 On behalf of the president of the Albert Einstein Society, Professor Otto unfortunately cannot be with us today. I would like to welcome all of you to the Einstein lectures 2023, especially I would like to welcome our young audience. So at various levels from students, but even maybe school kids, fans of math, Olympians, and whatsoever, we welcome those who are attending today, but also those who will watch the podcast later, because they cannot come today. This is the 14th edition of the Einstein lectures, which are organized jointly by the Albert Einstein Society and the University of Bern, and which are in turn dedicated to physics, mathematics, and philosophy. My name is Christiane Treta. I'm professor of mathematics here at the University of Bern, and it is my great pleasure today to introduce the fifth Einstein lecturer in mathematics, Professor Marina Wiersowska from the Ecole Polytechnique Fédérale de Lausanne. With just 39 years, since two days, if I found out correctly, Marina is the youngest Einstein lecturer we've ever had. And also in other respects, her life and career stand out. She was born in Kiev, and there her high talent in mathematics was discovered and supported early in school. She competed in several domestic and international math olympiats, wrote her first research paper when she was just 21. Then she obtained a master's degree in Germany from Kaiserslautern in 2007. And then did a PhD back in Kiev at the National Academy of Sciences of Ukraine in 2010. And she got another one from the University of Bern in 2013. Only three years later, well-prepared, of course, from the PhD, in March 2016, she had completed, after some longer research work before I assume, the two research papers that got her to the Olympus of Math six years later. Within a week in March, she posted her solutions of the sphere packing problem in dimensions 8 and 24 on Archive, which is the free mathematical preprint server available to everyone. And with these solutions, she stunned the entire mathematical community because they were so ingenious and elegant and, of course, because also so many other much more experienced researchers had tried to solve these problems for her. Then a whole other launch of prizes followed, among them the Salem Prize in 2016, the Clay Research Award in 2017, a New Horizon in Math Prize in 2018, the Fermat Prize in 2019, a Prize of the European Math Society EMS in 2020, and then in 2020 to the Fields Medal, the highest award in math for scientists under the age of 14. This Fields, the Fields Medal, there are four each four years, they are awarded by the International Mathematical Union, usually at the International Congress of Mathematicians. As a treasurer of the Swiss Math Society, I had the great pleasure to be a member of the Swiss delegation to the IMU, and Marina was another delegate by chance. But of course, nobody knew what would then happen. And so because I was there, this took place in Helsinki, but not within an ICM because the ICM was scheduled for July 2022 for St. Petersburg. But then Finland was so generous to jump in and hold the ceremony and at least the IMU assembly. So this was how I came to be part of this award ceremony. And it is a memorable moment in my career to be there when Marina was awarded the prize. And of course, I took the chance after the award ceremony to invite her. I'm not sure if at the time she was aware it's three lectures and not just one. But anyway, we are very, very happy that you are here, that you accepted the invitation, and we are very much looking forward to your talk on the sphere packing problem. The stage is yours. Christina, thank you very much for inviting me here. Also, thank you for the introduction. And so today I will speak about the sphere packing problem. And so let me start with explaining in very general words what this means. So imagine that you have a very, very big box, maybe, and an infinite supply of balls. And then the question would be, how many balls can you fit into the box? And then if the box is much, much bigger than the size of each of these equal balls, then this becomes a question really of not of the exact number, but rather the density. So then we would know that the number of balls inside of the box, it will be roughly speaking the volume of this box divided or multiplied by some number, the density constant. And so what we are interested in in determining, because there are many different shapes and sizes of boxes that we can consider, but we would like really to focus only on this one number, the density. And so again, to clarify what the rules of our games are, that all balls, they're hard balls, they all have the same size, and they're allowed to touch each other and not allowed to intersect. And so here in this picture, you see this intuitive three-dimensional picture that we have, and now an interesting component that I would like to introduce is that also the notion of dimension. So what we can do, we can consider this question not only in our favorite familiar three-dimensional space, but in different dimensions. And so before we get to the dimension D, let me start it easy with dimensions 1, 2, and 3. As those are the dimensions we all know and understand. And so what is the dimension in a very simple word? This would be the number of degrees of freedom to move in the space. And so if we imagine that we live on a line, then we have only one degree of freedom. We can move only forwards or backwards. And so on a line, what is a ball for? What is a