 Je veux remercier les organisateurs de cette conférence de m'avoir invité à parler. C'est un immense plaisir de parler en honneur de Luc et de Lusie, qui a été toujours pour moi une figure grand paternelle. D'un voyant, c'est un peu intimidant. Donc, l'ambition de l'English, ça s'est passé, je n'ai pas de résultats récents. C'est-à-dire qu'il y a de l'intérêt. Donc, j'ai décidé de parler de quelque chose de personnel qui m'a élevé pour moi-même. Ce qui peut être vu, c'est quelque chose d'exercice dans l'élusile classique, dans l'azubergeometry, dans l'oxylongole. Mais j'espère que c'est quelque chose d'exercice. Ça peut être... Donc, j'ai envie de parler des bandons d'exercice. J'aimerais vous présenter, si vous avez une nouvelle présentation de l'exercice de l'élusile classique avec des bandons d'exercice. Donc, j'ai... J'ai une motivation. Il y a deux formes. L'une est... L'une est... Quand j'étudie l'art space, l'art space de l'azubergeometry, quand je fais la calculation, c'est-à-dire que j'ai besoin de choisir les coordonnées pour qu'on puisse trivialiser les bandons d'art. L'art space a des bandons sur les variétés base. Et vous voulez vraiment trivialiser les bandons, pour faire la calculation. J'ai envie de comprendre quelles sont les structures, les structures de l'intersec qui permettent d'utiliser les trivialisations pour faire la calculation des bandons d'art. Donc, la deuxième motivation est complètement différente. Ça vient de mon travaillant sur la vibration de des variétés basées et dans ce travail, j'ai eu des rencontres avec Bougoumoulouf qui parlent des formes symétriques. J'ai eu le temps de comprendre ce qu'il signifie par les formes symétriques. J'ai essayé de défendre les formes symétriques. J'ai essayé de défendre les formes symétriques. C'est la motivation. Mais nous allons commencer avec une définition très basée. J'ai travaillé avec K qui est un sens de caractéristique 0. Quelques places peuvent être des caractéristiques, mais je ne veux pas être complémenté par ce problème. J'ai commencé à regarder le catégorie d'Arctinium K qui est le catégorie local d'Arctinium K qui est un sens résiduel de K. J'ai K, J'ai K et d'Arctinium K de la maximale idée. Maintenant, quand X est une variété de variété majoritaires de K, il y a de la définition de l'Arctinium. Et pour chaque Arctinium K N, l'Arctinium K puis on peut défendre le JRX qui est le foncteur sur le T-angébres. Pour le T-angébres A, c'est le T-exe, A-KR. C'est une restriction de X de R à K. Mais parce que j'ai cette map de R à K, c'est des fibres de X. C'est un bundle de X, parce que j'ai la map de X à A, par la map de R à K. JRX à X, c'est le jet bundle. C'est un cas de très familial, c'est la r1, c'est le rang de douleur des douleurs. JR1 est un jeu de tangent en X à O, Mais en général, donc quand l'axe est smooth, alors jRx sur l'axe est smooth. Et il y a des types de 5 bundles, mais ce n'est pas un bundle vector. Il n'y a aucun moyen d'adresser deux différents objectifs, si vous êtes dans le cas de R1. Mais en tout cas, il y a une calculation familiale dans la théorie de formation. Quand vous avez R, vous avez un objectif nocturne et vous avez un idéal, une dimension idéale. C'est une dimension idéale pour choisir la base pour identifier cet idéal. Ce type de R est un type d'axe et un R. Vous avez un Rx et un Rx primaire. Quand l'axe est smooth, c'est un objectif, c'est smooth. Et c'est une calculation de formation théorie que c'est un objectif sous le bundle tangent. Quand vous avez deux points de JR, deux jets, c'est le même objectif. Et la différence est un objectif sous le bundle tangent. C'est une calculation de formation théorie que vous savez. C'est le premier factif que vous allez utiliser. Je veux qu'il y ait des questions. Je veux aussi vraiment explorer les structures de l'ingénie actinienne. Quand vous avez un R1 et un R2, deux ingénie actinienne, vous pouvez définir qu'il y a des casques. C'est le roulage de l'ingénie actinienne du R1 et du R2 consistant d'un élément qui se met entre les mêmes ingénieurs. Vous avez vu que le roulage est un enjeu local. C'est un mod de mod M1. C'est le même que le mod de mod M2 qui est un enjeu dans le catégorie de l'ingénie actinienne du local de l'ingénie actinienne. Vous avez un r1 et un R2 de l'ingénie actinienne. Vous pouvez utiliser les produits de l'ingénie actinienne du R1 et du R2 en x. Si le r1 est un R2 dans le r1 des r1 et un R2 est le de l'ingénie actinienne du R1 du R1 T2, T2 square. Et en ce cas, le Jfq2x est un bandon tendant à produire un bandon tendant avec itself en X. C'est la première structure. La seconde structure est de toute façon plus intéressante. Quand vous regardez le R1, la tendance est en R2. Et ensuite, vous pouvez voir que le Jfr1, le temps de payage de R2, est le Jr1 à la jette de la jette. Et donc, nous avons deux différents idomorphismes. Ce qui est le Jr2, le Jr1. Donc, nous commençons avec plus d'intéressions. Alors, quand vous... Donc, j'ai trouvé le plus intéressant objet de cette histoire. C'est ce rang de Rn, qui est le R1, le temps de payage de Rn. Donc, ce que vous avez, c'est le temps de T1, T2, Tn. Ce n'est pas de la square du maximum d'idômes, mais seulement de T1 square, T2 square, Tn square. Donc, c'est