 I would like to welcome you again, or for the first time for the second Einstein lecture 2023. And I would like to welcome you also on behalf of the President of the Albert Einstein Society, Professor Otto, who cannot be with us unfortunately today. And I would especially like to welcome our honored guests today. I think it has never happened before that the rector of the university, a vice rector, who will be the next rector, Christian Leumann and Virginia Richter are here at the same time. So a special welcome from us. Otherwise the audience yesterday was, I was really positively surprised about the average age. We were very happy to have so many young people here because enthusiasm for mathematics, it's very important to get people when they are still very young. So just to recall, this is the 14th Einstein lecture. We've started out in 2009. It's the fifth in mathematics. And the Einstein lectures are organized jointly by the Albert Einstein Society and the University of Bern. My name is Christiane Treta. I'm professor of mathematics here in Bern. And it's a special pleasure and honor to introduce our Einstein lecturer 2023, Professor Marina Wiezovska, who holds a chair in number theory at EPFL, Ecole Polytechnique Fédérale de Lausanne. So Marina sets not just one record in our series of Einstein lectures. She's the youngest ever Einstein lecturer. She various other things that I will list maybe not now. But since quite a number of people weren't here yesterday, I will be a bit more detailed than I had planned. Marina was born in Kiev and we had a number of people who met her in Kiev, even people who attended the same school in the audience today. And it's quite impressive how early she started her career there, supported by a dedicated teacher who was a researcher before, if I remember correctly. And then she made her academics first steps also in Germany, back in Kiev, a PhD there, another PhD in Bonn. And then came the memorable month of March 2016, when she had completed the solution of the Kepler conjecture in dimension 8, the sphere packing problem in dimension 8 and posted her single author paper on the free math preprint server archive. And from the Laudatio that I heard when Marina received a big prize, I learned what the reaction in the community was, especially from those people who had tried to solve this problem themselves. So an avalanche of prizes followed that Marina asked me not to list again. It's quite many, but I have to list the most important one, the Fields Medal in 2022. And so we are extremely happy that you agreed to give these three Einstein lectures. And so we are very much looking forward to your second one. Marina has changed the title slightly because I think she reacted also a little bit to the audience, but she will say more about that herself. Please, Marina. Christiana, thank you very much for another introduction, also this time. So today I will continue speaking about the sphere packing problem and today I am excited to shift my second lecture to interchange them and first to speak about the proofs. And the last lecture will be about very high dimensions and also what are the applications of sphere packing. And so the proofs. So this was something we already discussed yesterday and since there are many questions actually related to the proofs, I have a question. So I have decided to go more into this topic, the topic of proofs. So today's lecture might be a little bit technical, but I hope it still will be entertaining. So the proofs. So here the concept of mathematical proof is actually very ancient and so this is probably one of the first written mathematical proofs. This is a papyrus which contains a part of Euclid's elements written in Greek and so here we also can see that this papyrus contains a proof of a geometric statement. We also see a geometric sketch here. And of course the whole idea of mathematical proof is I want to say a backbone of mathematics and very important and also very controversial subject. And through history our definition of what is what constitutes mathematical rigor, what is the necessary degree of rigor has also changed. And so for example we can go as far as formal mathematical proofs when we want to just operate with mathematical level of mathematical logic and prove everything so that the proof becomes not like a text, not a conversation between people but more or less like a computer program. And so here for example you can see a proof of the fact that 1 plus 1 is 2. So of course the whole notation definition of this symbol that was given on the previous pages of this book and so now we come to this formal proof that 1 plus 1 is 2. And of course mathematical logic I think it's a fabulous subject but also one of the most terrifying subjects I studied in mathematics because mathematical logic first it comes with all this labour which we need to put even to be able to prove very simple things but then also as we go deeper and deeper we would discover all kind of paradoxes and counter-intuitive statements. And so for me somehow the course in which we learned from Gödel's theorem it was one of the most maybe biggest traumas of my life. So that's why today somehow modern mathematicians often do we find somehow middle ground between the rigor but also there somehow I think mathematics is not so much about this code like statements it's more about ideas and so how to somehow convey mathematical ideas that are still rigors but appealing to humans without turning into physicists or poets and so to demonstrate what is mathematical proof for me or how mathematical proofs can work so what I've decided to do today is to give a proof of the pecking problem in dimension 2 and I think this is a good choice because the pecking problem in dimension 2 is easier than the pecking problem in dimension 3 and of course it's still much more interesting than pecking problem in dimension 1 and also the