one-dimensional ball? One-dimensional ball is an interval. And the line can be covered almost perfectly with intervals. So we can think of these intervals that do not overlap but only touch each other, and they will just cover all our line. And so the density of this one-dimensional packing would be one, which is somehow the maximal possible density we could even hope for. And in dimension 2, we have two degrees of freedom. We can move, so to say here, northeast and southwest. And what is a two-dimensional ball? A two-dimensional ball, it's a disk. So here imagine that our two-dimensional space, it would be a surface of this table. And as a disk here you can imagine, for example, one-frank coins. And that we are piling this one-frank coin so that we would like to put as many coins as we can on the table and to cover as much of the surface of this table as we can. And so after a little bit of experimenting, you can come to at least a hypothesis that probably this is the best way to pack disks on the surface, so into this regular triangular grid. Sometimes it's also called honeycomb lattice. And so if we put our balls on the surface, like our disks on the surface of a table like this, then they will cover not, this time not 100%, but a bit more than 90% of the surface. And we will have this exact, if you like mathematics and like mathematical constants, we will have this exact formula for this approximate number. And so in dimension three, what can we do in dimension three? Probably you've all been to a grocery store and you have seen there are oranges piled in pyramids. And now if you think of these pyramids, if they will be prolonged infinitely to the whole store, then what we will get would be this perfect sphere packing. And its density is about 74%, or this exact constant, pi divided by square root of 18. And so historically, the sphere packing in dimension three, it was known as Kepler's conjecture. So it was a long time ago when people became interested in this geometric question. And the question of packing sphere seems very natural. And I guess there are some ancient Indian manuscripts with text and pictures that show us that even in ancient India, people thought about this question. I'm sure that this question came over again and again through the course of history, even though it's not my expertise demand, but in our modern European history, this question is known as Kepler's conjecture. And so there is an interesting long story behind it. So from what we know that at the 16th century, British aristocrat had a great idea of invading Spain, or at least somehow fighting seriously with Spain. And so for this, if he wants to fight with another country, he would need a lot of cannons and he wanted to pack them into a ship as efficiently as possible. So he asked his scientist, Thomas Ariot, to tell him what is the most efficient way to pack cannonballs inside of a ship. And Thomas Ariot, he did not take this task slightly. He actually thought about it. And so we know about all this because the letters were preserved and he actually developed this theory of a closed sphere packing. He gave descriptions of what the best configurations would be and he found not one, but actually two different ones. So he described all this in his letter. But also this made him think about so packing cannonballs inside of the ships. This made him think about condensed matter and this ancient idea that maybe matter around us consists of atoms. And maybe atoms are packed inside of matter in something like cannons inside of a cannonball. So he was thinking about all this and he wrote a letter to his colleague, Johann Kepler. And Kepler also thought that this is such a beautiful idea. And the Christmas was coming and Kepler was supposed to give a present to his friends. And so to one of his friends, he gave the following present. So it was an essay of six corners snowflake. And so here's the text he has written to his friend. I think it's a nice coincidence that we are also meeting around Christmas time. So maybe if you are short in money and cannot buy a great present for your friends, maybe you can just write an essay that would be quoted 400 years later. So here you see somehow this is the dedication and also how the letter starts. So Kepler says, I'm not a historian. I'm not a specialist in the Renaissance time or habits people had that time. But it seems like an apology. So he could not somehow give him a present which would be of monetary value. However, he wrote an essay. And I think this is what he refers as nothing. So he tells like, I just give you nothing. And nothing is 14 pages of text. So I will just give you some quotes. So it was the main topic. So like now joking apart, let's get back to business. And the business is the following. So Kepler was walking on this outside and they have seen snow and snowflakes. And he made an observation that all snowflakes, they all are look like starlets and they have six corners. So he could not find a snowflake that has five or seven, but almost all of them had six corners. And he was thinking, why is it like this? And again, as a person from Renaissance age, he would think that probably this has nothing to do with the properties of water or properties of vapor because vapor does not have anything. There is nothing hexagonal about vapor. It's shapeless. Even though we know this, he knew that time that snow comes from vapor. And so he realized that probably there should be an agent, some agent which really likes the beauty of hexagon