 Thank you to the organizers for the invitation to speak and I apologize, I have to speak in English out of necessity. So, and so while somebody remarked just after Michael's talk that so my job was now to just wrap everything up instead of explain how it all hangs together. And so somehow luckily I had decided from the beginning not to attempt to do this. And so instead I wanna give a talk which is maybe a little different again to some of the previous talks. There'll be some, the relation with history will just be my own naive view of history. So my view of history of mathematics is the heroic view where Galois are supposed to inspire us and we sort of derive maximal inspiration from him and from the work that followed. And at the same time I want to try and illustrate in some detail with some concrete examples some of the concepts that have been discussed in a way that I hope will be useful to at least some people here. So certainly not to any of the experts but maybe to some other people. So my own understanding which has sort of developed as I was thinking about this meeting is that the theory of Galois representations seems to begin with Galois himself. So Galois considers a polynomial of prime degree say P and he says that well we let G be the Galois group of F. I shouldn't say this is irreducible. And then he says that G is solvable and hence the polynomial is solvable by radicals if and only if we have an embedding of G into this group of matrices sitting inside Geo2FP where well this G acts on the P roots of F so this G naturally sits inside the symmetric group on P letters but this group acts on Fp, via affine linear transformations and this is also has P letters and so in fact this group sits inside Sp as well. And so this was Galois so-called Galois theorem and it produces an example of a Galois representation over a finite field. So something I didn't know until quite recently was that of course the many people he didn't know was that Galois also studied another example of or introduced another example of Galois representations into Geo2FP so again Galois and now we'll use the modern language says if we consider an elliptic curve and I have an example in mind so let me write down my example. So y squared, this is an example of an elliptic curve so that's a cubic equation in x and y and it has some graph in the xy plane and because it's a cubic curve we have a relation which is somewhat hard to draw. This is a somewhat realistic depiction of the graph but it's rather flat so it's a little hard to draw but when we draw a line a line will meet in three points a straight line will meet this cubic in three points. Now this cubic has one distinguished pointed infinity so all the way up here there's a pointed infinity and so there's a ternary relation on the points of this cubic and there's a pointed infinity and so we can define an operation an operation is a ternary relation we have to begin with two things and produce another but we can think of it as a three things being related and so the relation is that p plus q plus r equals zero so the pointed infinity will be the zero if and only if p, q and r are collinear. So the points of this elliptic curve form a group and so this has been explained in the lecture of Housel and in the lecture of Bost. Of course Gawar and his contemporaries at least in our modern view is probably for me easiest to say they labored under the language of elliptic integrals which made it harder for them to discuss this group law but they could certainly understand it and discuss it and what they found is that of course these points are, I've drawn a real curve but as most people or everyone here knows in fact if you plot the complex points of this curve you get a torus which we can describe for example a C modulo ellatus so in Bost notation you wrote gamma so this is a lattice in C and then it's easy to see the torsion points so the p torsion points in our elliptic curve well as a group this is as a space and as a group it's a product of two circles and so the p torsion points are just if you like there's even a pz plus even a pz or if you want a two dimensional vector space over fp but a typical thing that seems to happen in number theory is that rather simple objects get presented in rather interesting ways so that if you try to for example with this curve that's not so hard but already minimally painful to compute by hand the two torsion points if you try to compute the three torsion points you suffer more the four torsion points is as far as I was personally willing to go by hand it's not so easy to actually find the equations so the equations on this curve that describe the x's and y's that live in this set of order p squared these are the equations of multiplication that Jacoby and Abel and Galois were trying to find that Bost was discussing as he said they're these marvelous formulas one could also say horrendous formulas so it's not so it's not so easy to actually describe this rather simple group concretely in terms of these x and y and so one reflection of that is that so we write e p q bar for the points that are p torsion so the x and the y will be algebraic numbers and because the equation had coefficients in the rational numbers the Galois group acts we're solving an equation over q we have irrational solutions but they have the Galois symmetry and so this is another source of maps of the Galois group of q into g o 2 of f p so those are two ways in which Galois introduced representations of the Galois group of q into g o 2 of a finite field and so the title of my talk was Galois representations and so Galois deformations or deformations of Galois representations so what is the theory of deformations about? So primer-facie if we begin with some row bar from a homomorphism from g q into g o 2 of p what we mean by deforming this means lifting row bar to representations whose whether entries of the matrix live modular higher powers of p say modular instead of just being having matrices to find modular p try and find representation with matrices to find modular p to the n which when you reduce modular p recovers the original row bar so