proof itself it does not require knowledge outside of high school geometry so of course also the proof that I give it would not be at this level of details I would rather explain you the ideas behind and so here what we can see on this slide this is a lake in California which was covered with shadow balls so here the idea is that somehow the lake was evaporating and it was bad for various reasons like there would be less water if too much water gets too hot then also some disease might appear there so one of the possible solutions to solve the problem was to cover lake with balls and of course when balls are on the surface of a lake so we can effectively speak of two-dimensional pecking problem and here you can see that somehow the balls themselves arrange into this honeycomb configuration so if you look closely at this picture we can see the small triangles and hexagons so this would be this hexagonal pecking or regular triangular pecking or another name it has in mathematics, honeycomb lettuce pecking and so as I hope you remember from our lecture yesterday so the density of this pecking is almost 90% even slightly more than 90% so more than 90% of the area of the lake in this case is covered by balls and therefore water would not evaporate and the water would be preserved even though later I learned that this idea of shadow balls at the moment I think it was considered as not very ecological and sustainable and eventually it did not work for preventing ecosystem but I think it does work very nicely in creating these spectacular images and so this is the theorem which an original mathematician Axel Tore he announced the proof at the end of the 19th century in 1890 and so the theorem tells us that the densest pecking of discs on the plane of equal discs on the plane is the honeycomb lettuce pecking so the pecking we have seen on the previous slide and so what is the idea? as I told you I will not somehow try to do it through rigorous mathematical proof but I will try to explain the idea behind and the idea is rather simple so suppose that we have any pecking which means it must be optimal or might be also not optimal the only rule is that the discs are not allowed to intersect only allowed to touch each other and so given such a pecking we will divide our plane into regions and we will make our division so that in each of the regions the density of a pecking in that region should be at most this number which is the density of the honeycomb lettuce pecking and so what I mean by the density of region I mean that I look at my region and I compute the area of the region itself and I compute the area of this region which is covered by discs and take the ratio this will give me a density of a pecking of a region and so now how do we do this? so what Axel Tuer have done he have suggested the following procedure and when I prepared this talk I did not read the original paper of Tuer instead I wrote, I was reading a review article by Thomas Hales so I think there he explains this geometric argument very nicely so let's start with an arbitrary pecking so here of course I don't draw all of the balls which lie on the plane only a few of them suppose that we start like this so here are these black dots they are the centers and these are the discs of radius one and so now what we first step what we are doing we will cover our original blue discs with larger discs with discs of radius which case this number would work slightly bigger than one which is 2 divided by square root of 3 and so here an important geometric fact is that after we do so because we started with a pecking we cannot have a point which is an interior of three large discs so this green large discs they can intersect however it will be genetically forbidden for three discs to have one common point so what can happen the worst case is that these three large discs they are limiting circles, they are boundary circles they can have one common point on a boundary and this will happen exactly when all three balls they form this somehow we have three discs that touch each other the same way as they do in the honeycomb pecking and so here somehow I do I will not prove this fact for you but I will leave it to you as an exercise and so the next thing that we will do so now if we have two this bigger discs large discs that intersect each other then we will look at their centers and we will connect those centers with a segment so in this picture we see that these two green discs intersect so we connect them with a segment this, this and this and for example for this green disc it does not intersect with any of them so that's why it stays unconnected okay and now what we do next we do the following thing so now we look we choose two green discs that intersect each other and what we are going to do we are going to look at the the the this segment that connects to centers and at this intersection point of limiting circles of these two discs and so with these two points we add the third one and we draw a here I draw it in red, a red triangle and we do it for each pair of green discs that intersect and so now what we have achieved like this we have actually divided our plane into three different okay so here something did not render well okay so we have divided our plane into three different regions so I've told you that our strategy is to divide plane into different regions and to show separately that density in each of these regions is at most the density of the honeycomb packing and so here we have three different regions so one of them are all the first region are the points that lie outside of all this big green discs and of course in this white region we don't have any blue discs left and so there the density would be just zero and zero is smaller than the density of honeycomb packing so here we are good the second region is what happens inside of this of the green discs and outside of all red triangles and