that forms snowflakes like this. And so I leave it for you to decide how close does this bring us to atomistic theory or to the somehow formational principles of matter. And I think many people who study the work of Kepler, they interpret this in the following way, but maybe I will just leave it to you. And so let's better come to mathematics. So here the question is what is so special about hexagon? And so one thing that Kepler thinks is special about hexagon are these geometric optimization problems, the pecking problems. So here he starts thinking about pecking problems. And so this is how one of the additions of his essay looked like. And so what would be important for us are this, oh, sorry, is this picture, this diagram. So here are the things about how he could peck balls and he thinks of his diagram A, it shows us this rectangular grid or square grid and diagram B shows us this triangular grid. And so now out of these layers, he will produce the dense pecking of balls. And so this is what he writes. Again, it's interesting that in the 17th century of course people did not have all the mathematical machinery we have now. There was no algebraic notation, there was no established language to speak about geometry, so it's all really more rather poetry than mathematics. And so at first what he suggests to do is so let's look at this diagram A and let's build our sphere pecking from these flat layers. And the first thing to do is just to put the one square layer on the top of another so that each ball touches exactly one ball which is beneath and widen ball which is below. And so this is what Kepler describes. Here, so if we proceed like this, of course he sees if he understands that we will not get the densest pecking because we can apply somehow more pressure to all the structure and then the all layers could slide and become denser together. And so what he does, he suggests here another arrangement so that to put each ball of the higher configuration not at the top of another ball but rather in a deep hole. And so here he writes that then somehow then the pecking will be the tightest possible so that if no other arrangement could be pellets could be stuffed into the same container. So he just claims that this is then, if we do it like this, this would be the densest pecking. And this is actually the end. So Kepler is just very confident that this is the densest pecking. And again by the standards of science at his time, this was an absolutely reasonable thing to say. So at that time people did not think so much about mathematical proofs and the rigor of mathematical proofs. And this seems to be such a self-obvious statement. And so what, so here are some pictures. So here's the pecking which Kepler suggested to build is to take here are these balls in layers and each layer is arranged as this square grid. And now for, as you see for each ball of the next layer we put it in this biggest hole between smallest layers. Between sort of previous smallest hole, we put it into these holes that are created in the layer below. And so here, what we see here is actually another construction. And this construction looks in the following way. So now you see that if you want to arrange our disks on a plane, then the strangler grid, it will be better than the square one. It will be denser. And so maybe if we start building our pecking not from rectangular layers, but from triangular layers, maybe we can achieve something better. And so here one problem comes that in the picture here, if we put these balls of a next layer to these big holes of a previous layer, then somehow miraculously there will be just enough space for if we just put this ball as low as we can, there will be just enough space for the next ball in the hole here. And for the strangler grid, it is not the case. Here we cannot put one more ball into each hole so that then there will be not enough place for them to touch the previous layer. So actually what we have to do, we have to choose every second hole. And so this way we see that we have these two constructions. So this construction number one, which I already described for you. And so here each layer is less dense but the layers are very, very close together. And the construction here when each layer is very dense but the layers stay far away from each other. And what I do sometimes at this part of the lecture, I ask audience to vote, which you think is denser. Is this denser or is this denser? So who thinks that this one is more dense? Okay. And who would vote for this one? And who thinks that they actually equally dense? Okay, so people who think that they are equally dense, they are right. And there is actually more to this story because these two packings, they're not only equally dense, they are geometrically identical. And it seems somehow difficult to see it from these pictures even if we try imagining the packings for ourselves. So let's convince you that they are did the same and these both constructions, they will lead to the same sphere packing. So called this configuration is called the face, the central cubic, sorry. Cubics centered face letters, sorry. Faced central cubic letters, FCC letters, yes. Faces central cubic, yes. So what the name comes from? So we have this, we take a usual cubic letters and how this one works. So we think of our spaces filled with cubes in a natural way. We just put cubes one onto another and then put them in first like in a perfect square grid and then extend also in all the dimensions. And so in the