that's what I mean by lifting row bar to a row n and so we want to study these lists or these deformations and one thing we observe is they form a space what does this mean? Well what does it mean to have a space? You have a space if you can say whether two points in the set are close or not and if we have two if we have for example two lifts of row bar modular p to the 39 we could agree they were somewhat close if they agreed modular p to the 38 maybe still reasonably close but not quite as close if they agree modular p to the 20 primer-facie they have to agree modular p because they're both lift to a bar but we can so we can talk about how congruent different lifts are and hence our set of lifts is a space and the kind of basic goal given row bar understand the space of lifts and then an important point that I have to come back to is perhaps with conditions so we can put conditions on our lifts and the kind of conditions that we might want to put are related to the conditions that Michael described in the near the end of his talk ramification conditions so but to explain why we would do that I have to talk about the kind of true goal the more kind of verified goal of the theory is to establish the conjectured relationship between our representations and automorphic representations so that's so this theory of deforming our representations was introduced by Barry Mazer so the whole thing I have introduced introduced by Barry Mazer inspired by work of Hitta in part which Hitta's work had its origins in the theory of modular form to the relation with our representations and Mazer introduces theory as a way to try and understand what Hitta was doing but more generally to see more deeply how we would understand the relations between automorphic forms and Gaoua representations so that's sort of been the goal of the theory from the beginning and it's the goal of the practitioners currently so when I speak about Gaoua representations satisfying certain or lifting satisfying certain conditions I have conditions in mind which are going to be local conditions at various primes on the Gaoua side which is supposed to match in the sense of Michael's matching of row p's and pi p's that are supposed to match with conditions on the automorphic side but I think what I would like to do is to try and illustrate explain a Gaoua deformation problem an actual honest one which will illustrate some conditions some liftings, some conclusions and it's going to be in a very special case but I think it's sort of indicative of how the general theory goes so let me turn to a special Gaoua deformation problem or I should say a very particular one well in fact let me sort of as a so the problem I want to describe will be GL2 but let me begin with a preliminary with GL1 so this is an example that's been mentioned many times already because we can't hope to discuss it it's Gauss's theory of cyclotomy but let me describe it in a certain way it's not exactly how Gauss described it I think so I will use the language that Jean-Marc Fontaine introduced in his talk so P a can be any commutative dream and we have the following function that we can apply to a so mu P to the n of A we define to be the elements of A which are P to the n fruits of one and so there's a functorial construction in the ring A and we produce a group of course if this is a subgroup of the multiplicative units inside A so we have functorially construct the group from A and we construct the group by solving an equation in A so that's called a group scheme this is in fact a finite flat group scheme so this is an example well this symbol is mu P to the n is a finite flat group scheme so it's a group that depends functorially on A constructed by solving equations and if you have such an object you can apply it to Q bar we can apply it to the algebraic closure of Q and on the one hand it's a functor so we get an action of the Galois group of Q or more concretely we're solving equations with coefficients in Q even in Z with values in Q bar so we have an action of the Galois group of Q of course in this particular case it's not a mystery what the values are the P to the n fruits of unity form a cyclic group of order P so this is isomorphic to Z mod P to the n and so this action gives a representation maybe I'll call chi sub n because it's a character into G O 1 of Z mod P to the n and this is called the mod P to the n cyclotomic so we can look at the mod P cyclotomic character the chi 1 and then all these chi n's are deformations of that chi 1 so this is an example of a collection of compatible lifts of a one-dimensional mod P Galois representation and they have a property which is that they are finite flat they're all constructed by solving a finite flat group scheme so this finite flat is an example of a vermification condition so we've seen various vermification conditions occur in the previous talks we've seen unremifed away from S we've seen crystalline and deram at S so this is a very special a very special condition it's a special case of crystalline and it has a very concrete thing it just means you're solving some equations and so with this warm-up done now we can move to n equals 2 keep that we move to n equals 2 okay so now I want to begin with a row bar a two-dimensional row bar and then I'll write down some row n's and so well a good place to start is is a Galois theorem because that gave us some two-dimensional row bars so we just have to choose an f we have to choose a solvable polynomial with prime degrees so let me choose a famous one so I take f of x equals x squared plus one in Galois context and he produces for me a two-dimensional Galois representation I have the Galois group of q of course the roots of this we know they're plus or minus i so we factor through the Galois group of q a joint i over q which is a fairly simple group it has the identity it has complex conjugation which will embed into f2 star f2 1 0 but f2 stars are fairly simple groups which just will one goes to the identity and c goes to the only non-identity matrix which is one one zero one so that's an example of a two-dimensional Galois representation one thing I want