 Thank you for the introduction and thanks to the organizers, Ahmed and Fabrice, for the invitation. I want to talk about positive characteristic analogs of the San Sebastianis theorem. But before I get to that, let me just recall the classical story of the complex numbers. Let F, a germ of an olemophic function at 0, having 0 as an isolated critical point, value 0 at 0. In this situation, if you consider a small ball b around 0 in cm plus 1, then the inverse image of the small disk around 0 is a topological vibration outside 0 with typical fiber Mf minus 1 of t. This is also b, which is known by Minner to have the homo to be type of the bouquet of spheres of dimension m in number equal to mu. Mu, the so-called Minner number, which is the dimension of c of 0 xm divided by the ideal generated by the partial. So if you consider the reduced homology of this Minner fiber with coefficient in z, what reduced homology means co-kernel of the hq of a point to hq. This is 0 for q different from m. And it is z to the mu for q equal to m. This in quotient notation is, and also in tachy notation, is rq phi for this map f and coefficient z at 0. Now a positive generator of the fundamental group of the punctured disk acts by an automorphism t on this group, which is called the monodromy operator. And this monodromy operator is quasi-unipotent, as was conjectured by Minner and proved by Rothenlich soon afterwards. Now suppose that g is the second germ of homomorphic functions, say cn plus 1, 0 to c0, having also 0 as an isolated critical point. Then you can consider the so-called sum map f plus g on cn plus n plus 2 to c0 defined by f plus g of x, y is f of x plus g of y. And it is immediate to see that this f plus g also has 0 as an isolated critical point. And also on Minner's formula, mu f equals this quotient, you see that mu f plus g is mu f times mu g. Now what Tom and Sebastian he proved in 1970 is the following theorem. There exists an isomorphism, construct an isomorphism, from our m phi f z, tensor r n phi g of z to the similar group here are n plus n plus 1 phi f plus g of z. In particular, this formula gives that the rank here is a product of the ranks. And compatible with the monotomy operators, the sense that tf tensor tg corresponds to tf plus g. Their proof was ontological using the fact that the Minner fiber for f plus g is a join of the Minner fibers for f and g. So this is the story of the c. And now I want to pass to the story of the field k, characteristic p, mainly interested in the case where p is positive, but you can have p equal to 0. And for simplicity at the beginning, assume that k is algebraically closed. Now suppose you have two polynomials, fi for i equal 1, 2 from a and i plus 1 k to a1 k. And assume that, again, fi has 0 as an isolated critical point, or isolated point of non-smoothness. Then take n equal to n1 plus n2. Then you can, again, consider f1 plus f2, which can be obtained by first taking the product over k to a1 k cross over k1 k. And then, so this is f1 over kf2. And then a is a sum map, which sends xy to a of xy, x plus y. And the previous f1 plus f2 is the composition af. So then, again, you have a critical point for this sum map. And the question is to compare vanishing cycles for f1, f2, and the sum map. So vanishing cycles, I mean I have two vanishing cycles in the etal sense. So I have to fix some coefficient ring lambda. So here, I will not take a finite field characteristic L. I will just take, let's say, z model nu. Possibly, you could take zl. Or you could take, of course, a ql. You could take ql bar. Could take something intermediate here, some finite extension of ql. Or you could take the ring of integers of the extension. But I'd be mostly interested in the case of finite coefficient like this. So if you consider our phi fi of lambda, and you shift it by ni, then this is my gabber. This is the perverse sheaf. And because 0 is an isolated critical point, it is near 0. It is concentrated at 0. So it's a perverse sheaf, which is concentrated at a point. So it has to be also in degree concentrated in degree 0, which means that our phi itself is concentrated in degree minus i. So you have our q phi of fi of lambda is 0 for q different from ni. And some power of lambda, let's say lambda are i for q equal to ni. So now the question is, and of course, the same is true for the sum map, af of q phi af of lambda, because, again, 0 is an isolated critical point. It's 0 for q different from 0. And it's lambda to the r for q equal to n plus 1. So the question is, can one find an isomorphism in the form of q phi of fi lambda at 0 tensor here, ni phi of fi at 0 tensor r, sorry, n1, n2 phi f2, 2 r phi af at 0, which would have certain naturality in the sense that, if you assume that your situation over k comes by extension of scalar from situation over subfield k0 of k, then this isomorphism should be compatible with the isomorphism of k over k0. Well, the answer, in general, is no. There are two kinds of obstructions or problems, if you like, two