 Okay, so welcome everyone to the afternoon session. Our first speaker, Gerard Duchamp, will report on Cleese Stars and Chappellabba. Thank you dear Gleb. I must first of all thank Maxim and the IHES team and my co-organizers Gleb Koshevoy and Hognok Min. And of course you all for attending this talk. And I will speak about Cleese Stars and Chappellalgebras. Why a tangle tail? Because you will see that when you begin to calculate within non-competitive series, you have to manage with superpositions. And this is a notion we made clear with Gleb Koshevoy and Christophe Tolleux in our publication in L'Ottar angiant, twisting and perturbations. And we will speak mainly about perturbations and its combinatorics. So now what is the goal of this talk? We have three goals today. First I will take some time for motivation, explain how this is motivated and what is the root from the very first motivation that was to extend the polylogarism and to some mat overflow post which made the collaboration with Dary Greenberg possible. And after, we will side the main theorem and it is a general theorem. We have two proofs but maybe due to lack of time I will not go into detail. But this result takes place into three variations of the general theme which is the linear independence of character on a co-algebra. And this is the subject of our publication. So we will see, this is generality but we will go into detail. This is more or less generality because it is done on enveloping algebras. The main result will be generality and it has been discussed, condition has been discussed. And after, we will see also examples and examples are made mainly on by algebras formed from the free algebra by adding some co-multiplication. And after, we will end with some remarks about the structure of Hausdorff groups. Hausdorff groups, this term is inherited from Borbaki. It is a group of group like. Borbaki calls Hausdorff groups a group of primitive series. So this will be the group of characters, more general than group likes in this context. And we will see how we can devise a local system of coordinates and have some identities that cannot, in my opinion, be proved otherwise. Of course, parts of this work rest on iterated integrals, so it is linked with Dyson series. Now, we go to the initial motivation. Initial motivation is from hyper-logarism. You have, in the complex plane, you take points, different points or different singular, which will be seen as singularities a1, a2, an, and you see that the functions that are integrated, it is 1 over z minus aj are all homomorphic within the plane without these points. Of course, as you have a monodromy phenomenon, you must base yourself on a path. So if you take a list of singularities, possibly with repetitions, all singularities are different, but your list can be with repetitions and a path, avoiding all singularities from z0 to z, and you form this. Of course, you see that this quantity does not, does depend on the path, but it does not depend on the, it does depend only on the homotopy class, meaning that if you take two paths that are homotopic, of course, the result is the same now. As the result depends on the homotopy class, it can be seen as a function on the universal covering, or you can work, sometimes it is easier, on the cleft plane, which is a section of this universal covering, you take the singularities, and you take a direction from which the singularities are seen different, and you take half-race in order to avoid the logarithm phenomenon, which consists in turning around a singularity. So now, how do we change this with our concern today? As I said, these functions take a list of singularities and a path. As we have said, we work on a simply connected domain, so only the two, the the extremal point of the path, the initial point and the base point and the end point, and we will shift from a list of singularities to words because we said that to each singularity we have attributed a letter, so now you have a word. So you have here, you have done a mapping, which takes an end point, I will make this clear here, because your base point is given, transfer all, and it takes the end point and a word. So now it depends only on a word with variable coefficients. So this is a series, which is the generating series of all your computations, of your computations that you can imagine, taking a word and as monomial, and it is a non-commutative generating series of your hyperlogarism. Now, why do we take this generating series? Because it has a lot of nice features, it fulfills, it is a solution of non-commutative differential equation with left multiplier. I will not go back to this because I went many, many times here and elsewhere since 2017, in Cap 17, Cap 18 and Cap 19 on the study of this series. And the left multiplier will be proved as primitive and as you know, if you have an outdoor group, it is a sort of lig group with the algebra, the primitive series. So we can prove as this left multiplier belongs to the algebra of our group, as at one point, which is Z0, it is a group line, it can be said to be group line for all Z and it is a shuffle morphism. We have seen also the second feature, which is linear independence of the coordinates. With respect to C, it is not difficult to prove this with monotromy, but it can be seen to be independent of larger sets of scalars, larger rings or larger even fields of functions, which will not be the subject of today. But if you go to the slides of Cap 19, you will see this study and which will be our concern today, it will be at the end of this talk, will be the factorization of this character in elementary characters. And as it is once it is factorized, once this series is factorized, you can delete because you see that when you approach the singularities, you can have some divergences. So you can re-normalize by deleting some factors of this factorization, which is this point. And we will see at once what is the problem of extension to some series because you see that here your quantity depends on a word. So you can linearize this mapping and say, oh, it depends on non-commutative polynomials, but what will it be if you, instead of a word or a polynomial, you put inside a series an infinite sum. So you cannot put every infinite sum avoiding the divergences, you have to put infinite sums that are in its domain. We will see at once what will contain the domain. And in