to remark is that this two-dimensional Galois representation is again finite flat so this is an equation we can also obtain by solving equations of course we built it by solving an equation but that equation doesn't realize the representation as being finite flat if we think about how the Galois group acts on the roots of this equation we get a permutation representation on a set of order two I'm trying to make a linear representation on a vector space over f2 of dimension two but so this robot nevertheless it's finite flat and I can just write down the function for you so let me write it down so I'll call it g sub one in anticipation of having some g sub n's in a moment so g sub one is going through again be a functor that takes a ring and produces a group so I start with a ring and I'm going to produce a group and it has to have something to do with a square root of minus one and you'll see that it does so I take my ring and I adjoin a variable i so that i squared equals minus one and then I look for elements so not just in a but elements in a adjoin i such that a naught plus a one i squared equals one and also such that a naught times a one equals zero and now so these are square roots of unity but not inside a but they're square roots of unity inside a adjoin i so they're naturally multiplicative group and you can check the g1 of q bar this is for any a this is a group of order two which is constructed so all the elements have order two so when we plug in q bar we're going to get some f2 vector space and in fact uh... it's spanned by you just check it's spanned by minus one plus zero i and little i so this little i is the actual square root of minus one inside q bar and this capital i is a formal square root of minus one that we adjoin to q bar so it's uh... yes zero plus i times i so those two elements span g1 of q bar is an f2 vector space and you compute how gawa acts and it acts through the representation row bar and so that's for example what it means to realize this row bar as a finite flat representation and having done that we can deform it so deform it in a kind of obvious way if we had the second use of unity we could lift them to fourth and then eighth and sixteenth use of unity and so on so we can play the same kind of a game here so we define gn of a to b so i have a variable now zn so this is an element of a adjoin zn and i want to satisfy the equation that this element now to the uh... two to the nth power equals one i again ask that the the products of the distinct coefficients vanish and here this zn the formal variable satisfying the relation instead of just adjoining a square root of minus one we adjoin a two to the nth root of minus one and so again this is some very explicit group scheme that we can write down so again this is thought of as sitting inside this multiplicative group so it's a attaches a a group to every a we get a group the exponent is two to the n and we can map these groups one to the other by squaring so we we square and we set zn squared equals zn minus one and z one of course will be capital i you see that you get a map you get maps of these groups and so we've written some group schemes that form a kind of a group scheme of exponent two and that exponent four and exponent eight which are compatible we can take the q bar points and we get a family of representation so we set rho n to be g n of q bar this is a two dimensional representation of gq over now z mod two to the n and these rho n lift rho bar compatibility so so one reason I'm writing down these equations is just so in the uh... maybe the first day Howard Edwards in his talk commented that gawa's theory was in principle constructible constructive but in practice one couldn't write down the equations and this is certainly the feeling one has when one works with elliptic curves in the torsion elliptic curves so I think that somehow at least for me psychologically useful to see some family of finite flat group schemes some family of equations which is written down which you can just see by hand is is there is not a complicated thing as this concrete is more concrete than what gawa is writing in the premier memoir but it's somehow pointing at the same time to the general theory to the theory of schemes to the theory of deformations to the theory of gawa representations so what what can we say now about this family so is this family of any uh... is this family of any merit at all well this family has a very prescribed ramification because it's finite flat over as we say these group schemes of a flat over z so this is over z so this is a finite flat group scheme over z and that's a very very tight ramification and it's furthermore the only kind of prime that's getting into the action is a prime two and that's a very small prime so michael explained in his talk on the automorphic side that if you restrict ramification you get a final statements and he said that such things are less obvious on the gawa side but we should remember there are some final statements on the gawa side that we know well so we know minkowski's theory which tells us that there are no unamified extensions of q we know the finiteness of class groups and then by class field theory we know finiteness of abelian extensions if you restrict the ramification and so similarly this this set of group schemes kind of is witnessing a certain finiteness statement because the ramification is so tightly constrained and so what one can show is that the row n the unique list of row bar that a finite flat over z and so that's a typical example of a theorem in deformation theory you start with a row bar that's perhaps of interest you impose conditions that reflect ramification you have you hopefully have an example of gawa representations you'd like to understand and then you try to identify your you try to make your example exhaustive enough so make your examples are sufficiently exhaustive in relation to the conditions you imposed to see that all the examples you know is a draft from some all the examples you know coincide with all the possible examples that nature knows and so so of course the difficulty is if you can if you