vertices of our cubes, we will locate our balls and we will make the radio of our balls, okay, so the radio of our balls, we'll think about them a bit later, we'll fix them later, but this configuration of the of the vertices of cubes, this is what will give us the cubic letters. And so we would like to put a red ball and of some radios, which we will choose later and the center in here. And, but also we would like to make our configuration a bit more interesting. So we'll look at all the faces of all our cubes and we will, for each face, we will choose its center and we will put a blue ball with a center which coincides with the center of the face. And so this way we will, now we will also think of this configuration as being continued in all somehow six directions. So here at up, down to the east to the south, to the north to the south to the east and to the west. And so now we would have, how can we see that this lattice packing is actually coincides with the two packings we have constructed before. So I suggest doing this by coloring the balls in a different way. And then we will see that it will coincide with two configurations that I proposed before. So one way, so this, let's take our cubes and draw the main diagonal in our cube. So take two points which are at the biggest possible distance. So we choose one of these main diagonals in the cube and we will draw the planes which are orthogonal to this main diagonal and we will color our balls, again, in green, in red and blue as I've done it in my previous picture. And so then you can see that this picture here actually is the same as this picture here only rotated in the space. And now it's the same, but configuration is the same, it's only a different coloring. And now what we can do, we can do yet another coloring. So now instead of looking at this main diagonal, we again look at our cube and now we will look at the planes which are parallel to one of the faces of our cube. And so this way you will see that the squares will emerge here. Here it is this rotated square letters. And so we'll see that this picture, it actually coincides with this construction here. And so these two configurations, they're actually the same only rotated. So maybe you can try to imagine how you rotate it. And then you can also see that it's not that easy to do. So we, first of all, we should give credit to Kepler and to Thomas Ariot for describing these configurations to finding them so long time ago. And maybe it also should not be that angry at mathematicians that it took them 400 years to show that this configuration is optimal. And so, okay, so here you can see it's another beautiful sentence from the essay by Kepler. So here he writes this somehow that triangular pattern is impossible without the, the triangle is impossible without the square and vice versa. So of course, at that time, probably what the Kepler would imply by this sentence is something like what I showed you before. So we start with triangles and then we inevitably build squares in other direction. And if we start with squares, we will, by doing this kind of packing, we will still, we will build the triangles, but in yet another direction. So, and so now a few more demonstrations to show you that the geometric equation, optimization geometric equations can be very difficult and counterintuitive and not that easy for mathematicians to solve. And so another problem, which is actually an easier question, easier than this fair packing is so called the question about the kissing number. So here again, the name of the subject comes from another epoch. So you see that people used to romanticize geometry a lot. And so here the question is the following. So let's consider a ball. For example, this is a two dimensional example. So red ball, remember that two dimensional balls are discs. So we take a red ball and once around our red ball of radius one with blue balls of radius one. And so the rule is that each blue ball is supposed to touch the red one. So this is the condition that they always have to touch. Each of the blue balls has to touch the red one. And the blue balls are not allowed to intersect and they are only allowed to touch each other. And so then the kissing number would be the maximum number of blue balls that can surround a red ball like this. And so in dimension six, it's not a very interesting problem. So here in dimension two, it's not a very interesting problem. And here we obviously have six blue balls touching blue discs touching the red one. And already in dimension three, the question becomes much more interesting. So this was a famous discussion between Issa Akhenyuton and David Gregory. And so Issa Akhenyuton thought that in dimension three, the kissing number is 12. And the 12, it's easy to construct. So what we do, maybe you know a platonic body that has exactly 12 vertices. So this is the icosidron. And so what we can do, we can take the icosidron and put a red ball into its center. So we choose a red ball so that its center coincides with the center of icosidron. And then we put blue balls so that their centers will coincide with each of the 12 vertices. And so then we also choose the radius of our balls to be so that the red ball inside will just touch each of the blue balls around it. And so this way it's not a kind of an easy computation to see that if all blue balls touch the red one, then they will not touch each other. They will not also intersect. So they will be just comfortably sitting apart. And this is a configuration for 