kinds of obstructions. So no in general, I mean, in certain cases, it would be possible. So the first obstruction is y-grammification. So in fact, with Takeshi, we have just seen that. In fact, in this case, contrary to what happens in other complex numbers, the Miller number for fi, which is defined similarly as before, is in general larger than r i and strict inequality is possible. In fact, as Takeshi recall, we have the delin formula, which says that mu fi is not the dimension of phi ni, but the total dimension of r ni i lambda at 0. So that is r i plus s i, where s i is the swan conductor. And again, we have mu for af is mu f1 times mu f2. But this translates into the formula r 1 plus s 1 times r 2 plus s 2 is r s, and not r 1 plus r plus s, sorry, but not r 1 r 2 equal r. Second obstruction is that even in the best case you could dream of tame ramification, both for fi, f2, and some map, then you can't have something natural with respect to Galois automorphisms. Where is the? Yeah. So perhaps I should. So even in the other simple example, following a simple example, so suppose you take n1 and 2 is 1, and you take f1 is f2 from a1 to a1, and f1 of x is x, fi of x is x square. It means n1 plus 1 from a2 maybe is 0. And then 1 equals here. The medial notation is n plus 1. It's n. And you want to correct it? Oh, yes, yes, yes, yes. n1 and n2 equals 0, yes. Yes, sorry, thank you. So then in this case, f1 plus f2 is af from a2 to a1. It's just x1, x2 goes to x1 square plus x2 square. And then in this case, suppose, in fact, instead of working over k, start over just sp. And in this case, our phi, in fact, concentrated in degrees 0 because relative dimension is 0 for fi, is just long that we stood by some team of ramified character, chi of the Galois group, which is defined by chi of sigma is sigma of square root of t divided by square root of t. So by doing plus and minus 1, the team of ramified. While you get r1, phi for af, well, you get the inertia, in fact, actually, really. But you get something arithmetically non-constant. You get lambda of minus 1. So here is the state twist. Then there are some other ramified character epsilon, which comes from, which is the character of Galois, fp bar over sp, and which sends the Frobenius sp to minus 1p. This is seen because you want to write x1 square plus x2 square as a product, then you have to extract the root of minus 1. So then if you take a tensor product, then this kills that. So you get just lambda. While here you have something non-trivial arithmetically. So in 1981, the link proposed a solution that you should replace a tensor product by a local convolution product. Consider the initialization of a1, so maybe a h, initialization of a1 at 0. You have the close point 0 and generate point eta, and inclusion j. So suppose mi4i1,2 is a sheaf of a lambda module over eta, free of finite type, say. So this corresponds to a representation of Galois veta bar over eta into free module, finite type. Then you can consider the product here over k, and take again its initialization. I will denote it by a2h. And the sum map, or the map which is induced by the sum to a h. Now you can extend m1 and m2 by 0 onto a h. And then you can take the external tensor product over the product and localize it here. And then you apply our phi a at 0 so that you get something in theory dbc of a h and lambda. And in fact, it turns out that our q phi a of Galois tensor Galois shriek m2. So this is not on the, this is just on the close point of a h because it's the our phi. Oh yes, yes, 0, yes, sorry. Thank you. Then this is 0 for q different from 1. And this is something in dimension 1, which I will denote by m1 star 1, m2. This convolution was introduced first by DeLing, and then widely studied by Lomo in his paper on Fourier transform and product formula for the constants. And in Lomo notation, it was star. But later I want to take complexes and 1 and m2 constructible complexes. And then I want to take r phi or r psi. And I will use the star. So then I will here use this notation. So in any case, the link formula was that in fact, if you replace tensor product by disk convolution, then this is correct. Then rn1 phi f1 lambda at 0 star 1 rn2 phi f2 lambda 0 is isomorphic to rn plus 1 phi a f lambda at 0. He gave a proof in 1981 in a seminar. I got some notes, but it went very fast. My notes were not so good. I could never reconstruct the argument. It was a complicated compactification and deformation argument. And when I asked him recently, he said that he didn't remember. So in any case, before I say more about that, let me say the difference between star and tensor. So we'll see later that in the case case algebraically closed, let's say. And ramification is same. Then star is isomorphic to tensor product, actually. And general not. And this star actually takes care of the arithmetic input I explained here. In fact, in the early 2000, Foulet thought about the problem and gave a proof of the link formula, even a slightly more general situation, using