the following so now what we can do we can apply our interpolation formula to the complex Gaussian so the Gaussian with yes yes yes also I so what we do now we want to so so we now in applying interpolation formula which is now in the middle upper blackboard to the following functions to the complex Gaussian p with index tau of x it would be e to the minus pi i tau Euclidean norm of x squared and then what we will see so that it is equivalent to the following expression which involves our generating functions capital f and capital tilde it's equivalent to saying that 2 e pi i tau times the Euclidean norm of x squared equals to the f capital f of tau x plus the following and now because the Fourier transform acts nicely on Gaussians so here as a from the last two terms in our interpolating formula we will get also not exactly the capital f tilde but the function which is closely related to it so it's the function which is obtained by capital f tilde by a simple modular transformation function like this so we have got a functional equation which looks like this and we secretly actually would have also two other functional equations for these two functions so secretly we also have we also know that this both functions they are so to say linearly periodic we know that if we take f of tau plus 1 x minus 2 f of if we take the second difference with respect to tau so this will vanish and the same will happen for capital tilde for the capital f tilde oh yes minus one over tau think minus one over tau and so now we have this free equations for f and f tilde and now these three equations they actually they allow us to find these functions explicitly and this is something we will speak about on the on our next lecture tomorrow so somehow somehow because we want our interpolation formula to to be convergent so this gives us already some gross assumptions on f and these functional equations together with gross assumptions they will allow us found f and f tilde explicitly and so I think I have one minute left so I can maybe show you how to instead of proving two inequalities how can we prove only one inequality so now we can see that actually if we prove this inequality it will also apply this one and it follows from the functional equation and the fact that the Fourier transform of capital f with respect to the second argument will be just f tilde in equal this inequality f of i will equal 0 for it implies is actually bounded by e to the minus p again for all and so how do we see it we see it from first we know that if you take f capital this would be just tilde and this formally this follows from the properties of the functions and the interpolating basis it will follow from from this property and second what we just have to use we have to use the functional equation so so we know that f of i alpha x it will be equal to the e minus pi alpha the Euclidean norm of x squared minus this number and so here we use the fact that f tilde is actually equal to the Fourier transform of f and here we use the fact that our d is divisible by 8 the dimension is divisible by 8 and so here at this step we can use our assumption which we have made so we see that this is bounded by the exponential by the Gaussian this is exactly what we needed to prove so instead of proving two inequalities we actually need to prove only one of them so thank you very much okay so I did not formulate it like this in the theorem but it but it is actually the property yes yes okay so they actually have to do with dimension right so it's amazing that this lattice comes out special if I follow your experience I would say I would guess that the other light is that one might have looked at like 8 to twice the 8 or the 16 or the 23 nemyo lattices don't satisfy this so they will not work yes yes yes so they do it by 8 is just a small part of the story of course of course I think it's and also we still hope that something is something you know something is going on with for example in dimension 2 and also in dimension 1 so yeah so arithmetic properties are important but I think they're quite subtle here question about interpolation formula so are there other Dean and not for which it works I think I actually it works for every dimension with some okay so maybe of course like this and zero it will always do depend on it on a dimension and this number it can be actually like how exactly do depends it can be described by dimension of some space of modular forms and also like what's a bit special about in this case which might not always be true is that like for example here we can start our formula from the same number and zero everywhere and in general it could happen that yeah like like the total deficit in dimension is not divisible by 4 and then we have to divide maybe they take I don't know more values on free a site or more derivatives so is it possible to find explicitly this function f capital for any dimension yes there are also some I think in the sense that we have to make some choices like for example to choose where to start your summation and after that it is possible so you only need to specify these two pairs because there are examples of the latest yes yes because there are also some maybe not so pleasant technical details details which you did not want to work them in like in a full generality but yeah but we concentrated on these two cases only because because of our application