le rang de dimension 2 à la jette. Et puis, quand vous regardez ce Jfrn, le temps de payage, c'est le temps de la tendance, le temps de payage. Donc, le temps de Tn. Et autre chose que vous pouvez observer, c'est que dans le rang de Rn, vous avez toute cette action des groupes symétriques. Mais aurait-être qu'il y a des祭res d'induction, ce sont du rang d'induction, c'est le rang de la jeteur, ce sont des confinements pas qu'il y a, et d'autres, c'est du rang de la jeteur. Et d'autres, c'est d'autres, c'est des confinements, d'un facteur infactuel, d'un facteur possible d'identification. Par exemple, un autre point, les tangents square of X ont une option naturelle de S2. Donc, c'est l'observateur qu'on va vraiment utiliser dans le long de cet état. Mais maintenant, let's look at the action of the just kind of very elementary algebra. I look at the action of Sn over Rn and I claim that if I if I know Sn is this actinian ring kU mod U to the n plus 1. So this is the ring that is that is the Eugengeski so to speak. So we have the that is the Eugengeski so J of N X is the the Eugengeski so J of Sn of X and the kind of the fact is that Sn is isomorphic with the fixed point of Sn on Rn. But here I really had to use the characteristic of K is 0. So this kind of curious fact is going by just map U into T1 plus Tn. Of course, because you want something that is invariant and if you do the formula then you what you said you and we're going to map to n factorial of T1 Tn. So we better to have a calculation K to not 0 to be 0 to not to be have point with Rn factorials. So in generally without this condition that you can as any count is P with a calculated divided power. So we can develop by replacing the Eugengeski by the Gerski with a calculated divided power but I'm not going to do this. I just stick myself to count this 0 case. And so by functorialities of the Gerski you can identify Jn just as a JSN of X. So the Eugengeski with the fixed point of of Jrn of X SN. So the Eugengeski that is the on the map from this KU divided by Un plus spectrum of this SN into X can be identified with SN fixed point in the range of space. This is one of the first observations I will use it to do some to do the construct of the formal flow this is going to happen to leave some kind of nice algebraic construction of the formal flow. So in the elementary differential geometry when you have a vector field then the pica theory is going to associate with some flows whose tangent vectors is your given vector fields but now I want to do it on the 400 vectorities and you cannot have what you can have is the formal flows. Okay, so let's try to work out the formal flow so it is actually quite so you have the Jn of X it is to an arc space which is the J of SN of X and as SN can put into when you take the limit of SN then you get the ring of the formal series so as a reason you have the limit of Jn of X in the J infinite of X so are the X point values in the space X point is value in the formal series and this is the even arc space alright, so what I want to to construct a canonical map on the section of the tangent mandons to the section of the arc mandons so I claim that it is such a map so the map is this so when you have when you have a map from X to the tangent of X this is the vector fin V then by functoriality of the tangent mandon you can go again of T of V to T square of X and so on on the way to Tn of X and this is Tn minus 1 of V and I call V to the N it's kind of differential stress in N time by the same vector fins it's the Tn minus 1 the composition of all this so when you look at differential operators you really just the composition N times the same vector fins and this V of N is the section of X into J of R N of X which is the tangent N X 10 times and moreover this thing is symmetrical it is some kind of direct calculation that can V N is fixed under the action of S N so V N is actually a section of the of the use of that scheme and then they are compatible that is give right to V infinity's section of of your arcs so this means that every time you have a vector fins you can glue them together to have really an arc at every point of rights so this is the first you know the first use of this formalism and just scheme to reinterpret things so the next thing I want to talk about is a fight connection a fight connection can also be interpreted very nicely in this language of jet bundles I just need actually here I need arc 2 which is so the tensor twice of the new one numbers so I have K of T1 T2 divided by T1 square T2 square I also have this Q2 so the arc 2 is R1 tensor with itself I also have EOK yes yes and I have Q2 which is the sum of R1 plus itself et now I have K of T1 T2 divided by the maximal ideal so this is portion of this T1 square T2 square T1 T2 so as a result I have this diagram so this is J of arc 2 of X and it map to J of Q2 of H which is nothing but the tangent bundle time itself we have two maps with two tangent bundles and think map to X so I have two different map P1 and P2 and this is PR1 which is PR2 and I have involution the obvious involution nothing on both Q2 and R2 so this let's go on this tau of Q and this tau of R so let's remember as I already mentioned JR2 of X is not a