 OK, thank you everyone for joining today. This week, I am pleased to announce our TSVP lecture is going to be given by Professor Kyoji Saito. Professor Saito has had a very illustrious career in Japan. So he first studied as an undergrad and master student in the University of Tokyo. He received his PhD from Gottingen University in Germany and has since had several research positions at the University of Tokyo and at RIMS in Kyoto University. RIMS is the Research Institute of Mathematical Sciences, which is very famous, at least here in Japan, and probably quite famous in mathematics abroad as well. Currently, he's a professor emeritus of RIMS and also associated with IPMU at the University of Tokyo. His prizes include receiving the Geometry Prize of the Japan Math Society in 2011. So we're very happy to be joined by Professor Saito, who is currently working sort of at the interface of algebraic geometry and some mathematical physics applications, I think. So please join me in welcoming Professor Saito. Is it on now? OK, thank you very much for the introduction. And so I'm going to talk about what I call Platon's Symmetry. This is some geometric talk. And before starting, I should give two free slides. One is this is mainly about some history of mathematics related to this Platon's symmetry. And as far as possible, I try to be elementary for non-mathematical audiences. But in some places, I'm necessary to talk mathematical terminology or mathematical concepts or some results. And if you don't understand this mathematical part, don't worry and ignore it. See, my main purpose is how some idea, mathematical idea, appeared at one point. Then how it developed and influenced each other. And sometimes they split, sometimes they unify, and so on. This stream continues from Platon time, even present time. I want to show that type of history. So even you don't understand mathematical technical details, you may follow this history. That is one apology. The other apology, this is my first time to use TEF for the lecture. I use, of course, TEF for the writing book or writing paper. For the lecture, I don't know. I'm not very skillful for that. So sometimes it may look strange, but please apologize. I use some colors. This is some content. I use some colors in this lecture, blue and red. Most of them, no meaning. But maybe red means something related to some historical mathematician or his statement. Blue color usually means more mathematical concept or some mathematical statement. But this is not always blue. But roughly, this is a sketch of talk. And we start with Greek time. And after 2,000 years in the Galois time, that is the beginning of 19th century. And 40 years later, some crimes will come killing or Felix crimes work up here. And 60 years later, that means the beginning of the 20th century, some work by Duval and Coxter. And again, after 40 years, around 1970, Bruton League, Briskon work up here. And after that, present time, what we are now. And if time allows, I come to here. But it doesn't. Maybe we should stop on the way. It depends on how the lecture develops. So let's start with the definition of this platonic solid. Platonic solid or more mathematical terminology, it is called regular polyhedron. That is convex polyhedron whose all faces are congruent, regular polygons, which are assembling in the same way around each vertex. This is defined already, Greek time in the Euclid book. And this definition is still valid nowadays. But practically, this means these five figures. This is tetrahedron, cubic, octahedron, dodecahedron, and hexahedron. And there are no more. And already in the Greek time, by, for instance, in the famous book of Euclid, the Stokeia written three or four centuries before, already all these figures are given and attributed to some previous mathematicians like Pythagoras or other people. But these figures nowadays are more mainly called platonic solid for the reason that these figures in the mythology, in the story, I don't know whether this is the real history or not. But in Platon's academia, they exhibit these all figures. And at the gate, it was written that those who don't understand geometry should not enter academia. For this reason, these pictures are called platonic solid nowadays. So but in my talk, I'm not mainly considering this geometric figure. But my interest is more talking about symmetry. That means in modern terminology, the group, the concept of group in mathematics appears early 19th century Cauchy or these people here. And this concept of group as established by the work of Gallo, that I shall explain later. But already in these pictures, we see some symmetry. That means you consider that these pictures are embedded in the three space. And you put the center of the figure at the origin of the three-dimensional space. Then you are going to consider some set of rotations of the three-spaces centered at the origin. And this rotation, if rotation brings the figure to itself, let us call it this isomorphism. And this assembling of this all isomorphism makes so-called group. Group means in mathematical terminology, some object which admit composition of the operation, inverse of the operation. And actually, all isomorphism of such figure form group. In this case, for so-called tetrahedral group, cubic group, octahedral group, dodecahedral group, but if you look at everything from the viewpoint of group, these two are dual. They give the same group for the reason if you look at their dual to each other in the sense, if you consider the center point of each face, then combining all these points in the center, you obtain octahedral. Oppositely, if you take the central point of this octahedral and combine