12. However, in this we will see that there is still, it seems like there is still plenty of space left. So maybe if we choose configuration which is not so symmetrical, maybe we can shift the balls, blue balls around and still find a place for the 13s. And so David Gregory, he thought that maybe if somebody tries really hard, they could squeeze the 14s ball in. However, he could not produce this configuration himself, but he thought that maybe if somebody spends enough of time of it, it still would be possible. And so now we know that Isaac Newton was right. After all, we know that Newton did have very good intuition for mathematics. At the same time, the rigorous proof for this geometric problem, it took quite a long time to find a proof like this. And the first in somehow maybe still incomplete proofs were found in 19th century. And then the rigorous proof was written down in the 1950s, so already the last century. And so here's another explanation of what is so hard about the sphere packing problem. And it's also maybe something which is a bit counterintuitive. It's that I've showed you these two different packings and then I demonstrated that they actually coincide and they are geometrically identical. However, in dimension three, we have uncountably many sphere packings which attain this the same maximal density. And how it works, here again, our method of building packings from layers would work. And so you remember at some point when we built our packing from this triangular layers, I have told you that we cannot put balls of a new layer into each hole between the balls of a previous layer. We always have to make a choice. So we always have to make this shift. We could put them either like this or like this. And these choices, we were making them at each level. And by making these different choices, we can produce many, many packings which would be geometrically unequivalent to each other. And so it also explains why the sphere packing problem is mathematically a hard problem. That's because we don't have only unique, great solution. We have many equally, seemingly equally great solutions. And when optimization problem has many solutions, this is a difficult problem. And so now let me tell you a few words about mathematical proof. So as I already told you, if Kepler and Thomas Heriot, they did not need any proofs. They could just somehow, it was obvious to them that this is the best they can do. They tried, they are so smart and probably they would not have missed a better configuration. And of course, as mathematicians, we want somehow more rigor in our argument. And so for many years, what was an example of mathematical rigor was the elements written by Euclid. And so here is one of the oldest texts that we have that contain some elements of a mathematical proof. And so of course, and so as mathematicians, when we want to prove something, we don't want just to write somehow a poem or a text that would look so convincing that everybody would be convinced. We have very strict rules on what constitutes mathematical proof. And maybe it's also an experience you have at some point in high school, when you do study plenometry or geometry, that inside of geometry often have this argument on how to prove something, how to prove a theorem. So as we proceed in mathematics, what we usually have to do, we have to establish some axioms, so to say ground rules, what constitutes a mathematical argument and what does not, and also if we work with certain interesting objects, like in geometry, this would be points, lines, and planes, and the balls. We have to define, we have to first somehow to give the properties which are axioms, and then for things which are not the initial objects, we have to give the proper definitions, for example, for the sphere packing problem. As a mathematician, what I would have to do, I would have to really define what is the density of a sphere packing. And then we would realize that, actually not every configuration of balls has a well-defined density, and I would have to play around with it, but probably I will not do it in this series of lectures. So you would just have to believe me that we work with nice configurations where the density is well-defined. And when we have our axioms and definitions, we are good to prove theorems. And so of course what mathematicians wanted to do for many centuries is to turn this one sentence in Kepler's essay into a theorem. And there was a lot of effort and ideas and failed attempts. And so finally the Kepler's conjecture was resolved in 1998 by American mathematician Thomas Hales. And this also in itself, this was an interesting story. As for long time people thought that this theorem is still out of reach. And then the strategy in 1960s, more or less the strategy was developed on how to tackle the problem. However it still required a lot of computational power. And what Thomas Hales did, he somehow pushed this, really pushed on this strategy. And so he reduced the sphere packing problem in dimension three to a number of problems that can be resolved computationally, either combinatorial problems or optimization problems. And the issue of these problems could be used by computer. And so his proof, it was one of the first examples of a computer assisted proof. And then it took mathematical community quite a long time and a big discussion on whether such proof should be accepted. Is it still a proof? How much computer assistance do we allow? Another interesting thing is that referees were not