 finish at 5.30 if you scoot over there. And the movie is only about an hour long, so then there'll be plenty, it'll be early to finish before dinner. Tomorrow is usually a half day off, and on Thursday evening we have the closing dinner. It's going to be in the main dining tent, and it'll be catered, so there'll be food more than the welcoming reception that we had, and I think it's BYOB. And I think in terms of that, and the other cross-program activities this week I wanted to remind you of, so on Thursday and Friday at our normal time, 3.15, there'll be presentations by the undergraduate group, the Experimental Math Lab, so there'll be presentations. Please come and be encouraging for them. So anyway, let me get started for today, so we're extremely pleased to have Marina Vyazovska. She's a woman who almost needs no introduction, but I'll give her a little one. So first of all, let me point out that she is our other Clay senior scholar, so the Clay Mathematical Institute sponsors two great people in the field. We've already had a talk from our other Clay scholar, Hendrik Lenstra, on week one, and Marina is the second person sponsored by the Clay Mathematical Institute, so we're very grateful to Clay for sponsoring her visit. In any case, as I say, she almost needs no introduction, but let me say it anyway, she's hot off the press the most recent field to medal winner. I don't know which of the four you're the last one, then you're the most recent, but anyway, we're extremely pleased to have Marina Vyazovska, who is a great expert in spear packing, spherical design and approximation theory. She has many great accomplishments to her name and many more coming in the future, and we're really pleased to have her here at PCMI this week. So thank you very much for the introduction, and I'm happy that I could finally make it and be here, and for these two days I do enjoy this summer school a lot. So today I will speak about sphere packing, my favorite topic, and as I will try to make this talk interesting for everyone, for those who already knows a lot about sphere packing, and for those who don't know much about sphere packing yet, so here's a picture which explains a little bit what we will be talking about. So this is actually the shade balls on a lake, which is supposed to be somewhere in California, not that far away from here, and you see that balls on a lake, they arrange themselves almost into this perfect hexagonal lattice, and we think that they arrange them like this because this is the best packing in dimension two. So what is this sphere packing problem? The sphere packing problem is the following one. So suppose that we have a big d-dimensional box, which is big in all of its d-dimensions, and we have an infinite supply of balls of radius one, and so the radius of the ball is much smaller than the size of our box, and then our question would be how many balls can we put into the box, and if our box is very big, then what happens at the boundary of the box is not that important for us, and the only important thing is that how many balls per volume can be fit inside, and so for each dimension we will denote by delta D the maximal possible density of the balls inside of the box, and the density means that the portion of the of this box covered by the balls, and so of course this is, I'm a bit, for those who is a mathematician you see that I'm a bit hand-waving here, and of course to give an exact definition we need to do a bit of analysis, but in the Euclidean space sphere packing problem it's a very nicely behaving problem, and so whatever effort we make we will usually arrive at the reasonable answer, so I will not go into details and just leave us with this intuition, and so let's look at the dimensions that we all know and love, the dimensions 1, 2, 3 which we can experience in our everyday life, so in one dimensional Euclidean space is just a line, and the ball in one dimensional space is just an interval, and we can cover our line with unit intervals, or actually if it's radius 2 with intervals of lens 2 with almost no blank space left, and so the density of sphere packing in dimension 1 is a trivial, it's trivially 1, so case is solved here, it's a trivial problem, and in dimension 2 situation is a bit more interesting, so two dimensional Euclidean space it's an Euclidean plane like the stables that are in front of you, and ball in two dimensional space it is a disk, so you can think of a coins lying on a surface of a table, and they would like to put this coin so that they cover as much of the surface as possible, and so by if we are playing a little bit we will see that the best we can do is probably this hexagonal lattice, and so the hexagonal sphere packing covers slightly more than 90% of the surface, and nothing better can be done, and this is not a trivial but still a rather easy problem, and so it was solved, essentially the solution was obtained at the edge of 90s and 20th century, and but then it was in the middle of the 20th century like more rigorous argument was found, and so now we know that this is indeed the correct solution, and in dimension 3 so the best solution is this pyramidal shape so the way we stuck oranges in a supermarket, and so this configuration it covers about 74% of the volume, and this problem about packing three-dimensional balls in three-dimensional space it is known as Kepler's conjecture, and it has a very long history, and so it started back in 17th century, and actually probably this problem was considered long long time ago, and I'm sure that every Asian civilization would