all these figure points, you obtain this cube. That means these are essentially isomorphic from the viewpoint of symmetry. Same thing happened for this dodecahedron and hexahedral. If you take the central point of all faces and connect all these central points, you get this ecosahedral. Oppositely, if you take the central point of this ecosahedral and connect these central points, you obtain this octahedral. In this sense, these two are same from the viewpoint of symmetry. So later on, I shall not distinguish. Oh, I'm sorry. Yeah, distinguish. Okay, let's go to further. Ah, I'm sorry. These are the figures. Of course, later made. This is the figure of Platon and this is the figure of Euclidate. But anyhow, let's call it. After Euclidate, or the Greek time, there are not much essential progress understanding this. Platonic. I'm sorry. I'm sorry. Some of my friends are calling me. Okay, let's go further. And then also I remark, also in the nature, in the crystals, you find such figures. But I don't want to go much in this direction. Or even in the nature, some virus has such figures. And also this morning, some of my researchers in this institute visited my office and show me some figure related to her work, which shows also octahedral and not cubic symmetry. So such symmetry appears several times in our study. And one strange example is by Kepler. He tried to explain the planet system using this regular polyhedron. But from the modern viewpoint, his try was completely stupid. The figure is quite beautiful, but unfortunately this has no meaning. More meaningful work was done by Euler. Euler gave very remarkable equality like this, where you consider the number of the vertex, number of edges, and the number of faces, and you make plus, minus, plus of these. Then for all tetrahedron cubes, octahedron, dodecahedron, mucosahedron, you have this number, give two. And this phenomenon is now considered Euler characteristic, some very initial, very early appearance of some topological structure in geometry. And nowadays this Euler characteristic is one of the key techniques to understand varieties. But this is the direction I don't want to discuss today, so I'll speak more. And then I jump to the Galois time. That is the beginning of the 19th century. And what happened in that time? Maybe I should talk first about solving equations for the unknown number. So suppose you have an unknown number, x, and you want to know this x, what is that? By knowing the relation which x satisfies. For instance, if you have known number A and B, such that Ax plus B equals 0, then you can, it is elementary arithmetic that x is solved with. Also, in the high school you learn, if you are unknown number, x satisfies that quadratic equation, Ax squared plus B times x plus C equals 0, then x has a famous formula for the quadratic equation. And cubic case, I don't like the explicit formula, but already this formula was known, for instance, in the Islam time. I forgot his name, but famous poetry and famous scholar in the Islam time, he gave this formula. Also, in the Renaissance time, they found also quadratic equation also known in the Renaissance time. I forgot Altadia or somebody in that time. But I don't want to write explicit formula too complicated here. Then how about the higher degree equation? For instance, for given number N, you can consider the equation A0 times x to the power N plus A1 times x N minus 1 plus AM called 0. Then, do you know what is x? Someone writes down explicitly what is it? There is famous theorem Gauss that such number, if you are this given number, so-called complex numbers, then always there are N number of solution of this polynomial. But then only Gauss said, only existence. Someone write down explicitly like this formula, only using so-called addition, multiplication, subtraction, using only N-th root. Someone write down the x. And this was known until the Arbel and Galois time. That is really the beginning of 19th century. And complete answer was given by Galois. Maybe I showed you. Unfortunately, he died when he was 21 because of this. I don't know how to say in English, but I'm fighting. And what he has done was really revolutionary. Even nowadays, many people admire him. And I'm sure he is one of the greatest mathematicians in our history. And what he has done in the following, instead of, by the way, you don't need to read the text. Most of the text I will explain already. And if necessary, I show by this point at some point. So, please just hear what I'm saying. So, Galois's case, his viewpoint is different than the predecessor. Predecessor was looking some direction, the number you want to know. But Galois's case, he don't look at just each individual number, but he begins to look at the system of all the numbers who are related to this equation. This is nowadays called field in the mathematical terminology, field. That means you look at the system of numbers which is closed by addition, construction, multiplication, division. And such system is number. For instance, all rational numbers are filled. If you add some number, solution of algebraic equation only, this forms some algebraic number field, et cetera. All real numbers are again filled like this. And Galois considers if you start with some number filled which are related to the equation and you begin to add step by step, square root, cube root or whatever necessarily, and look at the extended field. And this extension system was completely described in terms of so-called Galois group. Group in this case appears as the isomorphism of the field. So before I was talking isomorphism of atomic solid and you consider the effect of all rotation which preserved the same figure. And