somehow, scientists who were supposed to read the paper and to say that it is correct. After many years of work, they could write, okay, we think that this paper is 99% correct, but we cannot really follow all the details, especially the things which are done by programs. It's too hard for us, but 99% is correct, which is not so nice for mathematics. And then somehow what Thomas Hales did is he think it's another great achievement in history of mathematics, he would write formal proof of his theorem. And so this is another interesting thing that exists now is that, so as I told you that mathematicians, we do want to be like Euclid, we want to be very rigorous. We want to start with axioms and definitions and then very rigorously prove all our theorems. And now we are at the point where we can actually write our proof as a computer program. And then the compiler could check that each step is indeed correct because for ever is they just may get old before they read the paper to the end. At the same time, and so until now, I think Thomas Hales was one of the first people who would write a major modern mathematical result in this fashion. And now it seems that with all the technical advances, this movement is getting somehow more momentum, more attention, which is also exciting and interesting. And so what Tom Hales did, he actually he wrote a formal proof of the sphere packing problem and the whole project took him like almost 10 years. So it was a really major, major thing because one of the important steps also to prove to write the formal proof of his theorem, he had to somehow formalize all the prerequisites because as you know, mathematicians were usually reliant work of other people. And if we want to write this in a mode to have proof to be checked by a computer, we also have to explain to the computer all the previous work that has been done. And so, okay, so now we are, as I told you, the sphere packing problem in dimension one is trivial. In dimension two, it's not trivial but still somehow simple and it was resolved long time ago. What about in dimension three, it was a really big adventure for mathematicians for some very hundreds of years. But okay, one, two, three, seems like we ran out of dimensions, right? We have no more degrees of freedom as human beings. And so here what comes to rescue is the mathematical obstruction. And so here maybe let me a little bit justify to you why should we look in this higher dimensions and also explain what are these higher dimensions to me as a mathematician. And so here maybe as a metaphor, I would like to use this illustration to a book about Alice through the looking, who went through the looking glass. Actually, the book which was also written by a mathematician, maybe that's why. It feels to me so much. So we have somehow our experience from the everyday world that we are intelligent creatures and of course we use our intellect to think about it to make our lives easier. And then as we think about our everyday experience, we can also come up with obstructions. I think people are also very good to coming up with obstructions and some of them are the obstructions I am interested in are the mathematical ones. And so we can bring our concepts of freedom of real life into an abstract form and I would describe this process as going through a looking glass. So we started with something which was real then we think of its possible reflection in the world of ideas and now another step we do. So now we try as mathematicians we start living on this side and this world of mathematical obstructions it is in some sense reflection of course of all our human experience. At the same time as in the book about Alice it is slightly different. The laws that govern there are different and so if as a human I have only three degrees of freedom to move myself on this side I might have more. I just have to find a correct mathematical obstruction for that. And so with dimensions, with mathematical dimensions it's actually very easy. So as I already told you the dimension it's a number of degree of freedom and what it means it means that if I want to describe a point in space I need three real numbers to do it three coordinates. So you can think of it as a GPS coordinates for example. So here this is the point with coordinates so this is the origin and this is the point with coordinates minus one, one, one. And now if I think of a point as this somehow this row of its the list of its coordinates now nothing stops me from adding simply adding more numbers to this list. And this is how dimension of Euclidean space is defined in mathematics. And so here another thing which somehow often I think confuses people is that for example if we are physicists then our whatever we are working with is supposed to have physical meaning. So very often in physics it's useful to introduce extra dimensions like Einstein. He introduced this fourth dimension which would be time. And as mathematician I don't somehow I have more freedom I have luxury to say I don't care about what force dimension is. The force dimension it could be a time or I could add two more dimensions or I could say that maybe my dimensions there are just some numbers on excel sheet with salaries of people working in the same organization and it would be totally normal for mathematician. I don't need to think about an interpretation I just think about the logical structure of things. And it turns out that the sphere packing problem can be easily generalized to higher dimensions. And so here