make your constraints too tight I mean you may get something a nice statement but perhaps not so much interest so so I want to push this example a little further make it a little more interesting by introducing an elliptic curve so well for me that's kind of interesting enough to have an elliptic curve hopefully for some other review as well and so which elliptic curve do I want to bring into the picture so I'll use that one so let me just recall it so we have x y squared plus x y plus x is equal to x cube minus x squared minus x minus fourteen now I guess thank you so so this is actually a rather interesting elliptic curve it's not sort of the first elliptic curve in nature but it's maybe the third so so Huzal and Burst both explained in their talks that the in the 1830s people were studying these modular equations and so this is the equation that relates all the elliptic curves with varying modulus and all the isogenous elliptic curves for some prime p so so the kind of modern notation for the kind of the modular equation for p isogenous so something we can do that was harder for for the ancients to do is we don't have to write an equation we can just give the object a name so our notation is x naught p and when we write x naught p so as Burst explained you have a choice of variables j and j prime k and k prime k to the one quarter and k prime to the one quarter when we write x naught of p we're not committed to any particular variables in fact we can choose them as we want so I'm going to choose these x and y and this is x naught seventeen so that's the curve x naught seventeen so it's describing the modular it's describing the relation between elliptic integrals and then seventeen times the elliptic integral although I couldn't tell you now how to compute x and y in terms of k and k prime but one could in principle even in practice with a little calculation so so it's a very natural curve and so as a and so as I described at the beginning its torsion gives rise to Galois representations and now what kind well there'll be to compute the torsion on this curve we have to solve equations we have to draw lines and figure out when we add something to itself so many times whether we get zero or not so we have to solve equations so we get a group scheme and what kind of a group scheme do we get well we almost get a finite flat group scheme except a world be finite flat over z in fact there's an a priori reason for this so there's a theorem of growth and dig which said I think it's due to growth and dig which says that if all the n torsion group schemes for an elliptic curve for every n a finite flat over z then the elliptic curve would have good deduction at every prime over z but this elliptic curve doesn't have good deduction as chronicle already knew at seventeen at seven so so most at most primes when you reduce it it looks like a nice cubic curve but at seventeen it has it has a node so this curve has a node at seventeen in fact I can even tell you the point so you can check so this has a node at the point seven thirteen mod seventeen you can just compute the derivatives there and you'll see they both derivatives finish and so what happens is if you take torsion points if you write down the group scheme of torsion points it will be finite flat away from seventeen but typically what happens so you'll see this first when you look at the fourth division of points is that when you reduce the module at seventeen some of the points hit this node when you reduce the module at seventeen and that means they escape from the group so this nodal curve is not a group the node is if you remove the node it's a group at the node there's no group law and so some of the torsion points escape into the node when you reduce mod seventeen so so it's a group scheme of a z but it's finite flat just away from seventeen what happens what happens is that at the prime seventeen when you look at say the the two to the n torsion and you have z mod two to the n cause z mod two to the n and you reduce mod seventeen it will turn out that exactly one of the two dimensions doesn't go to the node and everything else does so kind of so to speak only half of the dimension of the gawa representation sort of survives in character 617 so we say that semi-stable so this group scheme so this gawa representation on the torsion points here they are finite flat away from seventeen and semi-stable at seventeen and now but not that's not completely the case because there's a little mercy which is that if you look at just the two torsion points you can compute them there's the origin eleven over four and minus fifteen over eight there's minus one minus two i and then there's minus one plus two i minus i these two have to be conjugate and so the two torsions to find over q adjoint i so the gawa action on the two torsion it's not trivial but it factors through the the gawa group which is the same gawa group that roba factors through and in fact you can check this is isomorphic to roba so this elliptic curve the two torsion doesn't know about this node in characteristics seventeen the two torsion is finite flat over z but as you look at if you were to compute the four and the eight and higher two-power torsion eventually it discovers the node and so uh... what do we have so we could look at we can consider deformations of roba that a finite flat over z seventeen inverted and semi-stable at seventeen and the determinant equals the cyclotomic character it was a member as uh... as both explained for an elliptic curve the determinant of the uh... the uh... the uh... the intersection pairing on homology gives you a pairing on the two to the n torsion but it takes values in roots of unity and so the determinant is cyclotomic character so we could look at that defamation problem and then we have two solutions so so the row n coming from my group scheme is gn a one solution to this defamation problem because they're even finite even at seventeen they're better than semi-stable and then i take the two to the m torsion points on this elliptic curve and those are another solution and again it turns out that these are the the unique for the