so here we can see that actually we can compute what is the density here and the density here it would be actually the ratio of the area of a blue disc to the area of a green disc and this would be three quarters because the radius of the green discs was square root of three divided by two so this would be the ratio and it is smaller than our density that we would like to prove that it's optimal and so now the most somehow interesting region would be what happens inside of these red triangles and so inside of the red triangles and again somehow it would be exercise that I leave for you that inside of red triangles actually the density could be at most this number and so the density is this optimal density is achieved actually when two blue discs they touch each other and also you can think about what happens in a critical situation when we have the honeycomb packing and so if we do have the honeycomb packing then we would see that we will not have any white space left and we will not have so everything will be covered by this by the triangles so in the honeycomb case what happens is that all this somehow all the plane will be covered by this green large discs so some points will be covered only once some points will be covered twice and will be those individual isolated points where three green discs intersect and no point will be overlapped by all three and so this is the intuition between behind the two as proof and then somehow it was different slightly different versions of this proof were written and there was a quite different proof suggested by Laszlo Feestoth where instead of considering these coverings by triangles he rather studied what is called the Voronoi cells around blue balls and so now in dimension three somehow so this is the dimension two it was a playing ground and then the dimension three it was famously solved by Thomas Hales and so what happened for the proof of Thomas Hales so he somehow he's in a very very big picture he did he tried to follow similar strategy also to divide three-dimensional space into regions where he could bound the density however in three-dimensional space it turned out to be much much more difficult so and at the end he's this plan of dividing space into somehow this nicer chambers where we can control everything the path turned out to be much much longer and actually if you are interested in this so you think it's worth to read this short paper it's an IMS notices paper so it's a paper which is written not for colleague mathematicians but rather for general audience and here actually the strategy is described for the three-dimensional case but now things are already so complicated in dimension three so what can be somehow still use this strategy to higher dimensions for example to dimension eight or to dimension 24 and here I should say that in principle nothing forbids that from being possible and maybe this is possible however nobody was brave enough to trying to apply this technique to such a big dimensions I think mostly people are discovered discouraged by this experience in dimension three where everything turned out to be so complicated and also yes so in here instead of course instead of working with triangles or polygons as nice as they are in a plane we would have to work with this polytops and higher dimension and those we don't really understand how they behave so a new idea is needed and so what this new idea is actually this new idea it existed in geometric optimization for quite some time and the idea is that what we should do we can use not only direct geometric approach we can also use functions and we can use the duality between points in a space and functions which are defined on that space and what we can use we can use the whole power of harmonic analysis and so here is this method of harmonic analysis which was somehow the these ideas they existed in mathematics for quite a long time and so mathematicians who were able to apply these ideas to the sphere-packing in Euclidean space are the Henry Kohn and Norm Elkis and so here unfortunately unlike the proof of Axel II probably I will not be able to explain fully analysis in one popular lecture so here I allow myself even more hand-waving than before so the idea is actually the following so let's define a special function on Euclidean space instead of trying to divide a space into this like nice regions or nice chambers what we will try to do we will try to define a function on a space and we want this function to satisfy certain constraints for example our function that Kohn and Elkis have thought about it should be non-positive outside of a big ball and so how the way I can think about it is the following suppose that we have our configuration of points which are centres of our packing and so this each of the points it sends a signal to all other points and the signal depends only on the distance between from our original point to another one and somehow if another point is too far away if another point is far away then the signal would be some real number and if another point is too far away then the signal would be okay so maybe so this is our point it sits in Euclidean space and it sends this signal to all its neighbours and somehow if the point is far enough then it sends some negative number it's just like okay you are far enough you are okay and if a point is close then it could send it's also something positive saying that no you are too close to me and so this is what this condition in the Kohn-Elkis theorem is about and so but that's not enough so here this is where the Fourier analysis comes into play and so this we had a signal like this function f and now we turn it into something different another function called f hat which is the Fourier transform of f and what this function is about it actually tells us which like this perfect signals perfect harmonic signals contribute to this function and so this condition of