particular, this domain contains easy to compute with series. And these series are expressed in a very useful language, very easy to compute in language, which is linked to automata theory. And this is a clean star. If you take a series without constant term, you can take the series 1 plus s plus square plus until infinity. And it is 1 minus s inverse for respect to the concatenation in the algebra of non-commutative series. So now it is a detail, but the Dapo-Dalilevsky was reading from left to right and reading from right to left. And of course it is to match with a more recent bibliography. Now what do we have? We have done the hyper-logarism. I recall that hyper-logarism takes you a word for a non-commutative polynomial and it gives you a holomorphism function. So we have evoked argument of the proof such that this hyper-logarism which takes a word and then a non-commutative polynomial is a shuffle morphism. If you put in as an argument the shuffle of two polynomials, you get the product of these two hyper-logarisms. And how to extend this? We have here a space which is a space of formal power series. For these spaces, we like to decompose a series into a homogeneous slice which is the sum of all coefficients for all words with the same length n. And as our alphabet is finite, the number of singularities is finite, this SN is a polynomial. So you can substitute this SN into hyper-logarism. Now you have to consider as in your source, you have your series that can be like that. It is just not other than the sum of its homogeneous components. You have to consider this series in the target. So this series is in the target, may not converge. So we ask that it converge unconditionally means in French they say convergence commutative and sometimes in the Anglo-Saxon world, say commutatively convergent and unconditionally means it converges whatever the terms are. And if it converges whatever the terms are for the compact convergence, which is convergence on every uniform convergence on every compact, it is a standard topology on H of omega in order to preserve an electricity. We say that this series is admissible as an argument of HL if and only if this series is unconditionally convergent. And this is why we ask this. It is because H of omega shares with finite dimensional topological vector spaces. We are on C here. The nice property that if a series is unconditionally convergent for this compact convergence, then it is absolutely convergent. And we need this to prove that the shuffle property is preserved in the new domain. So now you have your polynomials. You have your holomorphic functions. It is of course a subspace of all holomorphic functions in omega. And you can extend these two series, which are members of the domain. The domain is then, as we said, the series such that if you decompose the series as in these homogeneous components, this series converges commutatively. Now, so if the main result is if you take two series in the domain, their shuffle is in the domain. Of course, one is in the domain. So you have extended this into the domain such that the shuffle property is preserved. And it is a character with values in H of omega with values in commutative algebra. So now, as we want to give examples, we take the polylogaried. The polylogaried is a special case of deeper logarithm with some asymptotic condition. I don't go into detail. So please admit that it is a special case of hyperlogarism. So it is the same as it was for hyperlogarism. The domain is the set of series such that if you decompose the series, then this series converges unconditionally. And now what can you substitute? You can substitute as we, it is a motivation of our title, clean star. Each time you take an expression or a series and you of course check that it is without constant term, the clean star of it is justice. So what will it be here? You take x0, you have two singularities which are 0 and 1 in this example instead of a1, a2, an. And you have x0 and x1, the letters indicating to which around which singularity you integrate. And x0 star is just 1 plus x0 plus x0 squared plus x0 cubed and so on and so on. And this you can prove of course it is an easy exercise that it is in the domain. x1 star is in the domain and you get these values when you compute and which will be seen here and it is why it is called star of the plane because you have x0, x1 which are two independent letters. The two dimensional space alpha x0 plus beta x1 is a plane and star of the plane is exactly these guys and these guys we will see how they can be seen as characters. And it can be proved that if you consider alpha x0 and beta x1 in the previous slide, you have the general formula is this one. It is the star of alpha x plus beta alpha x0 plus beta x1. So as it can be decomposed and as this guy can be seen as alpha x0 star shuffle beta x1 star, the ellie of this is not other than this product of the ellie. And one can prove it is a lot of easy computations. If you put alpha here, you have z to the alpha. If you put here, you have z to minus beta. So it is a product formula which was seen here. Now we go to the next. It will be more and more algebraic. The first motivation was analytic but it will be more and more algebraic. Every common character is of this form. Of course, you have only one character due to the fact that we are in the free algebra and you have only one character such that to each letter you may correspond alpha x. But it is an easy but not immediate exercise to prove that it is a star of this linear combination. And now you have shuffles with perturbations as after the word we call them with Gleb Koshivei and Christophe Tolu, it is a perturbation of the shuffle product. The shuffle product is only based on this recursion with the same unit. And if you can perturbate it with the shuffles, it takes two letters. You can be linearized it and from this quality of this operation defined, you can prove that it's associative or commutative and so on. So you have the shuffle with no perturbation. The shuffle with the perturbation which is given by the addition of indices. The twill filtration which gives a bi-algebra which is not enveloping algebra. These two give enveloping algebras but this gives bi-algebra which is not enveloping algebra. Due to the lack of time, I will not go into detail for Hadamard. And this is a table of shuffle products that we see in the