Foulet transform, local Foulet transform of Le Mans, which had the miraculous property of transforming convolution into tensor product. Plus, Gabber formula for rpsi tensor rpsi. So what do I mean by that? Is that Gabber had proven that the formation of the nearby cycle is compatible with external product over a tray. So a product, something like this, here, a tray. He put a prep in an archive, which should appear soon. I put it, I think, two years ago. But at the same time, maybe in 2011 or 2012, the link proposed some generalization of this formula, which involves complexes and more general spaces than affine space. So suppose you have X1 over there, let's say, A H, so Fi, so this is a finite type. And suppose you have complex Ki in the CTF of lambda. If lambda is ZL, then you could just put DBC, like in Takeshi's lecture, if lambda is FL. And then you can consider the, suppose you have some point here, geometric point Xi, like this, over 0 here. Then the link conjecture, let's say D, assume that Fi Ki is universally, locally acyclic outside Xi. Then you have natural isomorphism, Fi F1 Ki1. So this means the right version of this star 1, previously. So it means that you extend by 0. So this is at X1. You extend by 0, take the tensor product, external tensor product, and push down by the sum map. But in the derived sense. So pushing down by the sum map, you have here this A2 H, or there A H, corresponds to taking upside A at the point 0. But since you work with our Fi, which are 0 at 0, then taking upside our Fi 0. So you can just take the previous formula with our Fi. And you get this, this. And then this should be isomorphic to upside to our Fi. For this map, A F, define as before. So X1 cross X2. Then you have F1 times F2 to A1, A cross A1 to A H cross. And then you extend it to, you pull back by A2 H, and then compose with the sum map. So here external tensor product, I mean internal tensor product here. And then I take pull back here by A2 H here. And then I push by A. So do you need to take a fiber at X1 and X2? Yes, thank you. And here X is X1, X2. Why did you put our Fi at Fi A, and what did you write there in the substrate? Well, you take the, you push that by our Fi for this addition map from the A2 H. So you see... You call it F, you don't have F. F is F1, F2. I am sorry. No, no, that's okay. Yes? Yes? So F is F1, F1 cross F2. Maybe I should write it here. So is it clear or not? It's clear. So you take the product and then you apply the sum, so you get the sum map. So this AF is somehow the F1 cross F2. So my result is that D is true, and in fact this assumption is superfluous. Well, the proof is a bit stupid. I just retained from Foulet's proof the fact that our Fi, our Psi, our Psi commutes with external products. But here when you say that you are a little vague. So our Psi commutes with external products over a tray. But here it's not a product over a tray that we are performing. So here X1 over A1, X2 over A2, X2 over A1. And you perform the product over A1 cross A1. So this is a two-dimensional basis. So in fact you would need a two-dimensional base. It would be, what you need actually is a compatibility of our Psi with external products but in a slightly more general form like here. But then you are in trouble because you land into a known territory. So nearby cycles over a base of dimension two. So, but then you have this marvelous formalism that Takeshi mentioned, nearby cycles over general base. And then you can apply that and you can expect a Q-net formula and then everything should come out. And this is indeed the case. But I will say a few more, a little more about that. So three short review of nearby cycles over general basis. This is the construction of the lean, 1981 I think. If you have two morphisms of schemes say X to F and G to Y, then you can construct what is called an oriented product of X and Y over S. Which is not a scheme but a two-poss category of sheaves on some site. With two maps P1 and P2, projection to X and Y. And well between morphisms of two-posses you can have maps So here you have a two map tau from GP2 to FP1. So it's morphism from G lower star P2 star to F lower star P1 star. Which is universal in the obvious sense. That means that any time you have the top of T and projection Q1, Q2 and the map T from G Q2 to FP1. Then there exists a unique map H making the two upper triangles commute. And such that T is obtained by composition with H from tau. The two-poss is constructed at sheaves over a site consisting of triple UZW where U is a tile over X, D is a tile over Y, and W is a tile over S. Y like this. With a certain topology I have no time to recall now. A point by universal property, what is the point of this two-poss? Then you take T to be punctual two-poss. So then you get a point of X, a point of Y, and a map from the image from G by G from the point of Y. So points of this oriented product of the triples, a point X of X, a point Y of Y. So I mean geometric point and a map between points, that is a specialization