have an ideas about how to pack balls, and there are even proofs to that, but in so to say in what we called in a modern history, so probably the first person who started thinking about this problem was Thomas Harriet, and his reason to start working on this problem were very practical because his sponsor, British aristocrat, asked him a very practical question how many cannonballs can he put inside of a ship, and so Thomas Harriet as a British scholar of that time who actually spent a lot of time here in the United States and made some considerable contribution to the study of American continent, he started thinking about this question, and he came up with several, not with one, but with several solutions that cover this 74% of the space, and he described it in his letter to his patron, and as we already yesterday that at 70th century people still did not have archive, they could not post their great ideas and immediately share them with everyone, so at that time people wrote letters to their colleagues, and Thomas Harriet wrote a letter to Johannes Kepler where he described his discoveries about the sphere packing and three-dimensional space, and he also thought about maybe it is this properties of this packing, maybe they are somehow related to the atomistic theory which was a hot topic at 17th century, and that maybe the condensed bodies we see around us there also consist of atoms and those atoms are packed in a similar way as the oranges in a supermarket, and Johannes Kepler really liked his ideas and he wrote, actually he already published an essay which he dedicated to his patron of that time, and of course he did not mention Thomas Harriet and his essay at all, but he also wrote about the packing in three-dimensional space and where we can see such structures in the nature, he wrote about the shape of snowflakes and how snowflakes always have exactly six edges and maybe this is related to the packing problem in two dimensions, and he wrote about beehives and how they are similar to this orange stacking configuration we have seen on previous slides, and he also wrote about atoms and that maybe atoms inside of condensed bodies they also are packed like cannonballs in a ship, and our days we actually understand condensed matter much much better and we know that these ideas they were very revolutionary for 17th century but still rather naive and this is not how atoms behave inside of condensed body also actually I think recently I heard the talk about how snowflakes are formed and why they have exactly six edges and it also has nothing to do with packing problem so all these were really great ideas but and very important way of thinking but now as much as we know about universe about us and as much we know about physics so they do not apply directly to those physical problems and so but the question whether this particular configuration of balls is it really the densest configuration in dimension 3 it remained opened and of course neither Thomas Ariot neither Johannes Kepler could have asked that question in the 17th century because at that time people did not think much about rigorous mathematical proofs however this question remain became famous as Kepler's conjecture and as mathematics advanced people started thinking more and more about proofs and about exact statements but this theorem it turned out to be not an easy one and so to explain you why it is a difficult problem maybe I would like to explain you another version of it I would say a much easier local version of a packing problem which is a problem of kissing spheres and so what are the kissing sphere configuration it's a configuration which consists of one red sphere of radius one and several blue spheres also of radius one and the rules of our problem are the following so the old all blue spheres they have to touch simultaneously touch the red one and they don't should not intersect they might touch each other but they're not allowed to intersect so all this blue spheres they're kind of kissing the red one simultaneously and in dimension two as you know this is an easy problem so we see that we have one red disc in the middle then it can be surrounded by six other discs and it's not difficult to see that seventh one is impossible we cannot put one more disc here and in dimension three it is not such an easy problem so here's one possible configuration is that we put one red ball inside and we surround it by 12 blue balls and the centers of those blue balls are in the vertices of icosidron and so this way we see that we can have a kissing configuration with 12 blue balls however is it possible to put one more and here it's really not non obvious that we cannot put one more and it's because all this blue balls they all touch the red one and they do not touch each other so for example if you make them a bit bigger they still will not intersect with each other and this gives an idea that maybe if we rearrange them a little bit maybe there is enough of space for the teens one and so this was a famous dispute between Isaac Newton and David Gregory and Isaac Newton thought that 12 is the maximum possible while David Gregory thought that 13 is still possible we just need to think really hard how to put it there and so maybe this explains a little bit why we all know so much about Isaac Newton and remember now so little about David Gregory so David Gregory was wrong and Isaac Newton was right but the rigorous mathematical proof of this statement was obtained only at the 19th century so long time after the question raised and so you see that even this one much simpler