in this Galois case, more abstract, the system of numbers is called field whether one system is isomorphic to the other system. Just you don't ask which number it is. Just you ask whether addition, multiplication or these structures are preserved from one to the other. This viewpoint is quite close to the modern viewpoint of category. In this last century, this sort of idea is quite quickly developing in mathematics. But Galois's idea contained already the idea of category. But anyhow, he looked at the group of isomorphisms. For instance, for the given equation, you consider just some number field related to this only these coefficients and rational numbers. And then look at again extension adding possible solution for this equation and so on. Then I'm very roughly explaining don't worry that I'm not precise. But anyhow, then he entered a group called he introduced isomorphism such group is called Galois group and some structure if the Galois group has some particular structure which is nowadays called solvable, then this equation can be solved only by using nth group. And even now nowadays people found another formula for instance using hyper geometric function or many other elliptic function using such particular function there is some formula to describe this solution. But if you restrict yourself only algebraic operation like in the truth Galois group should be so-called solvable group some very particular structure. That was his answer. Then for the polynomial equation of degree less than 4 are all solvable. But the equation for degree 5 this unsolvable group appears. This unsolvable group there are several possible unsolvable groups but I'm now going to talk about A5. A5 is some unsolvable group of rotation permutation of 5 elements corresponding to 5 solution of degree 5 and this A5 is strangely appearing in the platonic solid. That is the viewpoint of the next stage done by the Galois. Not Galois but Felix Klein so let's come to Felix Klein. Before coming to Felix Klein I should talk about with Galois' idea he as I told you he died early but his work was published by I think it was Elmite or somebody who collected his papers and published in the I think published in 1846 but his work was hard to understand even at the same time Galois or Cauchy the biggest mathematician at that time Yako didn't understand unfortunately but nevertheless his work began to influence on geometry and analysis and 40 years later than after Galois this works of Klein and Lee appeared but before coming I just want to mention four items this platonic symmetry and the Kleinian similarity Hoxter, Dinkin, Diagonal I shall explain all these items later more carefully but and simply these are mathematical objects platonic solids, Kleinian similarity, Hoxter, Dinkin, Diagonal simply and later on I show all these four items from the viewpoint of the category they are equivalent objects that means platonic solid symmetry appears in all these other things and that is one main subject what I'm going to talk and my question is what is the next thing but that is another question I shall discuss later so let's first come to the case of Felix Klein and Sophie Lee and later by Wilhelm Kirin so before coming to the description of what Klein and Lee did I just of this is a figure this is Felix Klein one of the biggest leaders in this time Sophie Lee is one of the founder of the most basic concept, the algebra and even now there is one of the very quite important subject and Kirin is some person who make available his theory to the modern style I shall discuss later and I want to just first look at the some introduction of the book written by Felix Klein when he wrote his famous book, before coming his book on the Icosahedron he wrote very well even now people are reading this is more than 100 years old still people are interested again published and referred and there are many works still concerning this Icosahedron the reason the following this the symmetry of this Icosahedron I was talking at the beginning if you look at the symmetry group of Icosahedron that group is in the modern terminology between A5 euthymetric not permutation index 2 subgroup of permutation group of A5 this A5 this Icosahedron group is exactly the group of some Galois group of degree 5 so they are the same group using this fact Klein was writing very careful study of geometric connection algebra and geometry that was this Icosahedron but in his introduction he was writing the following my indebtedness to professor Lee dated back to the year 1869 to 1770 when we were spending the last period of our student life in Berlin and Paris together in intimate camaraderie at that time we jointly conceived the scheme of investigating geometric or analytic form perceptible of transformation by means of groups of changes this purpose has been of direct directing influence in our subsequent labors though these though these may have appeared to lie for some time I primarily directed my attention to groups of discrete operations and was thus led to the investigation of regular solid and their relation to the theory of equations that is what I just mentioned professor Lee attacked the more decondite theory of continued groups of transformation and their with of different equations, differential equations okay so in this way Felix Klein described they got the same idea to use the idea in geometry and analysis but they go different direction in the sense that Klein goes to the discrete direction and Lee goes to the continuous direction but I shall explain this more carefully later that actually they their work after 60 years again meet by the other mathematicians which I call Cochster and Duval and much later by