map like in Takeshi's lecture from G of Y to F of X. There are two special instances of this oriented product which are of interest. The first one, which is the main object of into consideration here, the vanishing topos where you take Y to be S and G the identity vanishing topos. And another one is S cross S Y, the co-vanishing topos. So this was in fact appeared first in some work of Faltings, and later was studied by Abess and Groh in their work on Simpson's correspondence. But are we interested in this? Actually, to get a feeling about this, suppose X is just a point. Then X cross say S cross SS, so S here is a geometric point. So this is just a strict localization of S at S. And now if you have a geometric point X of X with image S, then the map here corresponds to the map induced by F of the strict localizations. Now here in the vanishing topos, we have those two projections on S and here X, P1, P2, and here it's F. So if I look here at the following diagram, so this is commutative. So by universal properties, it defines the map Psi F, which in fact gives rise to the right factor, nearby cycle factor, Psi F, D plus of X lambda to D plus of X cross SS lambda. In the case S is a tray with close point S and eta, then this XS cross S eta is the familiar topos in the SGA 713 consisting of sheaves on the special fiber together with an action of the Galois group of eta bar over eta, compatible with this action on XS. And in fact in this case, it's easily seen that Psi F of K restricted to this S eta is the classical usual Psi. And similarly, we have also some R Phi because this triangle here gives rise to map from P1 per star K to Psi FK and a cone R Phi K and again the R Phi K restricted to this is the usual R Phi. Now, not much time. How is this object good or bad? Then you have a notion of finiteness on this topos. I mean, given a sheave of lambda modules on this vanishing topos, then you can say when it is constructable. So if F is a sheave on lambda modules on this oriented product, you have notion of constructable. So you can write X as a region union of X alpha and S as beta so that this restriction is least. And this is a good notion. Sixth category of the category of lambda modules and then it gives rise to a good derive category DBC of this. You mean this locally constant, fired locally constant, not just this constructable? This constructable, yes. Yes, yes, thank you. Now, the general notion of constructability. Yes, yes. So here I should say that to be safe I was a little careless here. But to be on the safe side, suppose S is no Syrian and this is a finite type. This is the only case of interest for me. And then you have a good notion. Well, already constructability on general schemes, you have to be a little careful what you mean by closed. So then the definition, we said that fk is a psi good if psi fk is in DBC. So for k, k in DBC, a constructable complex in the sense of Takishi of X cross S lambda and base change compatible. Actually, by theorem, the second condition implies the first one. But here are some examples. S is the dimension at most one. Then any fk is a psi good. This is due to the linear. The case of dimension zero, by the way, is not trivial. This is the local acyclicity or map over field. Second, if fk is universally locally acyclic, then fk is psi good actually. And in fact, phi fk is zero universally. As Takishi pointed out, this uses an argument in SGA 4 and a half theorem finitude appendix. And three, which was of use to him, suppose it's universally acyclic outside sigma in X, which is quasi-finite over S. Then fk is a psi good. And this implies the flatness of phi in his notation. So actually, the bad cases are blow-ups or maps with hidden blow-ups like Sabba constructed. But the main theorem is that after some modification, all losses are restored and so is end. That is, our psi becomes constructable and compatible with base change. We had theorem of Augusto, which is that given, so X was an S, a finite type, the Serian here. And lambda, of course, is Z mod ln. It's mod n nu and l is invertible on S. Then there exists a modification, let's say g from S' to S such that, well, S' k' with the obvious notation that is base change by g is psi f' good. And this is the main result I will use in my proof of theorem one. So now four, I briefly describe the Q-ness formula. Now suppose you have a very general situation. Suppose you have the situation X1 over Y1, S, Y2, X2. Then you get X over Y is the product, X1 cross X2 over Y, S. And if you have k1 here and k2 here, then by general nonsense you get a map from psi f1, k1. So which lives on the vanishing topos of this map, tensor product of psi f2, k2 into psi f of k, where k is the external tensor product. And theorem two is that f1, f2 and f, is it okay? So theorem two is that assume f1, k1 psi good, fI, kI psi good. So this is always the case if Y is Y1 and Y2 are dimension one, which is the case of interest for me in the affine line. Then star is an isomorphism and in fact fk is