literature beginning with Ri and Hoffman and Kostermans who is a student of Mien. Mien and Mien considered a shuffle for their computations. These two were considered in order to filter the superposition terms of the shuffle. And as was evoked, we said that we considered twistings and perturbations. And here it is a non-twist expression and you have this perturbation and this is the term which is added. I passed the others, save the one with Marchion maybe who considered the generality with perturbation which is a law of algebra associative with unit and it can be non-commutative and then the five shuffle associated with this is not commutative. So the common pattern is the following. You have the beginning of the recursion of shuffle with perturbation. Now what is the initial motivation? It is a post I cast a lot of flow and of course this post is what is written in clear here and it is the following. You take a Li algebra for Irving without zero divisors. It has been discussed that this condition is important for the conclusion and you take the enveloping algebra and you consider the standard decreasing filtration which is just the product of the term in the augmentation ideal. Now you have this filtration which is decreasing. You take the orthogonal filtration with a little shift of index and this is in the dual of this enveloping algebra and you can show that this filtration is a filtration of convolution with respect to convolution. So as it is increasing and the filtration with respect to convolution, the union of all the sectors is a convolution sub-algebra of a few stars. It is not in general in the polynomial examples it is not all dual. So we see that you have a commutative as I said. So I pass this or maybe not. It is a way given to us by Gleb Kochewoy which is very nice how to compute the fish apple. You take your two words. The first word you take you put it vertically. The second word you put it horizontally from left to right and you allow from this leftmost low corner to this rightmost upper corner you allow passes with east, north pass and diagonal pass and to each pass you can evaluate each pass as a term. For example if you have a diagonal term I will take five of the two letters and if I have a horizontal pass I put the letter in the horizontal word. If I have a north step I will put a letter of the vertical word. So now I can this is another example. I pass this which is Radford theorem. I pass as it is a dualizability here. So we have considered a fish apple and we can say that K of X with this fish apple providing that five is associative algebra and we can prove that due to the recursion if you endow this algebra with a co-product which is a d concatenation which is dual concatenation and epsilon which takes a constant term of every polynomial you have a b algebra. In fact due to the fact that this delta is nilpotent delta plus you take delta plus and you repeat this procedure due to the fact that this delta is nilpotent then this algebra is in fact the hope for algebra because you can compute the inverse of the identity because you can put identity equals epsilon plus i plus i plus and then you compute the inverse with the usual series. Now you want to go to the dual so if you want to go to the dual your decon concatenation will take the place of the concatenation and your fish apple will take the place of a delta fish apple. So you have a technical condition on the structure constant in order to warranties. I will not develop but the slides are available on the site. What is interesting is that if you have a fish apple the characters on this algebra are stars of the plane which means that you take a linear combination of the letters if your alphabet is infinite this linear combination can be infinite it's just a linear form and you can you put a star on it it is star of the plane and the as you know in every d algebra the characters of the the algebra itself compose with respect to the dual law of the delta. So you compose these stars of the plane with the fish apple itself and this is what is not so easy to prove combinatorially. I mean in a pedestrian way but if you take the into account the fact that the fish apple of two characters is again a character you just have to test this formula on letters and you can prove that this follows the recursion means that if you have this as you have firstly the first term the second term and the perturbation and you have an identity which can serve to analyze groups housed of groups of of staffel for example which are not so and this is now our main result with Darish Greenberg and Hong Nocmin means that this this property holds for every by algebra but in you know in enveloping algebra the monoid of character in is a group here for by algebra you have only a monoid of character so we have to take this into account so we start with a b algebra and we we make the standard decreasing filtration as was the case for enveloping algebra I take the orthogonal filtration which is increasing with no surprise it is a filtration which is compatible with a convolution product and the the conclusion is the set if this can be done on whatever the ring of coefficients but now you have to take an integral domain and for an integral domain this group of characters is linearly independent with respect to this convolution algebra so as an example I can give an example which is a which is with only an univariate polynomial so you take the b algebra one the algebra itself is the algebra of polynomials you take only one variable delta is a delta of x is x tensor one plus one tensor x means that x is non-primitive and epsilon is that to take the constant term of polynomials so as a not clean starts of the plane it is clean starts of the line because you have only one dimension here so the characters are of the this form so it is easily checked that you have this it is a particular case of the formula that was given previously and the the monoid is isomorphic to the abelian group in particular it is a group but it is a case for every envelope in algebra and this is an envelope in algebra and now let us give a non-trivial example you take the of the of the rational in order to have the roots you could take also the extension of q by the roots and you take all the prime numbers and you take these roots of the prime numbers and this is a character of k of x this is not difficult but now what the