local version of sphere packing problem in dimension 3 is not easy and so let's look now closer into this dimension of the sphere packing in dimension 3 so how can we construct how do we construct let's look back at this pyramid of oranges how do we construct it so here what we do first we let's construct our configuration in layers so this seems like a natural way so we have imagined that we have our infinite table and we put our oranges on a table so that their centers which would form and hexagonal lattice so this is the best packing in dimension 2 and at the next step so now we are putting next layer of this was the first layer of blue oranges and now we've put the next layer of green oranges and we put green oranges into a holes between blue ones but as you can see here we cannot put a new orange into every hole between the blue oranges because the distance will not be enough and so we have to choose but we have to put a green orange into every second hole and then they again will form the centers of this green oranges they will lie a bit higher but all the centers they will again be on one in one surf in one plane and they will form a hexagonal lattice in this plane and so this way we continue and this is how we build our packing but there is another way how to how we can build a packing so here's another way so let's again build our packing in layers but this time we will not work with a hexagonal packing but we will work with a square packing so we put our oranges and like first our blue oranges into this square in a square lattice and now as we are putting the next layer and now the because our packing is not so dense anymore our flat like packing is not so dense anymore it means that the holes between oranges became bigger and now we can actually fit new ball into every every hole and we continue like this so now it's of course now each layer is not as dense as it was in the previous slide however layers now are closer to each other and this now we have constructed these two different configurations and so let's decide which one is denser so maybe we can vote here so if you think that this one is denser you can raise your hand okay and you think that this one is denser you can raise your hand and if you think that two configurations are equally dense you can raise your hand okay so I should tell you that these two configurations they are not only equally dense but if we if here if we put our layers correct here they will be even identical to each other so these two configurations which I've constructed into different ways they are actually geometrically identical after a rotation and three-dimensional space and so how can we see this we can see this in the following way so let's consider this configuration which is called a face central cubic lattice and so why does it have this name so we are taking this usual cubic lattice or if you are a mathematician we are taking a lattice z cubed so just the lattice of all points for example with integer coordinates here I'm not I don't pack with balls of unit volume I will maybe readjust the size of a ball and so now I put one ball into a center of every lattice point of this cubic grid z3 and also I will place one blue ball into a center of every face because I have these cubes cubes have faces and these faces have centers and blue balls they have they are centers coincide with the centers of the face and I adjust the size of my ball so that red balls and blue balls they have equal radio and they don't intersect and only touch each other and so this is so called face central cubic or FCC lattice and so now let's see that this lattice is actually the same configuration as I have constructed by layers and so to see that it's the same let's color this balls in two different ways so first let color them like this so we'll take our plane to be so the plane which is so to say the diagonal plane of our our cube so it's orthogonal to the main diagonal of our cube and so you see that each niche layer I will have this perfect hexagonal lattice and so you see that this configuration is actually the same as the layered configuration I have constructed on my first slide and so now we can color the same configuration but again but differently so now I color I take my layers to be parallel to one of the faces of my cube and this way I get I see you can see that this FCC lattice is actually the same lattice as I have constructed in my second layered example and so as you see even three do like three-dimensional space we all live in it can be quiet complicated and counter-intuitive and so to say a bit more to explain you why this is a difficult problem is that in dimension three actually this FCC lattice it's the optimal solution but as we know now but it's not the only optimal solution and actually we have uncountably many equally dense sphere packings in dimension three and they will how we can we achieve that so as I've told you when we are building our configuration by the first method I have shown you I've also told you that we cannot put a ball of the new layer into every hole between the balls of the lower layer and so at each step you have a choice which which holds to choose and as we can do this choices at every layer we will we can construct geometrically different configurations but they all will have the same density because the density of each layer is the same and the distance between layers is the same and so this partially explains why sphere packing is in dimension three is such a complicated geometric problem and so but despite being so difficult this problem was fire