 In 2005, I was young, a young postdoc, younger, and I was invited to give a talk in Strasbourg, but my train was delayed, and that day, Thomas offered to pick me up at the station so that I had time to have a quick lunch and a coffee before my talk, and we never met. It was the first time that we had contact, and I was grateful to him that he did that for me. And in 2012, I moved to Strasbourg. I was recruited there, and when you moved to a new place, you arrived at the June fight for the attribution of the lecture, and for my settling, Thomas did many things to ease this, and also he just gave me his lecture that he was doing the previous care, so when I did my first year in the real life of teaching, I had a nice lecture to give, and I'm also grateful, and I think many other colleagues in Strasbourg can add a similar story when they moved in. So, in the name of all your colleagues that you have, and in my name, I would like to thank you, and to wish you a happy 15th, nice, point, 98th birthday. Okay, so I would like to report on some joint work with Anna Wienard and with François Gaboris, Anna Wienard, which try to give what is a positive structure on a frag manifold, that is gmod p, and I will give a greater precise definition, but it's some orientability condition on this frag manifold, and our maybe first motivating example is the circle with its orientation, and when you have the orientation of the circle, you can define many other things, but the thing that will be relevant here is that when you take three points, x1, x2, and x3 in a circle, distinct, you can say when they are either positively oriented or negatively oriented, or negatively oriented, and Gator will explain what is a positive structure on a frag manifold, but it is really a set of triples that mimic the property of this positively oriented triples that you can find on the circle. So what can we prove with this definition that I didn't give so far is that we know all the frag manifold admitting a positive structure. So it's a classification result that I will be able to state completely after giving other, more many for example, that's just this circle, and when you have a positive structure, I will be able to define the notion of positive representation and establish properties on the positive one. So this is joint work with Anna Wienardt and the second result with François Gaboury. So I take a group that is either a free group or the fundamental group of a surface of hydrogenous, and for this example, I will not define what are positive representation. You can guess if you know a little about the boundary of this group, but I will give a later this definition, but a positive representation, so from this free group or surface group into G is a very nice discrete representation. So it's faithful, maybe there is only one line, has a discrete image and has many other relevant discreteness property. The set of positive representation in the set of all the monomorphisms from gamma to G is stable by deformation, so it's open, and under some condition we can show that a limit of positive representation is again positive. I don't know, there are infinitely many, yes it's probably semi algebraic, but I didn't think too much about this question. What is the third property? The third property is some closeness property, so if rho is a limit of positive representation, so it's my hypothesis of positive representation, and rho is a risky dense, then rho is positive, so maybe this can help answering a Frédéric question, it's connected component in some, a union of connected component in some algebraic. Okay, but in fact we expect that for surface group this is always a closed condition, so and this was established in some cases by Jonas Behrer and Beatrice Potseti, so it's a collaboration. So if gamma is a surface group and G is a SOPQ, so this is one group that is in the list of known group admitting positive representation, then three poles without the risky density, and colorally already from this point three that is conditional, but that is easier to obtain with the unconditional result of Beatrice Potseti and Jonas Behrer is that there are people constructing nice in the X-bundle parameterization of surface group, so gamma is again a surface group for this color, sorry, and the connected component that I can call generalized each in component that were understood by Brailleux, Collier, Garcia-Paler, Goten, and Oliver, so there are explicitly parameterized component of the set of the representation space of the surface group by some holographic object that are fiber bundled over some known base, but all of this consists of positive representation, so what I gave the result that I can stop my talk at any time, so the chairman is particularly happy, and I would like to give some classical instance of a flag manifold admitting a positive structure, so a notion of three poles that you want to call positive, and I will then explain what is axiomatic that you want for such a positive structure, and then give the classification result, and if this time permits I will give more combinatorial and algebraic property of this positive structure. If PQ is two-one, are there other positive representations than the particular ones? No, no, this really works here when Q is bigger than P plus one, and you want this at least and PS is two. This is a range, and the notion