also psi good. Do you assume that one of them is finite or dimension to make it bounded or it's not? So kI, I'm sorry, kI, dCTF, yes. Thank you, yes, thank you. So just to be sure to follow, so you are identifying the vanishing topos of f1, the product vanishing topos of f2. No, let me explain here. Or maybe because you are on the right-hand side here. So in fact you are here x1 goes to x1 cross Y1, Y1, right? By psi f1. And then here you have x and here x2, right? And here you have x cross Y, you have two projections here. And here it's x2 cross Y2, Y2. So you have psi f and psi f2. And so what I said in saying that you have the map from the psi here, psi here, external tensor product here into the psi here, right? And I claim that if you assume that fIk is psi good, then this map is an isomorphism. And also the psi f, the fk is also psi good. And in fact the proof is just a game using Augusto theorem. In fact the problem is that a priori you don't know whether this is psi good. If it was, then it means that you could do base change and reduce to a tray. So if you reduce to a tray, then you are reduced to Gabber theorem. So in fact what you do is by Augusto you make fk good by some proper hypercover. And then you have to use ecological descent. But then you have to use both classical ecological descent, like in Deline, 50 years ago. And also a more recent ecological descent, oriented descent by Gabber, which is also in this volume on his work. So then you get that. And then it's, I think, five minutes complicated. So in fact it's then formal to deduce theorem one from theorem two. So again you use some general nonsense. So theorem two implies theorem one. So suppose you have a composition like this, f, g and gf. You have x to x across y, y. And here you have x cross z, z. Here you have psi f. Here you have psi gf. And here you have some map induced by g, which is induced by g with an arrow above. And of course you have that psi gf is g lower star of psi f. So composition of vanishing cycle. So the effect of this g. So remember, suppose x is just a point. So what is this g with an arrow? It's a projection of the mineral ball at x and mineral ball at y. And somehow it's some kind of calculating direct image. It's calculating vanishing cycle. So vanishing cycle is vanishing cycle. And then you apply this to g equal to a, the sum map. And you get the formula. In fact, you get the formula first for r psi. That is r psi, r psi k is r psi k1, r psi k2. But you need the formula for r phi. So this is more complicated. So here you have to use two things. The sum map is locally acyclic. It's smooth actually. So a is locally acyclic. And also you have to use some property of local convolution. Namely that if you take something geometrically constant, over your tray, and you start with something extended by zero, something like this. So this is zero. So in fact, when you look at the vanishing cycles for the product map, f1 cross f2, then you have two components. You have the r phi tensor r phi. And you have an extra component which is mixed, k1 tensor r phi1 and r phi1 tensor k2. It's like when you have two filtered objects of length one and take tensor product, you get the filtered object of length two. So you have two parts. And you have to show that the r a lower star kills this remaining part, but this follows on this. So then you get the formula by looking at what happens at point. And you can even do that in families along the special fiber. So now how do you recover Tom Sebastiani, original Tom Sebastiani? Then it's only on the same case. So the same case, maybe my time is almost finished, but the same case of the k is k bar. Then star one is in fact tensor product, even for coefficients z mod ln. For ql bar, there is a proof in some paper by Foulet using Fourier transform. But in any case, it's not difficult to see that by using the universal tame cover, the Kuhmer cover. And also in this case, suppose you take g in item, z prime out of one. So you have v1, v2, right? So if you look at v1 and v2 at eta bar, so then of course monodromy acts on this torque and gives an automorphism. But as it is commutative, this is an isomorphism in fact of the sheaf itself. So then you can do it as an automorphism of the sheaf. But when you take this, then you have some endomorphism of this also. And in fact, is that g upper star to this turns out to be g, let's say g1 upper star, g2 upper star. So in other words, the action is the diagonal action obtained by g1 and g2 on the two components. And you can translate it into tensor product by this isomorphism. So you recover Tom Sebastian's theorem. And with a little more work, you recover variation morphism, but I have no time to talk about that. So thank you very much. Can you explain how precisely the star product appears because we stopped at the current. So in the current there is product. And in the application, so then the star product appears. Yes. So here you have this x1 cross x2 restricted to say a2h. And it goes to a2h and then to