was solved by Thomas Hales and he announced his solution in 1998 and this was actually a very long proof based on previous work of many of other people and also it can contains a lot of it was a maybe a first one of the first computer assisted proofs of important mathematical result and so this also was the reason why for a long time mathematicians hesitated whether to accept this as a mathematical proof or not and so what Thomas Hales did except of this doing like huge work of proving Kepler's conjecture he have also written a formally verifiable computer proof of it and this project took to an almost another decade but at the same time I think what was a big progress as on the way of doing this he also worked a lot on developing this computer verify software for verifying mathematical proofs and even for guessing and producing new mathematical proofs and and so as the Kepler's conjecture in dimension three is solved so let's now go to other dimensions and so in my first slide I've casually introduced sphere packing for you as a sphere packing in the dimensional Euclidean space but maybe at least for some of you it would be I hope it will be useful to know what is a d-dimensional Euclidean space because we have already seen the Euclidean space in dimensions one two and three so here's again the Euclidean space of dimension three and we know that every point in dimension three it is given by three coordinates but now as mathematicians what we can do we nothing stops us from introducing more coordinates and in a popular culture we know there are several the results of this idea of higher dimensions which I think is for people who want to do higher-dimensional geometry are very misleading right maybe you know about the flat land which is a two-dimensional land populated with rectangles with different number of space or about four-dimensional aliens of Kurt Wienigut who can see through time but of course as far as for mathematicians even though this are all very beautiful metaphors they don't help us doing our job because our view of mathematics of course is just purely formal so what if you want to introduce a d-dimensional Euclidean space we will just our points it would be this arrays of d real coordinates and now to make it an Euclidean space and other piece of information we need we also need to measure distance between points and so we are measuring distance according to Pythagorean rule we are just generalizing it to the dimensions in the most natural way and now we are actually we are now set to consider almost done to consider a sphere packing problem because now we have to define what is a ball and also one small thing which I do not define here but what we do have to do we have to define what is a volume in d-dimensional space but once we have done all this now our sphere packing problem also makes sense in d-dimensional space and so what do we know about sphere packing problem in d-dimensions and actually we know not that much so here's an old there's a nice paper by John Conway and Niels Lohn written in 1995 where they describe the best sphere packings in dimensions from one to ten and so here here I written so in green are those dimensions for which we actually know at least as of today we know what is the best solution and in black are those which are still cases which are still open so these are the best known configurations but we don't have the proof that they are the best and so you see it in dimension one of course the best is the like also here are the centers of balls should be should coincide with integer points and then of course the also the balls have to be scaled in the right way so that they don't intersect so and of course in dimension one it's trivial problem in dimension two it's an easy problem in dimension three it's a complicated problem which was solved by Thomas Hales and in dimension four we have a unique solution so called lattice d4 and we believe that this is the unique solution in dimension four and in dimension five it's actually similar to dimension three here also we have one lattice called d5 and we also have uncountably many other packings and in their nature they are very similar to this uncountably many packings which we have constructed for dimension three they also come from putting smaller dimensional layers in the right way and so similar behavior is we have with four dimensions five six and seven here also we have some lattice packings which are the densest packings but also we have uncountably many equally dense packings and in dimension eight we have a unique e8 lattice and in this case we can prove that it is the best for packing and now in dimensions nine and ten different wonders start to happen so here this pattern which at some point convent Sloan they thought that this would work in all dimensions as they write in their paper here this idea breaks and just putting old layers in a smart way is not working anymore so in dimension nine it's still possible to put together smaller dimensional pieces in the right way however they convent also discovered not only this uncountably many packings but he also discovered one continuous family of packing so what he could have done he could have he could take this particular lattice divide its balls into two different subsets and he calls them golden balls and silver balls and then continuously move them with respect to each other so that the new configuration is still a packing and it has the same density and in dimension ten it's the first dimension where the best known packing is so-called packing named after a mathematician with