makes sense. Okay, so for Q equals P plus one, there is a chain component, but it was known before that it consists of positive representation. Okay, so what are the cases that were previously understood of positive representations? They are really action on the circle, and I'm leading to this from a really algebraic point of view. This is a very rich subject, but I don't go into action by homomorphism. So here I have still a surface group, and I define a representation to be positive into PSL2R. If there is a map from the boundary of the group to the projective gain, so I call it beta, so from this circle that is the boundary at infinity of the surface group to RP1, that sends positive triple to positive triple. Okay, so sending a positive triple to positive triple, in a way it's orientation preserving, and yes, you want this map to have some reaction with a wall, and it has to be the equivalent with respect to the reaction. What is known about positive orientation? So depending on your background, it's either a theorem or a sequence of tautologies, but when I had a course in, I don't know in matrices or they are about logic, the lecturer says that every theorem is a tautology, so every proven theorem is a tautology. So representation is positive, if and only if it belongs to the Taichmuir space, so it is the autonomy of a hyperbolic structure on the surface. So the equivalent map is unique, it is continuous, so in fact it has a finer continuity property, but I would not like to describe them. The set of positive representation here is the Taichmuir space, which is a connected component of the representation variety, and a last property that is also well known is that the mapping class group, so here exactly the automorphism group of gamma has a nice action on the set of positive representations, so this action is, probably, continuous. Again, I think this is all that I would like to say, but this very classical case, I would like not to give one other well-known instance of positivity, which is still classical in a sense. It is a case of a speed group, three speed groups, so my example of a speed group today will be not s1, but s1 and plus 1, and it is a little more convenient to take this as the stabilizer of the anti-diagonal quadratic form with alternating sign, so it makes, apparently it is a life more complicated, but many formulas simplify. So associated to this speed group, you have a root system, so here that are, maybe I will not describe them, and you have a simple system, so maybe the root system, yes I can describe it, it's all the plus or minus 1 epsilon j, plus or minus 1 epsilon i for i less than j, where epsilon i is the standard basis of r to the n, which is a part of the algebra of s1 and plus 1, and the plus or minus 1 epsilon i for i between 1 and 2, and the simple system here is the generating formula for this sigma, it's alpha 1 up to alpha n, alpha n is epsilon n, and the other alpha i are epsilon i minus epsilon i. And what I would like to annoy you with other classical combinatorial data associated with this group, there is a vial group that is around here, so that I can see as a subgroup, exactly as a subgroup of g and r stabilizing this set. So it's a finite coaxial group, it's generated by the reflection s i that are associated with the simple root, so s i sends is a hyper plane reflection that sends i to minus i for i, and since you have this finite group and this generating family, you have a length function from w to n, and there is one element in w0, which is a minus identity seen as an element of g and r in this example, and w0 is the element of longest length. And what is the length of w0, I will denote it by capital N, but in this example it is equal to n to the square, so you can understand precisely this. And I will need further notation here, and then I will be able to state a result for the one parameter unipotential group associated with the f i, so I denote x i, the map from r to s1 and plus 1, that sends t to the unipotent matrix that looks like this, so this is for i less than n, and this is going i, and for i equal to n, so xn is a map that sends t to a little more complicated unipotent, that has this 3 by 3 centribrals, that is 1t t squared over 2, 1t and 1. Okay, so all of this has r. So the second t is on line 2n? This is 2n plus 1, and this is 2n maybe plus 2 minus i, okay, and a sequence E i, E n is like to belong to the sequence of the deck of w0, if so n is the length, capital N is the length of w0, if w0 is the product s i1, s i2, etc., to s i n. So there is maybe an old theorem due to this teach, so that I am stating in the particular case of s1 and plus 1, that says that the set u subscript positive, that is the set of product of the form x i1 of t1, x i2 of t2, etc., x i n of tn for t1, tn positive renumber. This set a priori could depend on the sequence you choose, but it does not, so it does not depend on the decomposition you choose of this longest world w0. In fact, there is more, there is the map from r positive to the n to u that is defined by this formula, the product of the x i of t1 is a normal morphism, so this set is a ball of dimension n, which is not at all obvious from the formula there, is that this set is a semi-group. And I must say that I am just quoting a few of the properties that Gustav established for this positive one. The ball of isomorphism