only two solutions to that defamation problem and so that's just another example of a defamation problem and a solution but it's already a little more interesting because it involves an elliptic curve it involves the characterization of an elliptic curve purely in defamation theoretic terms and so and so that's that's one of the so this is sort of one of the goals in general defamation theory to characterize certain defamations of interest perhaps especially attached to interesting objects such as elliptic curves in some gawa theoretic purely gawa theoretic terms and how does this relate to automorphic forms well so now let me just be a little griffer but this elliptic curve E it was I mean it was x0 17 so it somehow thought logically relates to a modular form actually so it relates to a cuss form which well I can write down the q expansion anyone guess it's it's a certain form of level 17 and what about these are these finite flat row n well they relate to something as well they there's an there's another modular form of level 17 the eisenstein series as this q expansion and you can check that for example minus one and one a congruent mod two and minus one and three a congruent mod two these are congruent mod two in fact even if you forget the constant term they congruent mod four and that reflects the fact that this row n and this e were deforming the same row bar so so the fact that you had two different families of gawa two different kind of solutions to the gawa defamation problem coming from the same row bar on the automorphic side that reflects that two automorphic forms were congruent and so the theory of the theory of congruences of automorphic forms is sort of a tool on the automorphic side which is the kind of automorphic mirror of the theory of gawa deformations so we know there has to be in the postmodern formulation each side has to kind of answer a question of maybe pose a question to the other side and so to the theory the question to the theory of gawa deformations poses to automorphic forms is the theory of congruences of automorphic forms so okay so that's sort of supposed to be an illustrative example that shows the some of the kind of technical features of the theory but I hope give some sense that it's the same mathematics the same part of mathematics that was being studied by gawa and arbol and jacobi it's the same chronicle gawa it's in the same tradition it's the same kind of questions we've hopefully just moved a little further in our understanding and what I'd like to close within the last few minutes is just to say using this as some kind of illustration what what happens in the general case in the theory of deformations so well ultimately what happens is that some spectacular theorems have been proved so the modularity conjecture for elliptic curves the sato by wiles and taylor wiles the sato tate conjecture for elliptic curves by clausel and hyresonship of baron and taylor the uh... and then another important result very important is the caisse conjecture sold by carry and ventabagé well so kissen uh... and roughly okay so those are those are some theorems but where does gawa how did our deformations enter and so roughly to prove these sort of theorems one needs four steps and so we can just briefly describe the four steps and see how they relate to these examples so the first thing is you need to construct gawa representations attached to automorphic forms so in this case I had only two automorphic forms in the picture one was an eisenstein series and these correspond to some characters on the gawa side they're billion class field theory and we had a class form that was attached to an elliptic curve and so that problem was solved but in general this somehow this this is the problem of shimua varieties the general the the uh... developments arising from considering chronic as jogantram to misquote langlands is this part of the story then one has to having shown that they are gawa representations attached to automorphic forms one is going to try and show that they fill out certain deformation spaces so one has to show that there are enough such gawa representations which now the question of showing there are enough automorphic forms so one has to somehow count automorphic forms and be able to produce automorphic forms then one has to be able to bound the size of the gawa deformation space from above one's bounding it from below by exhibiting automorphic forms and hence exhibiting gawa representations and one must find some other way to bound it from above and with luck the two bounds meet and then one's proved that all gawa representations of a certain kind are attached to automorphic forms and there so there so one basic difficulty which has sort of been alighted here is you don't know what gawa deformation space you have to consider a pro because the gawa deformation space depends on the row bar and if suppose i have a an elliptic curve that i'd like to prove is modular and i look at its p division points and that's the row bar and i'm going to look at the deformation space for that row bar and i'm going to try and fill it up with automorphic points we'll have to know there's some any automorphic points at all and the conjecture that they are such points was was ses conjecture and that's been proved but the exhibiting for a fixed row bar to say especially dimension greater than two to exhibit any automorphic forms at all that give rise to this row bar that live in its deformation space seems to be one of the hardest problems and maybe it's one of the row blocks to further progress at the moment so the generalization of ses conjecture is a fundamental issue once that's dealt with there's a question of exhibiting enough automorphic forms to fill out the deformation space and that has to be coupled so that's and then there's a problem of bounding the deformation space so it turns out those two problems become coupled in an amazing method of tailor and wiles and so the substance of the method roughly is that you have a very very complicated space and you can compute its tangent space you can compute the tangent space your deformation space at row bar but