last name best and it's what is interesting it's a non-lattice packing and we don't know any lattice which is as dense as this one so it's a periodic but non-lattice packing and yes and so this is then these are all the cases that we do know so we do know sir packing is solved in dimension one two and three and also I've in 2016 I shown that the e8 in dimension eight is the best possible sphere packing and working together with Henry Kohn, Abinav Kumar, Steven Miller and Daniel Ratschenko we have proven that the leach lattice in dimension 24 it is the best packing configuration in this dimension and maybe one interesting thing as you can see as as our dimension goes up then the densities of our packings decrease and so yes I don't think so maybe one has to look at the paper of Kohn and Elkis but I don't think it comes from stacking smaller dimensional layers yes so so one important thing you can see that the density decays and we know that this dimension grows then the density of the best packing it will decay exponentially however we don't know what this exponent should be and so what is like 90% is what we can achieve at dimension two but in dimension 24 we can achieve only this about two thousands of the space can be covered by balls and in dimension 24 it's actually it's a very good result and it's better than at least what we know at the moment we can achieve in dimension 23 and so let me but now let me speak a little bit about these two special configurations so as you see that dimensions 8 and 24 they are somehow special and the configurations we have here they are special they don't have this nasty behavior like three-dimensional best packings because here in these both cases we have this unique configurations and let me tell you about what is so maybe there's a there's a photograph of my collaborators so you've already seen Henry in this summer school and this is also Abinav, Danilo and Stephen and so and this is the E8 lettuce so this is what is what you can see here this is a zoom model of the shortest vectors of E8 lettuce so it at lettuce it has 240 shortest vectors and those 240 shortest vectors they solve the sphere packing problem in dimension 8 and dimension 8 it's also one of the few dimensions where sphere packing sorry not the sphere kissing problem is completely solved and so this lettuce is known for a long time so first it's so it's the E8 lettuce it's the unique even unimodular lettuce in dimension 8 and what it means it means that it like we have one on average we have one point of this lettuce per unit of volume and even means that if we take any distance between those two points then it would be a square root of an even number so length squared is always an even number and this turned out to be a rather restrictive property and so what was proven by Smith in the middle of 19th century is that there exists at least one even in unimodular lettuce in dimension 8 and it's also interesting that his proof it was not a construction it was really an existence proof because he have proven a mass formula which told us something about this even unimodular lettuce is on average and he shown that this the take some overall even unimodular lettuce is weighted by inverse of their automorphism group then in dimension 8 this was a positive number and therefore at least one lettuce has to exist but he did not give produce the lettuce and so the first explicit example was dined by by Russian mathematicians Korkin and Zlatarov six years later but E8 lettuce is actually a famous object in mathematics and one reason for that is that it is the root lettuce of the E8 root system and E8 root system appears in many areas of mathematics for example in classification of Lie group and Lie algebras and so it it also enjoys a lot of symmetries and nice mathematical properties and so another special lettuce is the leech lettuce and leech lettuce it's may in some sense it's it's also an extraordinary mathematical object so leech lettuce it's also an even unimodular lettuce in dimension 24 and in dimension 24 we know that we have exactly 24 such lettuces up to an isomorphism and if you have already told you leech lettuce it's an even lettuce and it means that all the distances between its points they are square roots of even numbers and so the smallest possible is square root of two and the leech lettuce is the only lettuce among those 24 which has no vector of length square root of two so it's shortest vector vector has lens two and this makes it a great candidate for for a backing problem and for leech lettuce it actually took longer for mathematicians to discover it and so first maybe important step in discovery of a leech lettuce was actually done by engineers so this is a portrait of Marseille Galle and in 1949 he discovered an object called Galle cot which i will speak about a bit later and so then the Canadian mathematician John leech used Galle cot to construct a leech lettuce from it and he immediately recognized that it would be a great great solution for sphere packing problem and so this is a John Conway and he studied automorphism groups of leech lettuce and turns out that this lettuce also has many symmetries and the symmetries of leech lettuce they're very closely related to the discovery of simple simple sporadic simple groups and in particular to the discovery of the biggest simple sporadic monster group and finally this is the portrait of Richard Borherz who proved the monstrous moonshine conjecture which is also related to the monster group and somehow indirectly related to the leech lettuce so it's also a lettuce