vector bundle of X however it is a tangent bundle of a tangent bundle so both P1 and P2 are vector bundles so you can add fibers and also the map from JR2 of X to J of Q2 of X so this is this critical map this is this one let me change the color this one this one is not a vector bundle but it is a torsos this is a tangent torsos so here really you have to use X is smooth otherwise the fiber can be empty right so here the fibers is not empty if X is smooth so so so this is kind of the structure on the number 2 you have kind of complicated structures so we have a tangent space cell product with cells and then we have a tangent bundle torsos over it right and if you look at carefully what you get here is actually this give right to a bi-extension of Tx times X by Tx it is exactly the bi-extension on the axiom that you need to to require here it is the axiom of the bi-extension alright and also you can also remark that the bracket of vector fin can be very nice in this diagram for instance if you have let me change the color otherwise become too complicated diagram so if you have 2 vector fin say I have v1 here and let me draw v2 on this side then my functorerity have a tangent v1 v2 on this side and tangent v1 on this side so when you go on one way and go to the other and somehow you have to switch by the evolution to make sure you map to the same point then you map to the same point the tangent the tangent of itself so as a result the difference is now is now a vector fin so let me write now the reinterpretation of the reinterpretation of the reinterpretation of the of the of the bracket of vector fin in this this is the tau tau dv2 dv1 dv1 bracket v2 so this kind of line you can define directly by geometrically the bracket without using the how it act on function so this is the first remark I think the secret of the remark is a bit more important if you look at how you can define a connection you see that a five connections so connection of this vector you know tangent bandons is exactly the same as the trivialization of the both of the by extension that mean you have to give yourself a section a section of this map that respect all the structures that you can require this is what has been admitted when you are over one TX but to other TX also and when you check the formula that really give you when you have this and look at this type small game which composing vector fin then you get the how you can have a word to derive one vector fin respect to the other because you have this type of section to do it this is really exactly the same how to arrest man defy connection but you have to carefully want to be additive on both variables and that mean you require that this trivialization to be by extension and the other remark which is if I really nice but in difference in geometry there is this concept that I don't have time to understand which is the toxin free toxin free of fire connection so there is some new in the theorem le vichimita theorem is required of fire connection to be to be compatible with the demand form and also be toxin free and in this language toxin free just mean it is symmetric and that mean the the section had to be compatible with its evolution so so that be that is the same symmetric trivialization of the by extension and for this you can see that you know if you want to actually to be smooth if you want to have this map to be to be subjective and and you can also see that the there plenty of diamond axis of fire existent of fire symmetric of fire connection of fire connection when x is smooth and the fire because the old structure will just believe in some H1 of x from in some quality so it doesn't have H1 alright so the next thing I want to say that in this in the you have now the the construction of what the you call the zero music in the in different geometries in this context and that mean if you have if you give yourself a symmetric of fire connection so in different geometry you start having a metric and then there is nothing that minimise distance but but you can formulate in pure language by saying that if you have a symmetric of fire connection then that give rise a canonical map from the tangent bandons into the jet bandons so not just a spray of section but really a morphism from the tangent bandon to infinity so how you can do this so the fire connection the symmetric of fire connection so what it does is to allow you to write the t square of x into tx time x time tx time tx 3 times so instead of having a tx torser over tx time tx the symmetric of fire connection allow it to to to align t square of x but the thing that you have to remember that these 3 copies of tx do not play the same roles you should have the the first one on the map with tx and the third coordinate is the trivialization of the torches and of course you can, while you have this you can just keep repeating t square of x and because you have the first formula you can just develop it completely into the many copies of tx and if you do so what you get is tnx it's just a product of many copies of tx over x but now the point had to figure out this is indexed by the subset so i is a subset of 1 2n so non empty subset and obviously you have option of symmetric group on this side and on