what we should do we have to define the D dimensional Euclidean space. And so our points there are these lists of coordinates. So if you have dimension D then we have D different coordinates. Each coordinates it's a real number. And so now I want to think about sphere packing problem we need a bit more information. So what if you want to define what a ball is so in dimension in each dimension the ball it's a set of points which are at a certain distance at the most certain distance from the center of the ball. And so we already know what the point is what we now we need to know what's a distance. And so here we introduce a distance between point in this way. So this would be the usual Euclidean distance and probably if you remember the Pythagoras theorem from school then you would recognize that this is exactly the Pythagoras. If we dimension to this would be exactly the Pythagoras theorem which also works in dimension three. But now we can actually introduce as many coordinates as we want as we have decided here. And so this would be the distance between these two points. And so maybe this is something of course Kepler would not have because he didn't at his time so the algebraic notation was not developed yet and it was not used systematically by scientists at his time but we do have this luxury now. And so now after we have defined what is a distance we can define what is a ball. So it would be a set of all points. So if you want to have a ball with centered point X and radius R it would be just the set of all points which are a distance at most R from point X. And also somehow another thing which we need for our to define the sphere packing problem is the notion of volume. It's a quite useful notion but I will not somehow hear things get maybe way way too complicated. I will not write it down but I will just tell you that there is a meaningful mathematical way how to define volume in higher dimensions. And so after we have defined this higher dimensions what we can do, we can also think of configurations and it turns out that mathematicians have this ability and in dimensions for example up to 10 they have already listed the number of configurations that they believe are optimal. So it's somehow for the rows which are in green these are the rows where we actually do know the answer in the sense that we have a mathematical proof that this configuration is optimal and those which are in black these dimensions are still open so we have candidates but we don't know whether they are optimal or not so maybe if there are young mathematicians in this room maybe they can choose one of their favorite dimensions and try to start working on that. And we also see that somehow what also this table shows that the sphere packing in higher dimensions has this very strange flavor is that each dimension it's always, it feels like a completely new problem. So in some dimensions like in dimension one in dimension two, like four, eight we have only one configuration, unique configuration which does the job which is the best sphere packing and there are many dimensions which are more like our favorite dimension three where we have this many many geometrically distinct configurations. So we can do something similar to for example choosing stacking layers in different ways and we will actually get not one answer but many typically uncountably many answers. And of course when dimension becomes more than 10 then we lose somehow our confidence sometimes we still have great configurations but we don't have this confidence to say that they are really the best. And so this is now the list of the dimensions where we exactly know what the optimal density is and so okay so as I told you the dimension one is trivial and dimension two it was not a very difficult question even though it was I think it was difficult for mathematicians is to decide which proofs are rigorous and which are not because some of the proofs they were quite old and maybe there was still some hand waving there. And probably I think in one of my lectures actually I plan to demonstrate a simple because these proofs they have been somehow discussed and simplified a lot and maybe now we are at the moment when it's actually possible to explain to somebody who knows only school geometry to explain the proof. So what I found in the literature for example there is a two page proof of this result. So my goal for this series of lectures actually to demonstrate one of these short proofs. Then there is dimension three which I already told you that dimension three is very complicated. It's a dimension where we have this uncountably many optimal solutions and there is a computer assisted proof by Thomas Hales. And then there are two more dimensions where we know the answer. So this is my own work in dimension eight. We have this unique configuration that was believed to be the best and actually was able to prove that it is the best and proof in dimension eight turns out it was much shorter. So for example the proof in dimension three it is at least a hundred pages text and also many, many lines of code while this proof it took only 20 pages and a small computer program to verify certain positivity condition. And then working together with Henry Cohen, Abinav Kumar, Stephen Miller and Daniel Rachenko we were able to solve this fair packing problem in dimension 24 because in dimension 24 we also have an exceptional object, so called rich lattice and it is the unique optimal configuration in dimension 24. And so here are my co-authors. So