with a lot of interesting mathematical structure in it and so that's all let me speak about again about Marseille Galle and about the relation between leech lettuce and coding theory and also relation between sphere packing and coding theory and so as I read it to I hope that I convinced some of you that sphere packing it's a fun mathematical problem but it's not the only way for studying it and actually sphere packing has some very practical applications and so these applications they were described by Claude Shannon and his foundational works on information theory and so here are there are also portraits of Richard Heming and Marseille Galle so they are people who also worked in coding theory and created first important examples of codes for example Richard Heming is very famous for Heming code and Heming code it's somehow derivative of the E8 lettuce which we've seen before and the Galle codes found by Marseille Galle it was actually the first hint for mathematicians to discover leech lettuce and so let me explain the concept to you so here is a very simplified version of signal transmission described in the paper by Claude Shannon so the problem he thought about was the following one so to pose that we have to first signal and we have to transmit this thing signal through a channel and in this channel we can have a noise and so it means that our signal what we receive is not the original signal but it's our signal plus error and the problem is then how to we have to design our system in such a way that this mistakes they can be spotted they can be detected and also if they are detected it's good if we are able to correct them and you can imagine that problems like this they rise in all kinds of real-world system like for example one of the reasons why Heming developed his Heming code was there he worked with very first lamp computers and they're very unreliable and so he they always made mistakes and then like misinterpreted zeros and ones and so he wanted to have slightly better system that if one lamp does not work he still doesn't have to rewrite all his program and maybe even the computer could correct it but the this diagram here it works maybe better for your our wi-fi connection you know if our you know if you have a storm coming that we can still here decode the this signals and so here's the model which was suggested by Shannon how can we use the sphere packing so he suggested that in our channel we do not send any words we are in our channel we are just picking certain number of code words which we are allowed to send and we choose this code word so that they are far away from each other and here too if we are thinking of some reasonable communication channel we also think that our errors will not be too big because if our errors are very big then of course this whole idea of using this channel is hopeless so we hope that our errors are small and having big errors this can only happen with small probability and so this was one of the mathematical models he suggested is so that around each of this code word we have so to say this error ball of possible errors and if these balls do not intersect then after obtaining a corrupted signal we still will be will know in which ball it to which ball it belongs and we could reconstruct our original problem and so and maybe one thing i should tell you that this error correcting codes they're still used in all kind of signals we are sending in our internet in mobile phones but and here's one nice example of actually of the using of goley code so goley code was used for example for a voyager mission and it was really important to have a good error correction to to be able to obtain enough information from from this apparatus but of course this apparatus was sent long time ago maybe even before i was born and so right now in the error correction the kind of sphere packings that are used are sphere packings in effectively infinite dimensional spaces so this brings us away from this cosy realm of small dimensions and forces us to study very big dimensions okay so maybe i will and a few words about what are other reasons for studying sphere packing so as i told you knowing the exact sphere packing is very useful and practical but it's actually this is not my main areas of what what i mainly work on i do not try to construct new sphere packings but i rather try to prove bounds on sphere packings i try to prove that for example for the leach lattice leach lattice is great but it cannot be improved and on one hand maybe the value of this work is of course that nobody is wasting their time on trying to improve leach lattice but also other interesting reasons for studying this bounds on sphere packings is that this kind of bounds they're used not only in euclidean geometry but they're used in other problems and so this is one recent work by a theoretical physicists hartman delimil matzak and now or nardorasteli so what they found they found that actually the constraints which are used to to prove bounds on sphere packing problem in euclidean geometry they also for example can be used to study two two deconformal field theories and so here i explained to you what are the what is what is the fear sphere packing problem and it is intuitively clear problem even though if we try to analyze it geometrically then it becomes a complicated mathematical problem and so two deconformal field theories they also of course can be formalized mathematically and here i will not be able to explain to you what what the idea behind two for two different formal conformal field theories is but it's somehow