ah. And we have the sum map here. So now I get half psi, let's say, half psi of k1, k2. So this is the external tensor product by my Euclid formula for f1, f2. And then how do I get the vanishing for this composition, af, right? So I explain that you apply a lower star. This will be the upside gf, af. So this is the rA lower star of this product here, upside k1, upside k2. Now take the stock of this at some point. Look at the stock of this at zero. So then in fact it will be the rA, write 00 star of this upside at zero, stock at zero, tensor upside at zero. And A00 is this, actually this localization here. So in fact this is just the definition of the star product, right? This is by definition, upside, star upside. Except that here I work with general k. In fact I'm more interested in the r phi. And then in fact I have something on eta and then I extend by zero, take that some product, but this is basically the thing. So in fact to understand the stocks of this upside, I should have said a little more, but I had no time. So in fact in the local situation, say X, SS. So then in this local situation, you have this psi to vanishing topos. And here the projection on SS. And here you have the localization of S. So this is the map from the small ball to the small ball. So this is a minor ball, minor ball. Now by an observation due to Gabber, there is in fact a section of this map, sigma. And in fact it has a property that sigma, it depends on the point X. Sigma star upside is just upside star. So if you take upside sigma star of, sorry, sorry, sigma star, nonsense. Sigma star of upside here, let's say, this is in fact the, so we take sigma star here or this, and then you get R, F, X, S, lower star. Yeah, so sigma star is in fact, here is the second projection and sigma star is P2 star. So then you take pullback is the same as projecting, then you get that. So this is the basic relation which enables you to understand the stocks. So I should have said also that I'm taking too much time but in fact if you take in general you have S here, maybe big dimension S and T point X. So what is the stock of upside F, let's say K, at a point as I explained X to S and here a specialization map like in Takeshi's expose. So this is in fact, well you will write, you would think first that it just a gamma here because XT or the XS, T, K, but this is not that actually. So the idea is that it is okay in dimension one but not in general. In general you have to take a whole tube here. So it's slightly blurred. So you have here your specialization map, you have S and you have T and the tube here is here and this is a community of minor tube. In fact what you get is a gamma of XS, XS cross S, S, S, T. And in the good case, I'm sorry, good, then it commutes the community of the tube which restricts isomorphism to the community of the special fiber. So in fact you can, somehow you have to think that this vanishing topos is a localization along S or along X if you prefer. And of course here I just explained for just punctual isomorphism but you can make it a family of whole special fibers so you can define the global convolutions global local convolutions so local downstairs but global and then you have a more general formula and then you need that for the variation morphism which is also tensor product proved by the linear and long ago in fact. So I think I talked too much, I'm sorry. Yes, oh sure, sure. So you use the isomorph. Yes, yeah. What do you get to do with isomorphism? Very good question. So then in fact, first of all without essentially no change you could get let's say for any group scheme in fact a smooth group scheme and a convolution at the origin. But you would be interested for example in convolution on GM but at zero. So at zero, well, so at zero you get something like this. So already for the constant shift you have vanishing cycles. So in fact it can be analyzed and you can look at the convolution and you can look at the fibers, there are two pieces and you can analyze but I've not gone very much there. But in fact, local convolutions at various point of GM have been studied by, so I turn the name of the person now I don't remember. But more recently we saw in studied these local convolutions several points GM also and also not just GM but also for elliptic curves. So which is a very subtle example. But all of these somehow go into the frame of this general formalism except that to calculate the local convolution is always hard even for the additive convolution and even for the Artin Scheyer-Schief. In the same case it's easy. But how about the wild case then by Le Monde you have estimates on the rank and the swan. You have formulas for rank and the swan. But already Artin Scheyer is not an elliptic help side. So what is it? So it's a rank three Schief with swan one. So three plus one is four which is two by two which is correct. But you have to write it explicitly as a sum. So with so indeed that and other more complicated examples also. Okay. So we found the speaker again. Thank you.