is a semi-group isomorphism or just a normal morphism? No, no, this cannot be a semi-over. So, once you have this property, you can define what is a positive triple in g mod p. So, f1, f2, f3 in g mod p to the cube where p is a Borel subgroup here or the minimal parabolic subgroup is site positive if it is of the form p u w0 applied to p and w0 applied to p for some u in this positive subgroup. And you want to saturate by the g action, so I will add a g in front of each of these factors. So, there is g mod p, what is g mod p in this case? What is g mod p in this case sort of a Lagrangian or whatever? In this SON and plus one case? So, those are the complete flags where fn is isotropic and the orthogonal of f i is f2n plus 1 minus i or 2n plus 2. So, maybe I don't need to put this isotropic condition in. Is it not just maximizing tropics? No, no, no, I need all the intermediate one. Yes, so there is for g in g and if gamma now is a free group or a fundamental group of a surface, our group that has a cyclic structure and it's boundary but there is not many other examples of these two families, then I can define a representation on gamma e2g to be positive if there exists a map beta from the boundary of the group to g mod p that has something to do with rho that is a rho equivalent and that sends a positive triple to positive three. Maybe the main result here is due to François Labourry in the for the g equal to s again r and to Falken Gottrauf in the general case, it says that the etching component consists of positive representation. So, a component that was built by etching and that we now call the etching component of the space of representation from gamma into g that is constructing through X-bundle via an explicit parametrization as all its representation satisfying a very nice geometric flavor that connects connect in a sense to the classical Taichman theory. So, the etching component consists of positive representation. So, the other classical situation that was also well studied in the literature and that I represent in maybe biased way is the case of Hermitian New Group. So, g is such Hermitian New Group. So, here I go back to maybe a more classical situation in this classical setting and didn't try to have some more complicated group. So, g is this v group. g mod p here is the Schiele boundary. So, it can be characterized in different ways but here for my explicit example it is the space of n-dimensional space of r to the n that are isotropic with respect to the simply kick form. So, it's the Lagrangian manifold and in this case I have an invariant that is the mass of number that is invariant with respect to the action on g and that produce out of three Lagrangian integer that is bounded by n. So, if you ask me I can explain what is this integer but I will be thankful if you do not ask and I define a triple is said to be maximal if its mass of number is equal to n. And this we play the role of the positive triples so I can define the representation to be positive. Maybe I will not write this definition, probably I have already guessed it. I need an equivalent map from the boundary of the group to the space of Lagrangian sending positive triples to positive triples. And I would like to write away this is not the triples is positive, the mass log is negative, the triples is positive, the mass log is maximal. It is said positive. Okay, so you use two words for the same. The triples is positive, its mass of number is maximal. So this is not the correct way to define this representation. But I will not to much into this but I will not like to give you the impression that this is the correct definition. Please have this in mind. Are you saying that it is a short definition that it's not useful or are you saying that it is not a definition that it is not? I don't understand the remark. Are you saying that you are cheating or are you saying that it is a cheap way that hides a lot of the theory. So and the main result due to Mark Berger, Alexander Yodzi and Anna Wienhardt. Again I stated only for a surface group but the theory is a return. First is that the set of positive representation is open and closed. So it's a number of connected components in the deformation space of representation of gamma into Sv2Nr. That positive representation are nice from a point of view of discrete group theory. So they are faithful. They have a discrete image and they also have a compact center laser. And furthermore I want to add that the mapping as group action on the set of positive representation is a property discontinuity. And strictly speaking this guy's point is a result of Anna and or François Labouille obtained independently. Good. No but this is true that in this situation for closed surface there is a unique such a map and it has more regularities. So it is either continuous. Usually it is not more. You cannot expect C1 or other. So now I would like to go back to a general situation and state what are the properties that you expect for a positive structure on a fragmented fold. So we take G now a simple really group P which is a parabola result group. So G mod P is my fragmented fold and maybe it's my definition. A positive structure on G mod P is a subset of the triples that will be qualified as positive that is a gene variant. So it is a list that you want to ask and there is a property kind of maybe co-cycle properties that