Griscon and Grotundi so first let me explain what they have done so Klein's case I mentioned already he was looking at this automorphism group of regular polyhedrons and it's a little bit again technical so you don't need to follow mathematical details but at least I should explain mathematically so he considers a group of rotation of three dimensions which preserves some regular polyhedrons then obviously this group acts not only the regular solid but the total space of three-dimensional space as a rotation here then maybe you know already run in the school that in such three space you have the Cartesian coordinate using the XYZ coordinate you can consider this group of rotation studied by Klein acts on the space of all polynomial functions on this space three-valve polynomial then Klein was as according to the guiding principle one should look at the invariance of this action and he came to that invariant polynomial in these three variables are not unique but there are many many but finally he found there are three basic invariance so that all other invariance are described by only addition or multiplication of these basic invariance so he got three basic invariance even a little bit confusing but let us call these three polynomials which are invariant by this regular group let us call XYZ then moreover Klein found these three polynomials are not independent they have relation in the modern terminology he was looking at the Shizuigi resolution of these groups but it's a little bit technical and I don't go to detail but he found relation between these basic invariance and these relation are the following the very for the exohedral group this is the relation between this octahedral group and as I said at the beginning in the classification of all regular polyhedron I said there are two dihedron and unihedron so unihedron and dihedron case and invariant and this look like not particular just elementary simple polynomial nevertheless in the sequel people found these describe the D theory but before coming come let us give table A means this these polynomials I should I am going to call this polynomial so I put not polynomial but polynomial for the reason that maybe I just explained for the reason if you have a polynomial in technically in the two variable case F equals 0 may have some singular point that means it is not there are some fast or some irregular point and that is called singularity so client polynomial three variable if you look at carefully the zero locus define a surface in the hypersurface in the free space since three variable you get hypersurface in the free space then you observe that surface I cannot draw picture it's too much complicated they are smooth except at the origin at the origin there are very complicated singularity which I don't describe for that reason these three these polynomials are called client singularities so at least we come some client platons solid in one to one correspondence client singular let us call this equivalence class table A in the D case is more difficult to explain it his work is more closely related to differential geometry contact structure or whatever and this is too I need one another lecture to explain this so I don't want to go detail in D case this continuous group he studied not client case discrete group jumping element and jumping but D case he was considered in group which is continuous then he look at some central point some unit element of the group and then he look at infinitesimally what happened in the neighborhood of the D group and this concept lead him to the so called D algebra and D algebra is one of the fundamental to in the mathematics even nowadays in physics and mathematics and also many deep theory representation theory are developing even nowadays but I don't want to go detail this D algebra in general too general but then 20 years later than D will have feeling come to the classification but nowadays people called simple D group that mean the group which doesn't have so called some normal subgroup I don't want to explain what is normal subgroup and how simple he begin to classify all simple objects in the D cell and he found all these simple group are classified by so called root system and I don't want to explain what is root system here but just few words I can this study of feeling looking at the combative subgroup subalgebra of the group and he look at the action of this abelian path on itself so called a joint action and look at the Eigen space Eigen equation then solution of this Eigen equation are called root in that sense he called root system and strange this system of roots describe completely classify all simple real algebra and later on I shall later on refer coxstar that this simple root system finite root system are described by classified by diagram so called coxstar diagram instead of writing explicit real algebra I'm going to list up all coxstar thinking diagram coxstar type abcdfg this is the list and later on if I have time I will explain what this diagram originally mean and what it shall imply later on but this is some standard data when nowadays people study real algebra ok this but then we have established some relationship between this coxstar thinking diagram and coxstar thinking diagram simple real algebra in one to one since they classified so let us call table B so I made two table table A and B table A do you remember it was this relation between Kleinian singularity and Kleinian or Platon solid this is the some list of the Platon solid and table B is the list of this simple real algebra and coxstar thinking diagram of course they look completely different shall one find any relation between table A and table B apparently they