preceding study shows that this set of character is algebraically independent over the polynomial and you can double check this considering that this is this formula the star is the inverse of one minus rather than the square root of b and x this time with with respect to the concatenation product and you have to use this which means that the shuffle power goes into the multiplicator of the character so you can double check that it is algebraically independent of course if you use many variables you have a non-trivial result so what we consider now is the group of characters and the group of characters you have exponentials and the logarithm it is you have a very very easy log x correspondence now I will dive into I will swoop into the example of the shuffle product the shuffle product as was explained here uses the addition of indices as perturbations so you have the beginning is the same as a shuffle and you have y index i plus k this term now if you take two as an application of our formula which is general here for shuffle you have this you have this star shuffle this star equals this plus this I could have put only the one index and alpha i plus beta i and this is the billionaire product and this is the phi so if you see this formula you can immediately imagine that if you take as a mental image your y y y i as a as a as a power like in overall style so you code this linear combination by this series means that you take the index and you put it as a power you get the following from 11 you get the following formula we will call phi umbra y this coding and then you have phi umbra y of s shuffle shuffle sorry phi umbra one of t uh shuffle sorry star and the shuffle is here equal phi umbra y of one plus s one plus t minus one so this will result in the fact that this is a one parameter row and as it is a one parameter row we can uh have identities which can uh double check by newton girard formula like for example uh this is the star of the plane so it is a character and a character is also an exponential of primitive element so the question is ah yes we have a character so it is the exponential of what it is a the same as it is so taking our our coding we can use this and take one letter for example uh renormalized by t and say that this is of course not a one parameter group it is a one parameter path drawn on the out of group but it can be a log uh what one can take the log of this one parameter pass and have this formula uh i uh so this formula can be double checked by other means maybe so uh let me conclude uh star of the plane property which is not difficult but it is worth to be means that uh in this uh by algebra which are of use combinatorially for many operations of on formal power series coding uh coding different things star of the plane property means that a character of this is exactly of this form so uh this formula uh eases the computation on the group of character and for con characters you can have matrix valued characters or non-commutative valued characters it is not important because the free algebra of non-commutative polynomial now if you consider this the composition of these characters you have an evolution equation that's why it is so linked to the project evolution equation in physics led by carol pennson now uh if you have a character which is this this time a value means that a is a k commutative algebra associative with you contrary contrary wise to the one you must have a commutative well it is not important so that because uh as this is commutative uh then uh the image is commutative but we will you see that we if you feed feed i4 deform this it is not necessarily commutative so you must we must start with this and this character uh will be taken as identity you give me a word i i i give you that word the same word it is identity and of course it has a shuffle property because you it is a shuffle morphism so now you have uh this is uh what we call the diagonal series it is the sum on all words that w tensor w and you see that it is an expression of identity so you can change uh the terms from w and w to to a basis in duality to orthogonal basis orthogonal between themselves so it can be proved it is a mrs factorization from melanson written over at sudson verge that this uh is the product of this exponential of course now if you change the character you can keep the identity the identity between the diagonal theory which is a formal identity identity and this and then you obtain a system of local coordinate i said that the host of group was a sort of league group so you know that if you take a basis of the li algebra of a league group you and take the exponentials you have a system of local coordinate at the neighborhood of the of the unity now so uh you see that uh uh this will give you a system of local coordinate of uh your your uh your host of group by decomposing every character as a product of elementary characters and to end with you have a different version of this factorization and of course uh it is rather straightforward this phi is is uh is associative but uh and commutative but i don't know yet if it is uh generalizable uh under which condition it is generalizable to a non-commutative uh uh perturbated shuffles we have to explore this so this factorization which is easy to to set and to clear and to state uh in the case of shuffles in fact holds of all for all enveloping algebras which are free as a as a as k module because of the prokary bierkovid formula you can draw uh derive formula prokary bierkovid basis and under certain condition technical condition you can have uh the orthogonal basis which is in in this b star uh convolution algebra so this uh can be uh set in uh on every enveloping algebra so we have evoked uh two or three times in our conversations with uh uh glem koshevoy that uh we could use this uh to uh factorize some characters uh on and all other enveloping algebras like uh christinis zamolochikov pardon me for the pronunciation le algebra and otherwise other ones uh we were evoking another uh le algebra uh which was uh commutative and uh coming from uh maxi so uh now i have said uh i have uh finished with my presentation and thank you for of course your attention uh with many bubbles thank you sir so we have uh one minute for question before me talks yes yes are there questions or remarks or comments no thank you glem and uh uh thanks thanks for your attention to the public