it's a problem which is even when it's states in mathematical language it's something very very different from sphere packing problem it's a mathematical object the medical object behind this problem so it's a mathematical object of very different nature however the same exactly the same methods they help us bound the density of sphere packing but they also help us to constrain the spectra of this 2d dimensional conformal field theories and so that's probably all i wanted to tell you so maybe now i will give you an opportunity to ask questions okay do we have questions when you talked about the error correcting code and having many dimensions i'm guessing that like in a practical sense you would have a different estimate of the error in each dimension you would make and so it's really is fear packing really what you want to do so i mean they have several parameters right and you asked you estimate you would do arrows in each of the dimension so you maybe like when you set a signal you set a signal with many many different parameters each corresponding to one of the dimensions you want to you want to study right yes yeah but okay i think in error correct there are several models that work with it and maybe like one thing where this for which actually sphere packing like the simcletian sphere packing is useful is probably the this what's called like the gaussian noise model where we here but but yeah in in this sense like in very so here this is i would say a very general picture and very often these balls they're not balls in euclidean space sometimes these are actually balls in what we call the hamming space so this is just i would say a motivational picture but so sometimes it does happen that we send signal where each coordinate it's a real number and our channel is modeled in such a way that actually euclidean distance matters and then euclidean packing becomes somehow relevant thank you anybody else yes okay so this is for example leech lattice and so this is a projection maybe not of the all lattice but only of the shortest vectors so the rather is like however it's like how like 196 560 shortest vectors of leech lattice and yeah so here the plane is of course this is like a carefully chosen plane so that the picture is nice and symmetrical but i think i did not try it anywhere which which exactly plane this is but yeah this is some nicely chosen plane so that i think what we see here we don't see all all of the vectors because some of them project into one i think some of them like projections for some of them coincide here in this particular representation oh there is a question there okay so there things become a bit complicated so there are some maybe there are some nice dimensions like dimension 16 there is there is a candidate and but i think in very high dimensions is just we know that little that people maybe don't want to conjecture and then to be proven wrong so four i think four is an obvious candidate but but at the same time it also seems quiet far away right now so but but four is like a nice candidate it's because there there is this unique configuration d4 and d4 is not as great as e8 or leach lattice but it's still quiet nice so i would bet for four may may i cannot say that it's not useful but i don't think it has somebody who has realized this idea so far so i don't see why not but but i but i think to you know like all the proof that exists right now they don't use that idea like for for example in like whether that is like what's interesting with for example e8 lattice and leach lattice if we because if we reduce this lattice is more like modulo 2 then after some essentially what we will like get we will get this codes like hamming code and golly code and those are known to be optimal as well and they also like this perfect codes on the other hand we don't know how to use this information to prove optimality of euclidean packing i'm not sure but maybe yes also this is certainly like this and e8 and leach lattice they arise in many algebraic problems so it's not difficult to construct them i think they can be constructed in many many ways and maybe even maybe it's maybe it's possible but i'm not i'm not 100% confident oh yes no i think it's still a bit of a mystery so it's just just so of course we can look at like our proof and then see why it works in dimension 8 but i don't have a really good question why i don't have a simple question of sort of simple answer to this question so i think it's as i understand i'm not very much like expert in simulations but as i understand like when dimension is small it's like up till some dimension simulations work great and if it becomes a bit better they become terrible so maybe that's what the situation is but i think it's actually making simulations is not that easy because like for gradient this i think this problem is very bad for gradient descent because there are this kind of like jammed configurations the configurations that are not optimal but they cannot be improved locally so i think this is a nasty problem for from that point of view so but of course it does it doesn't mean that there is no method maybe but but i think there is no very good method right now and also i think here there is some kind of like course of dimensionality that it could happen that in dimension 8 things work great and find it latest for you but since slightly bigger dimension everything already is ruined and there is no more confidence that and here also maybe like one of the problems is that it's difficult like what what can we compare with because our upper bounds are also