look completely different but after 40 60 years and half one century later I will explain the example shows that table A implied table B and even 40 years later Briskon shows table A table B implied table A so let me explain what are those implications by the way in this list of coxstar thinking diagram there are A, B, C, D, F, G so there are too many compare the Kleinian or Platonic solid table and the reason the following here we have diagram which contains not only simple edge but double edge or triple edge and this diagram with double edge or triple edge are obtained by folding this simple edge diagram I don't want to explain detail so we shall consider essentially only the simple edge diagram that means A type, D type and E type and then this of this diagram a number exactly fit with this of the Kleinian but that we shall see later let's go to the so I am going to explain this implication table A to table B and table B to table A so first come to the table A to B by the double and coxstar and these mathematicians are active in the first half of the 20th century and it really marvelous coincidence that they are together in the 33 or 4 I forgot in the Cambridge 20 college and they found that what they are studying that exactly the same object but let me explain what I mean unfortunately in the Wikipedia I couldn't find the figure so I took only the coxstar figure but anyhow these people are the next main actors let me explain what they are in the coxstar case he was studying the group of action on the Euclidean space generated by what is called reflection if there is a hyper plane then you make a mirror symmetry the left side goes to the right side this reflection is obviously an isomorphism of this Euclidean space then you are going to consider the group which are generated by this reflection then coxstar finds how to classify such groups he has a very nice technique I don't want to go too much detail but roughly speaking if you have a group generated by reflection and a suitable good action you consider all reflections which appear in the group that cut the Euclidean space in small pieces and then you choose just a component a connected component called chamber then wall of chamber corresponding to some system of reflection then the group is generated by that system and coxstar finds using some angles of these two reflections hyper plane you can describe the group in a way so I don't want to explain more detail but anyhow that technique allows him to disturb this diagram for the reflection group which appeared in the study of the region in this I skipped in the killing of the work I was not to explain details but killing was starting so-called root system we come also the reflection action defined by this root and this reflection group is now called wild group but this wild group okay anyhow so this is what coxstar described his technique using this reflection group this diagram he obtained that is a coxstar path but in the same time in the what is strange this more strange is the study of duval duval look at this what I call client singularity that means the hyper surface defined by the client polynomial it's too complicated so I cannot draw picture but in the it has singular point but in the early beginning of 20th century there were some group of Italian mathematicians who were studying surfaces beautiful study of surface by castle noob or many other people and they developed some very special technique to study the singular point of surface namely that is a resolution of the singularity if you have such surface singular point then you blow it up that means replace this point by a circle by a curve then you get some smooth surface so this is the original singular surface this is a blowing up this is the most elementary case but in general only one curve c is not sufficient blowing up singularity you need some system c1 to cn some system curve and then duval look at this client singularity and he lists up all these singularities then he want to describe this system of curves more symbolically and he came to the idea if you have a system of curves I hope I wrote something this system of curves c1 to ck these are so called exceptional curves then he make a diagram associated to this system namely for each curve ci you associate one vertex I thought to some where I describe this diagram construction so for each what ci you associate diagram vi then if c two curves ci and cj are crossing to each other intersection then you put some edge some edge between vertex vi and vj in this way for all client singularity duval lists up all these diagrams then things happen duval notice surprisingly that the free diagram what I just explained he obtain was the same three diagrams which were obtained by coxter coxter studying the reflection group he use diagram to describe the reflection group but duval was studying the resolution of singularity he use the diagram both diagram coincide that is some surprising that means client singularity starting client singularity using the resolution you obtain coxtending in the diagram so if you look at the London Journal of Mathematics written in the 1933 or I forgot I look at this book carefully then duval was writing that it may be notice that trees of curves which we have had to consider this is dreaming duval's diagram bear a strict homology resemblance to the theoretical simplex whose angle are some multiple of pi considered by coxter he is saying that duval's diagram coincide resemblance he said only resemblance of coxter's diagram and in the same volume of the London Mathematical Journal in the 1933 or four coxter also was writing in the same time in the 20 courage and duval wrote in the volume according