I decided to use this somehow. Opportunity to show them to you since we already seen Kepler and Heriot we can also see my collaborators. And so let me, I still have eight minutes left. So let me tell you a little bit about this higher dimensions and also a little, advertise the new lecture. So some of these questions I will stop more in the future lectures. So this is, so as I told you in dimension eight we have this very special object, the so called E8 lattice. And unfortunately it's very difficult to visualize things in dimension eight. So here I'm using the following trick. So for example you can see my hand it's a three dimensional object and if I rise it here in front of the projector it will create a shadow and the shadow is two dimensional. And also if I rotate my hand the shape of the shadow will be changing. And so what is here it's really, it's also, it's a three dimensional shadow of the shortest vectors of the E8 lattice. So what it means, so remember I told you about the kissing problem. And so if we have to hear this is a kissing configuration which was constructed from the E8 lattice so we choose one ball in the E8 ball backing and we also look at all the balls which touch this one. So we do not try to visualize all the E8 lattice but only one ball and all other balls that touch it. And it turns out that in dimension, so for example in dimension three we could have one disk could be touched by six others. In dimension three one ball can be touched by 12 others and in dimension eight one ball can be touched by 240 other balls. So this is somehow this strange behavior of high dimension. So this is what high dimensions are about is in high dimensions we just have much more space. And so now what is done here it's a projection from the eight dimensional space into some three dimensional subspace. So it's like a three dimensional shadow. And of course here we also have this choice of subspace where we are projecting but this one was chosen so that the result is somehow symmetrical and this model is created from a constructor which is called Zoom. Maybe if you have ever been to a math museum maybe you've seen it. So it's this kind of medium to create various three dimensional models. Okay, so here is another object, the one which lives in dimension 24 and here I could not find this three dimensional model of this. So this is a three dimensional, two dimensional shadow. And by the way if you go to, so as I told you like as with my hand if I rotate my hand the shape of the shadow will change. So for example if you go to YouTube there are many, many beautifully done videos of different projections for example of shortest vector of E8 lattice or shortest vectors of Lich lattice that are projected on for example on a plane and then the plane is changed. So it becomes a cartoon and some people are very skilled in drawing it beautifully and rendering it so it's a bit like a screen saver. So of course like on one hand this videos and pictures like this they can look very beautiful. At the same time I can tell you as a mathematician unfortunately these realizations they're not extremely useful for understanding the geometry. So you'll think the best way to understand these structures is the method of mathematical obstruction and visualization it's mostly for aesthetic purposes and for entertainment. And in dimension, in the latest actually one ball is touching almost 200,000 other balls so a bit less than that. So somehow you see that somehow the dimensions they really things in high dimension behave very strangely. And so maybe the last thing I would like to tell you also one thing I would like to advertise for the future is the history of for example discovery of Lich lattice. It's a very beautiful story and I think it is beautiful because this mathematical object which has a lot of great properties and is loved by a mathematician was actually first discovered by an engineer. So this is Marcel, oh not by it was discovered, sorry it was discovered actually it's Lich lattice it was discovered by John Lich who's a Canadian mathematician but the discovery of John Lich was only possible because of a work of Marcel Gallet and Marcel Gallet he was not really a mathematician he was an engineer and his motivation for studying such objects was very practical. He was studying the error correcting codes and so he created a code which was called the Gallet code and then his code was used by John Lich to construct the Lich lattice which we are now using for sphere packing and then somehow same object led to lot of discoveries in mathematics which I hope I will speak a little bit in one of my next lectures. And so another topic I also would like to cover is this connection between error correcting and sphere packings. So this would be the main heroes of my talk of my story and yes so this is somehow the error correcting means and how it is connected to the sphere packing so this all will come in the future lectures and so this is how the slides to tell you that error correcting is very important and many of the technical advances would not be possible without it. And then again maybe one thing I should say that I am a pure mathematician so my law for sphere packing does not come from this great real life applications it actually comes from the intellectual beauty of it and also my belief that whenever something is fundamental in science it will inevitably become useful in everyday life as well. So now I think I am ready to answer the questions. Thank you very much.