makes you able to construct other triples if you know that a quadruple is in a oriented position. So if you have X1, X2, X3, X4 in G mod P such that X1, X2, X4 is positive so it belongs to T and X2, X3 and X4 belongs to T then the triple X1, X3, X4 also belongs to T. So orientability or positivity is a way to say when X2 is between X1 and X3 or X1 and X4 and if you have X2 between X1 and X4 and X3 between X2 and X4 you expect that X3 is also between X1 and X4. It's like a cyclic ordering. I must say that I didn't check the definition of cyclic ordering so it may be exactly the same definition but I'm not sure. So maybe the definition should state in one sentence it's a G equivalent cyclic ordering but okay so the classification result is the following. That positive structure exists for much more example than the classical one that I get. So it exists exactly for the following list. So one which is the first classical example I gave so G is split and P is a borelzavrup. Two which is classical two. G is Hermitian. I think I forgot to give one condition that may put a different, a little more difference with. So I want this set to be open. I want a stability by deformation that is open. Okay sorry for so the set of positive triples is stable by a small deformation. G is Hermitian of tube type so it rolls out a number of Hermitian loop for which there exists a kind of rigidity result that we are first established by Togedo. So for and G mod P is a sheik of modality. Okay and there is one other family and a few exceptional groups. So the other family is SOPQ where P is bigger or equal to 2 and Q is bigger than P plus 2. Okay and now G mod P is the set of complete but one flag in SOPQ. So it's the set of flag F1 included in F2 etc up to Fp minus 1 where the dimension of Fi is equal to i and Fp minus 1 is isotopic. And my last example for is G is one of the simple rigidity group whose restricted root diagram is of type F4. Okay so there is the split form of F4 and there is one refer of E6, one of E7 and one of E8 that are there. Sorry there was a small interruption so I'm not totally sure when it was interrupted so maybe I'll repeat quickly what is the positive structure on G mod P. It's a set of triples that mimic the that have that we can call cyclic ordering thanks to remarking the audience that is G invariant and that is open. And I gave the statement of the classification statement of positivity there are the classical case there is one other family for SOPQ and four other groups and identified G mod P when the G group is of type F4 and I will just try to quickly give an hint on how you can prove such a result. So what is important here is egg the levy subgroup of P and the action on L and U which is the unipotent radical on our in term of Liege-Gebrach's unipotent radical on the Liege-Gebrach P. Okay so there is a decomposition on U into irreducible conditions that is well understood thanks to Carton so there is a decomposition that is similar to the decomposition of the Liege-Gebrach G in a restricted workspace so it's a kind of restricted root decomposition and in R plus you have as well so it's in general this is not the positive root of the root system it's a subset of the dual space of some things that has some nice properties but that is not a root system so there is a simple system that I record maybe a big D a simple system and the main proposition that is the key to the classification result and the key to understanding many of the properties of the positive semi-group that you can define mimicking the definition of glue stitch of the positive semi-group in the speed case is the following that G mod P admits a positive structure if and only if for every simple root in D so those are not really roots but it's convenient to keep this terminology as there exists there is a cone in this piece an invariant piece of the decomposition of U an invariant acute convex cone and this is the most important property because there is a result due to Carton and Hegel-San that characterizes the irreducible representation of reductive G-groups admitting an invariant convex cone and thanks to this characterization applied to this situation you can end to this list and I will stop here thank you very much so you didn't say that the safety when the positive structure exists is unique in the examples you said maximum yes yes and this is not really the correct definition because you you here I could just put an equality here so this is this is not but yes when you when you saturate by the direction and by this properties that induce the semi-group property and the cones there there is more questions inside the room also online if you're still there no Alex so the first example in which we don't have positive structure is SL2C yes yes it's two transitive and triple it's transitive too much transitive as this is what you mean yes yes yes but is the next example which is not abuse SL2-1 yes but SL2-1 is almost an example any I think any rank one group that is not a SITURA is an example so SL2C I understand why there is no positive structure because it's transitive on triples yes but I expect to give us a simple argument for SO14 stating that is yes I mean a geometric show that white white is not transitive on triple there so another question is why does this exist while I say to R okay so more questions no so thank you again thank you