positivity what it means it also has an interpretation in this picture so what it would mean it would mean that if we have take one point and count all the signals each point have sent to its neighbours and we compute the sum and then we take the average over all points and then this average so the condition that the function f hat is non-negative it means that this average signal that each of the points have sent has to be non-negative and so here we see somehow there is a contradiction between these two conditions it means that if all somehow points are far away from each other they send each other this signal there is a negative number and our world that means that you are far away from me at the same time the average number has to be non-negative and so this is one of the very fundamental facts of mathematics if we have a lot of negative numbers and compute the average it has to remain negative and so it means that if we could manufacture a signal like that with these two properties it will create somehow this contradiction for points being too far away and so if you are able to manufacture a signal like this this would mean that that the packings that are too good are simply impossible and so this is the proof of the Kohn-Elkis theorem as today's lecture is about proofs so let me tell you what somehow proof by mathematicians that they usually show to my colleagues in the conferences look like so it often it goes like this so here acts as our set of points and we assume that we were able to so we first assume that we have this special signal that we were able to construct somehow and that we also have a set of points which is a very good packing somehow which is too good to exist and so this is a technical condition which tells us that it will be easy for us to define what the average means and so then we start with some computations some horrible symbols and usually mathematicians are very happy when they see these formulas because then they would say ok yes now it's all clear how that works and so what the meaning of this formula is actually what I have just told you it just means that if our function the signal effort satisfies those special conditions it would mean that the average over all of these communications between different points has to be non-negative and so here we come to the somehow come to the contradiction so here we can bound this of the density of our configuration and so here is the picture again how that actually how I think about this point sending signals to each other and so the method suggested by Kon and Elkis it worked quite nicely so what they have done they have tried this method in various dimensions so they have tried it in dimensions from 1 to 36 so originally in 2003 but then somehow since this method exists then other computations were also performed later it became it was just a matter of ability to program and computer power so they were searching for the special signals also by a computer by a certain optimization process and what they found that this method is not bad in many dimensions it gave results which were better than previously known results and somehow interestingly in dimension 2 it worked almost perfectly in dimension 3 it did not give the perfect result it gave only some estimate for the sphere packing it was not as good as the theorem of Thomas Hale and then in many dimensions again there was a new bound was obtained it was better than other bounds and mathematicians who work in this kind of geometric optimization they are competing if we cannot solve the problem completely then we can compete on who can prove better bound and so on amazing discovery that they've made is that in dimension 8 and in dimension 24 their method gave almost perfect result so what they were able to prove that if in dimension 8 if we would be able to improve the E8 lattice packing then the improvement could be only by this much so if something says they could not completely exclude the possibility that nothing better can be done but then if it's better then it's only teeny tiny better and similar result was obtained in dimension 24 here also some other mathematics was not as good as in dimension 8 but still pretty good and so also these are results from the paper of 2003 and at the later years they were actually what they have done they have computed improved their computer program which gives these good signals and they could find the function which even they could find function which proves that nothing can be better than E8 lattice as one and then 60 zeros and only then some significant digits and so what was my contribution to the field it was that I was able to find this special function which is not only a numerical approximation but also satisfies the condition of Cornelki's theorem explicitly sorry, exactly so in the E8 lattice if we normalize it so that per unit of volume we will have only one point of a lattice then the distance between different points of a lattice would be square root of 2 and so we can pack balls of radius 1 over square root of 2 in dimension 8 and so this function it somehow it was able to replace this one and many many zeros so to say by a mathematical one and then we worked together with Henry Cohn, Abinav Kumar, Stephen Miller and Daniel Radchenko and we were able to prove a similar result for radial Schwartz functions sorry, for dimension 24 so here we obtained this function for the leach lattice and in the leach lattice the distance between two different points is always bigger or equal than 2 and so that's why this is the optimal condition in the Cornelki's theorem and so and this is how the function should look like so this is the profile this is the signal that points are supposed to send to each other depending on their distance so that we can prove that the density of the packing configuration cannot be bigger than the density of the E8 lattice and so now I can answer your questions