to a request of duval I will list up the groups generated by reflection so duval suggested to look at such diagram what are the groups and coxter gets the list which I mentioned already before so if you look at the paper of coxter you get this diagram not only this but he coxter gave more diagrams according to some other reason but anyhow in this way the implication happen that means Kleinian singularity determines coxter's diagram that means Kleinian regular or regular solid determines the real you got this direction this is the what happened in the 1934 in the 20 courage but at that time duval wrote only the strict resemblance that is only the symbolical coincidence of the diagram but 40 years later in the time early 70 or end of 69 this was not resemblance but really geometric background that was found by briskon and grotan let me explain what this is briskon and grotan by the way briskon was my mentor my teacher and let me explain what happened it's again mathematical technical thing so I cannot go too much detail but briskon was studying around the end of 1960s some resolution simultaneous resolution of the duval singularity that is a more careful study of this Ukrainian singularity and I cannot there are many small details but I don't want to go detail but briskon found in order to describe the resolution of the singularity he was necessary to use the data of the associated this simple real that was very strange for instance there is some covering construction I don't want more geometric construction he was necessary to use simple real and at that time he was also it was exactly the end of the Vietnam war time many people go down the way from US and in Paris briskon, gluttony these people are together and briskon consulted gluttony what happened, what is this and then after some time discussion finally they are successful to construct all the Ukrainian singularity using simple real the construction is a little bit technical and I don't want to describe but at least some implication simple real is using simple real you obtain Ukrainian singularity inside this real more precisely this simple real as a set it's a big vector space but inside that there is so called a nilpotent element nilpotent subvariety nilpotent subvariety is not smooth and singular but if you look at the generic point of this nilpotent variety you obtain the Ukrainian thing that is a structure but I will skip so in this way all these four objects in concept we started with a Kratonic theory symmetry and Felix Klein used this symmetry to look at the invariant polynomial and the invariant polynomial defined what is Ukrainian singularity some polynomial then Dubal studies the resolution of this Ukrainian singularity then get Hockster-Denkin diagram on the other hand this Hockster's work shows that this Hockster-Denkin diagram or Killing's work shows that they are classifying simple real algebra on the other hand Briskon showed Briskon-Grotan-Denkin showed simple real algebra construct are trying to in this way all these four objects are somehow equivalent data one determines the other and cyclically so this is a story happened around the beginning of 1970 and I want to add one more comment on this phenomenon this the relation between this Ukrainian singularity and Hockster-Denkin diagram I mentioned Dubal's work on this but this was using the resolution of the singularity but there is another approach using the smoothing of the singularity that is the following before when I was explaining I Dubal said was doing resolution on singularity let us remember what was the resolution if you have a singular variety just blowing up some singular point and you obtain smooth space this is a way and usually this is not a single curve but a system of curve that is the what Dubal does but there is another viewpoint what I meant to call smoothing so let us consider the same problem suppose you have here a singular point which is a singular point at the origin singular surface is a singular point at the origin then this support this singular variety defined by a polynomial f equals 0 then you are going to consider f equals 1 I am sorry I forgot to put f equals 1 and f equals f then this picture surrounding this smooth picture suppose it defined f equals 1 then this is somehow smoothing of this original singular point this is more from the viewpoint of deformation not the resolution but the deformation strangely in the deformation you look some homological cycles here which look like quite dual to what Dubal described in his resolution this cycle is called vanishing cycle and then this idea more in general was studied by Wilner he showed that in general in any low isolated singular point f equals 0 there is always some vanishing cycles of certain cycles I don't want to explain details but some vanishing cycles for the singularity and in this way he found some possible candidate which described the root system and at the Milner time it was not clear he studied this work in 1969 but 20 years later in the end of last century in the 1995 or 6 I forgot there are new trends in mathematical physics where these two objects resolution and deformation can be understood resolution is more complex geometric object and deformation this vanishing cycle are concepts of more symplectic geometry then mathematical physics suggested that there is some duality between symplectic geometry and complex geometry that is now called mirror symmetry and mirror symmetry one of the most mysterious symmetries which people still try to understand we don't have complete understanding nevertheless there are basic work by Konsebiic or Yao and these people they try to understand or categorical and we are still on the