quite bad at the moment so if you are constructing some configuration how do you know it is the best how to gain this confidence it's maybe also something that is not so clear yes we already know that it does not work in higher dimensions and yeah so this is maybe answer to like also related to the question before like why it works in 824 so actually this method of the the conelki's linear programming bound it gives us perfect result in dimensions 8 and 24 and it does not give us perfect result in other dimensions and we don't and we know that's the fact and in some dimensions we can even prove that it's the fact but why is still it's like a more philosophical question and i don't think we have satisfactory answer to that yes yes as i understand the question is about considering the packing with spheres of two different radius yes so this problem is considered and of course there it's maybe more difficult to analyze it in higher dimensions but at least in dimension 3 i think it is there are works like this mostly computational also some bounds yes so in general this problem is considered but mostly i would say from the computational point of view and i think this kind of problem is also very interesting for simulations because here even in small dimensions it's possible to get like new interesting results and it's not as badly behaved as this findings fair packings in higher dimensions any other questions not really so so this is i think this mostly believed that probably in very high dimensions the best packing would be chaotic but i think we know too little to support this also when you say that okay like in dimension 10 we have this non-letters configuration but then in dimension 24 we have each lettuce which has all this like wonderful symmetries so it's it doesn't seem to be a very regular pattern and yeah and i think if you have this like philosophy that there is a finite number of exceptional objects in mathematics then maybe maybe it means that in very high dimensions we exhaust all these exceptional tricks and we just have to live with chaos but yeah but that's maybe more of a philosophy than mathematics at this point and also like if i think for if anyone to construct something in very high dimensions actually symmetry seems like our only guide and symmetry what helps us to construct or to prove the existence of dense things so it's yeah not that easy to answer that question okay yeah sorry uncountably many other packings where yeah so so what what i think i told you everything i knew so maybe yeah maybe i can say it again so yes so so this uncountably many comes from the fact that as we are like stocking the this hexagonal lettuces upon each other at each for each step we have two choices and so it means that like we can index our packings by for example this like now by infinite words right but each time it's like either like a or b and we have uncountably many infinite words of two consisting of two letters and of course like there is maybe some subtle thing is that maybe we it can happen that we have constructed two packings in different way but then they turn out to be geometrically identical so this is a slightly subtle point but mathematicians still worked it out and they do confirm that we have uncountably many packings what was the original reason why you got interested in a sphere packing problem okay so i think it's a nice problem also and i started working on it i already knew about work of conan delkis and i think it was a beautiful paper where it seemed like you know there is they almost solved it and there was only last step to be done and i think it's very attractive right so any other questions yeah please go ahead yeah so similar methods that's like linear programming ideas they do apply to the trajectory even though because the trajectory doesn't have this rotational symmetry things become technically more complicated and actually here i don't know what is the state of the art and how far away are we from sharp bounds yes so in principle these ideas they still work but it's not clear whether they give sharp bounds or not okay any i think anything else yes so the connection is indirect it's somehow i think it's not it's clear that two problems are somehow related to each other and this sphere kissing it's like it looks like a local version of sphere packing and it's possible to fall to have some quantitative connection between those two but of course the solution of one problem does not imply the solution of another and vice versa so here i think it's quite remarkable that for and i think also not maybe not yet completely understood that so this linear programming method it helps to solve it solves the sphere kissing problem in dimensions eight and 24 and so this was done by livenstein and independently by adlishek and slon it's still in i think in 1979 and then again linear programming only a different one the conelkis linear programming solves packing problem in dimensions eight and 24 and this is of course yeah i mean and i think at this moment at least i don't know what what's the connection these two facts they seem to be of course related to each other on one hand on the other hand i don't know like any formal connection between formal medical connection between them so it's maybe it's something yet to be understood and yes i don't know about that some um yes i don't don't know okay well at the interest of time we've um we're at 415 so we'll stop here marina's around through the end of the week i believe so please catch her with more questions and again let's thank her very much for a nice