this side just acting by permuting your subset of 1 2n so now when you have this kind of completely you can split tnx you can have a map from tx to tx to tnx by this side that when you have a a tangent vector you just map it to v and copies for the i of 1 and just put 0 a v l s y for i for the subset of list 2 so that is how you have this kind of so that give you a map from tx which is obviously symmetric so this is fixed under sn so this give you a map from tx t snx and to the limit by going to the limit you have your map from your tangent bandon into the arc space so j infinity of x so basically I give you a what every vector not a vector field just a tangent vector then you can you can have an arc a formal arc but if the geode is a formal arc alright so this is the how we can interpret the geodesic you know the formal flows you can interpret a five connections and you can interpret geodesic in this term of jet bandons so now I want to you know to do some to go to approach this question what do you mean by differential of a symmetric form which is not clear at all look at he had some definition of closed symmetric differential form but I just cannot get why he call it closed so I try to at least try to define what is a differential of symmetric forms okay so you can have you can see the omega one so the sheet of one form you can be see as algebraic function from the tangent bandon to A1 which is GM equivalent regime GM up on the fiber of a tangent and up on my scalar to one and so you want to what I try to code I don't know if anybody have studied this kind of strange things but I code cubicle cubicle differential forms omega nx is going to be function on the jet bandon nx into A1 which is GM to the end equivalent so we see rn is this 2 to the n dimensional ring and you have all this action of GM to the end by scaling individually the coordinate to the end and you also have action of GM to the end on the A1 by the product this is this makes sense when you differentiate when you have f a function of tn to the x to A1 which belong to omega rn of x omega rnx then df which is a tangent of 10 nx to A1 will belong to the omega rn which was so this equivalent condition is simply the y1 to have to be the differentiation so but again all kind of the difficulty you just have many way to identify rn with the tn of x so you have some kind of simplification structure you have the different map from one you have you just have many map from ok, let me you have look at the map from one to so strictly increasing map and so you call this thing maybe then by i the one that missed that missed the i in its larger set so for every i you have you have a map from you have a map from d i d rn i from omega rnx differentiation into omega rn plus one of x and obviously you can form the unified d of rn is going to be d rn0 minus d1 rn plus etc and in that way you just have another complex of cubic forms from all x of omega r1 of x, we see just using omega r1 but then have omega l2 x etc which look like complex but which is way larger excuse me i know i have questions it could be better to write 0 n minus 1 to 0 n and then the user convention and it's implicit object because you have d0 minus d1 and d0 misses 0 but 0 is not there ok yes yeah i'm sorry hope we can make it change size some part ok so what is very nice is this again you have action of symmetric group so s2 of omega r2 and sn of omega rn and this is very complicated but it's in the coherent shift so omega rn of x a coherent shift over x which is again the point that is kind of so you can use if you now locally so locally for zhagis ketopology of x where you have these of five connections symmetric of five connections so that you can trivialize your on your jet bandons then then you can do some kind of you know, calculation some you can do calculation to try to write down what exactly this omega rn x is which is a bit complicated but it's not very much difficult then what you get in this I just write the result I don't want to bother you with the calculation so you have a direction over on part or the way that you can partition so i you can i2im is a partition of the set 1,2n and then for each partition you have this sigma of i is kind of the subgroup of sn which is normalized normalized the partition that means you can allow your permutation to permute also the partition so and then with this you can have this induce from sn to si into sn of omega 1 of x to the m so it's not very sympathetic because you know I have a number of partitions so it's a bit too big to be interesting somehow but at least what is really nice it is so because you have the action of symmetric group after all this omega of rnfx if you look at the the sign the part when symmetric group part by the sign so look at the direct factor rnfx let me go in minus where snf by the sign representation then by this formula just by this formula you can prove that you can prove that this is the usual differential forms so the the point that when you look at this formula the is very complicated but on the factors can have a lot of symmetric forms and very few alternative forms so there's only one alternative form just one of the factors which is omega nfx it's a very meta exercise with