it will turn out to be very high dimensional typically and when that happens you don't know whether it's high dimensional because the space is high dimensional or does because the space is highly singular and in fact what you want is a space to be low dimensional you want it to be as low as possible so you can fill it up with all the automorphic forms you know but your so your hope is that it's highly singular and low dimensional but your fear is that it's quite smooth but huge dimensional but so what what happens in the theory of tailor and wiles is that they choose carefully certain directions in which to relax the deformation problem and so it's as if that's as if you were in the following situation as as if you had this variety and you computed the tangent space at the origin and saw that it was two-dimensional but you didn't know that whether the variety was a surface or a curve I mean it looks like it should be a curve but perhaps there's a surface you know perhaps this is you know perhaps there's something strange happening it would be probably better if I added more variables but let me just keep it as it is and so the the argument of tailor and wiles is the following that you you eliminate this equation f1 you see what happens when you remove that equation but you carefully chose that f1 among all the possible elements in the ideal so that the tangent space did not enlarge so the tangent space is still two dimensional but now there's only one equation so we must have a smooth surface and then our original variety must have been obtained by imposing a non-trivial equation in a smooth surface so our original variety must have been a curve so that's the argument of tailor and wiles but you have to choose this equation very carefully to not change the tangent space and at the same time they want to fill up with automorphic points so at the same time you as you remove that equation you get more space now there are more points that solve the equation they choose these primes so that they can make sure there are a lot of automorphic points there and I'll just close by saying that I was struck by again something I saw in boss talk today that well in the end in the tailor wiles method you choose is relaxing the conditions the equations you choose are the equations that say finite flat at 17 so when I relax you know in my two deformation problems I did I relaxed the condition at 17 from finite flat to semi-stable this is the kind of thing they do they relax the condition by allowing ramification at a prime but very interesting that you sort of have x naught one you have x naught q this modular equation and then as both said you have x one q which looks at the the actual q torsion points on the elliptic curve and we saw in the letter of jacobi that this is unsolvable but this is abelian and this abelian extension is what makes it the tailor wiles method work and in the end they can argue they're doing non-abelian class field theory but in the end they can use the principles of abelian class field theory as they're captured in the theory of gala chronology and so I mean somehow I found that a very kind of inspiring thing to see that the most advanced method in the current technique is somehow echoed being you know discussed in some sense maybe in a very ephemeral sense but appearing in some way back in the first practitioners of the theory of equations that gives me a good sense of continuity in the field you know optimism for the future so I'll stop there the question the case you discussed at the end it's complicated enough but it's it's still the case where finally you prove that you have the isomorphism and you have a complete intersection right yes so it's even more complicated in the general case and in particular in those well well in the most I would say like the sort of there's a there's a more recent form of the tailor wiles method which one will call the tailor wiles kissing method which deals with an additional phenomena which is that when you when you relax the conditions at a prime to add automorphic forms you count how many automorphic forms are added and the number of automorphic forms you have new forms you obtain is linear in the level and that's enough to see that the dimension of the automorphic if you close up the automorphic forms the dimension of the space that they fill out is the right dimension but these spaces could have multiple irreducible components and so then one has to investigate the question of whether all the components are filled out by automorphic forms in fact it's there's a slightly naive rendition of what really happens but at that point the commutative algebra and the geometry become more sophisticated the complete intersection drops away a little bit but still this I think this is a kind of reasonable mental model for what's happening in the tailor wiles method so I have a non-mathematical question so if I understood correctly you explained that a major open problem was to describe more or less explicitly on the representations that lift a given one and for me this question has a eugntrum like flavors flavors so is this a delusion or is there I don't think so and in fact so I mean I maybe gave a slightly wrong impression that I mean not only it says conjecture be improved for n equals two but there's a potential says conjecture for a kind of high-dimensional Gal representations which is of vital importance in the progress inside to take in everything else but also of vital importance in the proof of the full says conjecture so and but that's an argument that uses geometry to construct the the lifts and so I mean if we can interpret eugntrum to mean the interaction between Gal representations and geometry I mean it's exactly seems to be that sort of a question not too difficult but eugntrum doesn't have a precise formula in fact maybe depends on the dreamer I don't know if there are questions in the answer to the question I don't know if you hear me or if there is someone in the meantime I would like to remind you that there is a cocktail at 19 hours at 45 products at the school Normal Superior in the historical room of the library of letters