way but anyhow there is some duality between dual approach and Milner approach that is the point I want to add so far I explained the history and this is next paragraph is what I have done after this but I'm afraid there is no more time I don't know if you give me 2-3 minutes at this meal I just explain so thank you very much so then I come to the new concept elliptic singularity and elliptic root system in the previous table C which I explained before see here we saw some equivalence of 4 objects singularities some symmetry group or cocter diagram simple real I'm going in the after shortly after this griscon grotendix work I came to construct some generalization of this crinium singularity which I call elliptic singularity for several reasons and this elliptic singularity I shall give later just in the table but then it is a natural question what other for corresponding other part point what other this platonic symmetry whether since all platonic symmetry are classified is there anything more and actually it exists corresponding to the elliptic singularity which I studied I show later this symmetry is planar symmetry that means this platonic symmetry the symmetry of rotation of spheres but there are symmetry group on the plane how plane axis symmetry and planar symmetry theory corresponds to the elliptic singularity and then I'm going to find this what are the diagrams for the root system and what are the simple real that let me show the table so around the 74 I came to these three polynomial that look like very similar to the Kleinian singularity but some power are a little bit bigger in the Kleinian singularity these are some 5, 6 or whatever but this become bigger and then for these correspond to the platonic symmetry correspond to the Heisenberg group symmetry Kleinian singularity correspond to this these I call these singularity elliptic then elliptic the singularity then most important now I want to explain then what are the hoxter-denkin diagram what are the root system since all classical finite root system are classically classified by Tilling and Carlton it should be the end but there are infinite root system which are still classified by generalization of hoxter-denkin diagram so I should give some tables these are the some tables of this elliptic diagram and if I have time I could explain detail this connection shows some what are the radical and what is happening and this shows very close connection also the platonic symmetry but I cannot explain now so quickly so I just show some tables of this all symmetry and this is the end ok thank you very much I am calling one thing and elliptic real liver are also just described last year using very strong radical extension from finite area but that is another subject which we show here any questions for Professor Saito so I wanted to ask one quick go ahead Mark how does this relate to the Makai classification of singularities a Makai case is this finite type you are talking classical Makai yeah classical Makai correspond means you look at the character for this regular procedural group and this basis of character describe exactly the coxter dinking that is a Makai then instead of that as I showed you here in the Makai case he used a regular polyshedar group of the symmetry of polyshedar but if you look at the Heisenberg group of very particular type which are used for the description elliptic singularities and you look at the character of this Heisenberg group then again you obtain diagram like this very another aspect I skipped Hi you mentioned mathematical physics towards the end I'm curious what kind of physical systems to these singularities arise in I don't say directly this singularity corresponds to some physical system but what I mentioned about this mirror symmetry I should better professor consult professor Hikaru but this appears in high energy physics and I know only mathematical size and at least I don't of course I know in the string models there are 26-dimensional space and compactification but in this model there are type B2 models and type D model might be connected to these singularities but better not ask me but more expect so can I ask a quick question about this classification so with the usual say simply algebras somehow type A is the most natural with the coxter groups type A is always the easiest to study so what is your your analogous thing is this type A so do you do you think this should somehow be easy to study these elliptic algebras elliptic groups anything that is one question hard question this is still just started I cannot answer definitely but our experience showed that strange that this rank one case in principle usually this sort of thing is the most simple case easy case but it often the case in the study of elliptic groups system rank one case like this or this one this one behaves very exceptional I don't know for instance mojira group elliptic mojira group acts on this root system that makes theory quite mojira invariant that means the elliptic mojira but this structure for this exceptional rank one case structure is more complicated I don't know the reason that is more experience of my collaborator Yoshisa Saito or Iohara Togumi what they think that they are very particular nevertheless of course I'm sure it should be quite interesting to look at carefully already in this case you may spend years to understand it should contain many interesting stuff but I can't the rank two Taipei case should be simpler or even crazier in some sense the general case is more some general framework of theory and this case needs some particular treatment thank you there are no further questions then let's thank Professor Saito again thank you all for joining