the patient symmetric groups and so from this you can see that the usual and also you can see that the your d from omega rnfx to omega rm plus 1 after you map the minus part to the minus part and so the the round complex the usual round complex is a direct factors of this of this of this cubicle complex so I really don't know what this cubicle complex is but at least the one part on the other side have plenty of symmetric forms so it seem to start to be in answer at least some kind of approach the question how to define a differential of symmetric forms so let so now let's try to figure out what is the symmetric part so of omega rm plus of x so where sn are trivoli just by the same the composition you can see that the cmn of omega 1 of x there is a kind of equal map is a direct factor of this so definitely then you can really define what is the difference of a symmetric form so there you can see the definition because this cmn is a direct factor of this but this one is a sub of this it's a sub from sub back door but this is just way bigger you can compute it then when n1 to 2 what you get is a sim2 a sim2 of omega 1 of x map to omega r2 plus of x and map to omega 1 of x so the the symmetric part of the cubicle complex is an extension of omega 1 by the sim2 of omega 1 and this look familiar because this is the same form as the p2 p2 is two principle part a single principle part we can define this principle part and you can prove in this in this case that this is the same so it would be really nice if this story continues but for n2 to 3 then this omega r3 just way bigger it's a bit bigger it's contain the p3 but there's another factor nn grows this is just really big I have no clue of what it is so at least from this analysis you can define a differential chemical differential of a symmetric differential forms but the image is something rather gigantic so I guess we have to project some factors and the projection is 0 that may correspond to the definition that I couldn't figure it out yet so I think I have to say about this I'm very sorry for this very elementary story but I think probably what I can and you can find something that may be of interest to look so that's it, thank you very much okay, any questions? comment? okay so I remember that a few years ago came also another approach for differential calculus but I don't remember exactly my souvenirs it's a little bit close, have you looked at this at their work? yes but it's different I think it's completely different I have a question what can we say part of the cohomology of some differential complex or do you have some explicit example? no I don't, I think it's interesting to learn to calculate but I don't know how to compute it does hope theory tell you something whether or not so I mean this notion of closeness you define is automatically for thinking exist most projective I mean I've got hope to connect this to hot theory but I know I'm not there yet this is the beginning of this topic I don't have so what kind of knowledge I mean what I can tell it just kind of similar population if you look at the differential of the two symmetric forms and it's just by formula it's the same at a costum sum that you have you to define the effect method associated to the Riemannian forms so it's a sign that this construction may be not meaningless but I really don't know what is exactly the relation with the connection in Riemannian geometry this is what you meant you meant to so you know you went in the proof of the I haven't I mean there is a connection but I haven't finished an exercise yet but at least in the proof of the Liberti Vita theorem and you start with with the symmetric forms second symmetric form then when the proof we make some kind of alternating sum of derivations and from that you can then you can derive the effect connection and this kind of alternating sum the costum sum exactly the derivation that I use here so what they call the Christoffel symbol in classical okay I just a remark about the last part n equals 2 reminds me of an unpublished note of Gaudendick who classified order 2 thickness of some of X in terms of such extensions I think I sent you his note thank you yes it's possible do you think it would be possible to classify thickness of order 3 etc. using your using this formalism yeah maybe but the kind of let's talk about order 3 because you know it contains the p3 but it contains some other things I just do not do that yes but in the classification maybe p3 is not enough so maybe this would be this would give the right formalism what do you think and also is there a relation with the bar construction some place big tens of products and then inside you find all these symmetric exterior etc and no idea so they simply should construction you know these are these are but I'm afraid we lost no hello I'm here but I just do not have an answer to your question no I am just okay there is a question from the audience it is is the exact sequence for symmetric power 3 symmetric exact sequence can it be written down I would read I don't remember but I don't remember it can tell you what it is that is a question from the audience any other questions if there is no further question thank you professor goh for his talk